Content-Type: multipart/mixed; boundary="-------------9906100918258" This is a multi-part message in MIME format. ---------------9906100918258 Content-Type: text/plain; name="99-224.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-224.keywords" coupled map lattice, interacting particle system, transfer operator, decay of correlations, percolation ---------------9906100918258 Content-Type: application/postscript; name="paperversion10july99.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="paperversion10july99.ps" %!PS-Adobe-2.0 %%Creator: dvipsk 5.55a Copyright 1986, 1994 Radical Eye Software %%Title: paperversion10july99.dvi %%Pages: 50 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips paperversion10july99.dvi -o %DVIPSParameters: dpi=600, compressed, comments removed %DVIPSSource: TeX output 1999.06.10:1555 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind 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y(in\014nite-dimensional)f(system,)g(acting)h(on)f(Banc)n(h-spaces)h (that)g(include)f(measures)g(whose)i(\014nite-dimensional)326 2362 y(marginals)23 b(ha)n(v)n(e)f(analytic,)i(exp)r(onen)n(tially)e(b) r(ounded)g(densities.)34 b(W)-6 b(e)22 b(pro)n(v)n(e)g(existence)h(and) f(uniqueness)g(\(in)g(the)326 2453 y(considered)k(Banac)n(h)g(space\))g (of)g(a)g(probabilit)n(y)g(measure)f(and)h(its)g(exp)r(onen)n(tial)g (deca)n(y)f(of)h(correlations.)118 2723 y Fx(0)135 b(In)l(tro)t (duction)118 2905 y Fw(In)33 b(this)f(pap)r(er)g(w)n(e)g(study)h (coupled)f(map)g(lattices)g(with)h(indep)r(en)n(ten)n(t)g(iden)n (tically)f(\(i.i.\))52 b(P)n(oisson-distributed)31 b(up-)118 3005 y(datings)c(at)h(the)g(individual)f(sites.)118 3105 y(A)g(deterministic)f(coupled)g(map)g(lattice)g(\(CML\))h(is)f(giv)n (en)f(b)n(y)h(a)g Fv(Z)2261 3074 y Fu(d)2294 3105 y Fw(-lattice)g(with) h(a)e(cop)n(y)h(of)g(the)g(same)g(Riemannian)118 3204 y(manifold)31 b(at)f(eac)n(h)g(lattice)g(p)r(oin)n(t)h(\(i.e.)g(the)g (state)f(space)g(is)g(the)h(pro)r(duct)g(of)f(these)h(manifolds)f(with) h(index)g(set)f Fv(Z)3833 3174 y Fu(d)3865 3204 y Fw(\))118 3304 y(and)g(a)g(map)h(on)f(the)h(in\014nite)g(space)f(that)h(can)f(b)r (e)h(decomp)r(osed)f(in)n(to)g(an)g(uncoupled)h(map)f(that)h(acts)f (individually)118 3403 y(on)i(eac)n(h)f(comp)r(onen)n(t)h(and)g(an)g (`in)n(teraction)f(step')h(where)g(the)g(c)n(hange)f(of)h(eac)n(h)f(co) r(ordinate)g(dep)r(ends)i(also)e(on)g(the)118 3503 y(other)c(sites.)118 3603 y(L.A.)39 b(Bunimo)n(vic)n(h)g(and)f(Y.G.)i(Sinai)f(pro)n(v)n(e)e (in)i([8])g(\(cf.)h(also)d(the)j(remarks)d(on)i(this)g(in)g([5]\))g (the)g(existence)g(and)118 3702 y(uniqueness)24 b(of)g(an)g(in)n(v)-5 b(arian)n(t)23 b(measure)g(and)h(its)h(exp)r(onen)n(tial)e(deca)n(y)g (of)h(correlations)e(for)i(a)g(one-dimensional)f(lattice)118 3802 y(of)33 b(in)n(terv)-5 b(al)33 b(maps)g(with)h(w)n(eak)e (coupling.)54 b(By)33 b(constructing)g(a)g(Mark)n(o)n(v)e(partition)i (they)g(relate)g(the)h(system)f(to)g(a)118 3902 y(t)n(w)n (o-dimensional)26 b(spin)i(system)f(whose)g(Gibbs)h(measure)e(corresp)r (onds)g(to)i(the)g(in)n(v)-5 b(arian)n(t)26 b(measure)h(of)g(the)h (CML.)118 4001 y(G.)g(Keller)e(and)h(M.)h(K)r(\177)-44 b(unzle)27 b(pro)n(v)n(e)f(in)h([21)o(])h(the)g(existence)f(and)g (uniqueness)g(of)g(an)g(in)n(v)-5 b(arian)n(t)27 b(measure)f(for)h(p)r (erio)r(dic)118 4101 y(or)c(in\014nite)i(one-dimensional)e(lattices)g (of)h(w)n(eakly)f(coupled)h(in)n(terv)-5 b(al)23 b(maps)h(b)n(y)g (studying)g(the)g(transfer)f(op)r(erators)f(on)118 4200 y(the)27 b(space)f(of)h(measures)e(whose)h(\014nite-dimensional)h (marginals)e(ha)n(v)n(e)g(densities)i(of)f(b)r(ounded)i(v)-5 b(ariation.)35 b(F)-7 b(or)26 b(small)118 4300 y(p)r(erturbation)h(of)h (the)g(uncoupled)f(map)h(an)n(y)f(in)n(v)-5 b(arian)n(t)26 b(measure)h(is)g(`close')g(to)h(the)g(one)f(they)h(found.)118 4400 y(J.)22 b(Bricmon)n(t)g(and)g(A.)h(Kupiainen)f(extend)g(in)h([4])f (and)g([5,)h(6)o(])g(the)f(result)g(of)h(Bunimo)n(vic)n(h)e(and)h (Sinai)h([8)o(])g(to)f Fv(Z)3578 4370 y Fu(d)3611 4400 y Fw(-lattices)118 4499 y(of)28 b(w)n(eakly)e(coupled)i(circle)f(maps)g (with)h(analytic)f(and)g(H\177)-42 b(older-con)n(tin)n(uous)26 b(in)n(teraction,)h(resp)r(ectiv)n(ely)-7 b(.)118 4599 y(They)39 b(represen)n(t)g(the)g(iterates)g(of)g(the)h(P)n(erron-F)-7 b(rob)r(enius)37 b(op)r(erator)g(for)i(\014nite-dimensional)g (subsystems)g(\(o)n(v)n(er)118 4699 y(\003)31 b Ft(\032)f Fv(Z)363 4668 y Fu(d)396 4699 y Fw(\))j(b)n(y)f(a)g(`p)r(olymer'-)f(or) g(`cluster'-expansion)g(that)i(giv)n(es)e(rise)g(to)h(a)g(represen)n (tation)f(of)h(the)h(corresp)r(onding)118 4798 y(in)n(v)-5 b(arian)n(t)30 b(measure)f(in)i(terms)f(of)h(a)f(\()p Fs(d)21 b Fw(+)f(1\)-dimensional)30 b(spin)h(system.)45 b(The)31 b(w)n(eak)f(limit)h(\(as)f(\003)e Ft(!)g Fv(Z)3519 4768 y Fu(d)3552 4798 y Fw(\))j(of)g(these)118 4898 y(measures)e(is)h (the)h(unique)g(\(in)g(a)f(certain)f(class\))h(in)n(v)-5 b(arian)n(t)29 b(probabilit)n(y)h(measure)f(and)h(it)h(is)f(exp)r(onen) n(tially)g(mixing)118 4997 y(with)e(resp)r(ect)g(to)f(\(wrt\))h (spatio-temp)r(oral)e(shifts.)118 5097 y(C.)21 b(Maes)f(and)h(A.)g(V)-7 b(an)21 b(Mo\013aert)f(in)n(tro)r(duce)g(in)h([25])g(for)f(a)g(similar) g(setting)h(as)f(in)h([4])g(a)f(simpli\014ed)h(`cluster'-expansion)118 5197 y(for)29 b(the)h(truncated)f(P)n(erron-F)-7 b(rob)r(enius)27 b(op)r(erator)g(and)j(sho)n(w)e(sto)r(c)n(hastic)h(stabilit)n(y)g(of)g (the)h(CML)f(under)h(sto)r(c)n(hastic)118 5296 y(p)r(erturbation.)p 118 5356 1512 4 v 209 5410 a Fr(\003)245 5433 y Fq(The)24 b(largest)g(part)g(of)f(w)n(ork)h(on)g(this)g(pap)r(er)g(w)n(as)f(supp) r(orted)i(b)n(y)f(the)h(EC)e(via)h(TMR-F)-6 b(ello)n(wship)22 b(ERBFMBICT-961157)1987 5682 y Fw(1)p eop %%Page: 2 2 2 1 bop 118 354 a Fw(In)30 b([2)o(])g(V.)g(Baladi,)g(M.)f(Degli)h(Esp)r (osti,)g(S.)f(Isola,)g(E.)h(J\177)-42 b(arv)n(enp\177)g(a\177)g(a)27 b(and)i(A.)h(Kupiainen)g(de\014ne)f(F)-7 b(rec)n(het)30 b(spaces,)f(and,)118 454 y(for)36 b Fs(d)i Fw(=)f(1,)i(a)d(Banac)n(h)f (space)g(and)i(transfer)e(op)r(erators)f(for)i(the)h (in\014nite-dimensional)f(systems,)j(considered)c(b)n(y)118 554 y(Bricmon)n(t)27 b(and)g(Kupiainen)h(in)f([4],)h(and)f(study)h(the) g(sp)r(ectral)f(prop)r(erties)g(of)g(these)h(op)r(erators.)118 653 y(In)d([13)o(])g(w)n(e)g(consider)f(analytically)f(coupled)i (circle)f(maps)h(\(uniformly)g(expanding)f(and)h(analytic\))f(on)h(the) g Fv(Z)3611 623 y Fu(d)3644 653 y Fw(-lattice)118 753 y(with)i(exp)r(onen)n(tially)e(deca)n(ying)g(in)n(teraction)g(and)g(in) n(tro)r(duce)h(Banac)n(h)f(spaces)f(for)i(the)g(in\014nite-dimensional) g(system)118 852 y(that)d(include)g(measures)e(whose)h (\014nite-dimensional)g(marginals)f(ha)n(v)n(e)g(analytic,)i(exp)r (onen)n(tially)f(b)r(ounded)h(densities.)118 952 y(W)-7 b(e)27 b(de\014ne)f(transfer)f(op)r(erators)g(on)h(these)g(spaces,)f (get)h(a)g(unique)h(\(in)f(the)h(considered)e(Banac)n(h)g(spaces\))h (probabilit)n(y)118 1052 y(measure)h(and)g(pro)n(v)n(e)f(its)i(exp)r (onen)n(tial)f(deca)n(y)g(of)g(correlations.)118 1151 y(CMLs)34 b(with)h(m)n(ulti-dimensional)f(lo)r(cal)g(systems)g(of)h(h)n (yp)r(erb)r(olic)f(t)n(yp)r(e)h(ha)n(v)n(e)e(b)r(een)i(studied)g(b)n(y) f(Y)-7 b(a.B.)34 b(P)n(esin)g(and)118 1251 y(Y)-7 b(a.G.)28 b(Sinai)g([26)o(],)g(M.)h(Jiang)d([17,)i(18)o(],)g(M.)g(Jiang)f(and)h (A.)g(Mazel)f([19)o(],)i(M.)f(Jiang)f(and)g(Y)-7 b(a.B.)28 b(P)n(esin)f([20)o(])h(and)g(D.L.)118 1351 y(V)-7 b(olevic)n(h)27 b([29)o(,)h(30)o(].)118 1450 y(F)-7 b(or)27 b(detailed)h(reviews)e(on)i (mathematical)f(results)g(on)g(CMLs)h(w)n(e)f(refer)g(to)g([2],)g([5],) h([7)o(])g(and)g([20)o(].)118 1621 y(An)k(in)n(teracting)e(particle)h (system)g(\(IPS\))g(is)g(giv)n(en)f(b)n(y)h(an)g(in\014nite)h(lattice)f (with)h(a)f(cop)n(y)f(of)h(the)h(same)e(state)h(space)118 1720 y(\(that)38 b(is)g(usually)f(a)h(\014nite)g(or)f(coun)n(table)g (set)h(but)g(can)g(also)e(b)r(e)j(a)e(Riemannian)g(manifold\))i(at)e (eac)n(h)g(site.)68 b(The)118 1820 y(up)r(dating)32 b(at)f(an)g (individual)g(site)h(is)f(a)g(deterministic)h(or)e(sto)r(c)n(hastic)h (map)g(\(e.g.)g(in)h(the)f(case)g(of)g(\014nite)h(lo)r(cal)f(state)118 1920 y(spaces)g(it)h(is)f(giv)n(en)g(b)n(y)g(a)g(sto)r(c)n(hastic)g (matrix)g(with)h(transition)f(probabilities)f(as)h(its)h(co)r (e\016cien)n(ts\))f(that)h(is)g(applied)118 2019 y(with)i(`exp)r(onen)n (tial)e(w)n(aiting)g(times',)j(i.e.)e(lik)n(e)g(the)g(w)n(aiting)g (times)g(for)f(jumps)i(in)f(a)g(P)n(oisson)e(pro)r(cess.)52 b(The)33 b(jump)118 2119 y(rates)f(for)g(the)h(up)r(dating)g(dep)r (ends)g(also)f(on)h(the)g(other)f(sites.)52 b(R.J.)33 b(Glaub)r(er)f(in)n(tro)r(duces)h(in)g([14)o(])g(\(a)f(sp)r(ecial)h (case)118 2232 y(of)6 b(\))33 b(the)g(sto)r(c)n(hastic)e(Ising)h(mo)r (del)g(as)g(a)g(mo)r(del)g(for)g(magnetism.)50 b(The)32 b(total)g(state)h(space)e Ft(f\000)p Fw(1)p Fs(;)14 b Fw(+1)p Ft(g)3427 2202 y Fp(Z)3469 2177 y Fy(d)3534 2232 y Fw(represen)n(ts)118 2332 y(the)29 b(spins)g(of)g(the)g(atoms)f(at)h (all)f(sites.)41 b(The)29 b(rate)f(for)g(a)h(\015ip)g(of)g(an)f (individual)h(spin)g(dep)r(ends)g(on)g(the)g(states)g(of)f(the)118 2431 y(neigh)n(b)r(our)33 b(sites.)55 b(F.)33 b(Spitzer)h([27)o(,)g(28) o(])g(and)f(R.L.)h(Dobrushin)f([9,)h(10)o(])f(study)h(more)f(general)f (systems)h(where)g(the)118 2531 y(individual)28 b(jump)g(rates)f(do)g (not)h(only)f(dep)r(end)h(on)g(the)g(nearest)e(neigh)n(b)r(ours.)118 2630 y(A)40 b(basic)f(problem)f(is)i(to)f(establish)g(the)h(existence)f (of)g(in\014nite)h(systems)f(with)h(async)n(hronous)d(up)r(datings.)72 b(The)118 2730 y(in\014nitely)25 b(man)n(y)e(jumps)h(in)h(a)e(\014nite) h(time-in)n(terv)-5 b(al)24 b(cannot)f(b)r(e)i(`listed',)g(i.e.)f (there)f(is)h(no)g(order)e(preserving)g(bijection)118 2830 y(b)r(et)n(w)n(een)28 b(the)g(time-ordered)e(set)i(of)f(jumps)h (and)g Fv(N)t Fw(.)118 2929 y(R.L.)g(Dobrushin)f(obtains)g(in)h([9])g (the)g(in\014nite)g(system)f(as)g(the)h(limit)g(of)g(subsystems)f(o)n (v)n(er)f(\014nitely)i(man)n(y)f(sites.)118 3029 y(By)32 b(using)f(a)h(p)r(ercolation)f(argumen)n(t)f(T.E.)i(Harris)f(pro)n(v)n (es)f(in)i([15)o(])g(that)g(for)f(systems)h(of)g(\014nite)g(range)f(in) n(teraction)118 3129 y(and)d(a)g(su\016cien)n(tly)g(small)g(time)g(in)n (terv)-5 b(al)28 b(the)g(history)g(of)g(an)g(individual)g(particle)g (dep)r(ends)g(on)g(only)g(\014nitely)g(man)n(y)118 3228 y(sites,)h(and)f(so)g(he)h(pro)n(vides)e(a)h(natural)g(de\014nition)h (of)g(the)g(in\014nite)g(system.)40 b(With)29 b(probabilit)n(y)f(1)g (the)h(set)g Fv(Z)3646 3198 y Fu(d)3708 3228 y Fw(splits)118 3328 y(in)n(to)e(\014nite)i(clusters)e(suc)n(h)g(that)h(eac)n(h)f(site) g(is)h(a\013ected)f(at)h(most)f(b)n(y)h(sites)f(in)h(the)g(same)f (cluster.)118 3427 y(R.)34 b(Holley)g(sho)n(ws)e(in)i([16])g(for)f (generators,)g(corresp)r(onding)f(to)h(one-dimensional)g(mo)r(dels,)i (and)f(T.M.)g(Liggett)f(in)118 3527 y([23)o(])24 b(for)f(the)g (d-dimensioanal)g(case,)g(that)h(these)f(op)r(erators)f(generate,)h(in) g(fact,)i(a)e(semigroup,)g(acting)g(on)g(con)n(tin)n(uous)118 3627 y(functions.)118 3726 y(Here)33 b(w)n(e)g(ha)n(v)n(e)g(only)g(men) n(tioned)g(di\013eren)n(t)h(metho)r(ds)g(to)f(establish)g(the)h (existence)g(of)f(the)h(in\014nite)g(systems.)54 b(F)-7 b(or)118 3826 y(detailed)37 b(reviews)g(on)g(IPSs)f(and)i(results)e(on) h(in)n(v)-5 b(arian)n(t)36 b(measures,)j(mixing)e(prop)r(erties,)i (phase)e(transitions)f(and)118 3926 y(applications)27 b(to)g(ph)n(ysics)g(and)h(other)f(sciences)g(w)n(e)g(refer)g(to)g([11)o (])h(and)f([24)o(].)118 4109 y(In)34 b(this)f(pap)r(er)g(w)n(e)g (consider)g(the)g(in\014nite)h(top)r(ological)e(pro)r(duct)i Fs(M)41 b Fw(=)32 b(\()p Fs(S)2580 4079 y Fo(1)2617 4109 y Fw(\))2649 4079 y Fp(Z)2694 4054 y Fy(d)2761 4109 y Fw(and)h(con)n(tin)n(uous)f(up)r(dating)i(maps)118 4209 y(for)28 b(the)h(individual)g(co)r(ordinates)e(that)i(are)f(of)h (\014nite)g(range)e(or)h(Lipsc)n(hitz-con)n(tin)n(uous)f(wrt)h(all)h (co)r(ordinates)e(with)i(a)118 4309 y(summable)g(family)g(of)g(Lipsc)n (hitz)g(constan)n(ts)f(\(cf.)h(Section)g(2.2)f(for)h(the)g (de\014nition\).)42 b(The)29 b(times)g(for)f(the)i(up)r(datings)118 4408 y(at)d(the)g(individual)g(sites)f(are)g(indep)r(enden)n(tly)i(P)n (oisson-distributed)c(with)k(the)f(same)f(constan)n(t)g(rate)g Fs(\025)d(>)g Fw(0.)36 b(F)-7 b(or)26 b(the)118 4508 y(\014nite)32 b(range)d(case)h(w)n(e)h(sho)n(w)f(that)h(with)h (probabilit)n(y)d(1)i(the)g(up)r(datings)g(at)g(an)n(y)f(\014nite)i (set)f(of)f(sites)h(and)g(an)n(y)f(\014nite)118 4608 y(time-in)n(terv)-5 b(al)33 b(dep)r(end)i(on)e(only)h(\014nitely)g(man) n(y)f(sites.)56 b(Our)33 b(pro)r(of)g(uses)g(time-)h(and)g(space-orien) n(ted)e(p)r(ercolation)118 4707 y(and)f(is)f(di\013eren)n(t)h(from)g (the)g(one)g(in)g([15)o(].)46 b(This)31 b(result)g(pro)n(vides)e(a)i (natural)f(de\014nition)h(of)g(the)g(in\014nite)g(dynamical)118 4807 y(system.)118 4907 y(F)-7 b(or)29 b(the)h(systems)g(with)g (in\014nite)g(range)f(in)n(teraction)f(w)n(e)i(sho)n(w)f(that)h(with)g (probabilit)n(y)f(1)g(it)h(is)g(w)n(ell-de\014ned)f(as)g(the)118 5006 y(net-limit)f(of)f(its)g(\014nite-dimensional)f(subsystems)g(with) i(arbitrary)d(b)r(oundary)h(conditions.)36 b(W)-7 b(e)27 b(com)n(bine)g(standard)118 5106 y(estimates)32 b(for)g(error)f(gro)n (wth)g(with)i(ideas)f(from)g(p)r(ercolation)f(theory)-7 b(.)51 b(The)33 b(limit)g(of)g(the)f(corresp)r(onding)f(Mark)n(o)n(v) 118 5205 y(k)n(ernels,)e(acting)g(on)h(con)n(tin)n(uous)e(functions,)j (exists)e(and)g(pro)n(vides)f(a)i(de\014nition)g(of)f(the)h(in\014nite) h(pro)r(cess,)e(di\013eren)n(t)118 5305 y(from)e(the)h(widely)g(used)g (generator)d(approac)n(h.)118 5405 y(Our)k(pro)r(ofs)f(still)i(w)n(ork) e(if)i(w)n(e)f(replace)f Fs(S)1452 5375 y Fo(1)1518 5405 y Fw(b)n(y)h(an)n(y)g(compact)g(Riemannian)g(manifold)g(or)g(sto)r(c)n (hastic)f(systems)h(with)1987 5682 y(2)p eop %%Page: 3 3 3 2 bop 118 354 a Fw(\014nite)33 b(state)f(spaces.)49 b(The)33 b(assumption)e(of)h(ha)n(ving)f(the)i(same)f(constan)n(t)f (jump)i(rate)e(at)i(all)e(sites)h(is)g(b)n(y)g(no)g(means)118 454 y(essen)n(tial)26 b(and)g(can)g(b)r(e)g(w)n(eak)n(ened)f(to)i(the)f (case)g(of)g(upp)r(er)h(b)r(ounded)f(individual)h(jump)g(rates)f(that)g (dep)r(end)h(on)f(other)118 554 y(states)h(as)g(w)n(ell.)37 b(Ho)n(w)n(ev)n(er,)26 b(w)n(e)h(do)g(not)h(consider)e(these)i (generalizations)d(in)j(this)g(pap)r(er.)118 653 y(In)35 b(a)f(setting)h(similar)f(to)g(that)h(of)g([13)o(],)i(i.e.)e(for)f (analytically)f(coupled)i(circle)f(maps)g(\(uniformly)h(expanding)f (and)118 753 y(analytic\))18 b(on)h(the)g Fv(Z)759 723 y Fu(d)792 753 y Fw(-lattice)f(with)h(w)n(eak,)h(exp)r(onen)n(tially)e (deca)n(ying)f(in)n(teraction)h(but)h(with)h(async)n(hronous)c(up)r (datings)118 852 y(as)38 b(describ)r(ed)h(ab)r(o)n(v)n(e,)h(w)n(e)f (de\014ne)g(transfer)f(op)r(erators)f(for)h(the)i(Mark)n(o)n(v)c(k)n (ernels)i(of)h(the)g(in\014nite)h(system.)71 b(The)118 952 y(op)r(erators)17 b(act)i(on)g(the)g(Banac)n(h)f(space)g Ft(H)1419 964 y Fu(#)1483 952 y Fw(\(in)n(tro)r(duced)h(in)g([13)o(]\)) h(that)f(includes)g(measures)f(whose)g(\014nite-dimensional)118 1052 y(marginals)29 b(ha)n(v)n(e)h(analytic,)i(exp)r(onen)n(tially)e(b) r(ounded)i(densities.)47 b(Using)31 b(`cluster-expansion'-lik)n(e)d (tec)n(hniques,)k(w)n(e)118 1151 y(represen)n(t)c(these)h(in)n(tegral)e (op)r(erators)g(in)i(terms)g(of)f(con\014gurations)f(and)i(pro)n(v)n(e) e(the)i(existence)g(and)g(uniqueness)f(\(in)118 1251 y Ft(H)188 1263 y Fu(#)232 1251 y Fw(\))g(of)g(an)f(in)n(v)-5 b(arian)n(t)27 b(probabilit)n(y)f(measure)h(and)g(its)h(exp)r(onen)n (tial)f(deca)n(y)g(of)g(correlations.)118 1351 y(The)i(pap)r(er)f(is)h (organized)e(as)h(follo)n(ws.)39 b(Section)29 b(1)f(pro)n(vides)f (de\014nitions,)i(notation)g(and)f(some)g(prop)r(ositions)g(ab)r(out) 118 1450 y(sto)r(c)n(hastic)f(pro)r(cesses)f(and)i(metric)g(spaces.)36 b(In)28 b(Section)g(2)f(w)n(e)h(de\014ne)g(the)g (in\014nite-dimensional)g(systems)f(for)g(\014nite)118 1550 y(range)34 b(\(Section)h(2.1\))g(and)g(in\014nite)h(range)e(in)n (teraction)g(\(Section)h(2.2\))g(and)g(the)h(corresp)r(onding)d(Mark)n (o)n(v)g(k)n(ernels)118 1649 y(\(Section)24 b(2.3\).)35 b(In)24 b(Section)f(3)g(w)n(e)h(study)f(the)h(transfer)f(op)r(erators)e (for)j(a)f(sp)r(eci\014c)g(class)g(of)g(in)n(teractions.)35 b(In)24 b(Section)f(4)118 1749 y(w)n(e)k(pro)n(v)n(e)f(the)h(mixing)h (prop)r(erties)e(of)h(the)h(in)n(v)-5 b(arian)n(t)26 b(measure)g(\(found)i(in)g(Section)f(3\))g(wrt)h(spatio-temp)r(oral)d (shifts.)118 2024 y Fx(1)135 b(Basic)45 b(De\014nitions)h(and)e (Examples)118 2205 y Fw(In)33 b(this)f(section)g(w)n(e)g(presen)n(t)g (de\014nitions)h(from)f(probabilit)n(y)f(theory)h(and)g(top)r(ology)f (and)h(also)g(in)n(tro)r(duce)g(most)g(of)118 2305 y(the)g(notation)g (used)g(in)g(this)g(pap)r(er.)49 b(W)-7 b(e)32 b(ha)n(v)n(e)f(tak)n(en) 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b(is)f(called)h(a)f Fn(sto)s(c)m(hastic)32 b(pro)s(cess)f(with)g(v)-5 b(alues)32 b(in)f Fw(\()p Fn(E)p Fs(;)14 b Ft(A)3210 2981 y Fl(2)3252 2969 y Fw(\).)243 3135 y Ft(\017)41 b Fw(If)32 b Fs(I)38 b Fw(=)30 b Fv(R)636 3105 y Fm(\025)p Fo(0)731 3135 y Fw(,)j([0)p Fs(;)14 b(T)e Fw(])31 b(or)g([0)p Fs(;)14 b(T)e Fw(\))31 b(for)h(some)f Fs(T)42 b(>)30 b Fw(0)i(the)g(pro)r(cess)f(is)h(called)g(a)f Fn(con)m(tin)m(uous)38 b(time)c(sto)s(c)m(hastic)326 3235 y(pro)s(cess)p Fw(.)243 3401 y Ft(\017)41 b Fw(F)-7 b(or)27 b(\014xed)g Fs(!)f Ft(2)d Fw(\012)28 b(the)g(map)g Fs(t)23 b Ft(7!)g Fs(X)1475 3413 y Fu(t)1504 3401 y Fw(\()p Fs(!)s Fw(\))28 b(is)f(called)g(the)h Fn(tra)5 b(jectory)30 b Fw(of)e Fs(!)s Fw(.)36 b(It)28 b(is)g(also)e(denoted)i(b)n(y)f Fs(X)3605 3413 y Fn(.)3636 3401 y Fw(\()p Fs(!)s Fw(\).)243 3567 y Ft(\017)41 b Fw(W)-7 b(e)31 b(consider)f(the)i(set)f Fv(N)41 b Fw(as)30 b(measurable)g(space)g(with)i(the)f(discrete)g Fs(\033)s Fw(-algebra.)45 b(F)-7 b(or)31 b(an)n(y)f(set)h(\003)g(w)n(e) g(denote)326 3667 y(b)n(y)c Fv(N)495 3637 y Fo(\003)578 3667 y Fw(the)h(pro)r(duct)g(space,)f(established)g(with)h(the)g(pro)r (duct)g Fs(\033)s Fw(-algebra.)326 3800 y(A)i(con)n(tin)n(uous)e(time)j (sto)r(c)n(hastic)e(pro)r(cess)f(with)j(v)-5 b(alues)29 b(in)h Fv(N)2298 3769 y Fo(\003)2383 3800 y Fw(and)g(with)g(index)g (set)g Fs(I)37 b Fw(and)29 b Fs(P)12 b Fw(-a.a.)29 b(of)g(whose)326 3899 y(tra)5 b(jectories)28 b(are)i(non-decreasing)e(\(i.e.)j(the)g (functions)g Fs(t)d Ft(7!)g Fs(\031)2365 3911 y Fu(q)2423 3899 y Ft(\016)20 b Fs(X)2554 3911 y Fu(t)2583 3899 y Fw(\()p Fs(!)s Fw(\))30 b(are)g(non-decreasing)e(for)i(all)h Fs(q)g Ft(2)d Fw(\003)326 3999 y(and)21 b Fs(P)12 b Fw(-a.a.)21 b Fs(!)k Ft(2)f Fw(\012.)35 b(`)p Fs(\031)1069 4011 y Fu(q)1106 3999 y Fw(')22 b(denotes)f(the)h(pro)5 b(jection)21 b(on)g(the)i Fs(q)s Fw(th)f(co)r(ordinate.\),)g(is)g(called)f(a)g Fn(coun)m(ting)k(pro)s(cess)326 4098 y Fw(with)j(v)-5 b(alues)27 b(in)h Fv(N)913 4068 y Fo(\003)996 4098 y Fw(W)-7 b(e)28 b(sa)n(y)e(that)i(suc)n(h)f(a)h(pro)r(cess)e(is)i Fn(of)j(\014nite)h(exp)s(ectation)27 b Fw(if)h(for)f(all)h Fs(t)23 b Ft(2)g Fs(I)35 b Fw(the)28 b(random)326 4198 y(v)-5 b(ariable)26 b Fs(!)g Ft(7!)821 4136 y Fk(P)909 4223 y Fu(q)r Fm(2)p Fo(\003)1049 4198 y Fs(\031)1096 4210 y Fu(q)1152 4198 y Ft(\016)18 b Fs(X)1281 4210 y Fu(t)1310 4198 y Fw(\()p Fs(!)s Fw(\))28 b(has)f(\014nite)h(exp)r (ectation.)118 4381 y Fn(Remark)j(1.1)142 b Fw(1.)41 b(W)-7 b(e)28 b(will)g(also)e(use)i(the)g(short-hand-notation)d Fs(X)2426 4393 y Fn(.)2484 4381 y Fw(for)i(a)g(sto)r(c)n(hastic)g(pro)r (cess)f(if)j(\012,)e Ft(A)3606 4393 y Fo(1)3671 4381 y Fw(and)h Fs(P)326 4480 y Fw(are)e(ob)n(vious)h(from)g(the)h(con)n (text.)220 4646 y(2.)41 b(The)30 b(term)f Fj(p)l(ath)37 b Fw(seems)30 b(to)f(b)r(e)i(more)e(common)g(than)h Fj(tr)l(aje)l (ctory)38 b Fw(but)30 b(w)n(e)g(will)g(denote)f(something)h(else)f (later)326 4746 y(on)e(b)n(y)g Fj(p)l(ath)p Fw(.)220 4912 y(3.)41 b(Finite)26 b(exp)r(ectation)g(means)f(that)h(with)h (probabilit)n(y)d(1)i(there)f(are)g(only)h(\014nitely)g(man)n(y)f (jumps)i(\(cf.)f(De\014nition)326 5012 y(1.2)g(b)r(elo)n(w\))i(in)g(ev) n(ery)e(\014nite)i(time-in)n(terv)-5 b(al.)118 5194 y Fn(De\014nition)31 b(1.2)40 b Fw(\(cf.)31 b([3)o(]\))g(Let)e(\(\012)p Fs(;)14 b Ft(A)1370 5206 y Fo(1)1408 5194 y Fs(;)g(P)r(;)g Fw(\()p Fs(X)1638 5206 y Fu(t)1668 5194 y Fw(\))1700 5206 y Fu(t)p Fm(2)p Fu(I)1808 5194 y Fw(\))30 b(b)r(e)g(a)g(con)n(tin) n(uous)e(time)i(coun)n(ting)g(pro)r(cess)e(with)i(v)-5 b(alues)30 b(in)g Fv(N)118 5294 y Fw(as)d(in)h(De\014nition)g(1.1)f (and)g Fs(!)f Ft(2)e Fw(\012.)36 b(W)-7 b(e)28 b(de\014ne)1987 5682 y(3)p eop %%Page: 4 4 4 3 bop 1169 545 a Fs(X)1245 510 y Fo(+)1238 566 y Fu(t)1300 545 y Fw(\()p Fs(!)s Fw(\))1502 498 y Fo(def)1515 545 y Fw(=)1676 428 y Fk(\032)1911 495 y Fs(X)1980 507 y Fu(t)2009 495 y Fw(\()p Fs(!)s Fw(\))214 b(if)84 b Fs(I)30 b Fw(=)23 b([0)p Fs(;)14 b(t)p Fw(])1780 594 y(lim)1895 606 y 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Fj(have)31 b(the)f(same)g(distribution.\))3806 3797 y Fh(2)118 4029 y Fn(De\014nition)h(1.4)40 b Fw(Let)23 b(\003)e(b)r(e)h(a)g(nonempt)n(y)g(set)g(and)f(\(\012)1874 4041 y Fu(q)1911 4029 y Fs(;)14 b Ft(A)2014 4041 y Fu(q)2051 4029 y Fs(;)g(P)2141 4041 y Fu(q)2178 4029 y Fs(;)g Fw(\()p Fs(X)2323 3990 y Fu(q)2316 4050 y(t)2359 4029 y Fw(\))2391 4041 y Fu(t)p Fm(2)p Fu(I)2499 4029 y Fw(\))2531 4041 y Fu(q)r Fm(2)p Fo(\003)2680 4029 y Fw(b)r(e)23 b(a)e(family)h(of)g (sto)r(c)n(hastic)f(pro)r(cesses)118 4129 y(with)28 b(v)-5 b(alues)27 b(in)h(\()p Fs(E)744 4141 y Fu(q)782 4129 y Fs(;)14 b Ft(A)885 4099 y Fu(q)921 4129 y Fw(\),)28 b(resp)r(ectiv)n(ely)-7 b(.)36 b(W)-7 b(e)28 b(set)1543 4390 y(\012)1686 4343 y Fo(def)1699 4390 y Fw(=)1869 4312 y Fk(Y)1860 4490 y Fu(q)r Fm(2)p Fo(\003)1997 4390 y Fw(\012)2057 4402 y Fu(q)2094 4390 y Fs(;)1675 b Fw(\(6\))1565 4617 y(~)1537 4638 y Ft(A)1686 4591 y Fo(def)1699 4638 y Fw(=)1860 4559 y Fk(O)1862 4737 y Fu(q)r Fm(2)p Fo(\003)2000 4638 y Ft(A)2066 4650 y Fu(q)2103 4638 y Fs(;)1666 b Fw(\(7\))1557 4865 y(~)1538 4886 y Fs(P)1686 4839 y Fo(def)1699 4886 y Fw(=)1860 4807 y Fk(O)1862 4985 y Fu(q)r Fm(2)p Fo(\003)2000 4886 y Fs(P)2053 4898 y Fu(q)2090 4886 y Fs(;)1679 b Fw(\(8\))1537 5134 y Ft(A)1686 5087 y Fo(def)1699 5134 y Fw(=)96 b(completion)28 b(of)2408 5113 y(~)2379 5134 y Ft(A)g Fw(wrt)2644 5113 y(~)2626 5134 y Fs(P)12 b(;)1078 b Fw(\(9\))1538 5283 y Fs(P)1686 5236 y Fo(def)1699 5283 y Fw(=)124 b(extension)27 b(of)2368 5262 y(~)2350 5283 y Fs(P)39 b Fw(to)28 b Ft(A)1140 b Fw(\(10\))1288 5433 y(and)83 b Fs(X)1574 5445 y Fu(t)1686 5386 y Fo(def)1699 5433 y Fw(=)96 b(\()p Fs(X)1968 5393 y Fu(q)1961 5454 y(t)2005 5433 y Fw(\))2037 5445 y Fu(q)r Fm(2)p Fo(\003)2164 5433 y Fs(:)1563 b Fw(\(11\))1987 5682 y(4)p eop %%Page: 5 5 5 4 bop 118 354 a Fw(In)29 b(\(6\))g(w)n(e)g(mean)f(the)h(cartesian)f (pro)r(duct)h(of)f(spaces,)g(in)i(\(7\))e(the)i(pro)r(duct)e (sigma-algebra)e(and)j(in)g(\(8\))g(the)g(pro)r(duc)118 454 y(measure.)118 554 y(Then)37 b(the)h(pro)r(cess)e(\(\012)p Fs(;)14 b Ft(A)p Fs(;)g(P)r(;)g Fw(\()p Fs(X)1221 566 y Fu(t)1251 554 y Fw(\))1283 566 y Fu(t)p Fm(2)p Fu(I)1429 554 y Fw(with)37 b(v)-5 b(alues)37 b(in)h(\()2023 491 y Fk(Q)2101 578 y Fu(q)r Fm(2)p Fo(\003)2242 554 y Fs(E)2303 566 y Fu(q)2340 554 y Fs(;)2377 491 y Fk(N)2469 578 y Fu(q)r Fm(2)p Fo(\003)2610 554 y Ft(A)2676 523 y Fu(q)2713 554 y Fw(\))f(is)g(called)g(the)h Fn(pro)s(duct)43 b(of)g(the)118 653 y(family)31 b(of)g(pro)s(cesses)p Fw(.)118 836 y Fn(Remark)g(1.2)142 b Fw(1.)41 b(Pro)r(ducts)23 b(of)i(sto)r(c)n (hastic)e(pro)r(cesses)g(as)h(in)g(De\014nition)h(1.4)f(exist.)36 b(F)-7 b(or)23 b(example)h(the)h(existence)326 935 y(of)i(the)h (non-completed)f(pro)r(duct)h(measure)e(follo)n(ws)h(from)g(standard)g (measure)f(theory)h(\(cf.)i([3)o(].\))220 1102 y(2.)41 b(F)-7 b(or)33 b(non-empt)n(y)-7 b(,)35 b(at)e(most)h(coun)n(table)f (\003)g(and)h(a)f(family)h(\(indexed)g(b)n(y)g(\003\))f(of)h(P)n 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y(;)c(k)2330 4901 y(T)p 2322 4938 V 2322 5014 a(N)2408 4840 y Fk(\023)3750 4957 y Fw(\(18\))118 5157 y(for)27 b Fs(N)32 b Ft(2)23 b Fv(N)29 b Ft(n)18 b(f)p Fw(0)p Ft(g)26 b Fw(and)h(1)c Ft(\024)g Fs(k)j Ft(\024)c Fs(N)9 b Fw(.)37 b(By)27 b(\(4\))h(w)n(e)f(ha)n(v)n(e) g(for)g Fs(i)22 b Fw(=)h(1)p Fs(;)14 b Fw(2:)995 5356 y Fs(P)26 b Fw(\()p Ft(f)p Fs(!)f Fw(:)51 b(jumps)28 b(at)g(\()p Fs(q)1717 5368 y Fu(i)1745 5356 y Fs(;)14 b(t)p Fw(\))27 b(for)g(some)g Fs(t)d Ft(2)f Fs(I)2374 5368 y Fu(N)s(;k)2490 5356 y Ft(g)p Fw(\))g(=)g(1)18 b Ft(\000)g Fs(e)2857 5322 y Fm(\000)p Fu(\025)2961 5299 y Fy(T)p 2958 5308 50 3 v 2958 5342 a(N)3750 5356 y Fw(\(19\))1987 5682 y(5)p eop %%Page: 6 6 6 5 bop 118 354 a Fw(and)28 b(so,)f(using)g(the)h(estimate)f Fs(e)1136 324 y Fu(x)1201 354 y Ft(\025)22 b Fw(1)c(+)g Fs(x)p Fw(:)648 637 y Fs(P)25 b Fw(\()q Ft(f)p Fs(!)g Fw(:)e Ft(9)p Fs(k)s Fw(;)14 b Fs(t)1083 649 y Fo(1)1120 637 y Fs(;)g(t)1187 649 y Fo(2)1247 637 y Ft(2)24 b Fs(I)1362 649 y Fu(N)s(;k)1506 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Ft(7!)g Fs(Z)2311 4357 y Fu(t)2340 4345 y Fw(\()p Fs(!)s Fw(\))118 4527 y(If)28 b Fs(!)e Ft(2)d(N)425 4539 y Fo(1)490 4527 y Fw(or)k(it)h(do)r(es)f(not)h(jump)g(at)g(\()p Fs(q)1395 4539 y Fo(0)1432 4527 y Fs(;)14 b(t)p Fw(\))28 b(for)f(an)n(y)g Fs(t)c Ft(2)h Fw(\(0)p Fs(;)14 b(T)e Fw(\))26 b(w)n(e)i(set)f Fs(Z)2514 4539 y Fn(.)2544 4527 y Fw(\()p Fs(!)s Fw(\))d(=)e(0)28 b(on)f([0)p Fs(;)14 b(T)e Fw(].)118 4627 y(Otherwise)27 b(there)g(is)h(a)f(maximal)g (sequence)1418 4826 y Fs(t)1448 4838 y Fm(\000)p Fo(1)1560 4779 y(def)1574 4826 y Fw(=)36 b Fs(T)e(>)22 b(t)1875 4838 y Fo(0)1936 4826 y Fs(>)g(t)2053 4838 y Fo(1)2114 4826 y Fs(>)g(:)14 b(:)g(:)23 b(>)g(t)2439 4841 y Fu(m)p Fo(\()p Fu(!)r Fo(\))3750 4826 y Fw(\(25\))118 4975 y(suc)n(h)k(that) 849 5175 y Fs(t)879 5187 y Fu(i)929 5128 y Fo(def)943 5175 y Fw(=)36 b(max)o Ft(f)p Fs(t)23 b Ft(2)g Fw(\(0)p Fs(;)14 b(t)1512 5187 y Fu(i)p Fm(\000)p Fo(1)1625 5175 y Fw(\))23 b(:)g Fs(!)30 b Fw(jumps)f(at)e(\()p Fs(q)2226 5187 y Fu(i)2254 5175 y Fs(;)14 b(t)p Fw(\))p Ft(g)83 b Fw(for)f(0)23 b Ft(\024)f Fs(i)h Ft(\024)g Fs(m)p Fw(\()p Fs(!)s Fw(\))p Fs(:)583 b Fw(\(26\))118 5324 y(`Maximal')23 b(means)g(that)h Fs(!)i Fw(do)r(es)d(not)h(jump)g(at)f Fs(q)1681 5339 y Fu(m)p Fo(\()p Fu(!)r Fo(\)+1)1948 5324 y Fw(in)h(the)g(time)g(in)n(terv)-5 b(al)23 b(\(0)p Fs(;)14 b(t)2802 5339 y Fu(m)p Fo(\()p Fu(!)r Fo(\))2960 5324 y Fw(\))24 b(and)f(so)g(the)h(sequence)f(\(25\))118 5424 y(cannot)31 b(b)r(e)g(extended.)48 b(\(In)n(tuitiv)n(ely)31 b(one)g(can)g(think)h(that)f(one)g(sits)g(at)g(time)g Fs(T)43 b Fw(at)31 b(site)g Fs(q)3058 5436 y Fo(0)3126 5424 y Fw(and,)h(going)e Fj(b)l(ackwar)l(ds)1987 5682 y Fw(6)p eop %%Page: 7 7 7 6 bop 118 354 a Fw(in)27 b(time,)h(w)n(aits)e(for)g(the)h(next)g (jump)h(of)f Fs(!)i Fw(at)e Fs(q)1621 366 y Fo(0)1685 354 y Fw(\(whic)n(h)g(happ)r(ens)g(at)f(time)i Fs(t)2598 366 y Fo(0)2635 354 y Fw(\),)f(then)h(jumps)f(\(instan)n(tly\))g(to)g Fs(q)3700 366 y Fo(1)3764 354 y Fw(and)118 454 y(w)n(aits)33 b(\()p 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y(extended)e(to)f(an)n(y)g(causal)g(path)g(\()p Fs(q)1241 1987 y Fo(0)1279 1975 y Fs(;)14 b(:)g(:)g(:)g(;)g(q)1501 1987 y Fu(n)1546 1975 y Fs(;)g(q)1620 1987 y Fu(n)p Fo(+1)1749 1975 y Fw(\).\))118 2074 y(W)-7 b(e)28 b(de\014ne:)243 2254 y Ft(\017)41 b Fn(P)m(ath)q Fw(\()p Fn(q)p Fs(;)14 b Fn(n)p Fs(;)g Fn(R)p Fw(\))32 b(to)g(b)r(e)g(the)h(set)f(of)g(paths)f (that)i(start)e(at)h Fs(q)s Fw(,)h(ha)n(v)n(e)e(exactly)g Fs(n)h Fw(steps)g(and)f(are)g(of)h(step)g(size)g(at)326 2354 y(most)27 b Fs(R)q Fw(.)243 2518 y Ft(\017)41 b Fn(P)m(ath)q Fw(\()p Fn(q)24 b Ft(!)f Fn(\003)p Fw(\))28 b(for)f(an)n(y)f Ft(;)d(6)p Fw(=)g(\003)f Ft(2)i Fv(Z)1520 2488 y Fu(d)1580 2518 y Fw(to)k(b)r(e)g(the)g(set)g(of)f(paths)h (starting)e(at)i Fs(q)i Fw(and)e(ending)f(in)h(\003.)243 2683 y Ft(\017)41 b Fn(P)m(ath)525 2695 y(c)571 2683 y Fw(\()p Fn(t)p Fs(;)14 b(!)s(;)g Fn(q)p Fs(;)g Fn(\003)p Fw(\))28 b(for)g Fs(q)e Ft(2)d Fw(\003)k(to)h(b)r(e)g(the)g(set)f(of)h (causal)e(wrt)i(\()p Fs(t;)14 b(!)s Fw(\))28 b(paths)f Fs(Q)c Fw(=)g(\()p Fs(q)3007 2695 y Fo(0)3067 2683 y Fw(=)g Fs(q)s(;)14 b(:)g(:)g(:)f(;)h(q)3416 2695 y Fu(n)3462 2683 y Fw(\))28 b(suc)n(h)f(that)577 2815 y(1.)99 b Fs(Q)27 b Fw(is)h(maximal)f(causal)f(and)i Fs(q)1705 2827 y Fo(0)1742 2815 y Fs(;)14 b(:)g(:)g(:)g(;)g(q)1964 2827 y Fu(n)2032 2815 y Ft(2)23 b Fw(\003)475 2914 y(or)k(2.)99 b Fs(q)778 2926 y Fo(0)815 2914 y Fs(;)14 b(:)g(:)g(:)g(;)g(q)1037 2926 y Fu(n)p Fm(\000)p Fo(1)1190 2914 y Ft(2)24 b Fw(\003)j(and)g Fs(q)1552 2926 y Fu(n)1621 2914 y Ft(2)c Fw(\003)1757 2884 y Fu(C)1812 2914 y Fw(.)243 3088 y Ft(\017)41 b Fn(P)m(ath)525 3100 y(c)571 3088 y Fw(\()p Fn(t)p Fs(;)14 b(!)s(;)g Fn(q)24 b Ft(!)f Fn(\003)1016 3058 y Fl(C)1074 3088 y Fw(\))d(for)f Fs(q)26 b Ft(2)e Fw(\003)19 b(to)h(b)r(e)g(the)g (set)f(of)h(causal)e(paths)i(\()p Fs(q)2535 3100 y Fo(0)2596 3088 y Fw(=)i Fs(q)s(;)14 b(:)g(:)g(:)g(;)g(q)2945 3100 y Fu(n)2990 3088 y Fw(\))20 b(suc)n(h)f(that)h Fs(q)3430 3100 y Fo(0)3468 3088 y Fs(;)14 b(:)g(:)g(:)f(;)h(q)3689 3100 y Fu(n)p Fm(\000)p Fo(1)3842 3088 y Ft(2)326 3187 y Fw(\003)27 b(and)g Fs(q)609 3199 y Fu(n)678 3187 y Ft(2)c Fw(\003)814 3157 y Fu(C)870 3187 y Fw(.)37 b(\(So)27 b(this)h(is)g(the)g(subset)f(of)h(elemen)n(ts)f(in)h Fn(P)m(ath)2450 3199 y(c)2497 3187 y Fw(\()p Fs(t;)14 b(!)s(;)g(q)s(;)g Fw(\003\))27 b(for)g(whic)n(h)h(case)e(2.)i (applies.\))118 3367 y Fn(Remark)j(1.4)142 b Fw(1.)41 b(W)-7 b(e)29 b(ha)n(v)n(e)f(de\014ned)h(the)g(prop)r(ert)n(y)e(of)i(b) r(eing)g(causal)e(for)h(general)g(paths)g(and)h(not)f(related)g(this) 326 3467 y(de\014nition)k(to)f(an)n(y)g(kind)h(of)g(in)n(teraction.)48 b(When)32 b(w)n(e)f(study)h(\014nite)g(range)e(in)n(teraction,)i(of)g (range)e Fs(R)i Fw(sa)n(y)-7 b(,)32 b(w)n(e)326 3566 y(will)27 b(consider)g(only)g(causal)g(paths)g(of)h(step)g(size)f(at)g (most)h Fs(R)q Fw(.)220 3731 y(2.)41 b(A)22 b(term)g(lik)n(e)f Fj(inverse)26 b(c)l(ausal)f(p)l(ath)k Fw(from)21 b Fs(a)h Fw(to)g Fs(b)g Fw(instead)g(of)g Fj(c)l(ausal)j(p)l(ath)k Fw(w)n(ould)21 b(actually)g(b)r(e)i(more)e(appropiate)326 3831 y(as)36 b(it)h(corresp)r(onds)d(to)j Fs(b)f Fw(a\013ecting)g Fs(a)h Fw(\(cf.)g(De\014nition)h(2.1\))e(but)h(not)f(necessarily)f(the) i(other)f(w)n(a)n(y)g(around.)326 3930 y(Ho)n(w)n(ev)n(er,)25 b(w)n(e)j(prefer)f(the)h(shorter)e(notion.)118 4110 y Fn(De\014nition)31 b(1.11)40 b Fw(\(cf.)29 b([3)o(]\))g(Let)f(\(\012) 1310 4122 y Fo(1)1347 4110 y Fs(;)14 b Ft(A)1450 4122 y Fo(1)1488 4110 y Fw(\))28 b(and)g(\(\012)1802 4122 y Fo(2)1840 4110 y Fs(;)14 b Ft(A)1943 4122 y Fo(2)1980 4110 y Fw(\))28 b(b)r(e)h(measurable)d(spaces.)37 b(A)29 b(map)f Fs(K)h Fw(:)23 b(\012)3358 4122 y Fo(1)3414 4110 y Ft(\002)c(A)3564 4122 y Fo(2)3625 4110 y Ft(!)24 b Fw([0)p Fs(;)14 b Fw(1])118 4210 y(is)28 b(called)f(a)g Fn(Mark)m(o)m(v)33 b(k)m(ernel)f(from)f Fw(\()p Fn(\012)1472 4222 y Fl(1)1514 4210 y Fs(;)14 b Ft(A)1617 4222 y Fl(1)1659 4210 y Fw(\))32 b Fn(to)g Fw(\()p Fn(\012)1941 4222 y Fl(2)1983 4210 y Fs(;)14 b Ft(A)2086 4222 y Fl(2)2128 4210 y Fw(\))28 b(if)g(the)g(t)n(w)n(o)f(follo)n(wing)f(conditions)h (are)g(satis\014ed:)251 4379 y Fn(MK1)99 b Fs(!)616 4391 y Fo(1)676 4379 y Ft(7!)23 b Fs(K)6 b Fw(\()p Fs(!)943 4391 y Fo(1)980 4379 y Fs(;)14 b(A)1079 4391 y Fo(2)1116 4379 y Fw(\))28 b(is)f Ft(A)1325 4391 y Fo(1)1363 4379 y Fw(-measurable)f(for)h(all)g Fs(A)2129 4391 y Fo(2)2190 4379 y Ft(2)c(A)2334 4391 y Fo(2)2372 4379 y Fw(.)251 4479 y Fn(MK2)99 b Fs(A)626 4491 y Fo(2)686 4479 y Ft(7!)23 b Fs(K)6 b Fw(\()p Fs(!)953 4491 y Fo(1)990 4479 y Fs(;)14 b(A)1089 4491 y Fo(2)1126 4479 y Fw(\))28 b(is)g(a)f(probabilit)n(y)g (measure)f(on)i Ft(A)2269 4491 y Fo(2)2334 4479 y Fw(for)f(all)g Fs(!)2628 4491 y Fo(1)2688 4479 y Ft(2)d(A)2833 4491 y Fo(1)2870 4479 y Fw(.)118 4650 y(If)k(\(\012)293 4662 y Fo(1)331 4650 y Fs(;)14 b Ft(A)434 4662 y Fo(1)471 4650 y Fw(\))23 b(=)g(\(\012)706 4662 y Fo(2)744 4650 y Fs(;)14 b Ft(A)847 4662 y Fo(2)884 4650 y Fw(\))28 b(then)g Fs(K)33 b Fw(is)28 b(called)f(a)g Fn(Mark)m(o)m(v)33 b(k)m(ernel)f(on)g Fw(\()p Fn(\012)2496 4662 y Fl(1)2538 4650 y Fs(;)14 b Ft(A)2641 4662 y Fl(1)2683 4650 y Fw(\).)118 4830 y Fn(Example)30 b(1.1)41 b Fw(Let)25 b(\()p Fs(Y)5 b(;)14 b(\045)987 4842 y Fu(Y)1044 4830 y Fw(\))26 b(b)r(e)f(a)g (metric)g(space)f(and)h Ft(B)1968 4842 y Fu(Y)2050 4830 y Fw(its)g(Borel)f Fs(\033)s Fw(-algebra.)35 b Ft(C)2831 4800 y Fo(0)2868 4830 y Fw(\()p Fs(Y)5 b(;)14 b(Y)k Fw(\))26 b(is)f(the)g(space)f(of)h(con)n(tin-)118 4930 y(uous)j(maps)h(from)f Fs(Y)48 b Fw(to)28 b Fs(Y)19 b Fw(.)40 b(It)29 b(has)g(a)f(uniform)h (metric,)g(de\014ned)g(b)n(y)f Fs(\045)2410 4945 y Fm(C)2449 4928 y Fi(0)2482 4945 y Fo(\()p Fu(Y)t(;Y)13 b Fo(\))2653 4930 y Fw(\()p Fs(g)2725 4942 y Fo(1)2762 4930 y Fs(;)h(g)2839 4942 y Fo(2)2876 4930 y Fw(\))25 b(=)g(sup)3148 4950 y Fl(y)q Fm(2)p Fl(Y)3310 4930 y Fs(\045)3353 4942 y Fu(Y)3410 4930 y Fw(\()p Fs(g)3482 4942 y Fo(1)3519 4930 y Fw(\()p Fn(y)q Fw(\))p Fs(;)14 b(g)3711 4942 y Fo(2)3749 4930 y Fw(\()p Fn(y)q Fw(\)\))118 5029 y(and)28 b(the)g(Borel)e Fs(\033)s Fw(-algebra)g Ft(B)1067 5044 y Fm(C)1106 5028 y Fi(0)1138 5044 y Fo(\()p Fu(Y)t(;Y)13 b Fo(\))1337 5029 y Fw(wrt)28 b(this)f(metric.)37 b(F)-7 b(urther)28 b(let)g(\(\012)p Fs(;)14 b Ft(A)p Fs(;)g(P)e Fw(\))28 b(b)r(e)g(a)f(probabilit)n(y)f(space)h(and)1638 5309 y Fs(S)g Fw(:)c(\012)83 b Ft(!)g(C)2120 5274 y Fo(0)2157 5309 y Fw(\()p Fs(Y)5 b(;)14 b(Y)19 b Fw(\))1372 b(\(28\))1768 5433 y Fs(!)85 b Ft(7!)e Fs(S)2122 5445 y Fu(!)1987 5682 y Fw(7)p eop %%Page: 8 8 8 7 bop 118 354 a Fw(a)27 b(measurable)g(\(wrt)g(the)h Fs(\033)s Fw(-algebras)e Ft(A)i Fw(and)f Ft(B)1661 369 y Fm(C)1700 353 y Fi(0)1732 369 y Fo(\()p Fu(Y)t(;Y)14 b Fo(\))1904 354 y Fw(\))28 b(map.)118 454 y(Then)1399 653 y Fs(K)1470 665 y Fu(S)1518 653 y Fw(\()p Fs(y)s(;)14 b(Y)1679 665 y Fo(1)1716 653 y Fw(\))1772 606 y Fo(def)1785 653 y Fw(=)36 b Fs(P)12 b Fw(\()p Ft(f)p Fs(!)25 b Fw(:)f Fs(S)2200 665 y Fu(!)2247 653 y Fw(\()p Fs(y)s Fw(\))g Ft(2)f Fs(Y)2505 665 y Fo(1)2543 653 y Ft(g)p Fw(\))1133 b(\(29\))118 798 y(for)27 b(all)g Fs(y)f Ft(2)e Fs(Y)5 b(;)14 b(Y)644 810 y Fo(1)704 798 y Ft(2)23 b(B)837 810 y Fu(Y)894 798 y Fw(,)28 b(de\014nes)g(a)f(Mark)n(o)n(v)e(k)n(ernel)i (on)g(\()p Fs(Y)5 b(;)14 b Ft(B)2126 810 y Fu(Y)2183 798 y Fw(\).)118 969 y Fn(Pro)s(of)65 b Fw(T)-7 b(o)29 b(v)n(erify)h Fn(MK1)f Fw(w)n(e)h(\014x)g(an)g Fs(Y)1427 981 y Fo(1)1491 969 y Ft(2)e(B)1629 981 y Fu(Y)1716 969 y Fw(and)i(sho)n(w)f(that)h(the)h(map)f Fs(y)f Ft(7!)f Fs(K)2855 981 y Fu(S)2902 969 y Fw(\()p Fs(y)s(;)14 b(Y)3063 981 y Fo(1)3101 969 y Fw(\))30 b(is)g(measurable.)43 b(First)118 1068 y(w)n(e)34 b(note)f(that)i Fs(S)j Fw(can)c(b)r(e)g (seen)g(as)f(a)h(measurable)e(map)i(from)g(\012)22 b Ft(\002)g Fs(Y)53 b Fw(to)34 b Fs(Y)18 b Fw(.)56 b(W)-7 b(e)35 b(write)e(it)i(as)e(the)h(comp)r(osite)g(of)118 1168 y(measurable)26 b(maps)i Fs(S)23 b Ft(\002)18 b Fw(id)996 1180 y Fu(Y)1081 1168 y Fw(and)28 b(the)g(`ev)-5 b(aluation)27 b(map':)1542 1367 y(\()p Fs(!)s(;)14 b(y)s Fw(\))22 b Ft(7!)i Fw(\()p Fs(S)1954 1379 y Fu(!)2002 1367 y Fs(;)14 b(y)s Fw(\))23 b Ft(7!)g Fs(S)2295 1379 y Fu(!)2343 1367 y Fw(\()p Fs(y)s Fw(\))p Fs(:)1276 b Fw(\(30\))118 1512 y(The)22 b(map)f Fs(S)11 b Ft(\002)6 b Fw(id)662 1524 y Fu(Y)741 1512 y Fw(is)21 b(measurable)f(b)n(y)i (assumption)e(and)i(the)f(de\014nition)h(of)g(the)f(pro)r(duct)h Fs(\033)s Fw(-algebra)d(of)j Ft(C)3474 1482 y Fo(0)3510 1512 y Fw(\()p Fs(Y)5 b(;)14 b(Y)19 b Fw(\))6 b Ft(\002)g Fs(Y)19 b Fw(.)118 1612 y(The)36 b(ev)-5 b(aluation)36 b(map)g(is)g(con)n(tin)n(uous)f(\(wrt)h(to)g(the)h(pro)r(duct)f(top)r (ology\),)h(hence)f(measurable)f(wrt)h(the)h(Borel)e Fs(\033)s Fw(-)118 1711 y(algebras.)g(So)26 b(the)h(comp)r(osite)g(in)g (\(30\))g(is)f(measurable)g(in)h(\012)17 b Ft(\002)f Fs(M)9 b Fw(.)37 b(It)27 b(follo)n(ws)f(that)h(the)g(map)g Fs(y)e Ft(7!)f Fs(P)12 b Fw(\()p Ft(f)p Fs(!)25 b Fw(:)e Fs(S)3663 1723 y Fu(!)3711 1711 y Fw(\()p Fs(y)s Fw(\))g Ft(2)118 1811 y Fs(Y)166 1823 y Fo(1)204 1811 y Ft(g)p Fw(\))k(is)h(measurable)e(\(cf.)i(Lemma)f(8.1)g(on)g(p.)h(159)f(in)g ([22)o(]\))h(and)g(so)f Fn(MK1)g Fw(holds.)118 1910 y(Next)h(w)n(e)f (sho)n(w)g Fn(MK2)p Fw(.)36 b(Consider)27 b(for)g(\014xed)g Fs(y)f Ft(2)e Fs(Y)46 b Fw(the)28 b(comp)r(osite)f(of)h(measurable)e (maps)1648 2110 y Fs(!)g Ft(7!)d Fw(\()p Fs(!)s(;)14 b(y)s Fw(\))23 b Ft(7!)g Fs(S)2212 2122 y Fu(!)2260 2110 y Fw(\()p Fs(y)s Fw(\))1382 b(\(31\))118 2255 y(that)34 b(maps)f(\012)g(to)g Fs(Y)19 b Fw(.)54 b(W)-7 b(e)34 b(see)f(that)g Fs(K)6 b Fw(\()p Fs(y)s(;)14 b Ft(\001)p Fw(\))34 b(is)f(the)g(image)g(of)g Fs(P)46 b Fw(wrt)33 b(this)g(map)h(and)f(so)g(a)f(probabilit)n(y)h(measure)118 2354 y(whic)n(h)28 b(w)n(as)e(to)i(b)r(e)g(sho)n(wn.)3808 2454 y Fh(2)118 2700 y Fn(De\014nition)j(1.12)40 b Fw(\(cf.)27 b([3)o(]\))f(Let)g Fs(K)31 b Fw(b)r(e)26 b(a)g(Mark)n(o)n(v)d(k)n (ernel)i(from)g(\(\012)2321 2712 y Fo(1)2359 2700 y Fs(;)14 b Ft(A)2462 2712 y Fo(1)2499 2700 y Fw(\))26 b(to)g(\(\012)2749 2712 y Fo(2)2786 2700 y Fs(;)14 b Ft(A)2889 2712 y Fo(2)2927 2700 y Fw(\))26 b(and)f Fs(E)3210 2670 y Fm(\003)3249 2700 y Fw(\()p Ft(A)3347 2712 y Fu(i)3375 2700 y Fw(\))h(\()p Fs(i)d Fw(=)f(1)p Fs(;)14 b Fw(2\))25 b(the)118 2799 y(set)k(of)g Ft(A)411 2811 y Fu(i)438 2799 y Fw(-measurable)f (functions)h(with)g(v)-5 b(alues)28 b(in)h([0)p Fs(;)14 b Ft(1)p Fw(].)40 b(Then)29 b Fs(K)34 b Fw(de\014nes)29 b(a)f(map)h(from)f Fs(E)3184 2769 y Fm(\003)3223 2799 y Fw(\()p Ft(A)3321 2811 y Fo(2)3359 2799 y Fw(\))h(to)f Fs(E)3588 2769 y Fm(\003)3626 2799 y Fw(\()p Ft(A)3724 2811 y Fo(1)3762 2799 y Fw(\))h(as)118 2899 y(follo)n(ws:)1385 3120 y(\()p Fs(K)6 b(f)j Fw(\)\()p Fs(!)1660 3132 y Fo(1)1697 3120 y Fw(\))1752 3073 y Fo(def)1765 3120 y Fw(=)1866 3007 y Fk(Z)1913 3196 y Fo(\012)1960 3204 y Fi(2)2010 3120 y Fs(K)d Fw(\()p Fs(!)2171 3132 y Fo(1)2208 3120 y Fs(;)14 b(d!)2340 3132 y Fo(2)2377 3120 y Fw(\))19 b Fs(f)9 b Fw(\()p Fs(!)2562 3132 y Fo(2)2599 3120 y Fw(\))1119 b(\(32\))118 3320 y(for)27 b(an)n(y)g Fs(f)32 b Ft(2)23 b Fs(E)619 3290 y Fm(\003)657 3320 y Fw(\()p Ft(A)755 3332 y Fo(2)793 3320 y Fw(\))p Fj(.)118 3495 y Fn(Example)30 b(1.2)41 b Fw(\(cf.)28 b([3]\))g(F)-7 b(or)27 b(the)h(c)n(haracteristic)d(function)j Fs(\037)2148 3507 y Fu(A)2198 3515 y Fi(2)2263 3495 y Fw(of)f(an)g Ft(A)2538 3507 y Fo(2)2576 3495 y Fw(-measurable)f(set)h Fs(A)3229 3507 y Fo(2)3295 3495 y Fw(w)n(e)g(get)1574 3694 y Fs(K)6 b(\037)1703 3706 y Fu(A)1753 3714 y Fi(2)1789 3694 y Fw(\()p Fs(!)1873 3706 y Fo(1)1910 3694 y Fw(\))23 b(=)g Fs(K)6 b Fw(\()p Fs(!)2214 3706 y Fo(1)2250 3694 y Fs(;)14 b(A)2349 3706 y Fo(2)2387 3694 y Fw(\))p Fs(:)1308 b Fw(\(33\))118 3869 y(No)n(w)27 b(w)n(e)g(consider)g(a)g(sp)r(ecial)g (case)g(of)h(Example)f(1.1.)118 4044 y Fn(Example)j(1.3)41 b Fw(Let)d Fs(S)45 b Fw(:)40 b Fs(Y)59 b Ft(!)40 b Fs(Y)57 b Fw(b)r(e)38 b(a)g(con)n(tin)n(uous)e(map)i(map)g(on)g(\()p Fs(Y)5 b(;)14 b(\045)2638 4056 y Fu(Y)2695 4044 y Fw(\))38 b(and)g(let)g(\(\012)p Fs(;)14 b Ft(A)p Fs(;)g(P)r(;)g Fw(\()p Fs(X)3492 4056 y Fu(t)3522 4044 y Fw(\))3554 4056 y Fu(t)p Fm(2)p Fu(I)3663 4044 y Fw(\))38 b(b)r(e)g(a)118 4144 y(coun)n(ting)27 b(pro)r(cess)g(with)h(v)-5 b(alues)27 b(in)h Fv(N)37 b Fw(and)28 b Fs(t)23 b Ft(2)g Fs(I)7 b Fw(.)118 4243 y(The)28 b(map)1581 4518 y Fs(S)1637 4484 y Fu(t)1632 4539 y(!)1703 4518 y Fw(:)c Fs(Y)101 b Ft(!)83 b Fs(Y)1637 b Fw(\(34\))1773 4643 y Fs(y)85 b Ft(7!)e Fs(S)2121 4608 y Fu(X)2175 4616 y Fy(t)2203 4608 y Fo(\()p Fu(!)r Fo(\))2303 4643 y Fw(\()p Fs(y)s Fw(\))p Fs(;)118 4818 y Fw(where)23 b Fs(S)410 4787 y Fu(X)464 4795 y Fy(t)491 4787 y Fo(\()p Fu(!)r Fo(\))615 4818 y Fw(denotes)g(the)g Fs(X)1118 4830 y Fu(t)1147 4818 y Fw(\()p Fs(!)s Fw(\)-th)h(iterate)e(of)i Fs(S)5 b Fw(,)23 b(is)g(w)n(ell-de\014ned)g(for)g(all)g Fs(!)j Ft(2)d Fw(\012.)36 b(F)-7 b(urther)23 b Fs(S)3233 4830 y Fu(!)3280 4818 y Fw(\()p Fs(y)s Fw(\))h(is)f(measurable)118 4917 y(wrt)28 b(\()p Fs(!)s(;)14 b(y)s Fw(\).)36 b(In)28 b(fact,)g Fs(S)872 4929 y Fu(!)948 4917 y Fw(dep)r(ends)g(just)g(on)f Fs(X)1617 4929 y Fu(t)1646 4917 y Fw(\()p Fs(!)s Fw(\))h(and)f(so)g(w)n (e)h(get)f(a)g(coun)n(table,)g(measurable)f(partition)i(of)f(\012:)1658 5192 y(\012)97 b(=)1994 5113 y Fk([)1976 5291 y Fu(n)p Fm(2)p Fp(N)2117 5192 y Fs(U)9 b Fw(\()p Fs(n)p Fw(\))1453 b(\(35\))1294 5433 y(with)83 b Fs(U)9 b Fw(\()p Fs(n)p Fw(\))1801 5386 y Fo(def)1815 5433 y Fw(=)96 b Ft(f)p Fs(!)25 b Ft(2)f Fw(\012)f(:)g Fs(X)2372 5445 y Fu(t)2401 5433 y Fw(\()p Fs(!)s Fw(\))g(=)g Fs(n)p Ft(g)1027 b Fw(\(36\))1987 5682 y(8)p eop %%Page: 9 9 9 8 bop 118 354 a Fw(W)-7 b(e)28 b(de\014ne)g(a)f(Mark)n(o)n(v)e(k)n (ernel)i(b)n(y)1321 624 y Fs(K)1398 589 y Fu(t)1392 644 y(S)1440 624 y Fw(\()p Fs(y)s(;)14 b(Y)1601 636 y Fo(1)1638 624 y Fw(\))1754 577 y Fo(def)1767 624 y Fw(=)97 b Fs(P)12 b Fw(\()p Ft(f)p Fs(!)25 b Fw(:)e Fs(S)2247 589 y Fu(t)2242 644 y(!)2290 624 y Fw(\()p Fs(y)s Fw(\))g Ft(2)h Fs(Y)2548 636 y Fo(1)2585 624 y Ft(g)p Fw(\))1091 b(\(37\))1767 765 y(=)2043 686 y Fk(X)1929 868 y Fu(n)p Fo(:)p Fu(S)2033 851 y Fy(n)2073 868 y Fo(\()p Fu(y)r Fo(\))p Fm(2)p Fu(Y)2245 876 y Fi(1)2291 765 y Fs(P)12 b Fw(\()p Fs(U)d Fw(\()p Fs(n)p Fw(\)\))1753 1037 y(=)1929 924 y Fk(Z)1975 1113 y Fo(\012)2040 1037 y Fs(dP)j Fw(\()p Fs(!)s Fw(\))i Fs(\037)2333 1049 y Fu(Y)2372 1057 y Fi(1)2427 1037 y Ft(\016)k Fs(S)2543 1003 y Fu(t)2538 1058 y(!)2586 1037 y Fw(\()p Fs(y)s Fw(\))118 1249 y(for)27 b Fs(y)f Ft(2)d Fs(Y)47 b Fw(and)27 b Fs(Y)694 1261 y Fo(1)755 1249 y Ft(2)c(B)888 1261 y Fu(Y)945 1249 y Fw(.)118 1419 y(W)-7 b(e)28 b(prepare)e(a)i(generalization)d(of)j(Example)f(1.3)f(with)i(a)g (de\014nition)g(and)f(a)g(tec)n(hnical)g(lemma.)118 1588 y Fn(De\014nition)k(1.13)40 b Fw(Let)27 b Ft(F)35 b Fw(b)r(e)28 b(the)f(set)g(of)g(\014nite)g(subsets)g(of)g Fv(Z)2149 1558 y Fu(d)2182 1588 y Fw(.)37 b(Consider)26 b(a)g(\014xed)h(\003)c Ft(2)h(F)h(n)16 b(f;g)p Fw(.)36 b(W)-7 b(e)27 b(de\014ne)g Ft(J)43 b Fw(to)118 1688 y(b)r(e)30 b(the)g(union)f(of)h(a)f(one-p)r (oin)n(t)g(set)g Ft(f)p Fs(j)1356 1700 y Fm(1)1426 1688 y Ft(g)h Fw(and)f(the)h(set)f(of)h(\014nite)g(sequences)f(\(\003)2717 1700 y Fo(1)2754 1688 y Fs(;)14 b(:)g(:)g(:)f(;)h Fw(\003)2996 1700 y Fu(n)3041 1688 y Fw(\))30 b(of)f(subsets)h(of)f(\003.)43 b(Then)118 1788 y Ft(J)g Fw(is)28 b(coun)n(table)f(and)g(w)n(e)g (consider)g(it)h(as)f(a)g(measurable)f(space,)h(established)h(with)g (the)g(discrete)f Fs(\033)s Fw(-algebra.)118 1887 y(Let)h(\(\012)p Fs(;)14 b Ft(A)p Fs(;)g(P)r(;)g Fw(\()p Fs(X)692 1899 y Fu(t)722 1887 y Fw(\))754 1899 y Fu(t)p Fm(2)p Fu(I)862 1887 y Fw(\))28 b(b)r(e)f(a)g(discrete)g(or)g(con)n(tin)n(uous)f(time)i (coun)n(ting)e(pro)r(cess)h(with)g(v)-5 b(alues)27 b(in)h Fv(N)3328 1857 y Fo(\003)3411 1887 y Fw(and)f(index-set)118 1987 y Fs(I)j Fw(=)23 b Ft(f)p Fw(1)p Fs(;)14 b(:)g(:)g(:)e(;)i(N)9 b Ft(g)27 b Fw(or)g([0)p Fs(;)14 b(T)e Fw(],)26 b(resp)r(ectiv)n(ely)-7 b(.)36 b(W)-7 b(e)28 b(de\014ne)g(a)f(map)1730 2256 y Fn(j)c Fw(:)g(\012)83 b Ft(!)g(J)1557 b Fw(\(38\))1834 2381 y Fs(!)85 b Ft(7!)e Fn(j)p Fw(\()p Fs(!)s Fw(\))118 2551 y(If)26 b Fs(X)268 2563 y Fn(.)299 2551 y Fw(\()p Fs(!)s Fw(\))g(is)f(non-decreasing,)f(has)h(only)g(\014nitely)i(man)n (y)e(jumps)h(and)f(at)h(most)f(jumps)i(of)e(size)h(1)f(then)h(w)n(e)f (de\014ne)h Fn(j)p Fw(\()p Fs(!)s Fw(\))118 2650 y(to)i(b)r(e)g(the)g (\(time-ordered\))f(sequence)g(of)g(jump)i(sets)e(of)h Fs(!)s Fw(.)36 b(Otherwise)27 b(w)n(e)g(set)h Fn(j)p Fw(\()p Fs(!)s Fw(\))23 b(=)g Fs(j)2992 2662 y Fm(1)3062 2650 y Fw(.)37 b(W)-7 b(e)28 b(de\014ne)g(for)f Fs(j)h Ft(2)c(J)15 b Fw(:)1602 2850 y Fs(U)9 b Fw(\()p Fs(j)c Fw(\))1794 2803 y Fo(def)1808 2850 y Fw(=)36 b Ft(f)p Fs(!)25 b Fw(:)e Fn(j)p Fw(\()p Fs(!)s Fw(\))h(=)e Fs(j)5 b Ft(g)1336 b Fw(\(39\))118 3019 y Fn(Lemma)30 b(1.1)40 b Fj(L)l(et)d Fw(\003)f Ft(2)g(F)c(n)23 b(f;g)36 b Fj(b)l(e)h(\014xe)l (d)f(and)i Fw(\(\012)p Fs(;)14 b Ft(A)p Fs(;)g(P)r(;)g Fw(\()p Fs(X)2176 3031 y Fu(t)2206 3019 y Fw(\))2238 3031 y Fu(t)p Fm(2)p Fu(I)2346 3019 y Fw(\))37 b Fj(a)h(c)l(ontinuous)e (time)h(c)l(ounting)g(pr)l(o)l(c)l(ess)g(with)118 3119 y(index-set)31 b Fs(I)h Fw(=)24 b([0)p Fs(;)14 b(T)e Fw(])29 b Fj(and)j(values)f(in)g Fv(N)1411 3089 y Fo(\003)1497 3119 y Fj(such)g(that)g(for)g Fs(P)12 b Fj(-a.a.)32 b Fs(!)i Fj(the)d(tr)l(aje)l(ctory)g Fs(X)2920 3131 y Fn(.)2951 3119 y Fw(\()p Fs(!)s Fw(\))g Fj(is)g(non-de)l(cr)l(e)l(asing,)h(has) 118 3219 y(only)f(\014nitely)f(many)h(jumps)g(and)f(at)h(most)f(jumps)g (of)i(size)e(1.)41 b(Then)31 b(the)g(map)g Fn(j)p Fj(,)g(as)f(de\014ne) l(d)h(in)f(De\014nition)h(1.13,)h(is)118 3318 y(me)l(asur)l(able.)118 3488 y Fn(Pro)s(of)54 b Fw(W)-7 b(e)25 b(consider)f(the)h(case)f Fs(I)30 b Fw(=)22 b([0)p Fs(;)14 b(T)e Fw(].)35 b(By)25 b(assumption)f Ft(N)35 b Fw(=)23 b Fs(U)9 b Fw(\()p Fs(j)2462 3500 y Fm(1)2533 3488 y Fw(\))25 b(is)f(measurable)g(and)h(has)f (measure)g(zero.)118 3588 y(W)-7 b(e)29 b(ha)n(v)n(e)e(to)i(sho)n(w)f (that)g Fs(U)9 b Fw(\()p Fs(j)c Fw(\))29 b(is)g(measurable)e(for)h(an)n (y)g Fs(j)h Fw(=)c(\(\003)2191 3600 y Fo(1)2228 3588 y Fs(;)14 b(:)g(:)g(:)f(;)h Fw(\003)2470 3600 y Fu(n)2515 3588 y Fw(\).)40 b(F)-7 b(or)28 b(an)n(y)g Fs(q)2955 3600 y Fo(1)2992 3588 y Fs(;)14 b(q)3066 3600 y Fo(2)3128 3588 y Ft(2)25 b Fw(\003,)k Fs(n)3368 3600 y Fo(1)3405 3588 y Fs(;)14 b(n)3492 3600 y Fo(2)3553 3588 y Ft(2)25 b Fv(N)30 b Ft(n)18 b(f)p Fw(0)p Ft(g)118 3687 y Fw(and)27 b Fs(t)c Ft(2)g Fw([0)p Fs(;)14 b(T)e Fw(])26 b(w)n(e)g(de\014ne)h Fs(A)1044 3699 y Fo(1)1082 3687 y Fw(\()p Fs(q)1151 3699 y Fo(1)1188 3687 y Fs(;)14 b(n)1275 3699 y Fo(1)1312 3687 y Fs(;)g(q)1386 3699 y Fo(2)1424 3687 y Fs(;)g(n)1511 3699 y Fo(2)1548 3687 y Fw(\))27 b(to)g(b)r(e)g(the)g(set)g(of)g(all)f Fs(!)g Ft(2)d Fw(\012)17 b Ft(n)g(N)39 b Fw(that)27 b(ha)n(v)n(e)f(at)h (least)f Fs(n)3412 3699 y Fo(1)3476 3687 y Fw(jumps)i(at)e Fs(q)3860 3699 y Fo(1)118 3787 y Fw(and)j(at)f(least)g Fs(n)628 3799 y Fo(2)694 3787 y Fw(jumps)h(at)g Fs(q)1082 3799 y Fo(2)1148 3787 y Fw(and)f(the)h Fs(n)1504 3799 y Fo(1)1541 3787 y Fw(th)g(jump)h(at)e 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y(The)j(pro)r(of)f(of)g(the)h(measurabilit)n(y)f(of)g(the)h(sets)f Fs(A)1722 4098 y Fo(1)1760 4086 y Fw(\()p Ft(\001)p Fw(\))h(uses)f (similar)g(argumen)n(ts.)1148 4285 y Fs(A)1210 4251 y Fm(\025)1266 4285 y Fw(\()p Fs(q)1335 4297 y Fo(1)1373 4285 y Fs(;)14 b(n)1460 4297 y Fo(1)1497 4285 y Fs(;)g(t)p Fw(\))1619 4238 y Fo(def)1632 4285 y Fw(=)36 b Ft(f)p Fs(!)25 b Ft(2)f Fw(\012)18 b Ft(n)g(N)36 b Fw(:)23 b Fs(\031)2266 4297 y Fu(q)2296 4305 y Fi(1)2352 4285 y Ft(\016)18 b Fs(X)2481 4297 y Fu(t)2510 4285 y Fw(\()p Fs(!)s Fw(\))23 b Ft(\025)f Fs(n)2789 4297 y Fo(1)2827 4285 y Ft(g)881 b Fw(\(40\))118 4427 y(is)27 b(the)g(set)f(of)h(all)f Fs(!)g Ft(2)d Fw(\012)17 b Ft(n)f(N)39 b Fw(that)27 b(ha)n(v)n(e)e(at)i (least)f Fs(n)1790 4439 y Fo(1)1854 4427 y Fw(jumps)h(at)f(site)h Fs(q)2389 4439 y Fo(1)2453 4427 y Fw(and)g(the)g Fs(n)2806 4439 y Fo(1)2843 4427 y Fw(th)g(of)f(these)h(jumps)g(happ)r(ens)g(at) 118 4527 y(the)h(latest)g(at)f(time)h Fs(t)p Fw(.)37 b(W)-7 b(e)28 b(de\014ne)g(the)g(set)f Fs(A)1585 4496 y Fu(<)1642 4527 y Fw(\()p Fs(q)1711 4539 y Fo(2)1748 4527 y Fs(;)14 b(n)1835 4539 y Fo(2)1872 4527 y Fs(;)g(t)p Fw(\))28 b(analogously)-7 b(.)118 4626 y Fs(A)180 4596 y Fm(\025)236 4626 y Fw(\()p Fs(q)305 4638 y Fo(1)343 4626 y Fs(;)14 b(n)430 4638 y Fo(1)467 4626 y Fs(;)g(t)p Fw(\))28 b(and)f Fs(A)817 4596 y Fu(<)874 4626 y Fw(\()p Fs(q)943 4638 y Fo(2)980 4626 y Fs(;)14 b(n)1067 4638 y Fo(2)1104 4626 y Fs(;)g(t)p Fw(\))28 b(are)f(measurable,)f(and)h(so)g (is)h Fs(A)2236 4638 y Fo(2)2273 4626 y Fw(\()p Fs(q)2342 4638 y Fo(1)2380 4626 y Fs(;)14 b(n)2467 4638 y Fo(1)2504 4626 y Fs(;)g(q)2578 4638 y Fo(2)2615 4626 y Fs(;)g(n)2702 4638 y Fo(2)2739 4626 y Fw(\))28 b(since)947 4825 y Fs(A)1009 4837 y Fo(2)1047 4825 y Fw(\()p Fs(q)1116 4837 y Fo(1)1153 4825 y Fs(;)14 b(n)1240 4837 y Fo(1)1277 4825 y Fs(;)g(q)1351 4837 y Fo(2)1388 4825 y Fs(;)g(n)1475 4837 y Fo(2)1512 4825 y Fw(\))24 b(=)1758 4747 y Fk([)1655 4928 y Fu(t)p Fm(2)p Fo([0)p Fu(;T)9 b Fo(])p Fm(\\)p Fp(Q)1967 4758 y Fk(\000)2005 4825 y Fs(A)2067 4791 y Fm(\025)2124 4825 y Fw(\()p Fs(q)2193 4837 y Fo(1)2230 4825 y Fs(;)14 b(n)2317 4837 y Fo(1)2354 4825 y Fs(;)g(t)p Fw(\))19 b Ft(\\)g Fs(A)2608 4791 y Fu(<)2664 4825 y Fw(\()p Fs(q)2733 4837 y Fo(2)2771 4825 y Fs(;)14 b(n)2858 4837 y Fo(2)2895 4825 y Fs(;)g(t)p Fw(\))2994 4758 y Fk(\001)3046 4825 y Fs(:)681 b Fw(\(41\))118 5064 y(No)n(w)31 b Fs(!)j Fw(b)r(elongs)d(to)h Fs(U)9 b Fw(\()p Fs(j)c Fw(\))31 b(if)h(and)g(only)f(if,)i(for)e(all)g(1)e Ft(\024)g Fs(k)k Ft(\024)c Fs(n)i Fw(and)g Fs(q)2409 5076 y Fo(1)2447 5064 y Fs(;)14 b(q)2521 5076 y Fo(2)2588 5064 y Ft(2)29 b Fw(\003)2730 5076 y Fu(k)2802 5064 y Fw(and)j Fs(q)3005 5076 y Fo(3)3072 5064 y Ft(2)e Fw(\003)20 b Ft(n)h Fw(\003)3356 5076 y Fu(k)3428 5064 y Fw(the)32 b(follo)n(wing)118 5164 y(holds:)243 5333 y Ft(\017)41 b Fw(If)29 b(for)f(exactly)h Fs(n)876 5345 y Fo(1)942 5333 y Fw(indices)g(1)24 b Ft(\024)h Fs(i)g Ft(\024)g Fs(k)32 b Fw(the)d(p)r(oin)n(t)g Fs(q)1990 5345 y Fo(1)2056 5333 y Fw(b)r(elongs)g(to)g(\003)2517 5345 y Fu(i)2573 5333 y Fw(and)g(for)f(exactly)g Fs(n)3201 5345 y Fo(2)3267 5333 y Fw(indices)h(1)c Ft(\024)g Fs(j)30 b Ft(\024)25 b Fs(k)326 5433 y Fw(the)j(p)r(oin)n(t)f Fs(q)722 5445 y Fo(2)787 5433 y Fw(b)r(elongs)g(to)h(\003)1245 5445 y Fu(j)1307 5433 y Fw(then)g Fs(!)e Ft(2)d Fs(A)1714 5445 y Fo(1)1752 5433 y Fw(\()p Fs(q)1821 5445 y Fo(1)1858 5433 y Fs(;)14 b(n)1945 5445 y Fo(1)1982 5433 y Fs(;)g(q)2056 5445 y Fo(2)2094 5433 y Fs(;)g(n)2181 5445 y Fo(2)2218 5433 y Fw(\).)1987 5682 y(9)p eop %%Page: 10 10 10 9 bop 243 354 a Ft(\017)41 b Fw(If)23 b(for)f(exactly)g Fs(n)857 366 y Fo(1)917 354 y Fw(indices)h(1)f Ft(\024)h Fs(i)f Ft(\024)h Fs(k)j Fw(the)d(p)r(oin)n(t)g Fs(q)1932 366 y Fo(1)1992 354 y Fw(b)r(elongs)f(to)g(\003)2439 366 y Fu(i)2489 354 y Fw(and)h(for)f(exactly)g Fs(n)3099 366 y Fo(3)3145 354 y Ft(\000)9 b Fw(1)21 b(indices)i(1)g Ft(\024)f Fs(j)28 b(<)23 b(k)326 454 y Fw(the)28 b(p)r(oin)n(t)f Fs(q)722 466 y Fo(3)787 454 y Fw(b)r(elongs)g(to)h(\003)1245 466 y Fu(j)1307 454 y Fw(then)g Fs(!)e Ft(2)d Fs(A)1714 466 y Fo(2)1752 454 y Fw(\()p Fs(q)1821 466 y Fo(1)1858 454 y Fs(;)14 b(n)1945 466 y Fo(1)1982 454 y Fs(;)g(q)2056 466 y Fo(3)2094 454 y Fs(;)g(n)2181 466 y Fo(3)2218 454 y Fw(\).)243 620 y Ft(\017)41 b Fw(If)d(for)e(exactly)h Fs(l)k Ft(\025)e Fw(0)e(indices)h(1)h Ft(\024)g Fs(n)1618 632 y Fo(1)1694 620 y Fs(<)g(n)1848 632 y Fo(2)1925 620 y Fs(<)g(:)14 b(:)g(:)39 b(<)g(n)2319 632 y Fu(l)2384 620 y Ft(\024)g Fs(n)e Fw(a)g(p)r(oin)n(t)h Fs(q)k Ft(2)e Fw(\003)d(b)r(elongs)g(to)h(\003)3628 632 y Fu(n)3669 640 y Fy(i)3736 620 y Fw(then)326 720 y Fs(!)25 b Ft(2)f(f)7 b Fw(~)-49 b Fs(!)25 b Ft(2)e Fw(\012)c Ft(n)f(N)35 b Fw(:)23 b Fs(\031)1014 732 y Fu(q)1070 720 y Ft(\016)18 b Fs(X)1199 732 y Fu(T)1251 720 y Fw(\()7 b(~)-49 b Fs(!)s Fw(\))23 b(=)g Fs(l)r Ft(g)p Fw(.)118 902 y(W)-7 b(e)28 b(see)f(that)h Fs(U)9 b Fw(\()p Fs(j)c Fw(\))28 b(is)g(the)g(in)n (tersection)e(of)i(\014nitely)g(man)n(y)f(measurable)f(sets)i(and)f (hence)h(measurable.)3808 1002 y Fh(2)118 1255 y Fn(Example)i(1.4)41 b Fw(W)-7 b(e)24 b(consider)e(a)h(generalization)f(of)h(Example)g(1.3.) 34 b(Let)24 b(\()p Fs(Y)5 b(;)14 b(\045)2631 1267 y Fu(Y)2688 1255 y Fw(\))24 b(b)r(e)g(a)f(measurable)f(space,)i(\003)f(a)g(non-)118 1355 y(empt)n(y)30 b(\014nite)h(set)f(and)h(\(\012)p Fs(;)14 b Ft(A)p Fs(;)g(P)r(;)g Fw(\()p Fs(X)1311 1367 y Fu(t)1341 1355 y Fw(\))1373 1367 y Fu(t)p Fm(2)p Fu(I)1481 1355 y Fw(\))30 b(a)g(coun)n(ting)g(pro)r(cess)f(with)i(v)-5 b(alues)30 b(in)g Fv(N)2844 1325 y Fo(\003)2930 1355 y Fw(that)g(has)g(\014nite)h(exp)r(ectation)118 1455 y(and)h(with)h Fs(P)12 b Fw(-almost)31 b(surely)g(only)h(jumps)g(of)g (size)g(at)g(most)g(1.)50 b(Let)32 b Fs(S)j Fw(=)30 b(\()p Fs(S)2635 1467 y Fo(\003)2680 1475 y Fi(1)2718 1455 y Fw(\))2750 1467 y Fo(\003)2795 1475 y Fi(1)2828 1467 y Fm(\032)p Fo(\003)2961 1455 y Fw(b)r(e)j(a)f(family)g(of)g(con)n(tin) n(uous)118 1554 y(maps)c(on)f Fs(Y)518 1524 y Fo(\003)567 1554 y Fw(,)i(suc)n(h)e(that)h Fs(S)1037 1566 y Fo(\003)1082 1574 y Fi(1)1147 1554 y Fw(c)n(hanges)f(at)g(most)h(the)g(\003)1961 1566 y Fo(1)1998 1554 y Fw(-co)r(ordinates,)f(i.e.)h(if)g Fn(y)2751 1566 y Fo(\003)2825 1554 y Ft(2)c Fs(Y)2970 1524 y Fo(\003)3048 1554 y Fw(and)k Fs(q)e Ft(2)e Fw(\003)18 b Ft(n)g Fw(\003)3546 1566 y Fo(1)3611 1554 y Fw(w)n(e)28 b(ha)n(v)n(e)118 1654 y(for)f(the)h Fs(q)s Fw(th)g(co)r(ordinate)f Fs(\031)990 1666 y Fu(q)1045 1654 y Ft(\016)18 b Fs(S)1156 1666 y Fo(\003)1201 1674 y Fi(1)1238 1654 y Fw(\()p Fn(y)1320 1666 y Fo(\003)1370 1654 y Fw(\))24 b(=)e Fs(y)1554 1666 y Fu(q)1591 1654 y Fw(.)118 1754 y(F)-7 b(or)26 b Fs(t)d Ft(2)h Fs(I)34 b Fw(and)26 b Fs(P)12 b Fw(-a.a.)26 b Fs(!)g Ft(2)d Fw(\012)k(with)g Fs(X)1377 1766 y Fu(t)1406 1754 y Fw(\()p Fs(!)s Fw(\))d Ft(2)f Fv(N)1681 1723 y Fo(\003)1763 1754 y Fw(w)n(e)k(ha)n(v)n(e)f(a)g(\014nite)h(sequence)g (of)g(jump-sets)g Fn(j)p Fw(\()p Fs(!)s Fw(\))c(=)g(\(\003)3518 1766 y Fo(1)3555 1754 y Fs(;)14 b(:)g(:)g(:)g(;)g Fw(\003)3798 1766 y Fu(n)3842 1754 y Fw(\),)118 1853 y(as)27 b(de\014ned)h(in)g (De\014nition)g(1.13,)e(and)i(it)g(dep)r(ends)g(measurably)e(on)h Fs(!)s Fw(,)h(as)f(w)n(as)f(sho)n(wn)h(in)h(Lemma)f(1.1.)36 b(W)-7 b(e)28 b(de\014ne)1157 2135 y Fs(S)1213 2101 y Fu(t)1208 2156 y(!)1279 2135 y Fw(:)83 b Fs(Y)1452 2105 y Fo(\003)1584 2135 y Ft(!)23 b Fs(Y)1757 2101 y Fo(\003)3750 2135 y Fw(\(42\))1393 2285 y Fn(y)1443 2297 y Fo(\003)1584 2285 y Ft(7!)g Fs(S)1741 2300 y Fl(j)p Fo(\()p Fu(!)r Fo(\))1864 2285 y Fw(\()p Fn(y)1946 2297 y Fo(\003)1997 2285 y Fw(\))2052 2238 y Fo(def)2065 2285 y Fw(=)36 b Fs(S)2217 2297 y Fo(\003)2262 2305 y Fy(n)2326 2285 y Ft(\016)18 b Fs(:)c(:)g(:)k Ft(\016)g Fs(S)2612 2297 y Fo(\003)2657 2305 y Fi(1)2694 2285 y Fw(\()p Fn(y)2776 2297 y Fo(\003)2827 2285 y Fw(\))891 b(\(43\))118 2468 y(W)-7 b(e)28 b(get)f(a)h(represen)n(tation)e(of)h Fs(K)1183 2438 y Fu(t)1177 2491 y(S)1225 2468 y Fw(\()p Fn(y)1307 2480 y Fo(\003)1357 2468 y Fs(;)14 b(Y)1442 2480 y Fo(1)1479 2468 y Fw(\),)28 b(similar)f(to)h(the)g(one)f(in)h(\(37\):)1279 2750 y Fs(K)1356 2716 y Fu(t)1350 2771 y(S)1397 2750 y Fw(\()p Fn(y)1479 2762 y Fo(\003)1530 2750 y Fs(;)14 b(Y)1615 2762 y Fo(1)1652 2750 y Fw(\))83 b(=)g Fs(P)12 b Fw(\()p Ft(f)p Fs(!)25 b Fw(:)e Fs(S)2233 2716 y Fu(t)2228 2771 y(!)2276 2750 y Fw(\()p Fn(y)2358 2762 y Fo(\003)2409 2750 y Fw(\))g Ft(2)g Fs(Y)2590 2762 y Fo(1)2628 2750 y Ft(g)p Fw(\))1048 b(\(44\))1767 2917 y(=)1915 2804 y Fk(Z)1961 2993 y Fo(\012)2026 2917 y Fs(dP)12 b Fw(\()p Fs(!)s Fw(\))i Fs(\037)2319 2929 y Fu(Y)2358 2937 y Fi(1)2413 2917 y Ft(\016)k Fs(S)2529 2883 y Fu(t)2524 2938 y(!)2572 2917 y Fw(\()p Fn(y)2654 2929 y Fo(\003)2705 2917 y Fw(\))1767 3113 y(=)2091 3034 y Fk(X)1915 3216 y Fu(j)s Fm(2J)12 b Fo(:)p Fu(S)2107 3224 y Fy(j)2137 3216 y Fo(\()p Fl(y)2203 3224 y Fi(\003)2245 3216 y Fo(\))p Fm(2)p Fu(Y)2355 3224 y Fi(1)2401 3113 y Fs(P)g Fw(\()p Fs(U)d Fw(\()p Fs(j)c Fw(\)\))118 3402 y(for)27 b Fn(y)295 3414 y Fo(\003)368 3402 y Ft(2)d Fs(Y)513 3372 y Fo(\003)591 3402 y Fw(and)j Fs(Y)800 3414 y Fo(1)860 3402 y Ft(2)939 3340 y Fk(N)1031 3427 y Fu(q)r Fm(2)p Fo(\003)1172 3402 y Ft(Y)7 b Fw(.)118 3595 y(W)-7 b(e)28 b(ha)n(v)n(e)f(seen)g(in)h(Example)e(1.4)h(that)h Fs(S)1444 3565 y Fu(t)1439 3616 y(!)1515 3595 y Fw(dep)r(ends)g(on)f Fn(j)p Fw(\()p Fs(!)s Fw(\))h(only)-7 b(.)118 3695 y(As)36 b(w)n(e)f(are)g(in)n(terested)g(in)h(spatially)e(extended)i(systems)f (w)n(e)h(need)f(some)g(de\014nitions)h(and)f(facts)h(ab)r(out)f (in\014nite-)118 3795 y(dimensional)27 b(systems.)118 3977 y Fn(De\014nition)k(1.14)40 b Fs(S)831 3947 y Fo(1)892 3977 y Fw(is)25 b(the)f(one-dimensional)f(sphere.)35 b(W)-7 b(e)25 b(de\014ne)f(it)h(to)f(b)r(e)h(isometric)e(as)h (Riemannian)g(manifold)118 4077 y(to)30 b Fv(R)p Fs(=)p Fw(2)p Fs(\031)s Fv(Z)p Fw(.)43 b(This)30 b(de\014nes)g(in)g (particular)f(a)h(metric)g Fs(\045)1866 4092 y Fu(S)1910 4075 y Fi(1)1976 4077 y Fw(on)g Fs(S)2150 4047 y Fo(1)2217 4077 y Fw(and)g(also)f(the)h(normalized)f(Leb)r(esgue)h(measure)f(on) 118 4177 y(the)f(\(completed\))g(Borel)f Fs(\033)s Fw(-algebra.)118 4276 y(The)h(diameter)f(of)g Fs(S)785 4246 y Fo(1)850 4276 y Fw(is)1586 4475 y Fs(c)1622 4487 y Fu(s)1681 4428 y Fo(def)1694 4475 y Fw(=)36 b(diam)1975 4487 y Fu(\045)2009 4507 y Fy(S)2047 4495 y Fi(1)2088 4475 y Fw(\()p Fs(S)2176 4441 y Fo(1)2213 4475 y Fw(\))23 b(=)g Fs(\031)s(:)1321 b Fw(\(45\))118 4625 y(\(It)22 b(seems)g(a)f(bit)h(redundan)n(t)f(to)h (in)n(tro)r(duce)f(the)h(constan)n(t)f Fs(c)2005 4637 y Fu(S)2075 4625 y Fw(instead)h(of)f(using)g Fs(\031)26 b Fw(in)21 b(the)i(follo)n(wing.)33 b(But)22 b(w)n(e)g(indicate)118 4724 y(that)k(the)g(pro)r(ofs)e(in)i(Section)f(2)g(w)n(ork)f(if)i Fs(S)1465 4694 y Fo(1)1528 4724 y Fw(is)f(replaced)g(b)n(y)g(an)n(y)f (compact)h(Riemannian)h(manifold)f(or)f(more)h(general)118 4824 y(b)n(y)30 b(a)g(b)r(ounded)h(metric)g(space)e(with)i(a)f(Borel)g (probabilit)n(y)f(measure.)44 b(F)-7 b(urther)31 b(w)n(e)f(use)g(the)h (letter)f(`)p Fs(\031)s Fw(')h(as)f(notation)118 4924 y(for)d(pro)5 b(jections.\))118 5023 y(W)-7 b(e)28 b(set)1777 5223 y Fs(M)1889 5176 y Fo(def)1903 5223 y Fw(=)36 b(\()p Fs(S)2092 5188 y Fo(1)2129 5223 y Fw(\))2161 5188 y Fp(Z)2206 5163 y Fy(d)3750 5223 y Fw(\(46\))118 5372 y(and)28 b(giv)n(e)e(it)i (the)g(pro)r(duct)g(top)r(ology)e(and)i(pro)r(duct)f(Leb)r(esgue)g (measure)g(on)g(the)h(\(completed\))g(Borel)f Fs(\033)s Fw(-algebra.)1966 5682 y(10)p eop %%Page: 11 11 11 10 bop 118 354 a Fw(F)-7 b(or)27 b(\003)c Ft(\032)f Fv(Z)497 324 y Fu(d)557 354 y Fw(w)n(e)27 b(denote)h(b)n(y)f Fs(\031)1109 366 y Fo(\003)1187 354 y Fw(the)h(pro)5 b(jection)26 b(on)i(the)g(\003-co)r(ordinates.)118 454 y(Note)h(that)g(the)f(pro)r(duct)h(of)g(the)f(Borel)g Fs(\033)s Fw(-algebras)e(is)j(the)g(same)f(as)g(the)h(Borel)e Fs(\033)s Fw(-algebra)g(for)h(the)h(pro)r(duct)f(space.)118 554 y Fs(M)36 b Fw(is)28 b(compact)f(and)g(metrizable)g(in)h(the)g (follo)n(wing)f(w)n(a)n(y:)118 736 y Fn(De\014nition)k(1.15)40 b Fw(Let)28 b(\()p Fs(b)p Fw(\()p Fs(q)s Fw(\)\))1128 753 y Fu(q)r Fm(2)p Fp(Z)1251 736 y Fy(d)h Fw(b)r(e)f(a)g(family)f(of)h (p)r(ositiv)n(e)f(n)n(um)n(b)r(ers)g(suc)n(h)g(that)1677 935 y(lim)1644 989 y Fu(R)p Fm(!1)1879 935 y Fw(sup)1840 1009 y Fm(k)p Fu(q)r Fm(k\025)p Fu(R)2057 935 y Fs(b)p Fw(\()p Fs(q)s Fw(\))c(=)g(0)p Fs(:)1377 b Fw(\(47\))118 1152 y(Then)28 b(the)g(metric)f Fs(\045)779 1164 y Fu(M)881 1152 y Fw(on)g Fs(M)9 b Fw(,)27 b(asso)r(ciated)g(to)g(\()p Fs(b)p Fw(\()p Fs(q)s Fw(\)\))1837 1169 y Fu(q)r Fm(2)p Fp(Z)1960 1152 y Fy(d)r Fw(,)h(is)f(de\014ned)h(b)n(y)1420 1352 y Fs(\045)1463 1364 y Fu(M)1536 1352 y Fw(\()p Fn(x)p Fs(;)14 b Fn(y)q Fw(\))1763 1304 y Fo(def)1776 1352 y Fw(=)49 b(sup)1877 1427 y Fu(q)r Fm(2)p Fp(Z)1999 1410 y Fy(d)2056 1352 y Fs(b)p Fw(\()p Fs(q)s Fw(\))14 b Fs(\045)2253 1367 y Fu(S)2297 1350 y Fi(1)2333 1352 y Fw(\()p Fs(x)2412 1364 y Fu(q)2450 1352 y Fs(;)g(y)2528 1364 y Fu(q)2564 1352 y Fw(\))1154 b(\(48\))118 1562 y(for)27 b Fn(x)p Fs(;)14 b Fn(y)25 b Ft(2)f Fs(M)9 b Fw(.)118 1745 y Fn(Remark)31 b(1.5)142 b Fw(1.)41 b(One)31 b(can)g(easily)f(sho)n(w)h(that)h Fs(\045)1856 1757 y Fu(M)1929 1745 y Fw(,)h(as)d(de\014ned)i(in)f (De\014nition)h(1.15,)f(is)h(in)f(fact)h(a)f(metric)g(and)326 1845 y(also)26 b(compatible)i(with)g(the)g(pro)r(duct)f(top)r(ology)-7 b(.)220 2011 y(2.)41 b(A)27 b(sequence)f(\()p Fn(x)840 1980 y Fo(\()p Fu(n)p Fo(\))938 2011 y Fw(\))970 2023 y Fu(n)p Fm(2)p Fp(N)1129 2011 y Fw(in)h Fs(M)35 b Fw(con)n(v)n(erges) 24 b(wrt)i(the)i(pro)r(duct)e(top)r(ology)g(i\013)h(it)g(con)n(v)n (erges)d(wrt)j(to)f(eac)n(h)g(co)r(ordi-)326 2126 y(nate,)h(i.e.)h(\()p Fs(x)746 2083 y Fo(\()p Fu(n)p Fo(\))746 2136 y Fu(q)844 2126 y Fw(\))876 2138 y Fu(n)p Fm(2)p Fp(N)1036 2126 y Fw(con)n(v)n(erges)d(for)i(ev)n(ery)f Fs(q)g Ft(2)e Fv(Z)1957 2096 y Fu(d)1990 2126 y Fw(.)37 b(The)27 b(same)g(holds)g (also)g(for)g(nets)h(\()p Fn(x)3197 2096 y Fo(\003)3247 2126 y Fw(\))3279 2138 y Fo(\003)p Fm(2F)3427 2126 y Fw(.)220 2292 y(3.)41 b(The)24 b(pro)r(duct)g(top)r(ology)e(do)r(es)i (not)g(distinguish)g(an)n(y)f(particular)g(sites)g(despite)h(the)h(the) f(fact)g(that)g(the)h(w)n(eigh)n(ts)326 2392 y Fs(b)p Fw(\()p Fs(q)s Fw(\))h(dep)r(end)g(on)f Fs(q)s Fw(.)37 b(Spatial)25 b(shifts,)i(lik)n(e)e Fn(x)e Ft(7!)1846 2391 y Fw(~)1841 2392 y Fn(x)k Fw(with)k(~)-47 b Fs(x)2152 2404 y Fu(q)2212 2392 y Fw(=)22 b Fs(x)2346 2404 y Fu(q)r Fm(\000)p Fu(r)2494 2392 y Fw(for)j(some)g Fs(r)h Ft(2)d Fv(Z)3028 2362 y Fu(d)3060 2392 y Fw(,)j(are)f(homeomorphisms.)220 2558 y(4.)41 b(The)27 b(space)g Ft(C)767 2528 y Fo(0)804 2558 y Fw(\()p Fs(M)t(;)14 b(M)9 b Fw(\))28 b(of)f(con)n(tin)n(uous)g (maps)g(on)g(\()p Fs(M)t(;)14 b(\045)2147 2570 y Fu(M)2221 2558 y Fw(\))28 b(is)g(complete)f(wrt)h(the)g(metric)f(de\014ned)h(b)n (y)1420 2791 y Fs(\045)1463 2806 y Fm(C)1502 2789 y Fi(0)1534 2806 y Fo(\()p Fu(M)s(;M)6 b Fo(\))1746 2791 y Fw(\()p Fs(f)t(;)14 b(g)s Fw(\))1958 2744 y Fo(def)1971 2791 y Fw(=)51 b(sup)2072 2860 y Fl(x)p Fm(2)p Fu(M)2240 2791 y Fs(\045)2283 2803 y Fu(M)2357 2791 y Fw(\()p Fs(f)9 b Fw(\()p Fn(x)p Fw(\))p Fs(;)14 b(g)s Fw(\()p Fn(x)p Fw(\)\))p Fs(:)948 b Fw(\(49\))326 3022 y(W)-7 b(e)28 b(denote)f(b)n(y)g Ft(B)906 3037 y Fm(C)945 3020 y Fi(0)977 3037 y Fo(\()p Fu(M)s(;M)6 b Fo(\))1216 3022 y Fw(the)28 b(Borel)f Fs(\033)s Fw(-algebra)f(wrt)h(this)h(metric.)118 3205 y Fn(Lemma)i(1.2)40 b Fj(L)l(et)29 b Fw(\(\012)p Fs(;)14 b Ft(A)p Fw(\))31 b Fj(b)l(e)f(a)g(me)l(asur)l(able)g(sp)l(ac)l (e)h(and)f Fw(\()p Fs(f)2078 3174 y Fo(\003)2127 3205 y Fw(\))2159 3220 y Fo(\003)p Fm(2F)6 b(nf;g)2473 3205 y Fj(b)l(e)30 b(a)g(net)f(of)i(me)l(asur)l(able)f(maps)1588 3487 y Fs(f)1638 3453 y Fo(\003)1710 3487 y Fw(:)24 b(\012)83 b Ft(!)g(C)2115 3453 y Fo(0)2151 3487 y Fw(\()p Fs(M)t(;)14 b(M)9 b Fw(\))1323 b(\(50\))1762 3611 y Fs(!)86 b Ft(7!)d Fs(f)2116 3577 y Fo(\003)2107 3632 y Fu(!)118 3794 y Fj(such)22 b(that)f(for)i(al)t(l)f Fw(\003)753 3806 y Fo(1)813 3794 y Ft(2)i(F)9 b(n)q(f;g)18 b Fj(and)k Fs(!)k Ft(2)d Fw(\012)f Fj(the)f(net)g Fw(\()p Fs(\031)1881 3806 y Fo(\003)1926 3814 y Fi(1)1964 3794 y Ft(\016)q Fs(f)2057 3764 y Fo(\003)2048 3815 y Fu(!)2106 3794 y Fw(\))2138 3806 y Fo(\003)2183 3814 y Fi(1)2216 3806 y Fm(\032)p Fo(\003)p Fm(2F)2437 3794 y Fj(c)l(onver)l(ges)h(\(as)g Fw(\003)g Ft(!)h Fv(Z)3179 3764 y Fu(d)3211 3794 y Fj(\))f(in)f Ft(C)3409 3764 y Fo(0)3446 3794 y Fw(\()p Fs(M)t(;)14 b Fw(\()p Fs(S)3688 3764 y Fo(1)3726 3794 y Fw(\))3758 3764 y Fo(\003)3803 3772 y Fi(1)3840 3794 y Fw(\))p Fj(,)118 3894 y(say)31 b(to)e Fs(\031)411 3906 y Fo(\003)456 3914 y Fi(1)512 3894 y Ft(\016)18 b Fs(f)613 3906 y Fu(!)661 3894 y Fj(.)118 3993 y(Then)1523 4193 y Fs(f)1564 4205 y Fu(!)r(;q)1664 4193 y Fw(\()p Fn(x)p Fw(\))1802 4145 y Fo(def)1816 4193 y Fw(=)71 b(lim)1917 4252 y Fo(\003)p Fm(!)p Fp(Z)2073 4235 y Fy(d)2116 4193 y Fs(\031)2163 4205 y Fu(q)2218 4193 y Ft(\016)18 b Fs(f)2328 4158 y Fo(\003)2319 4213 y Fu(!)2377 4193 y Fw(\()p Fn(x)p Fw(\))1259 b(\(51\))118 4376 y Fj(de\014nes)30 b(a)g(me)l(asur)l(able)g(map)1613 4658 y Fs(f)i Fw(:)23 b(\012)83 b Ft(!)g(C)2090 4624 y Fo(0)2127 4658 y Fw(\()p Fs(M)t(;)14 b(M)9 b Fw(\))1347 b(\(52\))1737 4783 y Fs(!)86 b Ft(7!)d Fs(f)2082 4795 y Fu(!)118 4965 y Fj(whose)31 b Fs(q)s Fj(th)f(c)l(o)l(or)l(dinate)h (function)e(is)h(given)h(by)f(\(51\).)1966 5682 y Fw(11)p eop %%Page: 12 12 12 11 bop 118 354 a Fn(Pro)s(of)58 b Fw(Fix)27 b Fs(!)f Ft(2)d Fw(\012,)k Fn(x)d Ft(2)f Fs(M)36 b Fw(and)26 b(a)h(metric)f Fs(\045)1620 366 y Fu(M)1721 354 y Fw(lik)n(e)h(in)g(De\014nition)g (1.15)f(.)36 b(W)-7 b(e)28 b(sho)n(w)e(that)h Fs(f)3156 366 y Fu(!)3230 354 y Fw(is)g(con)n(tin)n(uous)f(in)h Fn(x)p Fw(.)118 454 y(F)-7 b(or)27 b(that)h(let)g Fs(\017)23 b(>)f Fw(0)27 b(and)h(c)n(ho)r(ose)e Fs(R)1268 466 y Fo(0)1329 454 y Ft(2)d Fv(N)38 b Fw(suc)n(h)27 b(that)1823 653 y Fs(c)1859 665 y Fu(s)1908 653 y Fs(b)p Fw(\()p Fs(q)s Fw(\))c Fs(<)g(\017)1557 b Fw(\(53\))118 803 y(for)32 b(all)h Fs(q)i Fw(with)e Ft(k)p Fs(q)s Ft(k)e Fs(>)g(R)951 815 y Fo(0)988 803 y Fw(.)52 b(W)-7 b(e)33 b(note)g(that)g(the)g Fs(q)s Fw(th)g(co)r(ordinate)e(function)i(of)g Fs(\031)2775 815 y Fo(\003)2820 823 y Fi(1)2879 803 y Ft(\016)21 b Fs(f)2983 815 y Fu(!)3063 803 y Ft(2)31 b(C)3198 773 y Fo(0)3235 803 y Fw(\()p Fs(M)t(;)14 b Fw(\()p Fs(S)3477 773 y Fo(1)3515 803 y Fw(\))3547 773 y Fo(\003)3592 781 y Fi(1)3629 803 y Fw(\))33 b(is)f(the)118 902 y(same)27 b(as)g(the)h Fs(q)s Fw(th)g(co)r(ordinate)f(function)h Fs(f)1492 914 y Fu(!)r(;q)1619 902 y Fw(of)g Fs(f)1755 914 y Fu(!)1803 902 y Fw(.)118 1002 y(By)f(con)n(tin)n(uit)n(y)g(of)h Fs(\031)782 1014 y Fu(B)832 1022 y Fy(R)875 1034 y Fi(0)934 1002 y Ft(\016)18 b Fs(f)1035 1014 y Fu(!)1111 1002 y Fw(w)n(e)27 b(can)g(c)n(ho)r(ose)g(a)g Fs(\016)f(>)c Fw(0)28 b(suc)n(h)f(that)h(for)f(all)g Fn(y)e Ft(2)e Fs(B)2763 1014 y Fu(\016)2800 1002 y Fw(\()p Fn(x)p Fw(\))28 b(and)g(all)f Fs(q)k Fw(with)d Ft(k)p Fs(q)s Ft(k)22 b(\024)h Fs(R)3773 1014 y Fo(0)3810 1002 y Fw(:)1435 1201 y Fs(c)1471 1213 y Fu(s)1521 1201 y Fs(b)p Fw(\()p Fs(q)s Fw(\))14 b Fs(\045)1718 1216 y Fu(S)1762 1199 y Fi(1)1798 1201 y Fw(\()p Fs(f)1871 1213 y Fu(!)r(;q)1971 1201 y Fw(\()p Fn(x)p Fw(\))p Fs(;)g(f)2163 1213 y Fu(!)r(;q)2264 1201 y Fw(\()p Fn(y)q Fw(\)\))25 b Fs(<)e(\017:)1169 b Fw(\(54\))118 1351 y(F)-7 b(rom)27 b(\(53\))h(and)f(\(54\))g(w)n(e)g (conclude)h(that)g(for)f(all)g Fn(y)e Ft(2)e Fs(B)1950 1363 y Fu(\016)1987 1351 y Fw(\()p Fn(x)p Fw(\))1622 1550 y Fs(\045)1665 1562 y Fu(M)1739 1550 y Fw(\()p Fs(f)1812 1562 y Fu(!)1860 1550 y Fw(\()p Fn(x)p Fw(\))p Fs(;)14 b(f)2052 1562 y Fu(!)2100 1550 y Fw(\()p Fn(y)q Fw(\)\))25 b Fs(<)e(\017)1356 b Fw(\(55\))118 1699 y(whic)n(h)29 b(w)n(as)f(to)g(b)r(e)i(sho)n(wn.)40 b(Finally)28 b Fs(f)38 b Fw(dep)r(ends)29 b(measurably)f(on)g Fs(!)k Fw(b)r(ecause)c(it)i(is)e (p)r(oin)n(t)n(wise)h(limit)g(of)g(measurable)118 1799 y(functions)f(with)g(v)-5 b(alues)27 b(in)h(a)f(metric)h(space)f(\(cf.) h([22)o(],)g(p)g(117,)e(for)h(example\).)3808 1899 y Fh(2)118 2152 y Fn(Remark)k(1.6)142 b Fw(1.)41 b(Lemma)20 b(1.2)f(is)g(in)i(particular)d(based)h(on)h(the)g(compactness)f(on)h Fs(M)28 b Fw(wrt)20 b(the)g(pro)r(duct)g(toplology)-7 b(.)326 2285 y Fs(M)36 b Fw(is)27 b(not)h(compact)f(wrt)h(the)g (di\013eren)n(t)f(metric,)h(de\014ned)g(b)n(y)1616 2517 y(~)-49 b Fs(\045)1652 2529 y Fu(M)1725 2517 y Fw(\()p Fn(x)p Fs(;)14 b Fn(y)q Fw(\))1952 2470 y Fo(def)1965 2517 y Fw(=)36 b(sup)2068 2587 y Fu(q)r Fm(2)p Fo(\003)2205 2517 y Fs(\045)2248 2532 y Fu(S)2292 2515 y Fi(1)2329 2517 y Fw(\()p Fs(x)2408 2529 y Fu(q)2445 2517 y Fs(;)14 b(y)2523 2529 y Fu(q)2559 2517 y Fw(\))p Fs(:)326 2788 y Fw(In)27 b(this)h(case)f(the)h(conclusion)f(from)g(`lo)r(cal')g(to)h (`global')e(do)r(es)h(not)h(hold.)220 2954 y(2.)41 b(As)32 b Fs(f)40 b Fw(in)33 b(\(52\))e(is)h(\()p Ft(A)p Fs(;)14 b Ft(B)1093 2969 y Fm(C)1132 2953 y Fi(0)1165 2969 y Fo(\()p Fu(M)s(;M)6 b Fo(\))1376 2954 y Fw(\)-measurable,)32 b(the)g(map)g(\()p Fs(!)s(;)14 b Fn(x)p Fw(\))31 b Ft(7!)f Fs(f)2626 2966 y Fu(!)2674 2954 y Fw(\()p Fn(x)p Fw(\))j(is)f(\()p Ft(A)22 b(\002)f(B)3170 2966 y Fu(M)3243 2954 y Fs(;)14 b Ft(B)3335 2966 y Fu(M)3408 2954 y Fw(\)-measurable.)326 3054 y(W)-7 b(e)28 b(ha)n(v)n(e)e(pro)n(v)n(ed)g(this)i(fact)g(in)g (Example)e(1.1.)118 3329 y Fx(2)135 b(In\014nite-Dimensional)47 b(Systems)118 3510 y Fw(In)23 b(Example)f(1.4)g(w)n(e)g(used)h(a)f (coun)n(ting)g(pro)r(cess)g(with)h(v)-5 b(alues)22 b(in)h Fv(N)2243 3480 y Fo(\003)2321 3510 y Fw(\(for)g(\014nite)g(\003\))g (and)f(a)h(family)g(of)f(up)r(dating-maps)118 3610 y(on)30 b Fs(Y)302 3580 y Fo(\003)382 3610 y Fw(to)f(de\014ne)h(Mark)n(o)n(v)e (k)n(ernels)h(on)g(the)h(pro)r(duct)g Fs(Y)1955 3580 y Fo(\003)2004 3610 y Fw(.)44 b(These)29 b(k)n(ernels)g(act)h(on)f(the) h(pro)r(duct)g(space)f Ft(C)3583 3580 y Fo(0)3620 3610 y Fw(\()p Fs(Y)3719 3580 y Fo(\003)3769 3610 y Fw(\))h(of)118 3710 y(con)n(tin)n(uous)36 b(functions)h(\(cf.)g(De\014nition)g(1.12)e (and)i(Prop)r(osition)e(2.4\).)63 b(In)37 b(view)f(of)h(spatially)e (extended)i(systems)118 3809 y(lik)n(e)28 b(coupled)h(map)f(lattices)h (or)f(in)n(teracting)g(particle)g(systems)g(w)n(e)g(w)n(ould)g(lik)n(e) h(to)f(de\014ne)h(analogous)e(op)r(erators)f(for)118 3909 y(in\014nite-dimensional)20 b(systems)h(\(\003)i(=)f Fv(Z)1412 3879 y Fu(d)1445 3909 y Fw(\).)35 b(As)21 b(coun)n(ting)f (pro)r(cess)f(w)n(e)h(tak)n(e)g(the)h(P)n(oisson)d(pro)r(cess)i(\(\012) p Fs(;)14 b Ft(A)p Fs(;)g(P)r(;)g Fw(\()p Fs(X)3689 3921 y Fu(t)3719 3909 y Fw(\))3751 3921 y Fu(t)p Fm(\025)p Fo(0)3865 3909 y Fw(\))118 4024 y(with)28 b(parameter)e Fs(\025)e(>)e Fw(0)28 b(and)f(v)-5 b(alues)27 b(in)h Fv(N)1492 3994 y Fp(Z)1543 3969 y Fy(d)1576 4024 y Fw(.)118 4124 y(Recall)23 b(that)g(the)g(set)g Ft(N)870 4136 y Fo(1)907 4124 y Fw(,)h(de\014ned)g(in)f(De\014nition)g(1.6,)g(of)g(all) g Fs(!)j Ft(2)d Fw(\012)g(suc)n(h)g(that)g Fs(X)2729 4136 y Fn(.)2759 4124 y Fw(\()p Fs(!)s Fw(\))g(is)g(not)g (nondecreasing,)f(jumps)118 4224 y(at)31 b(time)h(0,)h(has)e(sim)n (ultaneous)f(jumps)i(or)f(jumps)h(of)g(size)f(greater)e(than)j(one,)g (has)f Fs(P)12 b Fw(-measure)30 b(zero.)48 b(So)31 b(w)n(e)g(ha)n(v)n (e)118 4323 y(to)d(consider)f(up)r(datings)h(only)f(at)h(single)g (sites.)37 b(They)28 b(are)f(giv)n(en)g(b)n(y)h(a)f(family)h(of)g(con)n (tin)n(uous)f(maps)h(\()p Fs(S)3486 4335 y Fu(q)3523 4323 y Fw(\))3555 4340 y Fu(q)r Fm(2)p Fp(Z)3677 4324 y Fy(d)3738 4323 y Fw(suc)n(h)118 4423 y(that)k Fs(S)353 4435 y Fu(q)418 4423 y Fw(:)e Fs(M)37 b Ft(!)30 b Fs(M)40 b Fw(c)n(hanges)29 b(only)i(the)h Fs(q)s Fw(th)g(co)r(ordinate)e(\(cf.) i(Example)e(1.4)h(for)g(a)f(de\014nition.\))49 b(W)-7 b(e)32 b(remark)d(that)118 4523 y(suc)n(h)e(a)h(family)f(naturally)g (giv)n(es)f(rise)h(to)h(up)r(datings)f(at)h(more)f(than)g(one)g(p)r (oin)n(t)h(at)g(the)g(same)f(time.)118 4622 y(A)j(problem)f(is)h(ob)n (viously)e(that)i(the)h(P)n(oisson)c(pro)r(cess,)j(restricted)f(to)g (an)n(y)g(\014nite)i(in)n(terv)-5 b(al)29 b([0)p Fs(;)14 b(t)p Fw(])29 b(of)h(length)g Fs(t)c(>)g Fw(0)k(is)118 4722 y(not)23 b(of)g(\014nite)g(exp)r(ectation)g(\(cf.)h(De\014nition)f (1.1)g(and)f(Remark)g(1.3.1\).)35 b Fs(P)12 b Fw(-a.s.)22 b(there)g(are)g(in\014nitely)i(man)n(y)e(jumps)i(and)118 4821 y(it)h(is)f(ev)n(en)g(imp)r(ossible)g(to)g(de\014ne)h(an)f(order)f (preserving)f(bijection)j(b)r(et)n(w)n(een)f(them)h(and)f Fv(N)t Fw(.)42 b(Ho)n(w)n(ev)n(er,)23 b(in)i(Subsection)118 4921 y(2.1)30 b(w)n(e)h(will)g(sho)n(w)f(for)h(systems)f(with)i (\014nite)g(range)d(in)n(teraction)h(that)i(for)e Fs(P)12 b Fw(-a.a.)30 b Fs(!)h Ft(2)f Fw(\012,)i(an)n(y)e Fs(q)i Ft(2)d Fv(Z)3475 4891 y Fu(d)3539 4921 y Fw(and)i Fs(t)e(>)f Fw(0)118 5021 y(the)i(site)f Fs(q)k Fw(is)c(a\013ected)g(in)h([0)p Fs(;)14 b(t)p Fw(])29 b(\(cf.)h(De\014nition)g(2.1\))f(b)n(y)g(only)g (\014nitely)g(man)n(y)g(sites,)h(so)e(that)i(maps)f(`)p Fs(\031)3460 5033 y Fu(q)3517 5021 y Ft(\016)19 b Fs(S)3634 4991 y Fu(t)3629 5041 y(!)3677 5021 y Fw(')29 b(from)118 5120 y Fs(M)36 b Fw(to)27 b(\()p Fs(S)424 5090 y Fo(1)461 5120 y Fw(\))493 5090 y Fm(f)p Fu(q)r Fm(g)625 5120 y Fw(and)g(then)h(also)e(`)p Fs(S)1220 5090 y Fu(t)1215 5141 y(!)1263 5120 y Fw(')h(from)g Fs(M)35 b Fw(to)27 b Fs(M)36 b Fw(can)27 b(b)r(e)g(de\014ned)g(in)h(a)e(natural)h(w)n(a)n (y)-7 b(.)35 b(The)27 b(pro)r(of)g(is)g(based)f(on)h(a)118 5220 y(p)r(ercolation)i(argumen)n(t.)45 b(P)n(ercolation)28 b(tec)n(hniques,)j(but)g(di\013eren)n(t)f(from)h(the)f(ones)g(presen)n (ted)g(here,)h(w)n(ere)e(already)118 5320 y(used)35 b(b)n(y)g(Harris)f (in)h([15)o(])g(for)g(pro)n(ving)e(the)j(existence)f(of)g(certain)f(in) n(teracting)g(particle)h(systems)f(of)h(\014nite)h(range.)118 5419 y(It)f(follo)n(ws)e(in)i(particular)e(that)h Fs(\031)1221 5431 y Fo(\003)1294 5419 y Ft(\016)23 b Fs(S)1415 5389 y Fu(t)1410 5440 y(!)1492 5419 y Fw(:)34 b Fs(M)43 b Ft(!)34 b Fw(\()p Fs(S)1878 5389 y Fo(1)1915 5419 y Fw(\))1947 5389 y Fo(\003)2032 5419 y Fw(for)f(\014nite)i(\003)f Ft(6)p Fw(=)g Ft(;)g Fw(is)g(the)h(limit)g(\(as)3245 5398 y(~)3237 5419 y(\003)f Ft(!)g Fv(Z)3507 5389 y Fu(d)3540 5419 y Fw(\))h(of)f(maps)1966 5682 y(12)p eop %%Page: 13 13 13 12 bop 118 354 a Fw(that)32 b(are)e(constructed)h(b)n(y)h(using)f (the)h(`cut)g(o\013s')f Fs(\031)1778 366 y Fo(\003)1849 354 y Ft(\016)21 b Fs(S)1968 324 y Fu(t)1969 376 y Fo(~)1963 391 y(\003)p Fu(;\030)r(;!)2128 354 y Fw(,)32 b(corresp)r(onding)e(to)h (a)g(\014nite)3124 333 y(~)3115 354 y(\003)f Ft(\033)f Fw(\003)i(and)g(b)r(oundary)118 469 y(conditions)24 b Fs(\030)t Fw(.)37 b(In)25 b(fact,)g(this)g(limit)h(also)e(exists)g(and) h(is)g(indep)r(enden)n(t)g(of)g(the)g(b)r(oundary)f(conditions)h(for)f (a)g(h)n(uge)h(class)118 568 y(of)h(in\014nite)h(range)e(in)n (teractions)g(as)h(w)n(e)g(will)g(sho)n(w)g(in)g(Subsection)g(2.2.)36 b(It)26 b(giv)n(es)g(rise)f(to)h(a)g(natural)f(de\014nition)i(of)f(the) 118 668 y(system.)36 b(But)27 b(w)n(e)f(also)f(note)h(that)g(for)g (in\014nite)h(range)e(in)n(teraction)g(eac)n(h)h(site)g(is)g(with)h(p)r (ositiv)n(e)e(probabilit)n(y)h(a\013ected)118 767 y(b)n(y)h (in\014nitely)i(man)n(y)e(other)g(sites.)36 b(So)28 b(w)n(e)f(cannot)g (use)g(the)h(same)f(de\014nition)h(as)f(for)g(\014nite)i(range)d(in)n (teraction.)118 867 y(In)j(Section)h(2.3)e(w)n(e)h(de\014ne)g(Mark)n(o) n(v)e(k)n(ernels)h Fs(K)1676 837 y Fu(t)1670 890 y(S)1746 867 y Fw(for)h(the)h(in\014nite)f(system)g Fs(S)2635 837 y Fu(t)2693 867 y Fw(and)g Fs(K)2933 837 y Fu(t)2927 904 y(S;)2994 889 y Fo(~)2988 904 y(\003)3066 867 y Fw(for)f(the)i (system)f Fs(S)3672 837 y Fu(t)3673 889 y Fo(~)3667 904 y(\003)3745 867 y Fw(that)118 1000 y(\014xes)d(the)454 979 y(~)446 1000 y(\003)504 970 y Fu(C)559 1000 y Fw(-co)r(ordinates)f (for)g(a)h(\014nite)1439 979 y(~)1431 1000 y(\003.)36 b(W)-7 b(e)27 b(sho)n(w)e(that)i Fs(K)2150 970 y Fu(t)2144 1023 y(S)2218 1000 y Fw(is)f(the)g(w)n(eak)f(limit)i(of)g Fs(K)3015 970 y Fu(t)3009 1037 y(S;)3076 1022 y Fo(~)3070 1037 y(\003)3144 1000 y Fw(\(as)3285 979 y(~)3277 1000 y(\003)c Ft(!)g Fv(Z)3525 970 y Fu(d)3558 1000 y Fw(\),)k(i.e.)36 b(the)118 1114 y(corresp)r(onding)26 b(op)r(erators)f(on)j(con)n(tin)n (uous)e(functions)i(con)n(v)n(erge)d(w)n(eakly)-7 b(.)118 1344 y Fg(2.1)112 b(Finite)36 b(Range)i(In)m(teraction)118 1497 y Fw(No)n(w)c(w)n(e)g(are)f(considering)g(an)h(in)n(teraction)g (of)g(range)f Fs(R)i Ft(2)f Fv(N)f Ft(n)23 b(f)p Fw(0)p Ft(g)p Fw(,)34 b(i.e.)58 b Fs(\031)2639 1509 y Fu(q)2698 1497 y Ft(\016)23 b Fs(S)2814 1509 y Fu(q)2850 1497 y Fw(\()p Fn(x)p Fw(\))36 b(dep)r(ends)f(only)e(on)h Fn(x)3688 1512 y Fu(B)3738 1520 y Fy(R)3786 1512 y Fo(\()p Fu(q)r Fo(\))3875 1497 y Fw(.)118 1597 y(\(Recall)28 b(that)f Fs(B)643 1609 y Fu(R)698 1597 y Fw(\()p Fs(q)s Fw(\))h(w)n(as)f (de\014ned)h(in)g(\(23\).\))118 1764 y Fn(De\014nition)j(2.1)40 b Fw(Giv)n(en)31 b Fs(R)h Fw(as)e(ab)r(o)n(v)n(e,)h Fs(q)s(;)20 b Fw(~)-48 b Fs(q)31 b Ft(2)e Fv(Z)1725 1734 y Fu(d)1757 1764 y Fw(,)j Fs(T)40 b(>)28 b Fw(0,)k Fs(!)f Ft(2)e Fw(\012.)47 b(W)-7 b(e)31 b(sa)n(y)f(that)j Fn(~)-50 b(q)36 b(a\013ects)g(q)g(wrt)g Fw(\()p Fn(R)p Fs(;)14 b Fn(t)p Fs(;)g(!)s Fw(\))31 b(if)118 1864 y(there)c(is)f(a)h(causal)e (path)i(from)g Fs(q)i Fw(to)k(~)-48 b Fs(q)30 b Fw(of)d(step)f(size)h (at)f(most)h Fs(R)q Fw(.)36 b(\(Recall)27 b(that)g(w)n(e)f(de\014ned)h Fj(p)l(ath)34 b Fw(etc.)27 b(in)g(De\014nitions)118 1963 y(1.8)f(to)g(1.10\).)36 b(If)27 b Ft(;)c(6)p Fw(=)f(\003)h Ft(\032)g Fv(Z)1055 1933 y Fu(d)1114 1963 y Fw(w)n(e)k(sa)n(y)e(that)j Fn(~)-49 b(q)31 b(a\013ects)g(\003)g(wrt)g Fw(\()p Fn(R)p Fs(;)14 b Fn(t)p Fs(;)g(!)s Fw(\))26 b(if)34 b(~)-49 b Fs(q)30 b Fw(a\013ects)d(at)f(least)g(one)h(p)r(oin)n(t)g(in)f(\003)h (wrt)118 2063 y(\()p Fs(R)q(;)14 b(t;)g(!)s Fw(\).)118 2162 y(W)-7 b(e)28 b(set)769 2429 y(A\013)880 2444 y Fo(\()p Fu(R;t;!)r Fo(\))1095 2429 y Fw(\(\003\))1300 2382 y Fo(def)1313 2429 y Fw(=)96 b Ft(f)6 b Fw(~)-48 b Fs(q)26 b Ft(2)d Fv(Z)1719 2395 y Fu(d)1774 2429 y Fw(:)30 b(~)-48 b Fs(q)30 b Fw(a\013ects)e(\003)f(wrt)g(\()p Fs(R)q(;)14 b(t;)g(!)s Fw(\))p Ft(g)1039 b Fw(\(56\))886 2588 y(and)82 b(\012)1162 2600 y Fu(R)1300 2541 y Fo(def)1313 2588 y Fw(=)96 b Ft(f)p Fs(!)25 b Fw(:)f Ft(9)p Fs(t)f(>)f Fw(0)p Fs(;)14 b(q)26 b Ft(2)d Fv(Z)2108 2553 y Fu(d)2168 2588 y Fw(suc)n(h)k(that)h Ft(j)p Fw(A\013)2669 2603 y Fo(\()p Fu(R;t;!)r Fo(\))2884 2588 y Fw(\()p Fs(q)s Fw(\))p Ft(j)23 b Fw(=)g Ft(1g)503 b Fw(\(57\))118 2755 y(where)27 b Ft(j)19 b(\001)f(j)28 b Fw(denotes)f(the)h(cardinalit)n(y) -7 b(.)1164 4099 y Ff(u)1164 3103 y(u)1496 4099 y(u)1496 3103 y(u)1828 4099 y(u)1828 3103 y(u)2160 4099 y(u)2160 3103 y(u)2492 4099 y(u)2492 3103 y(u)2824 4099 y(u)2824 3103 y(u)3156 4099 y(u)3156 3103 y(u)p 1825 3883 7 67 v 1828 3883 V 1821 3883 V 1795 3853 67 7 v 1795 3850 V 1795 3856 V 2157 3717 7 67 v 2160 3717 V 2153 3717 V 2127 3687 67 7 v 2127 3684 V 2127 3690 V 1160 3601 7 67 v 1164 3601 V 1157 3601 V 1131 3571 67 7 v 1131 3568 V 1131 3574 V 1825 3468 7 67 v 1828 3468 V 1821 3468 V 1795 3438 67 7 v 1795 3435 V 1795 3441 V 2489 3302 7 67 v 2492 3302 V 2486 3302 V 2459 3272 67 7 v 2459 3269 V 2459 3275 V 2821 3966 7 67 v 2824 3966 V 2818 3966 V 2791 3936 67 7 v 2791 3933 V 2791 3940 V 1826 4099 4 997 v 1494 3850 4 748 v 1826 3850 V 2158 3850 V 2491 3684 4 582 v 2823 3269 4 167 v 1496 3853 665 7 v 1496 3856 V 1496 3850 V 1496 3860 V 1496 3847 V 1496 3853 V 1496 3856 V 1496 3850 V 1496 3860 V 1496 3847 V 1828 3687 V 1828 3690 V 1828 3684 V 1828 3694 V 1828 3680 V 1496 3438 V 1496 3441 V 1496 3435 V 1496 3445 V 1496 3431 V 2160 3272 V 2160 3275 V 2160 3269 V 2160 3279 V 2160 3265 V 1131 4232 a Fs(q)1168 4244 y Fm(\000)p Fo(2)1463 4232 y Fs(q)1500 4244 y Fm(\000)p Fo(1)1795 4232 y Fs(q)1832 4244 y Fo(0)2127 4232 y Fs(q)2164 4244 y Fo(1)2459 4232 y Fs(q)2496 4244 y Fo(2)2791 4232 y Fs(q)2828 4244 y Fo(3)3123 4232 y Fs(q)3160 4244 y Fo(4)p 996 4099 4 997 v 965 4101 67 4 v 882 4116 a Fs(T)p 965 3935 V 882 3949 a(t)912 3961 y Fo(1)p 965 3852 V 882 3866 a Fs(t)912 3878 y Fo(2)p 965 3685 V 882 3700 a Fs(t)912 3712 y Fo(3)p 965 3569 V 882 3584 a Fs(t)912 3596 y Fo(4)p 965 3436 V 882 3451 a Fs(t)912 3463 y Fo(5)p 965 3270 V 882 3285 a Fs(t)912 3297 y Fo(6)p 965 3104 V 882 3119 a Fw(0)1962 2979 y(A\013)2073 2990 y Fi(\(1)p Fy(;T)t(;!)q Fi(\))2266 2979 y Fo(\()p Fu(q)2322 2987 y Fi(0)2354 2979 y Fo(\))1463 3045 y Fk(z)p 1500 3045 634 10 v 634 w(}|)p 2208 3045 V 634 w({)1517 4448 y Fw(Figure)27 b(1:)36 b(The)28 b(history)e(of)i Fs(q)2462 4460 y Fo(0)118 4636 y Fw(Figure)i(1)g(is)h(a)f(picture)h(of)g(A\013)1107 4651 y Fo(\(1)p Fu(;T)5 b(;!)r Fo(\))1324 4636 y Fw(\()p Fs(q)1393 4648 y Fo(0)1431 4636 y Fw(\).)47 b(W)-7 b(e)31 b(consider)e(the)i(\014nite)h(time-in)n(terv)-5 b(al)30 b(\(0)p Fs(;)14 b(T)e Fw(])29 b(and)i(nearest)f(neigh)n(b)r(our)118 4736 y(in)n(teraction)f(and)h(a)g(particular)f Fs(!)s Fw(.)44 b(F)-7 b(or)29 b(eac)n(h)h(jump)h(w)n(e)e(dra)n(w)g(a)h(cross)f (at)h(the)g(particular)f(p)r(oin)n(t)h(\()p Fs(q)s(;)14 b(t)p Fw(\).)45 b(There)30 b(are)118 4835 y(jumps)35 b(at)f(\()p Fs(q)549 4847 y Fo(2)587 4835 y Fs(;)14 b(t)654 4847 y Fo(6)691 4835 y Fw(\),)36 b(\()p Fs(q)851 4847 y Fo(0)889 4835 y Fs(;)14 b(t)956 4847 y Fo(5)993 4835 y Fw(\),)36 b(\()p Fs(q)1153 4847 y Fm(\000)p Fo(2)1243 4835 y Fs(;)14 b(t)1310 4847 y Fo(4)1347 4835 y Fw(\),)36 b(\()p Fs(q)1507 4847 y Fo(1)1545 4835 y Fs(;)14 b(t)1612 4847 y Fo(3)1649 4835 y Fw(\),)36 b(\()p Fs(q)1809 4847 y Fo(0)1847 4835 y Fs(;)14 b(t)1914 4847 y Fo(2)1951 4835 y Fw(\))34 b(and)g(\()p Fs(q)2254 4847 y Fo(3)2292 4835 y Fs(;)14 b(t)2359 4847 y Fo(1)2396 4835 y Fw(\))35 b(The)f(last)g(jump)h(at)f Fs(q)3170 4847 y Fo(0)3241 4835 y Fw(is)h(at)f(time)g Fs(t)3665 4847 y Fo(2)3703 4835 y Fw(.)56 b(W)-7 b(e)118 4935 y(dra)n(w)34 b(a)h(thic)n(k)f (horizon)n(tal)g(line)h(b)r(et)n(w)n(een)g(\()p Fs(q)1581 4947 y Fo(0)1618 4935 y Fs(;)14 b(t)1685 4947 y Fo(2)1723 4935 y Fw(\))35 b(and)g(\()p Fs(q)s(;)14 b(t)2098 4947 y Fo(2)2135 4935 y Fw(\))36 b(for)e(all)h(nearest)f(neigh)n(b)r(ours)f Fs(q)39 b Fw(of)34 b Fs(q)3395 4947 y Fo(0)3468 4935 y Fw(b)r(ecause)g(the)118 5035 y(up)r(dating)e(of)g Fs(q)611 5047 y Fo(2)680 5035 y Fw(dep)r(ends)g(also)f(on)g(these)h(sites.)49 b(So)31 b(w)n(e)g(ha)n(v)n(e)g(to)h(consider)e(the)i(`histories')f(of)h Fs(q)3278 5047 y Fo(0)3347 5035 y Fj(and)41 b Fw(its)32 b(nearest)118 5134 y(neigh)n(b)r(ours)27 b Fj(b)l(efor)l(e)36 b Fw(time)29 b Fs(t)1003 5146 y Fo(2)1041 5134 y Fw(.)39 b(Note)29 b(that)g Fs(q)1523 5146 y Fo(3)1585 5134 y Ft(2)c Fw(A\013)1775 5149 y Fo(\(1)p Fu(;T)5 b(;!)r Fo(\))1992 5134 y Fw(\()p Fs(q)2061 5146 y Fo(0)2099 5134 y Fw(\))29 b(and)f(it)h(is)f(up)r(dated)i(at)e(time)h Fs(t)3139 5146 y Fo(1)3205 5134 y Fw(\(and)f(so)g(a\013ected)h(b)n(y)118 5234 y Fs(q)155 5246 y Fo(4)223 5234 y Fw(for)h(example\))g(but)h(that) g(up)r(dating)g(has)f(no)g(in\015uence)h(on)f Fs(q)2180 5246 y Fo(0)2248 5234 y Fw(\(at)h(time)g Fs(T)12 b Fw(\).)45 b(W)-7 b(e)31 b(also)e(note)i(that,)g(for)f(example,)118 5333 y Fs(q)155 5345 y Fm(\000)p Fo(1)275 5333 y Fw(a\013ects)g Fs(q)571 5345 y Fo(0)639 5333 y Fw(\(wrt)h(\(1)p Fs(;)14 b(T)7 b(;)14 b(!)s Fw(\)\))30 b(but)h(not)f(the)h(other)f(w)n(a)n(y)f (around.)45 b(So)30 b(w)n(e)g(ha)n(v)n(e)f(to)i(consider)e(only)h(the)h (time-)g(and)118 5433 y(space-ordered)25 b(p)r(ercolation.)1966 5682 y(13)p eop %%Page: 14 14 14 13 bop 118 354 a Fn(Prop)s(osition)30 b(2.1)41 b Fw(\012)858 366 y Fu(R)942 354 y Fj(has)30 b Fs(P)12 b Fj(-me)l(asur)l(e)29 b(zer)l(o:)1798 554 y Fs(P)12 b Fw(\(\012)1955 566 y Fu(R)2010 554 y Fw(\))23 b(=)g(0)p Fs(:)1532 b Fw(\(58\))118 727 y Fn(Pro)s(of)59 b Fw(A\013)520 742 y Fo(\()p Fu(R;t;!)r Fo(\))735 727 y Fw(\()p Fs(q)s Fw(\))28 b(is)g(increasing)e(in)i Fs(t)f Fw(and)h(so)1300 926 y(\012)1360 938 y Fu(R)1438 926 y Fw(=)1535 848 y Fk([)1525 1026 y Fu(t)p Fm(2)p Fp(N)1694 848 y Fk([)1665 1031 y Fu(q)r Fm(2)p Fp(Z)1787 1015 y Fy(d)1816 926 y Ft(f)p Fs(!)d Fw(:)e Ft(j)p Fw(A\013)2115 941 y Fo(\()p Fu(R;t;!)r Fo(\))2330 926 y Fw(\()p Fs(q)s Fw(\))p Ft(j)g Fw(=)g Ft(1g)p Fs(:)1034 b Fw(\(59\))118 1166 y(So)28 b(it)h(is)f(su\016cien)n(t)h(to)f(sho)n(w)f(that)i(for)f (\014xed)g Fs(q)g Ft(2)c Fw(\003)29 b(and)f Fs(t)c(>)g Fw(0)k(the)h(set)f Ft(f)p Fs(!)f Fw(:)d Ft(j)p Fw(A\013)2752 1181 y Fo(\()p Fu(R;t;!)r Fo(\))2967 1166 y Fw(\()p Fs(q)s Fw(\))p Ft(j)h Fw(=)e Ft(1g)28 b Fw(has)g Fs(P)12 b Fw(-measure)118 1266 y(zero.)36 b(If)28 b(w)n(e)f(set)1387 1465 y Fs(A)1449 1477 y Fu(N)1536 1418 y Fo(def)1549 1465 y Fw(=)36 b Ft(f)p Fs(!)25 b Fw(:)f(A\013)1926 1480 y Fo(\()p Fu(R;t;!)r Fo(\))2141 1465 y Fw(\()p Fs(q)s Fw(\))g Ft(6\032)e Fs(B)2419 1477 y Fu(N)2482 1465 y Fw(\()p Fs(q)s Fw(\))p Ft(g)1122 b Fw(\(60\))118 1609 y(it)28 b(is)g(su\016cien)n(t)f(to)h(sho)n(w)e (that)1728 1808 y(lim)1690 1862 y Fu(N)6 b Fm(!1)1895 1808 y Fs(P)12 b Fw(\()p Fs(A)2054 1820 y Fu(N)2118 1808 y Fw(\))23 b(=)g(0)p Fs(:)1424 b Fw(\(61\))118 1985 y(If)28 b Fs(q)j Fw(is)d(a\013ected)g(b)n(y)g(some)33 b(~)-48 b Fs(q)27 b Ft(62)d Fs(B)1190 1997 y Fu(N)1253 1985 y Fw(\()p Fs(q)s Fw(\))k(wrt)g(\()p Fs(R)q(;)14 b(t;)g(!)s Fw(\))28 b(then)g(there)g(is)g(a)f(maximal)h(causal)f(path)h(of)f(step) i(size)e(at)h(most)118 2084 y Fs(R)g Fw(from)g Fs(q)i Fw(to)k(~)-48 b Fs(q)31 b Fw(with)d(at)f(least)g Fs(N)1194 2096 y Fo(0)1259 2084 y Fw(steps,)h(where)f Fs(N)1798 2096 y Fo(0)1862 2084 y Fw(is)h(the)g(smallest)f(in)n(teger)g(greater)e (than)3168 2052 y Fu(N)p 3168 2066 59 4 v 3172 2113 a(R)3237 2084 y Fw(.)118 2184 y(Consider)i(an)n(y)h(maximal)f(causal)g(path)i Fs(Q)23 b Fw(=)h(\()p Fs(q)1654 2196 y Fo(0)1715 2184 y Fw(=)g Fs(q)s(;)14 b(:)g(:)g(:)g(;)g(q)2066 2196 y Fu(n)2111 2184 y Fw(\))28 b(of)g(step)h(size)f(at)g(most)g Fs(R)h Fw(and)f(with)g Fs(n)c Ft(\025)g Fs(N)3580 2196 y Fo(0)3617 2184 y Fw(.)38 b Fs(Q)28 b Fw(is)g(a)118 2284 y(maximal)23 b(causal)f(path)h(wrt)g(\()p Fs(t;)14 b(!)s Fw(\))23 b(i\013)g(the)h(tra)5 b(jectory)21 b(of)i Fs(!)j Fw(wrt)d(the)g(P)n(oisson)e(pro)r(cess)h(induced)i(b)n(y)f Fs(Q)f Fw(\(cf.)i(De\014nition)118 2383 y(1.9\))j(has)g(exactly)g Fs(n)h Fw(jumps.)37 b(The)28 b(probabilit)n(y)e(of)i(this)g(is)f Fs(p)2052 2395 y Fu(\025)2096 2383 y Fw(\()p Fs(n;)14 b(t)p Fw(\))28 b(\(whic)n(h)f(w)n(as)g(de\014ned)h(in)g(\(4\).\))118 2483 y(W)-7 b(e)28 b(set)1721 2682 y Fs(c)1757 2694 y Fu(d;R)1889 2635 y Fo(def)1902 2682 y Fw(=)36 b Ft(j)p Fs(B)2089 2694 y Fu(R)2144 2682 y Fw(\()p Fs(q)s Fw(\))p Ft(j)p Fs(:)1456 b Fw(\(62\))118 2826 y(\(Recall)28 b(that)f Fs(B)643 2838 y Fu(R)698 2826 y Fw(\()p Fs(q)s Fw(\))h(w)n(as)f (de\014ned)h(in)g(\(23\))f(and)g Ft(j)19 b(\001)f(j)28 b Fw(denotes)f(the)h(cardinalit)n(y)-7 b(.\))118 2926 y(Then)1624 3125 y Ft(j)p Fw(P)n(ath)o(\()p Fs(q)s(;)14 b(n;)g(R)q Fw(\))p Ft(j)23 b Fw(=)g Fs(c)2283 3091 y Fu(n)2283 3146 y(d;R)3750 3125 y Fw(\(63\))118 3269 y(b)r(ecause)k(at)h (eac)n(h)f(step)g(in)h(the)g(path)g(one)f(can)g(c)n(ho)r(ose)g(b)r(et)n (w)n(een)g Fs(c)2247 3281 y Fu(d;R)2384 3269 y Fw(lattice-p)r(oin)n (ts.)118 3369 y(So)g(w)n(e)h(ha)n(v)n(e)831 3568 y Fs(A)893 3580 y Fu(N)979 3568 y Ft(\032)1109 3489 y Fk([)1067 3667 y Fu(n>N)1213 3675 y Fi(0)1525 3489 y Fk([)1341 3674 y Fu(Q)p Fm(2)p Fq(P)n(ath)q Fo(\()p Fu(q)r(;n;R)p Fo(\))1802 3568 y Ft(f)p Fs(!)d Fw(:)e Fs(Q)k Fw(is)h(maximal)f(causal) f(wrt\()p Fs(R)q(;)14 b(t;)g(!)s Fw(\))p Ft(g)564 b Fw(\(64\))118 3807 y(and)28 b(so)1300 4107 y Fs(P)12 b Fw(\()p Fs(A)1459 4119 y Fu(N)1522 4107 y Fw(\))83 b Ft(\024)1814 4028 y Fk(X)1785 4206 y Fu(n)p Fm(\025)p Fu(N)1931 4214 y Fi(0)1977 4107 y Fs(c)2013 4073 y Fu(n)2013 4128 y(d;R)2140 4107 y Fs(e)2179 4073 y Fm(\000)p Fu(\025t)2327 4051 y Fw(\()p Fs(\025t)p Fw(\))2469 4021 y Fu(n)p 2327 4088 189 4 v 2385 4164 a Fs(n)p Fw(!)3750 4107 y(\(65\))1637 4368 y Ft(\024)g Fs(e)1824 4334 y Fo(\()p Fu(c)1880 4343 y Fy(d;R)1976 4334 y Fm(\000)p Fo(1\))p Fu(\025t)2174 4368 y Fw(\()p Fs(c)2242 4380 y Fu(d;R)2351 4368 y Fs(\025t)p Fw(\))2461 4334 y Fu(N)2514 4342 y Fi(0)2622 4312 y Fw(1)p 2579 4349 128 4 v 2579 4425 a Fs(N)2646 4437 y Fo(0)2683 4425 y Fw(!)3750 4368 y(\(66\))118 4578 y(whic)n(h)29 b(con)n(v)n(erges)d(to)j(0)g(as)f Fs(N)1074 4590 y Fo(0)1136 4578 y Ft(!)e(1)j Fw(whic)n(h)g(w)n(as)e(to)i(sho)n(w.)40 b(F)-7 b(or)29 b(the)g(last)g(inequalit)n(y)f(w)n(e)h(ha)n(v)n(e)f (used)g(the)i(estimate)118 4677 y(for)d(the)h(Lagrange)d(remainder)i (in)h(T)-7 b(a)n(ylor's)25 b(form)n(ula.)3808 4777 y Fh(2)118 5021 y Fn(De\014nition)31 b(2.2)40 b Fw(Let)e(a)g(\014nite)g (range)e(in)n(teraction)h(\(i.e.)h(a)f(family)h(of)g(up)r(datings\))g (b)r(e)g(giv)n(en)f(b)n(y)g(\()p Fs(S)3462 5033 y Fu(q)3499 5021 y Fw(\))3531 5038 y Fu(q)r Fm(2)p Fp(Z)3653 5022 y Fy(d)3686 5021 y Fw(.)68 b(Fix)118 5134 y Fs(!)26 b Ft(2)d Fw(\012)16 b Ft(n)e Fw(\(\012)498 5146 y Fu(R)569 5134 y Ft([)h(N)707 5146 y Fo(1)745 5134 y Fw(\),)26 b Ft(;)d(6)p Fw(=)g(\003)f Ft(\032)1155 5113 y Fw(~)1147 5134 y(\003)h Ft(2)g(F)8 b Fw(,)27 b Fs(\030)g Ft(2)c Fs(M)35 b Fw(and)26 b Fs(t)d(>)f Fw(0.)36 b(Then)27 b Fs(!)h Fw(has)e(only)g(\014nitely)g(man)n(y)g(jumps)g(in)3557 5113 y(~)3549 5134 y(\003)15 b Ft(\002)g Fw(\(0)p Fs(;)f(t)p Fw(\),)118 5234 y(sa)n(y)27 b(at)g(\()p Fs(q)432 5246 y Fo(1)470 5234 y Fs(;)14 b(t)537 5246 y Fo(1)574 5234 y Fw(\))p Fs(;)g(:)g(:)g(:)g(;)g Fw(\()p Fs(q)860 5246 y Fu(n)905 5234 y Fs(;)g(t)972 5246 y Fu(n)1017 5234 y Fw(\))28 b(with)g(0)23 b Fs(<)f(t)1448 5246 y Fo(1)1509 5234 y Fs(<)g(:)14 b(:)g(:)23 b(<)g(t)1834 5246 y Fu(n)1902 5234 y Fs(<)g(t)p Fw(.)118 5333 y(W)-7 b(e)22 b(denote)g(b)n(y)g Fn(x)683 5343 y Fo(~)677 5358 y(\003)734 5333 y Ft(_)7 b Fs(\030)838 5343 y Fo(~)832 5358 y(\003)877 5342 y Fy(C)952 5333 y Fw(the)22 b(p)r(oin)n(t)g(in)g Fs(M)31 b Fw(that)22 b(has)f(the)h(same)2167 5312 y(~)2159 5333 y(\003)o(-co)r(ordinates)e(as)i Fn(x)g Fw(and)g(the)g(same)3351 5312 y(~)3343 5333 y(\003)3401 5303 y Fu(C)3456 5333 y Fw(-co)r(ordinates)118 5433 y(as)27 b Fs(\030)t Fw(.)1966 5682 y(14)p eop %%Page: 15 15 15 14 bop 118 354 a Fw(W)-7 b(e)28 b(de\014ne)1312 631 y Fs(S)1363 656 y Fu(q)r(;)1421 641 y Fo(~)1415 656 y(\003)p Fu(;\030)1539 631 y Fw(:)c(\()p Fs(S)1674 596 y Fo(1)1711 631 y Fw(\))1749 582 y Fo(~)1743 597 y(\003)1880 631 y Ft(!)87 b Fw(\()p Fs(S)2138 596 y Fo(1)2175 631 y Fw(\))2213 582 y Fo(~)2207 597 y(\003)3750 631 y Fw(\(67\))1423 796 y Fs(S)1474 821 y Fu(q)r(;)1532 806 y Fo(~)1526 821 y(\003)q Fu(;\030)1628 796 y Fw(\()p Fn(x)1716 806 y Fo(~)1710 821 y(\003)1760 796 y Fw(\))1876 749 y Fo(def)1889 796 y Fw(=)96 b Fs(\031)2103 806 y Fo(~)2097 821 y(\003)2165 796 y Ft(\016)18 b Fs(S)2276 808 y Fu(q)2313 796 y Fw(\()p Fn(x)2401 806 y Fo(~)2395 821 y(\003)2464 796 y Ft(_)g Fs(\030)2579 806 y Fo(~)2573 821 y(\003)2618 805 y Fy(C)2672 796 y Fw(\))118 972 y(and)971 1249 y(\012)g Ft(n)g Fw(\()p Ft(N)1209 1261 y Fo(1)1265 1249 y Ft([)h Fw(\012)1399 1261 y Fu(R)1453 1249 y Fw(\))24 b Ft(3)f Fs(!)90 b Ft(7!)d Fs(S)1955 1214 y Fu(t)1956 1266 y Fo(~)1950 1281 y(\003)q Fu(;\030)r(;!)2138 1249 y Ft(2)24 b(C)2266 1214 y Fo(0)2316 1249 y Fw(\()q Fs(M)t(;)14 b(M)9 b Fw(\))1157 b(\(68\))1311 1422 y Fs(S)1367 1387 y Fu(t)1368 1439 y Fo(~)1362 1454 y(\003)p Fu(;\030)r(;!)1527 1422 y Fw(\()p Fn(x)p Fw(\))1725 1375 y Fo(def)1738 1422 y Fw(=)96 b Fs(S)1950 1447 y Fu(q)1980 1455 y Fy(n)2021 1447 y Fu(;)2047 1432 y Fo(~)2041 1447 y(\003)p Fu(;\030)2161 1422 y Ft(\016)18 b Fs(:)c(:)g(:)k Ft(\016)g Fs(S)2447 1447 y Fu(q)2477 1455 y Fi(1)2510 1447 y Fu(;)2536 1432 y Fo(~)2530 1447 y(\003)p Fu(;\030)2631 1422 y Fw(\()p Fn(x)2719 1432 y Fo(~)2713 1447 y(\003)2763 1422 y Fw(\))h Ft(_)g Fs(\030)2930 1432 y Fo(~)2924 1447 y(\003)2969 1430 y Fy(C)3022 1422 y Fs(:)118 1617 y Fw(The)27 b(maps)g Fs(S)561 1587 y Fu(t)562 1639 y Fo(~)556 1654 y(\003)p Fu(;\030)r(;!)748 1617 y Fw(are)f(con)n(tin)n(uous)g(as)h (comp)r(osites)f(of)h(con)n(tin)n(uous)f(maps.)37 b(F)-7 b(urthermore)26 b Fs(S)3121 1587 y Fu(t)3122 1639 y Fo(~)3116 1654 y(\003)p Fu(;\030)r(;!)3308 1617 y Fw(dep)r(ends)h(only)g(on)118 1736 y Fs(!)176 1746 y Fo(~)170 1761 y(\003)247 1736 y Fw(\(i.e.)g(on)g Fs(\031)580 1746 y Fo(~)574 1761 y(\003)641 1736 y Ft(\016)17 b Fs(X)769 1748 y Fn(.)800 1736 y Fw(\()p Fs(!)s Fw(\)\))27 b(and)g(\(68\))g(giv)n(es)f(rise)h(to)f(a)h(coun)n (table,)g(measurable)f(partition)g(of)h(\012)18 b Ft(n)f Fw(\()p Ft(N)3348 1748 y Fo(1)3403 1736 y Ft([)h Fw(\012)3536 1748 y Fu(R)3590 1736 y Fw(\):)37 b Fs(!)30 b Fw(and)125 1835 y(~)-49 b Fs(!)34 b Fw(b)r(elong)d(to)g(the)h(same)f(set)g(of)g (this)h(partition)f(if)h(they)f(ha)n(v)n(e)f(the)i(same)f(list)g(of)g (jump)i(sites)e(\()p Fs(q)3225 1847 y Fo(1)3262 1835 y Fs(;)14 b(:)g(:)g(:)g(;)g(q)3484 1847 y Fu(n)3529 1835 y Fw(\))32 b(\(ordered)118 1935 y(wrt)c(the)g(jump)g(times\).)118 2035 y(No)n(w)f(let)435 2014 y(~)427 2035 y(\003)c Ft(\033)f Fw(A\013)706 2050 y Fo(\()p Fu(R;t;!)r Fo(\))921 2035 y Fw(\(\003\))28 b(and)f Fs(\030)g Ft(2)d Fs(M)36 b Fw(and)28 b(de\014ne)1167 2311 y Fs(\031)1214 2323 y Fo(\003)1283 2311 y Ft(\016)18 b Fs(S)1399 2276 y Fu(t)1451 2311 y Fw(:)23 b(\012)18 b Ft(n)g Fw(\()p Ft(N)1735 2323 y Fo(1)1791 2311 y Ft([)h Fw(\012)1925 2323 y Fu(R)1980 2311 y Fw(\))87 b Ft(!)g(C)2318 2276 y Fo(0)2355 2311 y Fw(\()p Fs(M)t(;)14 b Fw(\()p Fs(S)2597 2276 y Fo(1)2635 2311 y Fw(\))2667 2276 y Fo(\003)2716 2311 y Fw(\))p Fs(;)979 b Fw(\(69\))1623 2460 y Fs(\031)1670 2472 y Fo(\003)1738 2460 y Ft(\016)18 b Fs(S)1854 2426 y Fu(t)1849 2481 y(!)1897 2460 y Fw(\()p Fn(x)p Fw(\))2095 2413 y Fo(def)2108 2460 y Fw(=)96 b Fs(\031)2316 2472 y Fo(\003)2385 2460 y Ft(\016)18 b Fs(S)2501 2426 y Fu(t)2502 2478 y Fo(~)2496 2493 y(\003)p Fu(;\030)r(;!)2661 2460 y Fw(\()p Fn(x)2749 2470 y Fo(~)2743 2485 y(\003)2793 2460 y Fw(\))p Fs(:)902 b Fw(\(70\))118 2666 y(The)25 b(de\014nition)g(do)r(es)g(not)g(dep)r(end)g(on)g(the)g (c)n(hoice)f(of)1864 2645 y(~)1856 2666 y(\003)g(or)g Fs(\030)29 b Fw(b)r(ecause)24 b(the)i(righ)n(t-hand)d(side)i(\(rhs\))g (of)g(\(70\))f(dep)r(ends,)118 2766 y(b)n(y)j(de\014nition,)h(on)g(the) g(A\013)994 2781 y Fo(\()p Fu(R;t;!)r Fo(\))1209 2766 y Fw(\(\003\)-co)r(ordinates)e(of)i Fn(x)g Fw(only)-7 b(.)118 2865 y(F)g(urther)28 b(the)g(family)f(\()p Fs(\031)895 2877 y Fo(\003)964 2865 y Ft(\016)18 b Fs(S)1080 2835 y Fu(t)1075 2886 y(!)1123 2865 y Fw(\()p Fn(x)p Fw(\)\))1269 2880 y Fo(\003)p Fm(2F)6 b(nf;g)1581 2865 y Fw(is)28 b(consisten)n(t)f(in)h(the)g(sense)f(that)h(for)f(an)n(y)g Ft(;)22 b(6)p Fw(=)h(\003)3178 2877 y Fo(1)3238 2865 y Ft(\032)g Fw(\003)3384 2877 y Fo(2)3443 2865 y Ft(2)h(F)8 b Fw(:)1413 3065 y Fs(\031)1460 3077 y Fo(\003)1505 3085 y Fi(1)1556 2997 y Fk(\000)1594 3065 y Fs(\031)1641 3077 y Fo(\003)1686 3085 y Fi(2)1742 3065 y Ft(\016)18 b Fs(S)1858 3030 y Fu(t)1853 3085 y(!)1901 3065 y Fw(\()p Fs(x)p Fw(\))2012 2997 y Fk(\001)2074 3065 y Fw(=)k Fs(\031)2208 3077 y Fo(\003)2253 3085 y Fi(1)2309 3065 y Ft(\016)c Fs(S)2425 3030 y Fu(t)2420 3085 y(!)2468 3065 y Fw(\()p Fs(x)p Fw(\))p Fs(;)1148 b Fw(\(71\))118 3210 y(and)28 b(so)e(de\014nes)i(a)f(map)1591 3487 y Fs(S)1647 3452 y Fu(t)1642 3507 y(!)1713 3487 y Fw(:)c Fs(M)96 b Ft(!)87 b Fs(M)1563 b Fw(\(72\))1536 3636 y(\()p Fs(S)1624 3602 y Fu(t)1619 3657 y(!)1668 3636 y Fw(\()p Fs(x)p Fw(\)\))1811 3648 y Fu(q)1932 3589 y Fo(def)1945 3636 y Fw(=)96 b Fs(\031)2153 3648 y Fu(q)2208 3636 y Ft(\016)18 b Fs(S)2324 3602 y Fu(t)2319 3657 y(!)2367 3636 y Fw(\()p Fs(x)p Fw(\))118 3813 y(Finally)28 b(w)n(e)f(set)g Fs(S)708 3783 y Fu(t)703 3833 y(!)774 3813 y Fw(=)c(id)931 3825 y Fu(M)1033 3813 y Fw(for)k Fs(!)e Ft(2)f Fw(\012)1376 3825 y Fu(R)1449 3813 y Ft([)19 b(N)12 b Fw(.)118 3989 y Fn(Prop)s(osition)30 b(2.2)41 b Fj(The)31 b(map)f Fs(S)1206 3959 y Fu(t)1201 4010 y(!)1249 3989 y Fj(,)g(de\014ne)l(d)g(in)g (\(70\))g(and)h(\(72\))f(is)g(c)l(ontinuous)f(and)i(dep)l(ends)f(me)l (asur)l(ably)h(on)e Fs(!)s Fj(.)118 4166 y Fn(Pro)s(of)59 b Fw(The)27 b(net)h(\()768 4145 y(~)754 4166 y Fs(S)810 4136 y Fu(t)811 4188 y Fo(~)805 4203 y(\003)q Fu(;\030)r(;!)970 4166 y Fw(\))1008 4176 y Fo(~)1002 4191 y(\003)q Fm(2F)6 b(nf;g)1314 4166 y Fw(satis\014es)26 b(the)i(assumptions)e(in)i(Lemma)f (1.2)f(and)i(so)e(all)h(statemen)n(ts)g(of)g(Prop)r(o-)118 4280 y(sition)h(2.2)e(follo)n(w.)3808 4380 y Fh(2)118 4682 y Fg(2.2)112 b(In\014nite)37 b(Range)h(In)m(teraction)118 4835 y Fw(W)-7 b(e)28 b(extend)g(our)f(notion)g(of)h(`)p Fs(S)1111 4805 y Fu(t)1106 4856 y(!)1154 4835 y Fw(')f(to)h(in)n (teractions)e(that)i(are)f(not)g(necessarily)f(of)i(\014nite)g(range.) 118 4935 y(Consider)f(a)g(family)g(\()p Fs(S)870 4947 y Fu(q)907 4935 y Fw(\))939 4952 y Fu(q)r Fm(2)p Fp(Z)1062 4935 y Fy(d)i Fw(of)f(maps)f Fs(S)1485 4947 y Fu(q)1545 4935 y Fw(:)c Fs(M)31 b Ft(!)23 b Fs(M)37 b Fw(suc)n(h)27 b(that)h Fs(S)2345 4947 y Fu(q)2409 4935 y Fw(do)r(es)f(not)h(c)n (hange)e(the)i Fv(Z)3222 4905 y Fu(d)3274 4935 y Ft(n)17 b(f)p Fs(q)s Ft(g)p Fw(-co)r(ordinates)118 5035 y(and)23 b Fs(\031)322 5047 y Fu(q)370 5035 y Ft(\016)10 b Fs(S)473 5047 y Fu(q)532 5035 y Fw(:)23 b Fs(M)32 b Ft(!)23 b Fs(S)853 5004 y Fo(1)913 5035 y Fw(is)h Fj(Lipschitz-c)l(ontinuous)i (wrt)g(al)t(l)h(c)l(o)l(or)l(dinates)k Fw(and)23 b(the)h(Lipsc)n(hitz)g (constan)n(ts)e(dep)r(end)i(only)118 5134 y(on)h(the)h(relativ)n(e)e(p) r(ositions)h(of)g(the)h(sites,)g(i.e.)f(there)g(are)f(constan)n(ts)h Fs(w)r Fw(\()p Fs(r)r Fw(\))i(for)e(all)g Fs(r)h Ft(2)d Fv(Z)2927 5104 y Fu(d)2985 5134 y Fw(suc)n(h)i(that)g(for)g(all)g Fs(q)s(;)20 b Fw(~)-48 b Fs(q)26 b Ft(2)e Fv(Z)3865 5104 y Fu(d)118 5234 y Fw(and)k Fn(x)p Fs(;)14 b Fn(y)25 b Ft(2)e Fs(M)36 b Fw(with)28 b Fn(x)876 5251 y Fp(Z)922 5234 y Fy(d)950 5251 y Fm(nf)5 b Fo(~)-38 b Fu(q)s Fm(g)1112 5234 y Fw(=)23 b Fn(y)1250 5251 y Fp(Z)1295 5234 y Fy(d)1324 5251 y Fm(nf)5 b Fo(~)-38 b Fu(q)r Fm(g)1490 5234 y Fw(\(i.e.)28 b Fn(x)g Fw(and)g Fn(y)h Fw(di\013er)f(at)f(most)h(in)g(their)33 b(~)-48 b Fs(q)s Fw(-co)r(ordinates.\))1106 5433 y Fs(\045)1149 5448 y Fu(S)1193 5431 y Fi(1)1230 5433 y Fw(\()p Fs(\031)1309 5445 y Fu(q)1365 5433 y Ft(\016)18 b Fs(S)1476 5445 y Fu(q)1526 5433 y Fw(\()p Fn(x)p Fw(\))p Fs(;)c(\031)1724 5445 y Fu(q)1780 5433 y Ft(\016)k Fs(S)1891 5445 y Fu(q)1928 5433 y Fw(\()p Fn(y)q Fw(\)\))24 b Ft(\024)f Fs(w)r Fw(\()6 b(~)-48 b Fs(q)23 b Ft(\000)18 b Fs(q)s Fw(\))c Fs(\045)2552 5448 y Fu(S)2596 5431 y Fi(1)2632 5433 y Fw(\()p Fs(y)2710 5445 y Fo(~)-38 b Fu(q)2742 5433 y Fs(;)14 b(z)2823 5445 y Fo(~)-38 b Fu(q)2854 5433 y Fw(\))p Fs(:)841 b Fw(\(73\))1966 5682 y(15)p eop %%Page: 16 16 16 15 bop 118 354 a Fw(W)-7 b(e)28 b(further)g(assume)e(summabilit)n(y) i(of)g(the)g(Lipsc)n(hitz-constan)n(ts,)e(i.e.)1766 475 y Fk(X)1750 659 y Fu(q)r Fm(2)p Fp(Z)1872 642 y Fy(d)1915 554 y Fs(w)r Fw(\()p Fs(q)s Fw(\))e(=)f Fs(c)2228 566 y Fo(1)3750 554 y Fw(\(74\))118 791 y(with)28 b(a)f(p)r(ositiv)n(e)h (constan)n(t)e Fs(c)1054 803 y Fo(1)1092 791 y Fw(.)118 890 y(W)-7 b(e)28 b(need)g(the)g(follo)n(wing)e(tec)n(hnical)i(lemma.) 118 1067 y Fn(Lemma)i(2.1)40 b Fj(If)25 b Fw(\()p Fs(w)r Fw(\()p Fs(q)s Fw(\)\))931 1084 y Fu(q)r Fm(2)p Fp(Z)1055 1068 y Fy(d)1112 1067 y Fj(is)g(a)g(family)h(of)f(non-ne)l(gative)g(r)l (e)l(al)f(numb)l(ers)g(satisfying)i(\(74\))f(then)f(ther)l(e)h(ar)l(e)g (families)118 1167 y Fw(\()p Fs(w)209 1179 y Fo(1)247 1167 y Fw(\()p Fs(q)s Fw(\)\))383 1184 y Fu(q)r Fm(2)p Fp(Z)506 1167 y Fy(d)32 b Fj(and)e Fw(\()p Fs(w)821 1179 y Fo(2)859 1167 y Fw(\()p Fs(q)s Fw(\)\))995 1184 y Fu(q)r Fm(2)p Fp(Z)1118 1167 y Fy(d)i Fj(of)e(non-ne)l(gative)g(and)g(p)l (ositive)h(numb)l(ers,)f(r)l(esp)l(e)l(ctively,)h(such)f(that)1533 1444 y Fs(w)r Fw(\()p Fs(q)s Fw(\))84 b(=)f Fs(w)1989 1456 y Fo(1)2026 1444 y Fw(\()p Fs(q)s Fw(\))14 b Fs(w)2203 1456 y Fo(2)2241 1444 y Fw(\()p Fs(q)s Fw(\))86 b(for)27 b(all)g Fs(q)f Ft(2)e Fv(Z)2876 1409 y Fu(d)2909 1444 y Fs(;)818 b Fw(\(75\))1348 1506 y Fk(X)1333 1690 y Fu(q)r Fm(2)p Fp(Z)1455 1673 y Fy(d)1498 1585 y Fs(w)1557 1597 y Fo(1)1594 1585 y Fw(\()p Fs(q)s Fw(\))84 b Ft(\024)f Fw(2)p Fs(c)2008 1597 y Fo(1)2063 1585 y Fw(+)18 b(1)1562 b(\(76\))1070 1805 y(and)160 b(lim)1330 1858 y Fu(R)p Fm(!1)1527 1805 y Fs(a)p Fw(\()p Fs(R)q Fw(\))83 b(=)g(0)p Fs(;)1755 b Fw(\(77\))118 2015 y Fj(wher)l(e)31 b(the)e(p)l(ositive)j (function)d Fs(a)p Fw(\()p Ft(\001)p Fw(\))i Fj(is)f(de\014ne)l(d)g(by) 1231 2214 y Fs(a)p Fw(\()p Fs(R)q Fw(\))1426 2167 y Fo(def)1440 2214 y Fw(=)240 b(sup)1541 2288 y Fm(k)p Fu(r)1606 2296 y Fi(1)1638 2288 y Fm(k)p Fo(+)p Fu(:::)o Fo(+)p Fm(k)p Fu(r)1898 2296 y Fy(n)1939 2288 y Fm(k)p Fo(=)p Fu(R)2088 2214 y Fs(w)2147 2226 y Fo(2)2185 2214 y Fw(\()p Fs(r)2254 2226 y Fo(1)2292 2214 y Fw(\))18 b Ft(\001)h Fs(:)14 b(:)g(:)k Ft(\001)h Fs(w)2600 2226 y Fo(2)2637 2214 y Fw(\()p Fs(r)2706 2226 y Fu(n)2752 2214 y Fw(\))966 b(\(78\))118 2427 y Fj(\(The)31 b(empty)f(pr)l(o)l(duct)g(is)g(de\014ne)l(d)g(to)g (b)l(e)f(e)l(qual)i(to)e(1.\))118 2605 y Fn(Pro)s(of)59 b Fw(W)-7 b(e)28 b(can)g(c)n(ho)r(ose)e Fs(r)1005 2617 y Fo(0)1066 2605 y Fw(=)c(0)h Fs(<)g(r)1343 2617 y Fo(1)1403 2605 y Fs(<)g(:)14 b(:)g(:)23 b Ft(2)g Fv(N)38 b Fw(suc)n(h)27 b(that)1380 2725 y Fk(X)1335 2907 y Fm(k)p Fu(q)r Fm(k)p Fu()g Fw(0,)i(\003)e Ft(2)h(F)33 b Fw(and)26 b Fs(!)g Ft(2)d Fw(\012)15 b Ft(n)f(N)1374 899 y Fo(1)1411 887 y Fw(.)37 b(By)25 b(de\014nition)h (of)g Ft(N)2127 899 y Fo(1)2190 887 y Fw(\(cf.)h(\(21\)\))e Fs(!)k Fw(has)c(no)g(jumps)i(at)e(0,)h(no)g(sim)n(ultaneous)118 987 y(jumps)h(and)f(only)g(\014nitely)h(man)n(y)e(jumps)i(in)g(\003)15 b Ft(\002)g Fw(\(0)p Fs(;)f(t)p Fw(\),)27 b(sa)n(y)e(at)h(\()p Fs(q)2236 999 y Fo(1)2274 987 y Fs(;)14 b(t)2341 999 y Fo(1)2378 987 y Fw(\))p Fs(;)g(:)g(:)g(:)g(;)g Fw(\()p Fs(q)2664 999 y Fu(N)2727 987 y Fs(;)g(t)2794 999 y Fu(N)2857 987 y Fw(\))27 b(with)f(0)d Fs(<)f(t)3285 999 y Fo(1)3346 987 y Fs(<)g(:)14 b(:)g(:)23 b(<)g(t)3671 999 y Fu(N)3757 987 y Fs(<)g(t)p Fw(.)118 1112 y(W)-7 b(e)28 b(set)g Fs(t)421 1124 y Fo(0)481 1065 y(def)494 1112 y Fw(=)36 b(0)28 b(and)f(\014x)h(arbitrary)d Fs(\030)i Fw(=)c(\()p Fs(\030)t Fw(\()p Fs(t)1582 1124 y Fo(0)1620 1112 y Fw(\))p Fs(;)14 b(:)g(:)g(:)g(;)g(\030)t Fw(\()p Fs(t)1939 1124 y Fu(N)2002 1112 y Fw(\)\),)2123 1091 y Fn(~)2117 1112 y Fs(\030)28 b Fw(=)22 b(\()2309 1090 y(~)2300 1112 y Fs(\030)5 b Fw(\()p Fs(t)2403 1124 y Fo(0)2440 1112 y Fw(\))p Fs(;)14 b(:)g(:)g(:)g(;)2666 1090 y Fw(~)2657 1112 y Fs(\030)t Fw(\()p Fs(t)2759 1124 y Fu(N)2822 1112 y Fw(\)\))24 b Ft(2)f Fs(M)3078 1082 y Fu(N)6 b Fo(+1)3253 1112 y Fw(and)27 b Fn(x)p Fs(;)14 b Fn(y)25 b Ft(2)f Fs(M)9 b Fw(.)118 1237 y(W)-7 b(e)28 b(set)g Fn(x)p Fw(\(0\))570 1189 y Fo(def)584 1237 y Fw(=)36 b Fn(x)735 1249 y Fo(\003)803 1237 y Ft(_)19 b Fs(\030)913 1253 y Fo(\003)958 1236 y Fy(C)1011 1237 y Fw(\(0\),)28 b Fn(y)q Fw(\(0\))1349 1189 y Fo(def)1362 1237 y Fw(=)36 b Fn(y)1513 1249 y Fo(\003)1582 1237 y Ft(_)1664 1215 y Fw(~)1656 1237 y Fs(\030)1692 1253 y Fo(\003)1737 1236 y Fy(C)1790 1237 y Fw(\(0\))28 b(and)f(de\014ne)h(for)f(1)22 b Ft(\024)h Fs(i)g Ft(\024)f Fs(N)37 b Fw(recursiv)n(ely:)1103 1515 y Fs(x)1150 1527 y Fu(q)1187 1515 y Fw(\()p Fs(t)1249 1527 y Fu(i)1277 1515 y Fw(\))1332 1468 y Fo(def)1345 1515 y Fw(=)1447 1345 y Fk(8)1447 1419 y(<)1447 1569 y(:)1562 1414 y Fs(\031)1609 1426 y Fu(q)1664 1414 y Ft(\016)18 b Fs(S)1775 1426 y Fu(q)1812 1414 y Fw(\()p Fn(x)p Fw(\()p Fs(t)1956 1426 y Fu(i)p Fm(\000)p Fo(1)2069 1414 y Fw(\)\))84 b(for)e Fs(q)26 b Fw(=)d Fs(q)2587 1426 y Fu(i)1702 1514 y Fs(x)1749 1526 y Fu(q)1786 1514 y Fw(\()p Fs(t)1848 1526 y Fu(i)p Fm(\000)p Fo(1)1961 1514 y Fw(\))224 b(for)82 b Fs(q)26 b Ft(2)e Fw(\003)18 b Ft(n)g(f)p Fs(q)2756 1526 y Fu(i)2783 1514 y Ft(g)1750 1614 y Fs(\030)1786 1626 y Fu(q)1823 1614 y Fw(\()p Fs(t)1885 1626 y Fu(i)1913 1614 y Fw(\))272 b(for)82 b Fs(q)26 b Ft(2)e Fw(\003)2599 1583 y Fu(C)2890 1515 y Fs(:)837 b Fw(\(89\))118 1779 y(W)-7 b(e)28 b(de\014ne)g Fn(y)q Fw(\()p Fs(t)614 1791 y Fu(i)643 1779 y Fw(\))g(analogously)-7 b(,)25 b(using)i Fn(y)i Fw(and)1635 1757 y(~)1627 1779 y Fs(\030)j Fw(instead)27 b(of)h Fn(x)g Fw(and)f Fs(\030)t Fw(,)h(resp)r(ectiv)n(ely)-7 b(.)118 1879 y(Tw)n(o)35 b(p)r(oin)n(ts)g(in)g Fs(S)730 1848 y Fo(1)802 1879 y Fw(can)g(ha)n(v)n(e)f(distance)h(at)g(most)g Fs(c)1848 1891 y Fu(s)1919 1879 y Fw(=)h(diam)2200 1891 y Fu(\045)2234 1910 y Fy(S)2272 1898 y Fi(1)2312 1879 y Fw(\()p Fs(S)2400 1848 y Fo(1)2438 1879 y Fw(\).)60 b(F)-7 b(or)34 b(estimating)h(the)h (distance)f(b)r(et)n(w)n(een)118 1978 y Fs(x)165 1990 y Fu(q)202 1978 y Fw(\()p Fs(t)264 1990 y Fu(i)292 1978 y Fw(\))28 b(and)g Fs(y)555 1990 y Fu(q)591 1978 y Fw(\()p Fs(t)653 1990 y Fu(i)681 1978 y Fw(\))g(w)n(e)f(de\014ne)998 2207 y Ft(4)1073 2219 y Fu(q)1109 2207 y Fw(\(0\))1238 2159 y Fo(def)1251 2207 y Fw(=)1368 2185 y(~)1352 2207 y Ft(4)1427 2219 y Fu(q)1463 2207 y Fw(\(0\))1592 2159 y Fo(def)1605 2207 y Fw(=)1706 2089 y Fk(\032)1810 2156 y Fs(\045)1853 2171 y Fu(S)1897 2154 y Fi(1)1934 2156 y Fw(\()p Fs(x)2013 2168 y Fu(q)2050 2156 y Fw(\(0\))p Fs(;)14 b(y)2234 2168 y Fu(q)2270 2156 y Fw(\(0\)\))84 b(for)e Fs(q)26 b Ft(2)e Fw(\003)2074 2256 y Fs(c)2110 2268 y Fu(s)2492 2256 y Fw(for)82 b Fs(q)26 b Ft(2)e Fw(\003)2874 2225 y Fu(C)2994 2207 y Fs(;)733 b Fw(\(90\))118 2401 y(and)28 b(for)f(1)22 b Ft(\024)h Fs(i)f Ft(\024)h Fs(N)713 2804 y Ft(4)788 2816 y Fu(q)824 2804 y Fw(\()p Fs(i)p Fw(\))1000 2757 y Fo(def)1013 2804 y Fw(=)1174 2609 y Fk(8)1174 2684 y(>)1174 2708 y(<)1174 2858 y(>)1174 2883 y(:)1290 2600 y(P)1377 2687 y Fu(r)r Fm(2)p Fp(Z)1499 2670 y Fy(d)1546 2662 y Fs(w)r Fw(\()p Fs(r)f Ft(\000)c Fs(q)s Fw(\))c Ft(4)1942 2674 y Fu(r)1978 2662 y Fw(\()p Fs(i)19 b Ft(\000)f Fw(1\))82 b(for)h Fs(q)26 b Fw(=)d Fs(q)2668 2674 y Fu(i)1579 2803 y Ft(4)1654 2815 y Fu(q)1689 2803 y Fw(\()p Fs(i)c Ft(\000)f Fw(1\))371 b(for)83 b Fs(q)26 b Ft(2)d Fw(\003)18 b Ft(n)g(f)p Fs(q)2836 2815 y Fu(i)2864 2803 y Ft(g)1716 2944 y Fs(c)1752 2956 y Fu(s)2297 2944 y Fw(for)83 b Fs(q)26 b Ft(2)d Fw(\003)2679 2914 y Fu(C)3750 2804 y Fw(\(91\))729 3289 y(~)713 3310 y Ft(4)788 3322 y Fu(q)824 3310 y Fw(\()p Fs(i)p Fw(\))1000 3263 y Fo(def)1013 3310 y Fw(=)1174 3090 y Fk(8)1174 3165 y(>)1174 3190 y(>)1174 3215 y(<)1174 3364 y(>)1174 3389 y(>)1174 3414 y(:)1290 3168 y Fw(max)o Ft(f)p Fs(c)1522 3180 y Fu(s)1557 3168 y Fs(;)1594 3106 y Fk(P)1682 3193 y Fu(r)r Fm(2)p Fp(Z)1804 3177 y Fy(d)1851 3168 y Fs(w)r Fw(\()p Fs(r)f Ft(\000)c Fs(q)s Fw(\))2188 3147 y(~)2172 3168 y Ft(4)2247 3180 y Fu(r)2283 3168 y Fw(\()p Fs(i)g Ft(\000)g Fw(1\))p Ft(g)82 b Fw(for)h Fs(q)26 b Fw(=)c Fs(q)3013 3180 y Fu(i)1768 3295 y Fw(~)1752 3317 y Ft(4)1827 3329 y Fu(q)1862 3317 y Fw(\()p Fs(i)c Ft(\000)g Fw(1\))545 b(for)83 b Fs(q)26 b Ft(2)d Fw(\003)18 b Ft(n)g(f)p Fs(q)3182 3329 y Fu(i)3209 3317 y Ft(g)1889 3458 y Fs(c)1925 3470 y Fu(s)2643 3458 y Fw(for)83 b Fs(q)26 b Ft(2)d Fw(\003)3025 3428 y Fu(C)118 3658 y Fw(The)31 b(functions)g Ft(4)728 3670 y Fu(q)794 3658 y Fw(and)975 3636 y(~)958 3658 y Ft(4)1033 3670 y Fu(q)1100 3658 y Fw(dep)r(end)g(on)f Fn(x)p Fw(,)i Fn(y)h Fw(and)d(\003)h(but)g(w)n(e)f(do)g(not)h(refer)f (to)h(this)g(in)f(our)g(notation.)46 b(W)-7 b(e)31 b(ha)n(v)n(e)118 3757 y(in)n(tro)r(duced)g(them)h(for)f(estimating)g(the)h(di\013erence) g(b)r(et)n(w)n(een)f Fs(x)2188 3769 y Fu(q)2225 3757 y Fw(\()p Fs(t)2287 3769 y Fu(i)2315 3757 y Fw(\))h(and)f Fs(y)2585 3769 y Fu(q)2622 3757 y Fw(\()p Fs(t)2684 3769 y Fu(i)2711 3757 y Fw(\))h(\(cf.)g(\(92\)\))g(and)f(so)g(the)h (di\013erence)118 3857 y(b)r(et)n(w)n(een)h Fs(x)491 3869 y Fu(q)528 3857 y Fw(\()p Fs(t)p Fw(\))g(and)g Fs(y)863 3869 y Fu(q)899 3857 y Fw(\()p Fs(t)p Fw(\).)53 b(This)33 b(di\013erence)g(dep)r(ends)g(also)f(on)g Fs(!)k Fw(and)c(so)g(do)h (the)g(corresp)r(onding)e(estimates)h(for)118 3966 y Ft(4)193 3978 y Fu(q)264 3966 y Fw(and)449 3944 y(~)433 3966 y Ft(4)508 3978 y Fu(q)543 3966 y Fw(.)60 b(In)35 b(De\014nition)h(2.4)f(w)n(e)g(will)g(relate)f Ft(4)1880 3978 y Fu(q)1951 3966 y Fw(and)2136 3944 y(~)2120 3966 y Ft(4)2195 3978 y Fu(q)2266 3966 y Fw(to)h(families)g(of)g(random)g(v) -5 b(ariables)33 b(\()p Fs(Y)3531 3978 y Fo(\003)3582 3966 y Fw(\))3614 3981 y Fo(\003)p Fm(2F)6 b(nf;g)118 4081 y Fw(and)26 b(\()323 4060 y(~)310 4081 y Fs(Y)358 4093 y Fo(\003)408 4081 y Fw(\))440 4096 y Fo(\003)p Fm(2F)6 b(nf;g)724 4081 y Fw(,)27 b(resp)r(ectiv)n(ely)-7 b(.)36 b(F)-7 b(or)25 b Ft(4)1474 4093 y Fu(q)1536 4081 y Fw(w)n(e)h(\014nd)h(a)f(particularly)e(nice)j(expansion)e(\(cf.)i (\(93\)\).)36 b(F)-7 b(rom)26 b(this)h(follo)n(ws)118 4195 y(the)36 b(con)n(v)n(ergence)d(of)i Fs(Y)886 4207 y Fo(\003)972 4195 y Fw(to)g(zero)f(in)i(exp)r(ectation)f(\(as)g(\003)h Ft(!)g Fv(Z)2240 4165 y Fu(d)2273 4195 y Fw(\).)61 b(W)-7 b(e)35 b(will)h(sho)n(w)f(that)3118 4174 y(~)3105 4195 y Fs(Y)3153 4207 y Fo(\003)3239 4195 y Fw(is)g(b)r(ounded)h(b)n(y)f Fs(Y)3848 4207 y Fo(\003)118 4294 y Fw(and)g(decreasing)e(and)i(so)f (con)n(v)n(erges)e Fs(P)12 b Fw(-almost)34 b(surely)g(to)h(zero)f(b)n (y)g(the)h(Monotone)f(Con)n(v)n(ergence)f(Theorem)h(\(cf.)118 4394 y(Theorem)27 b(2.1\).)118 4577 y Fn(Prop)s(osition)j(2.3)41 b Fj(The)31 b(fol)t(lowing)h(holds)f(for)g Fw(0)22 b Ft(\024)h Fs(i)f Ft(\024)h Fs(N)9 b 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Fo(1)2171 935 y Ft(\000)k Fs(r)2291 947 y Fo(0)2329 935 y Fw(\))h Ft(\001)f Fs(:)c(:)g(:)19 b Ft(\001)f Fs(w)2637 947 y Fo(1)2675 935 y Fw(\()p Fs(r)2744 947 y Fu(n)2808 935 y Ft(\000)g Fs(r)2928 947 y Fu(n)p Fm(\000)p Fo(1)3059 935 y Fw(\))326 1303 y Fj(wher)l(e)30 b Fs(N)9 b Fw(\()p Fs(!)s Fw(\))30 b Fj(is)g(the)g(numb)l(er)f(of)h(jumps)g(of)h Fs(!)h Fj(in)e Fw(\003)18 b Ft(\002)g Fw(\(0)p Fs(;)c(t)p Fw(\))p Fj(.)118 1485 y Fn(Pro)s(of)59 b Fw(W)-7 b(e)28 b(pro)n(v)n(e)e(\(92\)) i(and)f(\(93\))g(b)n(y)h(induction)g(wrt)f Fs(i)p Fw(.)118 1585 y Fs(i)c Fw(=)f(0)p 118 1598 181 4 v(:)39 b(\(92\))28 b(holds)h(b)n(y)f(de\014nition)h(of)g Ft(4)1413 1597 y Fu(q)1448 1585 y Fw(\(0\))g(and)1762 1563 y(~)1746 1585 y Ft(4)1821 1597 y Fu(q)1856 1585 y Fw(\(0\))g(\(cf.)g(\(90\)\).) 40 b(A)n(t)29 b(time)h(0)e(no)g(jump)i(has)e(happ)r(ened)h(and)f(the) 118 1684 y(only)c(summand)h(on)g(the)g(righ)n(t-hand-side)e(in)i (\(93\))g(corresp)r(onds)d(to)j(the)g(empt)n(y)g(path)g(at)g(site)g Fs(q)i Fw(and)e(so)f(the)h(equalit)n(y)118 1784 y(in)j(\(93\))f(holds.) 118 1884 y Fs(i)18 b Ft(\000)g Fw(1)23 b Ft(!)g Fs(i)p 118 1904 330 4 v -1 w Fw(:)37 b(\(93\))28 b(holds)f(ob)n(viously)f(for) h Fs(i)g Fw(and)h Fs(q)e Ft(6)p Fw(=)d Fs(q)1800 1896 y Fu(i)1855 1884 y Fw(as)k(there)g(is)h(no)f(up)r(dating)h(at)f(site)h Fs(q)j Fw(and)1310 2083 y(P)n(ath)1484 2095 y(c)1525 2083 y(\()p Fs(t)1587 2095 y Fu(i)1615 2083 y Fs(;)14 b(!)s(;)g(q)s(;)g Fw(\003\))22 b(=)h(P)n(ath)2195 2095 y(c)2236 2083 y(\()p Fs(t)2298 2095 y Fu(i)p Fm(\000)p Fo(1)2411 2083 y Fs(;)14 b(!)s(;)g(q)s(;)g Fw(\003\))1043 b(\(95\))118 2232 y(A)n(t)28 b(site)g Fs(q)428 2244 y Fu(i)483 2232 y Fw(there)f(is)h(a)f(jump)i(at)e(time)h Fs(t)1383 2244 y Fu(i)1438 2232 y Fw(and)g(so)f(w)n(e)g(ha)n(v)n(e)1410 2432 y Ft(4)1485 2444 y Fu(q)1515 2452 y Fy(i)1545 2432 y Fw(\()p Fs(i)p Fw(\))c(=)1764 2353 y Fk(X)1748 2537 y Fu(r)r Fm(2)p Fp(Z)1870 2520 y Fy(d)1913 2432 y Fs(w)r Fw(\()p Fs(r)f Ft(\000)c Fs(q)2185 2444 y Fu(i)2213 2432 y Fw(\))c Ft(4)2334 2444 y Fu(r)2370 2432 y Fw(\()p Fs(i)k Ft(\000)g Fw(1\))1144 b(\(96\))118 2670 y(Using)27 b(the)h(represen)n (tation)e(\(93\))h(for)g Ft(4)1414 2682 y Fu(r)r Fm(\000)p Fu(q)1534 2670 y Fw(\()p Fs(i)18 b Ft(\000)f Fw(1\))28 b(and)f(the)h(fact)f(that)h(ev)n(ery)e(\()p Fs(q)2732 2682 y Fu(i)2760 2670 y Fs(;)14 b(r)2834 2682 y Fo(1)2872 2670 y Fs(;)g(:)g(:)g(:)f(;)h(r)3093 2682 y Fu(n)3139 2670 y Fw(\))23 b Ft(2)h Fw(P)n(ath)3447 2682 y(c)3488 2670 y(\()p Fs(t)3550 2682 y Fu(i)3578 2670 y Fs(;)14 b(!)s(;)g(q)3744 2682 y Fu(i)3771 2670 y Fs(;)g Fw(\003\))118 2769 y(can)27 b(b)r(e)h(\(uniquely\))h(split)f(in)n(to)f(\()p Fs(q)1207 2781 y Fu(i)1235 2769 y Fs(;)14 b(r)1309 2781 y Fo(1)1346 2769 y Fw(\))28 b(and)g(\()p Fs(r)1637 2781 y Fo(1)1675 2769 y Fs(;)14 b(:)g(:)g(:)f(;)h(r)1896 2781 y Fu(n)1942 2769 y Fw(\))23 b Ft(2)h Fw(P)n(ath)2250 2781 y(c)2291 2769 y(\()p Fs(t)2353 2781 y Fu(i)p Fm(\000)p Fo(1)2466 2769 y Fs(;)14 b(!)s(;)g(r)2632 2781 y Fo(1)2669 2769 y Fs(;)g 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Fu(i)p Fm(\000)p Fo(1)2930 2978 y Fw(\))k(and)g Fs(y)3193 2990 y Fu(q)3229 2978 y Fw(\()p Fs(t)3291 2990 y Fu(i)3319 2978 y Fw(\))d(=)e Fs(y)3505 2990 y Fu(q)3541 2978 y Fw(\()p Fs(t)3603 2990 y Fu(i)p Fm(\000)p Fo(1)3716 2978 y Fw(\).)39 b(So)118 3078 y(in)28 b(b)r(oth)g(cases)e(the)i(\014rst)g(inequalit)n(y)f(in)h(\(92\))f (holds.)118 3178 y(No)n(w)e(w)n(e)g(consider)f(the)h(site)g Fs(q)1073 3190 y Fu(i)1126 3178 y Fw(where)g(a)g(jump)h(happ)r(ens)f (at)g(time)h Fs(t)2282 3190 y Fu(i)2309 3178 y Fw(.)36 b(Using)25 b(\(73\),)h(assumption)e(\(92\),)i(for)e Fs(i)14 b Ft(\000)g Fw(1,)24 b(and)118 3277 y(\(96\),)j(w)n(e)h(get)921 3559 y Fs(\045)964 3574 y Fu(S)1008 3558 y Fi(1)1045 3559 y Fw(\()p Fs(x)1124 3571 y Fu(q)1161 3559 y Fw(\()p Fs(t)1223 3571 y Fu(i)1251 3559 y Fw(\))p Fs(;)14 b(y)1361 3571 y Fu(q)1398 3559 y Fw(\()p Fs(t)1460 3571 y Fu(i)1488 3559 y Fw(\)\))83 b Ft(\024)1798 3481 y Fk(X)1783 3664 y Fu(r)r Fm(2)p Fp(Z)1905 3648 y Fy(d)1948 3559 y Fs(w)r Fw(\()p Fs(r)22 b Ft(\000)c Fs(q)2220 3571 y Fu(i)2247 3559 y Fw(\))c Fs(\045)2336 3574 y Fu(S)2380 3558 y Fi(1)2417 3559 y Fw(\()p Fs(x)2496 3571 y Fu(q)2533 3559 y Fw(\()p Fs(t)2595 3571 y Fu(i)p Fm(\000)p Fo(1)2708 3559 y Fw(\))p Fs(;)g(y)2818 3571 y Fu(q)2855 3559 y Fw(\()p Fs(t)2917 3571 y Fu(i)p Fm(\000)p Fo(1)3030 3559 y Fw(\)\))656 b(\(97\))1635 3798 y Ft(\024)1798 3719 y Fk(X)1783 3902 y Fu(r)r Fm(2)p Fp(Z)1905 3886 y Fy(d)1948 3798 y Fs(w)r Fw(\()p Fs(r)22 b Ft(\000)c Fs(q)2220 3810 y Fu(i)2247 3798 y Fw(\))c Ft(4)2368 3810 y Fu(r)2404 3798 y Fw(\()p Fs(i)19 b Ft(\000)f Fw(1\))1635 4011 y Ft(\024)83 b(4)1858 4023 y Fu(q)1888 4031 y Fy(i)1917 4011 y Fw(\()p Fs(i)p Fw(\))118 4193 y(So)29 b(the)g(\014rst)g(inequalit)n(y)g(in)g(\(92\))f (is)h(pro)n(v)n(ed)f(for)g Fs(i)p Fw(.)41 b(The)29 b(second)g(follo)n (ws)f(immediately)h(from)f(\(91\).)41 b(So)29 b(statemen)n(ts)118 4293 y(1)e(and)h(2)f(are)g(pro)n(v)n(ed.)118 4393 y(Finally)f(\(94\))g (follo)n(ws)f(from)h(\(93\)\):)36 b Ft(4)1353 4405 y Fu(q)1389 4393 y Fw(\(0\))23 b(=)g(0)i(for)h Fs(q)g Ft(2)d Fw(\003.)37 b(So)26 b(w)n(e)f(only)h(ha)n(v)n(e)f(to)h(sum)h(o)n(v)n (er)d(paths)i(\()p Fs(r)3408 4405 y Fo(0)3469 4393 y Fw(=)d(0)p Fs(;)14 b(:)g(:)g(:)f(;)h(r)3820 4405 y Fu(n)3865 4393 y Fw(\))118 4492 y(that)28 b(end)g(in)g Fs(r)589 4504 y Fu(n)657 4492 y Ft(2)c Fw(\003)794 4462 y Fu(C)849 4492 y Fw(.)118 4609 y(In)k(particular,)e(if)i(w)n(e)g(set)f Fs(R)1043 4562 y Fo(def)1056 4609 y Fw(=)36 b Ft(k)p Fs(r)1236 4621 y Fu(n)1282 4609 y Ft(k)p Fw(,)27 b(then)1341 4869 y(~)1325 4891 y Ft(4)1400 4903 y Fu(r)1431 4911 y Fy(n)1474 4891 y Fw(\(0\))83 b(=)g Fs(c)1847 4903 y Fu(s)1882 4891 y Fs(;)1845 b Fw(\(98\))1113 5015 y(dist)1248 5032 y Fp(Z)1292 5016 y Fy(d)1325 5015 y Fw(\()p Fs(q)s(;)14 b Fw(\003)1492 4981 y Fu(C)1548 5015 y Fw(\))83 b Ft(\024)g Fs(R)q(;)1852 b Fw(\(99\))1517 5140 y Fs(R)83 b Ft(\024)g(k)p Fs(r)1890 5152 y Fu(n)1954 5140 y Ft(\000)18 b Fs(r)2074 5152 y Fu(n)p Fm(\000)p Fo(1)2204 5140 y Ft(k)g Fw(+)g Fs(:)c(:)g(:)k Fw(+)g Ft(k)p Fs(r)2624 5152 y Fo(1)2680 5140 y Ft(\000)g Fs(r)2800 5152 y Fo(0)2838 5140 y Ft(k)p Fs(;)806 b Fw(\(100\))118 5323 y(and)28 b(so)e(b)n(y)i(the)g(c)n(hoice) f(of)g Fs(w)1040 5335 y Fo(1)1078 5323 y Fw(,)g Fs(w)1187 5335 y Fo(2)1253 5323 y Fw(and)g Fs(a)p Fw(,)h(made)f(b)r(efore)h (De\014nition)g(2.3,)f(w)n(e)g(get)1966 5682 y(19)p eop %%Page: 20 20 20 19 bop 1013 533 a Fs(w)r Fw(\()p Fs(r)1143 545 y Fo(1)1200 533 y Ft(\000)18 b Fs(r)1320 545 y Fo(0)1358 533 y Fw(\))h Ft(\001)f Fs(:)c(:)g(:)19 b Ft(\001)f Fs(w)r Fw(\()p Fs(r)1737 545 y Fu(n)1802 533 y Ft(\000)g Fs(r)1922 545 y Fu(n)p Fm(\000)p Fo(1)2053 533 y Fw(\))1624 b(\(101\))1096 657 y Ft(\024)83 b Fs(w)1303 669 y Fo(1)1340 657 y Fw(\()p Fs(r)1409 669 y Fo(1)1466 657 y Ft(\000)18 b Fs(r)1586 669 y Fo(0)1624 657 y Fw(\))g Ft(\001)h Fs(:)14 b(:)g(:)k Ft(\001)h Fs(w)1932 669 y Fo(1)1969 657 y Fw(\()p Fs(r)2038 669 y Fu(n)2103 657 y Ft(\000)f Fs(r)2223 669 y Fu(n)p Fm(\000)p Fo(1)2353 657 y Fw(\))h Fs(a)p Fw(\()p Fs(R)q Fw(\))1096 782 y Ft(\024)83 b Fs(w)1303 794 y Fo(1)1340 782 y Fw(\()p Fs(r)1409 794 y Fo(1)1466 782 y Ft(\000)18 b Fs(r)1586 794 y Fo(0)1624 782 y Fw(\))g Ft(\001)h Fs(:)14 b(:)g(:)k Ft(\001)h Fs(w)1932 794 y Fo(1)1969 782 y Fw(\()p Fs(r)2038 794 y Fu(n)2103 782 y Ft(\000)f Fs(r)2223 794 y Fu(n)p Fm(\000)p Fo(1)2353 782 y Fw(\))h Fs(a)p Fw(\(dist)2615 798 y Fp(Z)2659 782 y Fy(d)r Fw(\()p Fs(q)s(;)14 b Fw(\003)2859 747 y Fu(C)2915 782 y Fw(\)\))p Fs(:)118 960 y Fw(Using)27 b(\(93\),)h(\(98\))f(and)h(\(101\),)e(w)n(e)i(get)f(\(94\))g(.)3808 1059 y Fh(2)118 1309 y Fn(Remark)k(2.2)40 b Fw(The)30 b(summing)f(o)n(v)n(er)f(causal)g(paths)h(in)h(Prop)r(osition)d(2.3)i (re\015ects)g(that)g(the)h(result)f(of)g(an)g(up)r(dating)118 1408 y(dep)r(ends)f(only)f(on)h(what)f(has)g(happ)r(ened)h(b)r(efore.) 118 1586 y Fn(De\014nition)j(2.4)40 b Fw(W)-7 b(e)24 b(de\014ne)g(t)n(w)n(o)e(families)i(\()p Fs(Y)1633 1598 y Fo(\003)1683 1586 y Fw(\))1715 1601 y Fo(\003)p Fm(2F)6 b(nf;g)2022 1586 y Fw(and)24 b(\()2225 1565 y(~)2212 1586 y Fs(Y)2260 1598 y Fo(\003)2310 1586 y Fw(\))2342 1601 y Fo(\003)p Fm(2F)6 b(nf;g)2649 1586 y Fw(of)24 b(random)e(v)-5 b(ariables)22 b(on)h(\012)10 b Ft(n)g(N)3681 1598 y Fo(1)3718 1586 y Fw(.)36 b(Let)118 1686 y(\003)c Ft(2)g(F)e(n)22 b(f;g)31 b Fw(and)i Fs(!)i Ft(2)d Fw(\012)22 b Ft(n)g(N)1161 1698 y Fo(1)1198 1686 y Fw(,)35 b(sa)n(y)d(with)h (exactly)f Fs(N)9 b Fw(\()p Fs(!)s Fw(\))33 b(jumps)h(in)f(\003)22 b Ft(\002)g Fw([0)p Fs(;)14 b(t)p Fw(].)52 b(If)34 b(w)n(e)e(c)n(ho)r (ose)g Fn(x)p Fs(;)14 b Fn(y)34 b Ft(2)f Fs(M)41 b Fw(with)118 1786 y Fn(x)168 1798 y Fo(\003)250 1786 y Fw(=)31 b Fn(y)396 1798 y Fo(\003)479 1786 y Fw(the)i(v)-5 b(alue)33 b(of)f Ft(4)1021 1798 y Fo(0)1057 1786 y Fw(\()p Fs(N)9 b Fw(\()p Fs(!)s Fw(\)\))34 b(\(as)e(de\014ned)h(b)n(y)g(\(90\))f(and)h(\(91\)\)) g(do)r(es)f(not)h(dep)r(end)g(on)g Fn(x)g Fw(or)f Fn(y)q Fw(.)53 b(W)-7 b(e)33 b(de\014ne)118 1885 y Fs(Y)166 1897 y Fo(\003)216 1885 y Fw(\()p Fs(!)s Fw(\))28 b(to)f(b)r(e)h(equal) f(to)h(this)g(v)-5 b(alue:)1645 2085 y Fs(Y)1693 2097 y Fo(\003)1743 2085 y Fw(\()p Fs(!)s Fw(\))1885 2038 y Fo(def)1899 2085 y Fw(=)36 b Ft(4)2075 2097 y Fo(0)2111 2085 y Fw(\()p Fs(N)9 b Fw(\()p Fs(!)s Fw(\)\))1339 b(\(102\))131 2210 y(~)118 2231 y Fs(Y)166 2243 y Fo(\003)244 2231 y Fw(is)27 b(de\014ned)h(analogously)-7 b(,)25 b(using)1313 2209 y(~)1297 2231 y Ft(4)1372 2243 y Fo(0)1408 2231 y Fw(\()p Fs(N)9 b Fw(\()p Fs(!)s Fw(\)\))28 b(instead)f(of)h Ft(4)2151 2243 y Fo(0)2187 2231 y Fw(\()p Fs(N)9 b Fw(\()p Fs(!)s Fw(\)\).)118 2410 y Fn(Remark)31 b(2.3)142 b Fw(1.)41 b(W)-7 b(e)19 b(remark)e(that)i Fs(Y)1480 2422 y Fo(\003)1549 2410 y Fw(dep)r(ends)g(measurably)e(on)i Fs(!)s Fw(.)33 b(In)19 b(fact)g(there)f(is)h(a)f(coun)n(table,)i(measurable)326 2509 y(partition)32 b(of)h(\012)22 b Ft(n)f(N)990 2521 y Fo(1)1061 2509 y Fw(suc)n(h)32 b(that)h Fs(!)j Fw(and)j(~)-48 b Fs(!)35 b Fw(b)r(elong)e(to)f(the)i(same)e(set)h(\(of)g(that)g 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3518 y Fw(~)1598 3539 y Fs(Y)1646 3551 y Fo(\003)1719 3539 y Fw(=)22 b(0)169 b Fj(for)31 b Fs(!)26 b Ft(2)d Fw(\012)c Ft(n)f Fw(\()p Ft(N)2545 3551 y Fo(1)2601 3539 y Ft([)g(N)2742 3551 y Fo(2)2780 3539 y Fw(\))p Fs(:)874 b Fw(\(104\))216 3784 y Fj(2.)43 b(The)30 b(limit)1623 4015 y Fs(\031)1670 4027 y Fo(0)1726 4015 y Ft(\016)18 b Fs(S)1842 3981 y Fu(t)1837 4036 y(!)1908 3968 y Fo(def)1921 4015 y Fw(=)71 b(lim)2022 4075 y Fo(\003)p Fm(!)p Fp(Z)2179 4058 y Fy(d)2221 4015 y Fs(\031)2268 4027 y Fo(0)2324 4015 y Ft(\016)18 b Fs(S)2440 3981 y Fu(t)2435 4036 y Fo(\003)p Fu(;\030)r(;!)3709 4015 y Fw(\(105\))326 4238 y Fj(exists)29 b(in)h Ft(C)704 4208 y Fo(0)741 4238 y Fw(\()p Fs(M)t(;)14 b(S)951 4208 y Fo(1)988 4238 y Fw(\))30 b Fj(for)g(al)t(l)h Fs(!)26 b Ft(2)d Fw(\012)c Ft(n)f Fw(\()p Ft(N)1696 4250 y Fo(1)1752 4238 y Ft([)h(N)1894 4250 y Fo(2)1931 4238 y Fw(\))p Fj(.)39 b(It)29 b(is)h(me)l(asur)l(able)h(in)e Fs(!)k Fj(and)d(do)l(es)g(not)g(dep)l(end)h(on)e Fs(\030)t Fj(.)216 4403 y(3.)43 b(Ther)l(e)30 b(is)g(a)h(set)e Ft(N)35 b(\032)23 b Fw(\012)30 b Fj(of)g Fs(P)12 b Fj(-me)l(asur)l(e)29 b(zer)l(o)h(such)g(that)g(we)g(c)l(an)g(de\014ne)g(maps)1585 4634 y Fs(\031)1632 4646 y Fu(q)1687 4634 y Ft(\016)18 b Fs(S)1803 4600 y Fu(t)1798 4655 y(!)1869 4587 y Fo(def)1883 4634 y Fw(=)109 b(lim)1984 4693 y Fu(q)r Fm(2)p Fo(\003)p Fm(!)p Fp(Z)2217 4677 y Fy(d)2260 4634 y Fs(\031)2307 4646 y Fu(q)2362 4634 y Ft(\016)18 b Fs(S)2478 4600 y Fu(t)2473 4655 y Fo(\003)p Fu(;\030)r(;!)3709 4634 y Fw(\(106\))326 4871 y Fj(for)30 b(al)t(l)h Fs(q)26 b Ft(2)e Fv(Z)780 4841 y Fu(d)842 4871 y Fj(and)31 b Fs(!)25 b Ft(2)f Fw(\012)18 b Ft(n)g(N)12 b Fj(.)326 5003 y(F)-6 b(urther)29 b(we)h(c)l(an)g(de\014ne)g(a)g(map)g Fs(S)1454 4973 y Fu(t)1449 5024 y(!)1520 5003 y Ft(2)24 b(C)1648 4973 y Fo(0)1684 5003 y Fw(\()p Fs(M)t(;)14 b(M)9 b Fw(\))30 b Fj(by)1680 5168 y Fk(\000)1718 5235 y Fs(S)1774 5201 y Fu(t)1769 5255 y(!)1817 5235 y Fw(\()p Fn(x)p Fw(\))1931 5168 y Fk(\001)1970 5268 y Fu(q)2030 5188 y Fo(def)2043 5235 y Fw(=)36 b Fs(\031)2191 5247 y Fu(q)2247 5235 y Ft(\016)17 b Fs(S)2362 5201 y Fu(t)2357 5255 y(!)2405 5235 y Fw(\()p Fn(x)p Fw(\))p Fs(:)1167 b Fw(\(107\))326 5433 y Fs(S)382 5403 y Fu(t)377 5454 y(!)454 5433 y Fj(dep)l(ends)31 b(me)l(asur)l(ably)f(on)g Fs(!)s Fj(.)1966 5682 y Fw(20)p eop %%Page: 21 21 21 20 bop 118 354 a Fn(Pro)s(of)59 b Fw(First)28 b(w)n(e)f(sho)n(w)g (that)1753 554 y(lim)1718 613 y Fo(\003)p Fm(!)p Fp(Z)1874 596 y Fy(d)1917 554 y Fs(E)5 b Fw(\()p Fs(Y)2063 566 y Fo(\003)2113 554 y Fw(\))24 b(=)e(0)1411 b(\(108\))118 775 y(W)-7 b(e)28 b(set)g Fs(R)477 728 y Fo(def)491 775 y Fw(=)36 b(dist)726 792 y Fp(Z)771 775 y Fy(d)r Fw(\()p Fs(q)s(;)14 b Fw(\003)971 745 y Fu(C)1027 775 y Fw(\).)37 b(Using)27 b(\(94\))h(w)n(e)f(get)934 1057 y Fs(E)5 b Fw(\()p Fs(Y)1080 1069 y Fo(\003)1130 1057 y Fw(\))2547 b(\(109\))1017 1224 y Ft(\024)1165 1111 y Fk(Z)1211 1300 y Fo(\012)1276 1224 y Fs(dP)12 b Fw(\()p Fs(!)s Fw(\))i Fs(c)1553 1236 y Fu(S)1615 1224 y Fs(a)p Fw(\()p Fs(R)q Fw(\))1179 1420 y Ft(\002)1476 1341 y Fk(X)1327 1511 y Fi(\()p Fy(r)1377 1523 y Fi(0)1410 1511 y(=0)p Fy(;r)1528 1523 y Fi(1)1560 1511 y Fy(;:::;r)1683 1519 y(n)1723 1511 y Fi(\))1267 1578 y Fe(2)p Fq(P)n(ath)1454 1592 y(c)1490 1578 y Fi(\()p Fy(t;!)q(;q)q Fe(!)p Fi(\003)1734 1566 y Fy(C)1783 1578 y Fi(\))1830 1420 y Fs(w)1889 1432 y Fo(1)1926 1420 y Fw(\()p Fs(r)1995 1432 y Fo(1)2052 1420 y Ft(\000)k Fs(r)2172 1432 y Fo(0)2209 1420 y Fw(\))h Ft(\001)g Fs(:)14 b(:)g(:)k Ft(\001)h Fs(w)2518 1432 y Fo(1)2555 1420 y Fw(\()p Fs(r)2624 1432 y Fu(n)2689 1420 y Ft(\000)f Fs(r)2809 1432 y Fu(n)p Fm(\000)p Fo(1)2939 1420 y Fw(\))1017 1721 y(=)83 b Fs(c)1201 1733 y Fu(s)1250 1721 y Fs(a)p Fw(\()p Fs(R)q Fw(\))1621 1642 y Fk(X)1436 1827 y Fu(Q)p Fm(2)p Fq(P)n(ath)q Fo(\(0)p Fm(!)p Fo(\003)1851 1810 y Fy(C)1900 1827 y Fo(\))1940 1721 y Fs(w)1999 1733 y Fo(1)2036 1721 y Fw(\()p Fs(r)2105 1733 y Fo(1)2162 1721 y Ft(\000)18 b Fs(r)2282 1733 y Fo(0)2320 1721 y Fw(\))g Ft(\001)h Fs(:)14 b(:)g(:)k Ft(\001)h Fs(w)2628 1733 y Fo(1)2665 1721 y Fw(\()p Fs(r)2734 1733 y Fu(n)2799 1721 y Ft(\000)f Fs(r)2919 1733 y Fu(n)p Fm(\000)p Fo(1)3049 1721 y Fw(\))1179 1957 y Ft(\002)p Fs(P)12 b Fw(\()p Ft(f)p Fs(!)24 b Fw(:)g Fs(Q)e Ft(2)i Fw(P)n(ath)1847 1969 y(c)1888 1957 y(\()p Fs(t;)14 b(!)s(;)g Fw(0)22 b Ft(!)h Fw(\003)2307 1922 y Fu(C)2363 1957 y Fw(\))p Ft(g)p Fw(\))118 2139 y(A)g(path)f Fs(Q)h Fw(=)g(\()p Fs(q)637 2151 y Fo(0)674 2139 y Fs(;)14 b(q)748 2151 y Fo(1)785 2139 y Fs(;)g(:)g(:)g(:)g(;)g(q)1007 2151 y Fu(n)1052 2139 y Fw(\))23 b(with)g Fs(q)1328 2151 y Fu(n)1396 2139 y Ft(2)h Fw(\003)1533 2109 y Fu(C)1611 2139 y Fw(is)e(causal)f(wrt)h(\()p Fs(t;)14 b(!)s Fw(\))23 b(\(i.e.)g Fs(Q)g Ft(2)g Fw(P)n(ath)2791 2151 y(c)2832 2139 y(\()p Fs(t;)14 b(!)s(;)g Fw(0)22 b Ft(!)h Fw(\003)3251 2109 y Fu(C)3307 2139 y Fw(\)\))g(i\013)g(the)g(P)n(oisson)118 2239 y(pro)r(cess)k(induced)h(b)n(y)f Fs(Q)g Fw(has)g(at)h(least)f Fs(n)g Fw(jumps.)38 b(So)27 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Fs(i)p Fw(\)\))1390 b(\(116\))1966 5682 y(21)p eop %%Page: 22 22 22 21 bop 118 354 a Fw(If)30 b Fs(q)f Ft(2)e Fw(\003)409 366 y Fo(1)445 318 y Fu(C)531 354 y Fw(then)j(\(116\))e(ob)n(viously)g (holds)h(b)r(ecause)1853 332 y(~)1837 354 y Ft(4)1912 324 y Fo(1)1912 375 y Fu(q)1948 354 y Fw(\()p Fs(i)p Fw(\))e(=)e Fs(c)2194 366 y Fu(S)2272 354 y Fw(is)k(an)g(upp)r(er)h(b)r (ound)g(for)3117 332 y(~)3100 354 y Ft(4)3175 324 y Fo(2)3175 375 y Fu(q)3211 354 y Fw(\()p Fs(j)5 b Fw(\).)43 b(F)-7 b(or)29 b Fs(q)g Ft(2)e Fw(\003)3737 366 y Fo(1)3803 354 y Fw(w)n(e)118 454 y(sho)n(w)g(\(116\))g(b)n(y)g(induction)h(wrt)f Fs(i)p Fw(.)118 554 y(If)32 b Fs(i)d Fw(=)g(0)i(then)h(\(116\))e(is)i (true)f(b)n(y)g(\(90\).)48 b(No)n(w)31 b(assume)f(that)i(\(116\))f (holds)g(for)f(all)i Fs(q)i Fw(and)d(a)g(particular)f Fs(i)f(<)g(n)p Fw(.)48 b(F)-7 b(or)118 653 y Fs(q)26 b Ft(2)e Fw(\003)318 665 y Fo(1)373 653 y Ft(n)18 b(f)p Fs(q)512 665 y Fu(i)p Fo(+1)623 653 y Ft(g)27 b Fw(w)n(e)h(ha)n(v)n(e) 1203 831 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b(follo)n(ws)f(for)h Fs(r)j Ft(2)f Fw(\003)3533 2063 y Fo(1)3592 2051 y Fw(from)d(the)118 2150 y(de\014nition)e(of)579 2128 y(~)563 2150 y Ft(4)638 2120 y Fo(1)638 2171 y Fu(r)693 2150 y Fw(and)861 2128 y(~)845 2150 y Ft(4)920 2120 y Fo(2)920 2171 y Fu(r)975 2150 y Fw(and)f(for)g Fs(r)26 b Ft(2)d Fw(\003)1444 2162 y Fo(1)1500 2150 y Fw(from)18 b(assumption)g(\(116\))f(and)i(the)f (fact)h(that)2952 2128 y(~)2936 2150 y Ft(4)3011 2120 y Fo(2)3011 2171 y Fu(r)3047 2150 y Fw(\()p Fs(j)5 b Fw(\()p Fs(i)p Fw(+1\))p Ft(\000)p Fw(1\))22 b(=)3583 2128 y(~)3567 2150 y Ft(4)3642 2120 y Fo(2)3642 2171 y Fu(r)3678 2150 y Fw(\()p Fs(j)5 b Fw(\()p Fs(i)p Fw(\)\).)118 2250 y(Using)27 b(the)h(de\014nition)g(of)971 2229 y(~)958 2250 y Fs(Y)1006 2262 y Fo(\003)1051 2270 y Fi(1)1088 2250 y Fw(\()p Fs(!)s Fw(\))g(and)1409 2229 y(~)1397 2250 y Fs(Y)1445 2262 y Fo(\003)1490 2270 y Fi(2)1527 2250 y Fw(\()p Fs(!)s Fw(\))g(\(cf.)g(De\014nition)g(2.4\),)f(w)n(e)h (conclude)1716 2428 y(~)1703 2449 y Fs(Y)1751 2461 y Fo(\003)1796 2469 y Fi(1)1833 2449 y Fw(\()p Fs(!)s Fw(\))c Ft(\025)2076 2428 y Fw(~)2063 2449 y Fs(Y)2111 2461 y Fo(\003)2156 2469 y Fi(2)2193 2449 y Fw(\()p Fs(!)s Fw(\))1397 b(\(119\))118 2598 y(whic)n(h)28 b(w)n(as)e(to)i(b)r(e)g(sho)n(wn.)118 2698 y(W)-7 b(e)26 b(ha)n(v)n(e)e(pro)n(v)n(ed)g(\(115\))h(and)h(that)f (\()1313 2677 y(~)1300 2698 y Fs(Y)1348 2710 y Fo(\003)1398 2698 y Fw(\))1430 2713 y Fo(\003)p Fm(2F)6 b(nf;g)1740 2698 y Fw(is)26 b(decreasing.)34 b(So)25 b(w)n(e)h(conclude)f(\(104\),) g(b)n(y)g(using)h(the)g(Monotone)118 2798 y(Con)n(v)n(ergence)f (Theorem.)118 2897 y(No)n(w)35 b(w)n(e)h(pro)n(v)n(e)e(the)i(second)f (statemen)n(t)h(in)g(Theorem)f(2.1,)h(using)g(the)g(\014rst)f(one.)61 b(First)36 b(w)n(e)f(note)h(that)g(for)f Fs(!)k Ft(2)118 2997 y Fw(\012)24 b Ft(n)f Fw(\()p Ft(N)367 3009 y Fo(1)428 2997 y Ft([)h(N)575 3009 y Fo(2)613 2997 y Fw(\))36 b(the)g(map)f Fs(S)1080 2967 y Fu(t)1075 3020 y Fo(\003)p Fu(;\030)r(;!)1276 2997 y Fw(is)g(con)n(tin)n(uous)g(since)g(it)h(is)f(the)h(comp)r(osite) f(of)h(\014nitely)g(man)n(y)f(con)n(tin)n(uous)f(\(cf.)118 3097 y(Lemma)27 b(2.2\))g(up)r(dating)h(maps.)118 3196 y(F)-7 b(or)27 b(\003)c Ft(\032)444 3175 y Fw(~)435 3196 y(\003)28 b(w)n(e)f(ha)n(v)n(e)1237 3395 y Fs(\045)1280 3410 y Fm(C)1319 3394 y Fi(0)1351 3410 y Fo(\()p Fu(M)s(;S)1507 3394 y Fi(1)1539 3410 y Fo(\))1569 3395 y Fw(\()p Fs(\031)1648 3407 y Fo(0)1704 3395 y Ft(\016)18 b Fs(S)1820 3361 y Fu(t)1815 3416 y Fo(\003)p Fu(;\030)r(;!)1980 3395 y Fs(;)c(\031)2064 3407 y Fo(0)2120 3395 y Ft(\016)k Fs(S)2236 3361 y Fu(t)2237 3413 y Fo(~)2231 3428 y(\003)p Fu(;\030)r(;!)2396 3395 y Fw(\))23 b Ft(\024)2552 3374 y Fw(~)2539 3395 y Fs(Y)2587 3407 y Fo(\003)2637 3395 y Fw(\()p Fs(!)s Fw(\))p Fs(:)930 b Fw(\(120\))118 3565 y(So)21 b(b)n(y)g(\(104\))f(the) i(net)g(\()p Fs(\031)899 3577 y Fo(0)942 3565 y Ft(\016)6 b Fs(S)1046 3535 y Fu(t)1041 3589 y Fo(\003)p Fu(;\030)r(;!)1205 3565 y Fw(\))1237 3580 y Fo(\003)p 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Fm(\025)1689 5373 y Ft(\002)13 b Fw(\012)g Ft(\002)g Fs(M)30 b Ft(3)23 b Fw(\()p Fs(t;)14 b(!)s(;)g(x)p Fw(\))24 b Ft(7!)f Fs(S)2563 5343 y Fu(t)2558 5394 y(!)2606 5373 y Fw(\()p Fs(x)p Fw(\))i(and)g(the)g(shifts)g Fs(\022)r Fw(\()p Fs(t)p Fw(\))h(on)e(\012,)h(de\014ned)1966 5682 y(22)p eop %%Page: 23 23 23 22 bop 326 354 a Fw(b)n(y)23 b Fs(X)513 314 y Fu(q)506 375 y(t)531 383 y Fi(2)568 354 y Fw(\()p Fs(\022)r Fw(\()p Fs(t)703 366 y Fo(1)741 354 y Fw(\))p Fs(!)s Fw(\))883 307 y Fo(def)896 354 y Fw(=)36 b Fs(X)1073 314 y Fu(q)1066 375 y(t)1091 383 y Fi(1)1124 375 y Fo(+)p Fu(t)1200 383 y Fi(2)1236 354 y Fw(\()p Fs(!)s Fw(\))11 b Ft(\000)g Fs(X)1518 314 y Fu(q)1511 375 y(t)1536 383 y Fi(1)1573 354 y Fw(\()p Fs(!)s Fw(\).)35 b(One)24 b(can)f(further)h(de\014ne)g (for)g Fs(P)12 b Fw(-a.a.)22 b Fs(!)k Ft(2)d Fw(\012)h(the)h(linear)e (op)r(erators)326 454 y(`)p Ft(\016)p Fs(S)447 424 y Fu(t)442 474 y(!)489 454 y Fw(',)35 b(acting)d(on)h(con)n(tin)n(uous)f (functions)i(and)f(the)g(corresp)r(onding)e(transfer)i(op)r(erators)e (`)p Ft(L)3351 424 y Fu(t)3351 474 y(!)3399 454 y Fw('.)54 b(So)33 b(one)f(has)326 554 y(op)r(erator-v)-5 b(alued)22 b(random)h(v)-5 b(ariables.)35 b(Ho)n(w)n(ev)n(er,)23 b(in)i(the)f(follo)n(wing)g(w)n(e)g(consider)f(only)h(the)h(a)n(v)n (eraged)c(\(wrt)k Fs(!)s Fw(\))326 653 y(op)r(erators.)220 819 y(2.)41 b(W)-7 b(e)22 b(ha)n(v)n(e)f(de\014ned)h(maps)g Fs(S)1197 789 y Fu(t)1192 840 y(!)1263 819 y Ft(2)h(C)1390 789 y Fo(0)1427 819 y Fw(\()p Fs(M)t(;)14 b(M)9 b Fw(\))22 b(for)g(\014nite)g(range)f(up)r(dating)h(in)g(\(72\))g(and)g(a)f(sp)r (ecial)h(class)f(of)h(in\014nite)326 919 y(range)e(up)r(datings)h(in)h (\(107\),)g(using)f(\(105\).)34 b(Note)21 b(that)h(the)f(second)g (class)g(do)r(es)g(not)g(include)h(the)f(\014rst.)35 b(It)22 b(migh)n(t)326 1019 y(b)r(e)27 b(in)n(teresting)g(to)g(\014nd)h (more)e(general)g(classes)g(of)i(\(single)f(site\))g(up)r(dating)h (functions)g(for)e(whic)n(h)i(the)f(limit)h(in)326 1118 y(\(105\))e(exists,)i(or)e(examples)h(where)g(it)h(do)r(es)g(not)f (exist.)118 1351 y Fg(2.3)112 b(Mark)m(o)m(v)38 b(Kernels)118 1504 y Fw(In)26 b(Section)g(1)f(w)n(e)h(de\014ned)g(the)g(P)n(oisson)e (pro)r(cess)h(\(\012)p Fs(;)14 b Ft(A)p Fs(;)g(P)r(;)g Fw(\()p Fs(X)2133 1516 y Fu(t)2163 1504 y Fw(\))2195 1519 y Fu(t)p Fm(2)p Fo([0)p Fu(;T)9 b Fo(])2408 1504 y Fw(\))26 b(with)g(parameter)f Fs(\025)h Fw(and)g(v)-5 b(alues)25 b(in)h Fv(N)3677 1474 y Fo(\003)3733 1504 y Fw(,)g(the)118 1603 y(measure)h(space)g(\()p Fs(M)t(;)14 b Ft(B)873 1615 y Fu(M)946 1603 y Fs(;)g(\026)p Fw(\))28 b(and)f(the)h(measurable)e(space)h(\()p Ft(C)2134 1573 y Fo(0)2171 1603 y Fw(\()p Fs(M)t(;)14 b(M)9 b Fw(\))p Fs(;)14 b Ft(B)2539 1618 y Fm(C)2578 1602 y Fi(0)2610 1618 y Fo(\()p Fu(M)s(;M)6 b Fo(\))2821 1603 y Fw(\).)118 1703 y(W)-7 b(e)23 b(ha)n(v)n(e)f(nets)g(\()p Fs(S)701 1673 y Fu(T)696 1726 y Fo(\003)754 1703 y Fw(\))786 1718 y Fo(\003)p Fm(2F)6 b(nf;g)1093 1703 y Fw(of)22 b(maps)h Fs(S)1451 1673 y Fu(T)1446 1726 y Fo(\003)1526 1703 y Fw(:)g(\012)9 b Ft(n)g(N)34 b(!)23 b(C)1949 1673 y Fo(0)1986 1703 y Fw(\()p Fs(M)t(;)14 b(M)9 b Fw(\))22 b(with)i(limit)f Fs(S)2718 1673 y Fu(T)2793 1703 y Ft(2)h(C)2921 1673 y Fo(0)2957 1703 y Fw(\()p Fs(M)t(;)14 b(M)9 b Fw(\),)24 b(and)f(the)g(follo)n(wing)118 1803 y(statemen)n(ts)k(hold:)220 1985 y(1.)41 b Fs(S)382 1955 y Fu(T)377 2008 y Fo(\003)461 1985 y Fw(and)28 b Fs(S)679 1955 y Fu(T)758 1985 y Fw(are)f(\()p Ft(A)p Fs(;)14 b Ft(B)1087 2000 y Fm(C)1126 1984 y Fi(0)1158 2000 y Fo(\()p Fu(M)s(;M)6 b Fo(\))1370 1985 y Fw(\)-measurable.)220 2151 y(2.)41 b Fs(S)382 2121 y Fu(T)461 2151 y Fw(is)28 b(the)g(p)r(oin)n(t)n(wise)f(limit)h(of)g(the)g(net)g(\()p Fs(S)1722 2121 y Fu(T)1717 2174 y Fo(\003)1774 2151 y Fw(\))1806 2166 y Fo(\003)p Fm(2F)6 b(nf;g)2090 2151 y Fw(.)220 2317 y(3.)41 b(F)-7 b(or)27 b(\014xed)g Fn(x)d Ft(2)f Fs(M)36 b Fw(the)28 b(map)g Fs(S)1328 2287 y Fu(T)1323 2340 y Fo(\003)p Fu(;)p Fn(.)1419 2317 y Fw(\()p Fn(x)p 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b(that)g(these)g(maps)f(are)f(a)i(priori)e(not)i(de\014ned)g (on)f(a)g(set)h(of)f Fs(P)12 b Fw(-measure)118 2911 y(zero.)36 b(F)-7 b(or)27 b(these)h(exceptional)e Fs(!)g Ft(2)d Fw(\012)28 b(w)n(e)f(de\014ne)h Fs(S)1785 2881 y Fu(T)1780 2932 y(!)1865 2911 y Fw(and)f Fs(S)2082 2881 y Fu(T)2077 2934 y Fo(\003)p Fu(;!)2218 2911 y Fw(to)h(b)r(e)g(equal)f(to)g(the)h (iden)n(tit)n(y)g(on)f Fs(M)9 b Fw(.)118 3024 y(Statemen)n(t)23 b(3.)e(follo)n(ws)g(from)h(measurabilit)n(y)f(wrt)h(\()p Fs(!)s(;)14 b Fn(x)p Fw(\))22 b(of)g Fs(S)2088 2993 y Fu(T)2083 3046 y Fo(\003)p Fu(;!)2197 3024 y Fw(\()p Fn(x)p Fw(\))h(\(Prop)r(osition)d(2.2)i(and)g(Remark)f(1.6.2)g(for)g (\014nite)118 3123 y(range)28 b(in)n(teraction)g(and)h(statemen)n(t)g (3.)f(of)h(Theorem)g(2.1)f(and)h(Remark)f(1.6.2)g(for)g(in\014nite)i (range)e(in)n(teraction\),)g(the)118 3223 y(fact)g(that)g(one-p)r(oin)n (t-sets)e(in)i Fs(M)36 b Fw(are)27 b(measurable,)f(and)i(F)-7 b(ubini's)28 b(Theorem.)118 3322 y(Lik)n(e)f(in)h(Example)f(1.1)g(w)n (e)g(set)1248 3605 y Fs(K)1325 3570 y Fu(T)1319 3625 y(S)1399 3605 y Fw(:)c Fs(M)28 b Ft(\002)18 b(B)1692 3617 y Fu(M)1852 3605 y Ft(!)87 b Fw([0)p Fs(;)14 b Fw(1])1520 b(\(122\))1422 3754 y Fs(K)1499 3720 y Fu(T)1493 3775 y(S)1551 3754 y Fw(\()p Fn(x)p Fs(;)14 b(A)p Fw(\))1848 3707 y Fo(def)1861 3754 y Fw(=)96 b Fs(P)12 b Fw(\()p Ft(f)p Fs(!)26 b Fw(:)d Fs(S)2341 3720 y Fu(T)2336 3775 y(!)2393 3754 y Fw(\()p Fn(x)p Fw(\))h Ft(2)f Fs(A)p Ft(g)p Fw(\))p Fs(:)118 3937 y Fw(The)28 b(corresp)r(onding)d(op)r (erator,)h(applied)i(to)f(an)h Fs(f)j Ft(2)24 b(C)1888 3907 y Fo(0)1925 3937 y Fw(\()p Fs(M)9 b Fw(\),)27 b(is)1350 4241 y(\()p Fs(K)1459 4207 y Fu(T)1453 4262 y(S)1511 4241 y Fs(f)9 b Fw(\)\()p Fn(x)p Fw(\))84 b(=)1939 4128 y Fk(Z)1985 4317 y Fu(M)2072 4241 y Fs(K)2149 4207 y Fu(T)2143 4262 y(S)2201 4241 y Fw(\()p Fn(x)p Fs(;)14 b(d)p Fn(y)q Fw(\))g Fs(f)9 b Fw(\()p Fn(y)q Fw(\))1084 b(\(123\))1791 4463 y(=)1939 4350 y Fk(Z)1985 4538 y Fo(\012)2050 4463 y Fs(dP)12 b Fw(\()p Fs(!)s Fw(\))i Fs(f)27 b Ft(\016)18 b Fs(S)2475 4429 y Fu(T)2470 4483 y(!)2527 4463 y Fw(\()p Fn(x)p Fw(\))p Fs(:)1045 b Fw(\(124\))118 4692 y(\(123\))27 b(is)g(the)h(de\014nition)g(\(cf.)h(\(32\)\),)e(and)h (\(124\))e(is)i(a)f(consequence)g(of)g(\(122\).)118 4792 y(W)-7 b(e)23 b(de\014ne)g(analogously)e(the)i(Mark)n(o)n(v)d(k)n (ernels)i Fs(K)1722 4762 y Fu(T)1716 4815 y(S;)p Fo(\003)1848 4792 y Fw(and)h(corresp)r(onding)e(op)r(erators)f(for)j(the)g(P)n (oisson)e(pro)r(cess)g(with)118 4904 y(v)-5 b(alues)27 b(in)h Fv(N)516 4874 y Fo(\003)572 4904 y Fw(.)118 5087 y Fn(Prop)s(osition)i(2.4)41 b Fs(K)875 5057 y Fu(T)869 5110 y(S)956 5087 y Fj(and)31 b Fs(K)1195 5057 y Fu(T)1189 5110 y(S;)p Fo(\003)1327 5087 y Fj(ar)l(e)f(b)l(ounde)l(d)h(line)l(ar)f (op)l(er)l(ators)h(on)f Ft(C)2548 5057 y Fo(0)2584 5087 y Fw(\()p Fs(M)9 b Fw(\))p Fj(.)118 5286 y Fn(Pro)s(of)60 b Fw(W)-7 b(e)28 b(giv)n(e)e(the)j(pro)r(of)e(for)g Fs(K)1288 5256 y Fu(T)1282 5309 y(S)1339 5286 y Fw(.)38 b(The)27 b(one)h(for)f Fs(K)1927 5256 y Fu(T)1921 5309 y(S;)p Fo(\003)2058 5286 y Fw(is)g(analogous.)35 b(Let)28 b Fs(!)e Ft(2)d Fw(\012,)28 b Fs(f)k Ft(2)23 b(C)3173 5256 y Fo(0)3210 5286 y Fw(\()p Fs(M)9 b Fw(\))28 b(and)g(\()p Fn(x)3636 5256 y Fo(\()p Fu(n)p Fo(\))3733 5286 y Fw(\))3765 5298 y Fu(n)p Fm(2)p Fp(N)118 5385 y Fw(a)f(sequence)g(in)h Fs(M)37 b Fw(with)28 b(limit)g Fn(x)p Fw(.)37 b(Then)1966 5682 y(23)p eop %%Page: 24 24 24 23 bop 1711 532 a Fw(lim)1682 582 y Fu(n)p Fm(!1)1869 532 y Fs(S)1925 498 y Fu(T)1920 552 y(!)1977 532 y Fw(\()p Fn(x)2059 498 y Fo(\()p Fu(n)p Fo(\))2157 532 y Fw(\))83 b(=)g Fs(S)2476 498 y Fu(T)2471 552 y(!)2528 532 y Fw(\()p Fn(x)p Fw(\))1067 b(\(125\))1222 701 y(and)27 b(so)125 b(lim)1554 751 y Fu(n)p Fm(!1)1741 701 y Fs(f)27 b Ft(\016)18 b Fs(S)1925 667 y Fu(T)1920 722 y(!)1977 701 y Fw(\()p Fn(x)2059 667 y Fo(\()p Fu(n)p Fo(\))2157 701 y Fw(\))83 b(=)g Fs(f)27 b Ft(\016)18 b Fs(S)2604 667 y Fu(T)2599 722 y(!)2656 701 y Fw(\()p Fn(x)p 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b(\()p Fs(K)2052 1918 y Fu(T)2046 1973 y(S)2104 1952 y Fs(f)9 b Fw(\)\()p Fn(x)p Fw(\))p Fs(:)118 2130 y Fw(So)27 b Fs(K)310 2100 y Fu(T)304 2153 y(S)362 2130 y Fs(f)36 b Fw(is)28 b(con)n(tin)n(uous.)36 b(Con)n(tin)n(uit)n(y)27 b(of)g(the)h(op)r(erator)e(follo)n(ws)h(from)g(\(124\))g(and)g (\(127\).)3808 2229 y Fh(2)118 2478 y Fn(Prop)s(osition)j(2.5)41 b Fj(The)31 b(net)e Fw(\()p Fs(K)1219 2448 y Fu(T)1213 2501 y(S;)p Fo(\003)1322 2478 y Fw(\))1354 2493 y Fo(\003)p Fm(2F)6 b(nf;g)1668 2478 y Fj(c)l(onver)l(ges)30 b(we)l(akly)h(to)f Fs(K)2476 2448 y Fu(T)2470 2501 y(S)2557 2478 y Fj(\(as)g Fw(\003)23 b Ft(!)g Fv(Z)2945 2448 y Fu(d)2978 2478 y Fj(\),)30 b(i.e.)i(for)e(al)t(l)h Fs(f)h Ft(2)23 b(C)3663 2448 y Fo(0)3700 2478 y Fw(\()p Fs(M)9 b Fw(\))p Fj(:)1675 2677 y Fw(lim)1640 2736 y Fo(\003)p Fm(!)p Fp(Z)1796 2720 y Fy(d)1839 2677 y Fs(K)1916 2643 y Fu(T)1910 2698 y(S;)p Fo(\003)2019 2677 y Fs(f)32 b Fw(=)22 b Fs(K)2256 2643 y Fu(T)2250 2698 y(S)2308 2677 y Fs(f)t(:)1333 b Fw(\(129\))118 2889 y Fn(Pro)s(of)59 b Fw(W)-7 b(e)28 b(ha)n(v)n(e)982 3188 y Ft(k)p Fs(K)1101 3153 y Fu(T)1095 3208 y(S)1152 3188 y Fs(f)f Ft(\000)18 b Fs(K)1380 3153 y Fu(T)1374 3208 y(S;)p Fo(\003)1483 3188 y Fs(f)9 b Ft(k)1575 3200 y Fm(1)1728 3188 y Ft(\024)1875 3075 y Fk(Z)1921 3263 y Fo(\012)1987 3188 y Fs(dP)j Fw(\()p Fs(!)s Fw(\))i Ft(k)p Fs(f)27 b Ft(\016)17 b Fs(S)2453 3153 y Fu(T)2448 3208 y(!)2524 3188 y Ft(\000)h Fs(f)27 b Ft(\016)18 b Fs(S)2791 3153 y Fu(T)2786 3208 y Fo(\003)p Fu(;!)2899 3188 y Ft(k)2941 3200 y Fm(1)3011 3188 y Fs(:)675 b Fw(\(130\))118 3420 y(Because)27 b(of)g(condition)h(2)f(on)g(page)g (23)g(and)g(\(127\))g(the)h(rhs)f(con)n(v)n(erges)e(to)i(0)g(\(as)h (\003)22 b Ft(!)h Fv(Z)2969 3390 y Fu(d)3001 3420 y Fw(\).)3808 3520 y Fh(2)118 3768 y Fn(Remark)31 b(2.5)40 b Fw(It)g(follo)n(ws)e (from)g(Remark)g(2.4.1)g(and)h(the)g(homogeneit)n(y)f(of)h(P)n(oisson)e (pro)r(cesses)g(wrt)i(time)g(that)118 3868 y(\()p Fs(K)227 3837 y Fu(t)221 3890 y(S)269 3868 y Fw(\))301 3880 y Fu(t)p Fm(\025)p Fo(0)443 3868 y Fw(is)28 b(a)f(semigroup.)118 4141 y Fx(3)135 b(T)-11 b(ransfer)44 b(Op)t(erators)118 4323 y Fw(In)23 b(this)f(section)g(w)n(e)g(de\014ne)h(transfer)e(op)r (erators)g(for)g(the)i(Mark)n(o)n(v)d(k)n(ernels)h(for)h(a)g(sp)r (ecial)g(class)g(of)g(up)r(dating)g(functions)118 4423 y(that)28 b(w)n(e)f(ha)n(v)n(e)g(already)f(studied)i(in)g([13)o(].)37 b(First)27 b(w)n(e)g(recall)g(some)g(de\014nitions)h(and)f(notations)g (from)g([13)o(].)118 4522 y(F)-7 b(or)27 b Fs(\016)f(>)d Fw(0)k(w)n(e)g(denote)h(b)n(y)f Fs(A)1054 4534 y Fu(\016)1118 4522 y Fw(the)h(ann)n(ulus)1443 4722 y Fs(A)1505 4734 y Fu(\016)1565 4675 y Fo(def)1578 4722 y Fw(=)36 b Ft(f)p Fs(z)26 b Ft(2)e Fv(C)44 b Ft(j)23 b(\000)p Fs(\016)i Ft(\024)e Fw(ln)14 b Ft(j)p Fs(z)t Ft(j)22 b(\024)h Fs(\016)s Ft(g)1136 b Fw(\(131\))118 4868 y(and)28 b(b)n(y)f(\000)g(its)h(p)r (ositiv)n(ely)f(orien)n(ted)g(b)r(oundary)-7 b(.)118 4968 y(F)g(or)27 b Ft(;)c(6)p Fw(=)f(\003)h Ft(\032)g Fv(Z)649 4937 y Fu(d)709 4968 y Fw(the)28 b(normalized)f(Leb)r (esgue-measure)e(on)i(\()p Fs(S)2162 4937 y Fo(1)2200 4968 y Fw(\))2232 4937 y Fo(\003)2309 4968 y Fw(is)g(denoted)h(b)n(y)f Fs(\026)2871 4937 y Fo(\003)2921 4968 y Fw(.)37 b(F)-7 b(or)27 b(\014nite)h(\003)f(it)h(is)f(giv)n(en)g(b)n(y)1381 5189 y Fs(d\026)1474 5155 y Fo(\003)1524 5189 y Fw(\()p Fn(z)p Fw(\))d(=)1846 5133 y Fs(d)p Fn(z)p 1752 5170 275 4 v 1752 5248 a Fw(\(2)p Fs(\031)s(i)p Fw(\))1937 5224 y Fm(j)p Fo(\003)p Fm(j)2046 5133 y Fw(1)p 2046 5170 43 4 v 2046 5246 a Fn(z)2121 5142 y Fo(def)2135 5189 y Fw(=)2245 5110 y Fk(Y)2236 5289 y Fu(p)p Fm(2)p Fo(\003)2384 5133 y Fs(dz)2466 5145 y Fu(p)p 2384 5170 121 4 v 2384 5246 a Fw(2)p Fs(\031)s(i)2542 5133 y Fw(1)p 2524 5170 77 4 v 2524 5246 a Fs(z)2563 5258 y Fu(p)2611 5189 y Fs(:)1075 b Fw(\(132\))118 5433 y(W)-7 b(e)28 b(also)f(use)g Fs(d\026)664 5403 y Fo(\003)714 5433 y Fw(\()p Fn(z)p Fw(\))i(as)d(a)i(shorthand)e(notation)h(for)h(the)g (righ)n(t-hand)e(side)h(of)h(\(132\))f(for)g Fn(z)c Ft(2)h Fs(A)3234 5403 y Fo(\003)3234 5457 y Fu(\016)3284 5433 y Fw(.)1966 5682 y(24)p eop %%Page: 25 25 25 24 bop 118 354 a Fw(In)29 b Fn(Assumption)h(I)f Fw(\(see)g(b)r(elo)n (w\))f(w)n(e)g(will)h(\014x)g(a)f Fs(\016)g(>)c Fw(0.)39 b(F)-7 b(or)28 b(\003)d Ft(2)g(F)36 b Fw(w)n(e)28 b(denote)h(b)n(y)f Ft(H)2969 366 y Fo(\003)3047 354 y Fw(the)h(space)f(of)h(con)n(tin)n (uous)118 454 y(functions)34 b(on)e(the)i(p)r(oly)n(ann)n(ulus)e Fs(A)1280 424 y Fo(\003)1280 477 y Fu(\016)1363 454 y Fw(that)i(are)e(holomorphic)g(on)g(its)i(in)n(terior)e(and)h(write)f Ft(k)22 b(\001)g(k)3250 466 y Fo(\003)3333 454 y Fw(for)32 b(the)i(uniform)118 570 y(norm)f(on)h Ft(H)533 582 y Fo(\003)583 570 y Fw(.)55 b(As)34 b(a)g(function)g(on)g Fs(A)1381 540 y Fo(\003)1381 593 y Fu(\016)1464 570 y Fw(is)g(also)f(a)g(function)i(on)e Fs(A)2317 540 y Fp(Z)2362 515 y Fy(d)2317 593 y Fu(\016)2429 570 y Fw(w)n(e)h(can)f(drop)g(the)i (index)e(\003)h(and)g(mean)f(the)118 669 y(uniform)j(norm)f(on)g(the)h (in\014nite-dimensional)g(p)r(oly)n(ann)n(ulus.)60 b Ft(H)37 b Fw(is)e(the)h(v)n(ectorspace)e(of)h(all)h(consisten)n(t)f (families)118 769 y Fs(\036)24 b Fw(=)e(\()p Fs(\036)359 781 y Fo(\003)410 769 y Fw(\))442 781 y Fo(\003)p Fm(2F)617 769 y Fw(of)28 b(functions)g Fs(\036)1119 781 y Fo(\003)1192 769 y Ft(2)23 b(H)1340 781 y Fo(\003)1390 769 y Fw(.)37 b(Consistency)27 b(means)852 1066 y(\()p Fs(\031)931 1078 y Fo(\003)976 1086 y Fi(1)1013 1066 y Fs(\036)1062 1078 y Fo(\003)1107 1086 y Fi(2)1145 1066 y Fw(\)\()p Fn(z)1251 1078 y Fo(\003)1296 1086 y Fi(1)1334 1066 y Fw(\))1449 1019 y Fo(def)1462 1066 y Fw(=)1623 953 y Fk(Z)1669 1141 y Fo(\()p Fu(S)1739 1125 y Fi(1)1772 1141 y Fo(\))1798 1123 y Fi(\003)1836 1135 y(2)1868 1123 y Fe(n)p Fi(\003)1937 1135 y(1)1991 1066 y Fs(d\026)2084 1032 y Fo(\003)2129 1040 y Fi(2)2162 1032 y Fm(n)p Fo(\003)2241 1040 y Fi(1)2278 1066 y Fw(\()p Fn(z)2352 1081 y Fo(\003)2397 1089 y Fi(2)2430 1081 y Fm(n)p Fo(\003)2509 1089 y Fi(1)2546 1066 y Fw(\))19 b Fs(\036)p Fw(\()p Fn(z)2720 1078 y Fo(\003)2765 1086 y Fi(1)2822 1066 y Ft(_)g Fn(z)2938 1081 y Fo(\003)2983 1089 y Fi(2)3016 1081 y Fm(n)p Fo(\003)3095 1089 y Fi(1)3132 1066 y Fw(\))545 b(\(133\))1462 1247 y(=)96 b Fs(\036)1672 1259 y Fo(\003)1717 1267 y Fi(1)118 1422 y Fw(for)26 b(all)g(\003)416 1434 y Fo(1)476 1422 y Ft(\022)d Fw(\003)622 1434 y Fo(2)682 1422 y Ft(2)g(F)35 b Fw(and)26 b Fn(z)1057 1434 y Fo(\003)1102 1442 y Fi(1)1163 1422 y Ft(2)d Fs(A)1303 1385 y Fo(\003)1348 1393 y Fi(1)1303 1447 y Fu(\016)1386 1422 y Fw(.)36 b(\(Note)27 b(that)g(w)n(e)f(use)h (the)g(same)f(sym)n(b)r(ol)g(`)p Fs(\031)2822 1434 y Fo(\003)2872 1422 y Fw(')g(for)h(pro)5 b(jections)25 b(of)h(functions)118 1522 y(and)i(pro)5 b(jections)26 b(of)h(co)r(ordinates,)g(for)g(example)g(from)g Fs(M)36 b Fw(to)28 b(\()p Fs(S)2221 1492 y Fo(1)2258 1522 y Fw(\))2290 1492 y Fo(\003)2340 1522 y Fw(.\))118 1622 y(F)-7 b(or)27 b(0)c Fs(<)f(#)h(<)g Fw(1)k(and)h Fs(\036)23 b Ft(2)h(H)k Fw(w)n(e)f(de\014ne)1441 1897 y Ft(k)p Fs(\036)p Ft(k)1574 1909 y Fu(#)1714 1897 y Fw(=)106 b(sup)1875 1966 y Fo(\003)p Fm(2F)2051 1897 y Fs(#)2100 1862 y Fm(j)p Fo(\003)p Fm(j)2203 1897 y Ft(k)p Fs(\036)2294 1909 y Fo(\003)2344 1897 y Ft(k)2386 1909 y Fo(\003)3709 1897 y Fw(\(134\))1377 2125 y Ft(k)p Fs(\036)p Ft(k)1510 2137 y Fu(v)r(ar)1701 2078 y Fo(def)1714 2125 y Fw(=)131 b(lim)1875 2185 y Fo(\003)p Fm(!)p Fp(Z)2032 2168 y Fy(d)2074 2012 y Fk(Z)2121 2201 y Fo(\()p Fu(S)2191 2184 y Fi(1)2223 2201 y Fo(\))2249 2184 y Fi(\003)2309 2125 y Fs(d\026)2402 2091 y Fo(\003)2470 2125 y Ft(j)p Fs(\036)2542 2137 y Fo(\003)2592 2125 y Ft(j)p Fs(:)1071 b Fw(\(135\))118 2356 y(W)-7 b(e)28 b(set)1423 2631 y Ft(H)1493 2643 y Fu(#)1621 2584 y Fo(def)1634 2631 y Fw(=)96 b Ft(f)p Fs(\036)23 b Ft(2)h(H)g Fw(:)f Ft(k)p Fs(\036)p Ft(k)2261 2643 y Fu(#)2328 2631 y Fs(<)f Ft(1g)1169 b Fw(\(136\))1398 2781 y Ft(H)1469 2747 y Fu(bv)1621 2734 y Fo(def)1634 2781 y Fw(=)96 b Ft(f)p Fs(\036)23 b Ft(2)h(H)g Fw(:)f Ft(k)p Fs(\036)p Ft(k)2261 2793 y Fu(v)r(ar)2391 2781 y Fs(<)g Ft(1g)1105 b Fw(\(137\))1398 2931 y Ft(H)1469 2897 y Fu(bv)1468 2951 y(#)1621 2884 y Fo(def)1634 2931 y Fw(=)96 b Ft(H)1866 2897 y Fu(bv)1953 2931 y Ft(\\)19 b(H)2097 2943 y Fu(#)2141 2931 y Fs(:)1545 b Fw(\(138\))118 3106 y(Then)28 b(\()p Ft(H)437 3118 y Fu(#)482 3106 y Fs(;)14 b Ft(k)j(\001)i(k)662 3118 y Fu(#)706 3106 y Fw(\))28 b(is)f(a)g(Banac)n(h)g(space.)36 b(F)-7 b(or)27 b Fs(\036)c Ft(2)h(H)1840 3076 y Fu(bv)1936 3106 y Fw(and)j Fs( )f Ft(2)e(C)2305 3076 y Fo(0)2342 3106 y Fw(\()p Fs(M)9 b Fw(\))28 b(w)n(e)f(de\014ne)1242 3403 y Fs( )1296 3415 y Fo(\003)1345 3403 y Fw(\()p Fn(z)1419 3415 y Fo(\003)1470 3403 y Fw(\))1585 3356 y Fo(def)1598 3403 y Fw(=)1759 3290 y Fk(Z)1805 3479 y Fo(\()p Fu(S)1875 3462 y Fi(1)1908 3479 y Fo(\))1934 3462 y Fi(\003)1972 3450 y Fy(C)2042 3403 y Fs(d\026)2135 3369 y Fo(\003)2180 3344 y Fy(C)2219 3403 y Fw(\()p Fn(z)2293 3419 y Fo(\003)2338 3403 y Fy(C)13 b Fw(\))19 b Fs( )s Fw(\()p Fn(z)2574 3415 y Fo(\003)2642 3403 y Ft(_)g Fn(z)2758 3419 y Fo(\003)2803 3403 y Fy(C)13 b Fw(\))820 b(\(139\))1127 3527 y Fk(Z)1173 3715 y Fu(M)1261 3640 y Fs(d\026)23 b( )e(\036)1585 3592 y Fo(def)1598 3640 y Fw(=)131 b(lim)1759 3699 y Fo(\003)p Fm(!)p Fp(Z)1916 3682 y Fy(d)1958 3527 y Fk(Z)2004 3715 y Fo(\()p Fu(S)2074 3699 y Fi(1)2107 3715 y Fo(\))2133 3699 y Fi(\003)2193 3640 y Fs(d\026)2286 3652 y Fo(\003)2349 3640 y Fs( )2403 3652 y Fo(\003)2467 3640 y Fs(\036)2516 3652 y Fo(\003)2566 3640 y Fs(:)1120 b Fw(\(140\))118 3876 y(Finally)30 b(w)n(e)g(recall)g(the)g(de\014nition)h(of)g(a)e (transfer)h(op)r(erator:)41 b(Let)c(~)-49 b Fs(\026)31 b Fw(b)r(e)f(a)g(measure)g(on)g(the)g(\(completed\))h(Borel)f Fs(\033)s Fw(-)118 3988 y(algebra)18 b(of)i(a)g(metric)g(space)1025 3967 y Fk(f)1015 3988 y Fs(M)29 b Fw(and)1290 3967 y Fk(e)1279 3988 y Fs(S)e Fw(:)1414 3967 y Fk(f)1404 3988 y Fs(M)k Ft(!)1632 3967 y Fk(f)1622 3988 y Fs(M)e Fw(b)r(e)20 b(a)g(non-singular)e(measurable)h(map.)34 b(The)20 b Fn(P)m(erron-F)-8 b(rob)s(enius)118 4108 y(op)s(erator)28 b Fw(\(or)f Fn(transfer)33 b(op)s(erator)p Fw(\))28 b Ft(L)1478 4121 y Fd(e)1469 4136 y Fu(S)1517 4108 y Fw(,)g(acting)f(on)g Fs(L)1989 4078 y Fo(1)2040 4016 y Fk(\020)2099 4087 y(f)2089 4108 y Fs(M)2179 4016 y Fk(\021)2228 4108 y Fw(,)h(is)g(de\014ned)g (via)f(the)h(equation)1462 4216 y Fk(Z)1516 4390 y Fd(f)1509 4405 y Fu(M)1596 4329 y Fs(d)7 b Fw(~)-49 b Fs(\026)19 b( )i Ft(\016)1855 4308 y Fk(e)1843 4329 y Fs(S)i(\036)h Fw(=)2078 4216 y Fk(Z)2131 4390 y Fd(f)2124 4405 y Fu(M)2211 4329 y Fs(d)7 b Fw(~)-49 b Fs(\026)19 b( )i Ft(L)2464 4342 y Fd(e)2455 4357 y Fu(S)2504 4329 y Fs(\036)1156 b Fw(\(141\))118 4554 y(that)28 b(m)n(ust)g(hold)f(for)g(all)h Fs( )e Ft(2)d Fs(L)1146 4524 y Fm(1)1230 4462 y Fk(\020)1290 4533 y(f)1279 4554 y Fs(M)1369 4462 y Fk(\021)1446 4554 y Fw(and)28 b Fs(\036)23 b Ft(2)h Fs(L)1816 4524 y Fo(1)1866 4462 y Fk(\020)1926 4533 y(f)1916 4554 y Fs(M)2005 4462 y Fk(\021)2055 4554 y Fw(.)118 4762 y(The)37 b(Mark)n(o)n(v)e(k)n (ernels)h(for)g(our)h(sto)r(c)n(hastic)f(systems)h(are)f(analogous)e (to)j(the)h(comp)r(osition)e(op)r(erator)f(`)p Ft(\016)p Fs(S)5 b Fw(')37 b(\(with)118 4861 y(deterministic)28 b Fs(S)5 b Fw(\),)28 b(acting)f(on)g(functions.)118 5037 y Fn(De\014nition)k(3.1)40 b Fw(W)-7 b(e)37 b(de\014ne)g(transfer)e(op) r(erator)f(for)i(a)g(Mark)n(o)n(v)e(k)n(ernel)h Fs(K)42 b Fw(analogously)34 b(to)i(\(cf.)h(\(229\)\))e(b)n(y)h(the)118 5136 y(equation)1403 5244 y Fk(Z)1456 5418 y Fd(f)1449 5433 y Fu(M)1536 5357 y Fs(d)7 b Fw(~)-49 b Fs(\026)24 b Fw(\()p Fs(K)6 b( )s Fw(\))18 b Fs(\036)24 b Fw(=)2029 5244 y Fk(Z)2082 5418 y Fd(f)2075 5433 y Fu(M)2163 5357 y Fs(d)7 b Fw(~)-49 b Fs(\026)23 b( )f Fw(\()p Ft(L)2444 5369 y Fu(K)2508 5357 y Fs(\036)p Fw(\))p Fs(:)1097 b Fw(\(142\))1966 5682 y(25)p eop %%Page: 26 26 26 25 bop 118 354 a Fn(Remark)31 b(3.1)142 b Fw(1.)41 b(In)28 b(the)g(cases)e(w)n(e)i(consider,)e(the)i(Mark)n(o)n(v)d(k)n (ernel)i Fs(K)2537 366 y Fu(S)2612 354 y Fw(is)h(giv)n(en)f(b)n(y)1517 608 y(\()p Fs(K)1620 620 y Fu(S)1668 608 y Fs( )s Fw(\)\()p Fs(x)p Fw(\))d(=)1980 495 y Fk(Z)2026 684 y Fo(\012)2091 608 y Fs(dP)12 b Fw(\()p Fs(!)s Fw(\))19 b Fs( )i Ft(\016)d Fs(S)2523 620 y Fu(!)2571 608 y Fw(\()p Fs(x)p Fw(\))p Fs(;)1004 b Fw(\(143\))326 833 y(where)26 b(\012)g(is)g(a)g(probabilit) n(y)f(space,)h Fs(S)1519 845 y Fu(!)1594 833 y Fw(dep)r(ends)h (measurably)e(on)h Fs(!)j Fw(and)d(the)h(map)f Fs(!)g Ft(7!)d(L)3275 845 y Fu(S)3316 853 y Fy(!)3389 833 y Fw(is)j(w)n(ell-de\014ned)326 933 y(and)h(in)n(tegrable.)36 b(Then)1084 1157 y Fk(Z)1137 1331 y Fd(f)1130 1346 y Fu(M)1217 1270 y Fs(d)7 b Fw(~)-49 b Fs(\026)23 b Fw(\()p Fs(K)6 b( )s Fw(\))19 b Fs(\036)83 b Fw(=)1830 1157 y Fk(Z)1883 1331 y Fd(f)1876 1346 y Fu(M)1964 1270 y Fs(d)7 b Fw(~)-49 b Fs(\026)p Fw(\()p Fs(x)p Fw(\))2182 1157 y Fk(Z)2229 1346 y Fo(\012)2294 1270 y Fs(dP)12 b Fw(\()p Fs(!)s Fw(\))24 b Fs( )d Ft(\016)d Fs(S)2731 1282 y Fu(!)2779 1270 y Fw(\()p Fs(x)p Fw(\))i Fs(\036)p Fw(\()p Fs(x)p Fw(\))639 b(\(144\))1682 1492 y(=)1830 1379 y Fk(Z)1883 1553 y Fd(f)1876 1568 y Fu(M)1964 1492 y Fs(d)7 b Fw(~)-49 b Fs(\026)p Fw(\()p Fs(x)p Fw(\))2182 1379 y Fk(Z)2229 1567 y Fo(\012)2294 1492 y Fs(dP)12 b Fw(\()p Fs(!)s Fw(\))24 b Fs( )s Fw(\()p Fs(x)p Fw(\))19 b(\()p Ft(L)2821 1504 y Fu(S)2862 1512 y Fy(!)2909 1492 y Fs(\036)p Fw(\)\()p Fs(x)p Fw(\))1682 1714 y(=)1830 1601 y Fk(Z)1883 1775 y Fd(f)1876 1790 y Fu(M)1964 1714 y Fs(d)7 b Fw(~)-49 b Fs(\026)p Fw(\()p Fs(x)p Fw(\))19 b Fs( )s Fw(\()p Fs(x)p Fw(\))2389 1601 y Fk(Z)2435 1789 y Fo(\012)2500 1714 y Fs(dP)12 b Fw(\()p Fs(!)s Fw(\))19 b(\()p Ft(L)2835 1726 y Fu(S)2876 1734 y Fy(!)2923 1714 y Fs(\036)p Fw(\)\()p Fs(x)p Fw(\))p Fs(:)326 1971 y Fw(So)27 b Ft(L)498 1983 y Fu(K)554 1991 y Fy(S)628 1971 y Fw(is)g(giv)n(en)g(b)n(y)1496 2226 y(\()p Ft(L)1585 2238 y Fu(K)1641 2246 y Fy(S)1687 2226 y Fs( )s Fw(\)\()p Fs(x)p Fw(\))e(=)1999 2113 y Fk(Z)2045 2301 y Fo(\012)2111 2226 y Fs(dP)12 b Fw(\()p Fs(!)s Fw(\))19 b(\()p Ft(L)2446 2238 y Fu(S)2487 2246 y Fy(!)2534 2226 y Fs(\036)p Fw(\)\()p Fs(x)p Fw(\))983 b(\(145\))220 2450 y(2.)41 b(The)30 b(op)r(erator)e(for)h(the)h (in\014nite)h(dimensional)e(system)h(that)g(w)n(e)f(are)g(going)g(to)g (consider)g(act)h(on)f(elemen)n(ts)h(of)326 2550 y Ft(H)396 2562 y Fu(#)475 2550 y Fw(that)36 b(do)f(not)g(in)h(general)e(corresp)r (ond)g(to)h(elemen)n(ts)g(of)h Fs(L)2387 2520 y Fo(1)2423 2550 y Fw(\()p Fs(M)9 b Fw(\).)61 b(Recall)35 b(\(see)g([13)o(]\))h (that)f Ft(H)3548 2520 y Fu(bv)3547 2573 y(#)3652 2550 y Fw(can)g(b)r(e)326 2649 y(iden)n(ti\014ed)h(with)f(a)g(subset)h(of)f Fs(r)r(ca)p Fw(\()p Fs(M)9 b Fw(\))36 b(\(or,)h(in)f(other)e(w)n(ords,) i(a)f(subset)h(of)f(the)h(Borel)e(measures\).)59 b(So)35 b(for)326 2749 y(example)c(in)h(Theorem)f(3.1)f(w)n(e)i(will)g(sho)n(w) e(that)i(the)g(equation)f(analogous)f(to)h(\(142\))g(holds)g(for)h Fs( )g Ft(2)f(C)3707 2719 y Fo(0)3743 2749 y Fw(\()p Fs(M)9 b Fw(\))326 2849 y(\(rather)26 b(than)i Fs(L)858 2819 y Fm(1)928 2849 y Fw(\()p Fs(M)9 b Fw(\)\))28 b(and)g Fs(\036)23 b Ft(2)h(H)1526 2819 y Fu(bv)1525 2872 y(#)1594 2849 y Fw(.)118 3031 y(No)n(w)31 b(w)n(e)f(consider)g(a)g(sp)r(ecial)h (class)f(of)h(in)n(teractions)e(\(cf.)j([13)o(]\),)g(namely)f(a)f (family)h(\()p Fs(S)2927 3043 y Fo(\003)2977 3031 y Fw(\))3009 3043 y Fo(\003)p Fm(2F)3188 3031 y Fw(of)g(maps)f(on)h Fs(M)39 b Fw(that)118 3131 y(can)27 b(b)r(e)h(written)g(as)1444 3413 y Fs(S)1495 3425 y Fo(\003)1568 3413 y Fw(:)23 b Fs(M)91 b Ft(!)84 b Fs(M)1675 b Fw(\(146\))1496 3538 y Fs(S)1547 3550 y Fo(\003)1596 3538 y Fw(\()p Fn(z)p Fw(\))94 b(=)e Fs(F)2006 3550 y Fo(\003)2074 3538 y Ft(\016)18 b Fs(T)2183 3550 y Fo(\003)2232 3538 y Fw(\()p Fn(z)p Fw(\))h Ft(_)g Fn(z)2473 3554 y Fo(\003)2518 3538 y Fy(C)118 3720 y Fw(where)1535 4003 y Fs(F)1588 4015 y Fo(\003)1661 4003 y Fw(:)k(\()p Fs(S)1795 3968 y Fo(1)1832 4003 y Fw(\))1864 3968 y Fo(\003)1997 4003 y Ft(!)83 b Fw(\()p Fs(S)2251 3968 y Fo(1)2288 4003 y Fw(\))2320 3968 y Fo(\003)3709 4003 y Fw(\(147\))1444 4127 y Fn(z)1486 4139 y Fo(\003)1559 4127 y Fw(=)23 b(\()p Fs(z)1718 4139 y Fu(q)1755 4127 y Fw(\))1787 4139 y Fu(q)r Fm(2)p Fo(\003)1997 4127 y Ft(7!)83 b Fw(\()p Fs(f)2236 4139 y Fu(q)2272 4127 y Fw(\()p Fs(z)2343 4139 y Fu(q)2380 4127 y Fw(\)\))2444 4139 y Fu(q)r Fm(2)p Fo(\003)118 4310 y Fw(and)1105 4592 y Fs(T)1154 4604 y Fo(\003)1226 4592 y Fw(:)24 b Fs(M)95 b Ft(!)88 b Fw(\()p Fs(S)1708 4558 y Fo(1)1745 4592 y Fw(\))1777 4558 y Fo(\003)3709 4592 y Fw(\(148\))1056 4791 y(\()p Fs(T)1137 4803 y Fo(\003)1186 4791 y Fw(\()p Fn(z)p Fw(\)\))1324 4803 y Fu(q)1445 4744 y Fo(def)1459 4791 y Fw(=)96 b Fs(z)1659 4803 y Fu(q)1723 4791 y Fw(exp)1863 4649 y Fk( )1929 4791 y Fw(2)p Fs(\031)s(i\017)2124 4688 y Fm(1)2097 4713 y Fk(X)2098 4891 y Fu(k)q Fo(=1)2231 4791 y Fs(g)2271 4803 y Fu(q)r(;k)2364 4791 y Fw(\()p Fn(z)p Fw(\))2470 4649 y Fk(!)2634 4791 y Fw(for)27 b Fs(q)f Ft(2)d Fw(\003)749 b(\(149\))118 5049 y(and)28 b Fs(f)321 5061 y Fu(q)385 5049 y Fw(and)f Fs(g)586 5061 y Fu(q)r(;k)706 5049 y Fw(satisfy)h(the)g(follo)n(wing)e(assumptions:) 118 5219 y Fn(Assumption)38 b(I)75 b Fs(F)12 b Fw(\()p Fn(z)p Fw(\))36 b(=)e(\()p Fs(f)1146 5231 y Fu(q)1183 5219 y Fw(\()p Fs(z)1254 5231 y Fu(q)1290 5219 y Fw(\)\))1354 5236 y Fu(q)r Fm(2)p Fp(Z)1477 5219 y Fy(d)j Fw(where)d Fs(f)1833 5231 y Fu(q)1904 5219 y Fw(:)h Fs(S)2018 5189 y Fo(1)2090 5219 y Ft(!)g Fs(S)2264 5189 y Fo(1)2335 5219 y Fw(are)f(real)g(analytic)g(and)h(expanding)f(\(i.e.)h Fs(f)3771 5189 y Fm(0)3762 5240 y Fu(q)3833 5219 y Ft(\025)118 5319 y Fs(\025)166 5331 y Fo(0)227 5319 y Fs(>)23 b Fw(1\))28 b(maps)f(that)h(extend)g(for)g(some)f Fs(\016)1457 5331 y Fo(1)1522 5319 y Fw(holomorphically)f(to)h(the)i(in)n(terior)d(of)i (an)f(ann)n(ulus)h Fs(A)3247 5331 y Fu(\016)3277 5339 y Fi(1)3314 5319 y Fw(.)37 b(In)28 b(Prop)r(osition)118 5418 y(3.1)33 b(and)h(3.2)g(of)g([13)o(])g(w)n(e)g(ha)n(v)n(e)f(sho)n (wn)g(that)i(the)f(holomorphic)f(extension)h(to)g(a)f(su\016cien)n(tly) h(thin)h(ann)n(ulus)f Fs(A)3771 5430 y Fu(\016)3842 5418 y Fw(is)1966 5682 y(26)p eop %%Page: 27 27 27 26 bop 118 354 a Fw(expanding)27 b(in)h(the)g(sense)f(that)h(the)g (preimage)f(of)g Fs(A)1805 366 y Fu(\016)1870 354 y Fw(wrt)g Fs(f)2063 366 y Fu(q)2127 354 y Fw(lies)g(in)h(the)g(in)n(terior)f(of)g Fs(A)2960 366 y Fu(\016)2997 354 y Fw(.)37 b(W)-7 b(e)28 b(\014x)g(suc)n(h)f(a)g Fs(\016)3611 366 y Fo(1)3648 354 y Fw(.)38 b(Then)118 454 y(for)32 b(ev)n(ery)f Fs(q)j Ft(2)d Fv(Z)692 424 y Fu(d)757 454 y Fw(the)i(P)n(erron-F)-7 b(rob)r(enius)30 b(op)r(erator)g Ft(L)1960 466 y Fu(f)1992 474 y Fy(q)2030 454 y Fw(,)k(acting)d(on)h Ft(H)2530 469 y Fm(f)p Fu(q)r Fm(g)2635 454 y Fw(,)i(has)e(a)f(simple)i(largest)e (eigen)n(v)-5 b(alue)31 b(1)118 554 y(with)d(eigen)n(v)n(ector)e Fs(h)787 566 y Fu(q)823 554 y Fw(,)i(suc)n(h)f(that)h Fs(\031)1288 569 y Fm(;)1327 554 y Fw(\()p Fs(h)1407 566 y Fu(q)1443 554 y Fw(\))c(=)e(1)28 b(and)f(the)h(restriction)f(of)g Fs(h)2500 566 y Fu(q)2564 554 y Fw(to)h Fs(S)2722 523 y Fo(1)2786 554 y Fw(is)g(p)r(ositiv)n(e)f(and)g(it)h(splits)g(in)n(to) 1721 753 y Ft(L)1778 765 y Fu(f)1810 773 y Fy(q)1871 753 y Fw(=)23 b Ft(Q)2027 765 y Fu(q)2082 753 y Fw(+)18 b Ft(R)2235 765 y Fu(q)2272 753 y Fs(;)1414 b Fw(\(150\))118 902 y(where)25 b Ft(Q)424 914 y Fu(q)485 902 y Fw(is)g(a)f(pro)5 b(jection)24 b(on)n(to)h(span)o(\()p Fs(h)1455 914 y Fu(q)1492 902 y Fw(\).)36 b(W)-7 b(e)26 b(assume)e(that)h(there)g(are)f (p)r(ositiv)n(e)g(constan)n(ts)g Fs(c)3236 914 y Fu(h)3304 902 y Fw(and)h Fs(c)3499 914 y Fu(r)3561 902 y Fw(suc)n(h)f(that)118 1002 y(the)k(follo)n(wing)f(t)n(w)n(o)f(estimates)i(hold)f(for)g(all)h Fs(q)e Ft(2)d Fv(Z)1763 972 y Fu(d)1796 1002 y Fw(:)1671 1284 y Ft(kQ)1781 1296 y Fu(q)1817 1284 y Ft(k)1859 1299 y Fm(f)p Fu(q)r Fm(g)2046 1284 y Ft(\024)83 b Fs(c)2230 1296 y Fu(h)3709 1284 y Fw(\(151\))1660 1492 y Ft(kR)1772 1457 y Fu(n)1772 1512 y(q)1817 1492 y Ft(k)1859 1507 y Fm(f)p Fu(q)r Fm(g)2046 1492 y Ft(\024)g Fs(c)2230 1504 y Fu(r)2266 1492 y Fs(\021)2310 1457 y Fu(n)3709 1492 y Fw(\(152\))118 1674 y(where)29 b Ft(k)20 b(\001)g(k)507 1689 y Fm(f)p Fu(q)r Fm(g)641 1674 y Fw(denotes)29 b(b)r(oth)i(the)f (uniform)f(norm)h(on)f Ft(H)2007 1689 y Fm(f)p Fu(q)r Fm(g)2142 1674 y Fw(\(for)g(this)i(w)n(e)e(migh)n(t)h(ha)n(v)n(e)e(to)i (tak)n(e)f Fs(\016)3348 1686 y Fo(1)3415 1674 y Fw(ev)n(en)g(smaller\)) 118 1774 y(and)f(the)g(induced)g(op)r(erator-norm.)34 b(W)-7 b(e)28 b(note)f(that)h(this)g(holds)f(in)h(particular)e(if)i Fs(f)2799 1786 y Fu(q)2863 1774 y Fw(do)r(es)g(not)f(dep)r(end)i(on)e Fs(q)s Fw(.)118 1874 y(W)-7 b(e)28 b(further)g(ha)n(v)n(e)1665 2073 y Ft(Q)1733 2085 y Fu(q)1769 2073 y Ft(R)1839 2085 y Fu(q)1899 2073 y Fw(=)23 b Ft(R)2057 2085 y Fu(q)2094 2073 y Ft(Q)2162 2085 y Fu(q)2222 2073 y Fw(=)f(0)1358 b(\(153\))118 2312 y Fn(Assumption)23 b(I)s(I)47 b Fw(F)-7 b(or)22 b(all)f Fs(q)27 b Ft(2)c Fv(Z)1219 2282 y Fu(d)1274 2312 y Fw(and)f Fs(k)k Ft(\025)c Fw(1)g(eac)n(h)f(map)h Fs(g)2050 2324 y Fu(q)r(;k)2165 2312 y Fw(extends)g(to)f(a)h (holomorphic)f(map)g Fs(g)3309 2324 y Fu(q)r(;k)3425 2312 y Fw(:)i Fs(A)3533 2269 y Fu(B)3583 2278 y Fy(k)3620 2269 y Fo(\()p Fu(q)r Fo(\))3533 2337 y Fu(\016)3563 2345 y Fi(1)3732 2312 y Ft(!)g Fv(C)118 2412 y Fw(\(recall)k (de\014nition)h(\(23\))f(of)h Fs(B)1074 2424 y Fu(k)1115 2412 y Fw(\()p Fs(q)s Fw(\)\))g(and)g(its)f(sup-norm)g(\(of)h(mo)r (dulus\))g(is)g(exp)r(onen)n(tially)f(b)r(ounded)h(b)n(y)1452 2611 y Ft(k)23 b Fs(g)1557 2623 y Fu(q)r(;k)1673 2611 y Ft(k)1714 2648 y Fu(A)1764 2614 y Fy(B)1807 2629 y(k)1843 2614 y Fi(\()p Fy(q)q Fi(\))1764 2668 y Fy(\016)1791 2680 y Fi(1)1948 2611 y Ft(\024)g Fs(c)2072 2623 y Fo(3)2123 2611 y Fw(exp)2264 2544 y Fk(\000)2302 2611 y Ft(\000)p Fs(c)2403 2623 y Fu(g)2441 2611 y Fs(k)2487 2577 y Fu(d)2526 2544 y Fk(\001)3709 2611 y Fw(\(154\))118 2801 y(with)29 b Fs(c)344 2813 y Fo(3)405 2801 y Fs(>)23 b Fw(0)28 b(and)g(`large')e Fs(c)1010 2813 y Fu(g)1073 2801 y Fs(>)d Fw(0.)38 b(\(In)28 b(sev)n(eral)f(statemen)n(ts)h(in)g(Section)g(3)g(and)f(4)h(a)g(lo)n(w) n(er)e(b)r(ound)j(for)e Fs(c)3489 2813 y Fu(g)3556 2801 y Fw(will)h(come)118 2901 y(out)c(of)g(our)f(computations.)35 b(The)24 b(idea)g(is)g(alw)n(a)n(ys)e(that)i(our)f(estimates)h(w)n (ork,)f(pro)n(vided)g Fs(c)3005 2913 y Fu(g)3067 2901 y Fw(is)h(bigger)f(than)h(a)f(certain)118 3000 y(constan)n(t.\))118 3171 y(F)-7 b(or)27 b(\003)c Ft(2)g(F)k(n)18 b(f;g)26 b Fw(w)n(e)h(denote)g(b)n(y)h Fs(h)1278 3183 y Fo(\003)1355 3171 y Fw(the)g(function)1620 3370 y Fs(h)1668 3382 y Fo(\003)1718 3370 y Fw(\()p Fn(z)1792 3382 y Fo(\003)1842 3370 y Fw(\))1897 3323 y Fo(def)1911 3370 y Fw(=)2020 3291 y Fk(Y)2012 3469 y Fu(q)r Fm(2)p Fo(\003)2148 3370 y Fs(h)2196 3382 y Fu(q)2233 3370 y Fw(\()p Fs(z)2304 3382 y Fu(q)2340 3370 y Fw(\))p Fs(;)1314 b Fw(\(155\))118 3605 y(where)27 b Fs(h)406 3617 y Fu(q)470 3605 y Fw(is)h(as)f(in)h (Assumption)f(I.)h(W)-7 b(e)28 b(set)g Fs(h)1614 3620 y Fm(;)1675 3605 y Fw(=)22 b(1)28 b(and)1624 3804 y Fs(h)1672 3821 y Fp(Z)1716 3804 y Fy(d)1772 3757 y Fo(def)1786 3804 y Fw(=)36 b(\()p Fs(h)1967 3816 y Fo(\003)2017 3804 y Fw(\))2049 3816 y Fo(\003)p Fm(2F)2220 3804 y Ft(2)23 b(H)q Fs(:)1317 b Fw(\(156\))118 3954 y(W)-7 b(e)29 b(further)f (de\014ne)g(for)g(a)g(\014xed)g Fs(\030)h Ft(2)24 b Fs(M)37 b Fw(and)28 b(\003)c Ft(2)h(F)h(n)19 b(f;g)27 b Fw(and)h(\003)2287 3966 y Fo(1)2348 3954 y Ft(\022)c Fw(\003)k(the)g Fj(up)l(dating)j(at)g (the)f Fw(\003)3299 3966 y Fo(1)3336 3954 y Fj(-sites)g(with)h(\014xe)l (d)118 4053 y(b)l(oundary)g(c)l(onditions)f Fs(\030)908 4070 y Fo(\003)953 4053 y Fy(C)42 b Fj(outside)31 b Fw(\003)c(\(or)g Fj(cut-o\013)45 b Fw(of)27 b Fs(S)1964 4065 y Fo(\003)2009 4073 y Fi(1)2046 4053 y Fw(\):)1289 4335 y Fs(S)1340 4347 y Fo(\003)1385 4355 y Fi(1)1418 4347 y Fu(;)p Fo(\003)1510 4335 y Fw(:)c(\()p Fs(S)1644 4301 y Fo(1)1681 4335 y Fw(\))1713 4301 y Fo(\003)1846 4335 y Ft(!)83 b Fw(\()p Fs(S)2100 4301 y Fo(1)2138 4335 y Fw(\))2170 4301 y Fo(\003)3709 4335 y Fw(\(157\))1671 4460 y Fn(z)1713 4472 y Fo(\003)1846 4460 y Ft(7!)g Fs(\031)2059 4472 y Fo(\003)2128 4460 y Ft(\016)18 b Fs(S)2239 4472 y Fo(\003)2284 4480 y Fi(1)2321 4460 y Fw(\()p Fn(z)2395 4472 y Fo(\003)2463 4460 y Ft(_)h Fs(\030)2573 4476 y Fo(\003)2618 4460 y Fy(C)12 b Fw(\))p Fs(:)118 4643 y Fw(And)28 b(for)f Fn(z)469 4658 y Fo(\003)p Fm(n)p Fo(\003)593 4666 y Fi(1)654 4643 y Ft(2)d Fw(\()p Fs(S)821 4612 y Fo(1)858 4643 y Fw(\))890 4612 y Fo(\003)p Fm(n)p Fo(\003)1014 4620 y Fi(1)1079 4643 y Fw(w)n(e)k(de\014ne)877 4925 y Fs(\031)924 4937 y Fo(\003)969 4945 y Fi(1)1025 4925 y Ft(\016)18 b Fs(S)1136 4937 y Fo(\003)1181 4945 y Fi(1)1214 4937 y Fu(;)p Fo(\003)1283 4925 y Fw(\()p Ft(\001)h(_)g Fn(z)1473 4940 y Fo(\003)p Fm(n)p Fo(\003)1597 4948 y Fi(1)1635 4925 y Fw(\))k(:)g(\()p Fs(S)1824 4891 y Fo(1)1861 4925 y Fw(\))1893 4891 y Fo(\003)1938 4899 y Fi(1)2058 4925 y Ft(!)84 b Fw(\()p Fs(S)2313 4891 y Fo(1)2350 4925 y Fw(\))2382 4891 y Fo(\003)2427 4899 y Fi(1)3709 4925 y Fw(\(158\))1851 5049 y Fn(z)1893 5061 y Fo(\003)1938 5069 y Fi(1)2058 5049 y Ft(7!)g Fs(\031)2272 5061 y Fo(\003)2317 5069 y Fi(1)2372 5049 y Ft(\016)18 b Fs(S)2483 5061 y Fo(\003)2528 5069 y Fi(1)2561 5061 y Fu(;)p Fo(\003)2630 5049 y Fw(\()p Fn(z)2704 5061 y Fo(\003)2749 5069 y Fi(1)2805 5049 y Ft(_)h Fn(z)2921 5064 y Fo(\003)p Fm(n)p Fo(\003)3045 5072 y Fi(1)3083 5049 y Fw(\))p Fs(:)118 5232 y Fn(Remark)31 b(3.2)142 b Fw(1.)41 b(The)24 b(map)g(de\014ned)h(in)f(\(158\))f(is)h(the)h (cut-o\013)f(of)g Fs(S)29 b Fw(wrt)24 b(\003)2644 5244 y Fo(1)2705 5232 y Fw(and)g(b)r(oundary)f(conditions)h Fn(z)3669 5247 y Fo(\003)p Fm(n)p Fo(\003)3793 5255 y Fi(1)3842 5232 y Ft(_)326 5332 y Fs(\030)362 5344 y Fo(\003)412 5332 y Fw(.)45 b(So)31 b(w)n(e)f(can)g(use)h(the)g(sp)r(ecial)f (represen)n(tation)f(in)h(terms)h(of)f(in)n(tegral)g(k)n(ernels)f(for)h (its)h(transfer)e(op)r(erator,)326 5431 y(restricted)e(to)g Ft(H)867 5443 y Fo(\003)912 5451 y Fi(1)949 5431 y Fw(,)h(for)f(the)h (prop)r(osition)e(b)r(elo)n(w.)1966 5682 y(27)p eop %%Page: 28 28 28 27 bop 220 354 a Fw(2.)41 b(The)30 b(family)g(\()p Fs(S)838 366 y Fu(q)875 354 y Fw(\))907 371 y Fu(q)r Fm(2)p Fp(Z)1030 355 y Fy(d)1063 354 y Fw(,)h(de\014ned)f(b)n(y)g (\(146\),)h(satis\014es)e(conditions)h(\(73\))g(and)g(\(74\))g(as)g (one)g(can)g(see)f(from)h([13]:)326 454 y(The)d(partial)g(deriv)-5 b(ativ)n(es)27 b(are)f(estimated)i(in)g(the)g(pro)r(of)f(of)g(Prop)r (osition)f(3.1)h(there.)118 637 y Fn(Lemma)j(3.1)40 b Fj(L)l(et)34 b Fw(\003)d Ft(2)g(F)e(n)22 b(f;g)32 b Fj(b)l(e)i(the)h (disjoint)g(union)f(of)h Fw(\003)2206 649 y Fo(1)2277 637 y Fj(and)g Fw(\003)2501 649 y Fo(2)2538 637 y Fj(.)51 b(The)36 b(tr)l(ansfer)e(op)l(er)l(ator,)i(r)l(estricte)l(d)f(to)118 736 y Ft(H)188 748 y Fo(\003)233 756 y Fi(1)270 736 y Fj(,)h(of)f(the)g(map)g Fs(S)814 748 y Fo(\003)859 756 y Fi(1)892 748 y Fu(;)p Fo(\003)992 736 y Fw(:)d(\()p Fs(S)1135 706 y Fo(1)1172 736 y Fw(\))1204 706 y Fo(\003)1285 736 y Ft(!)g Fw(\()p Fs(S)1488 706 y Fo(1)1525 736 y Fw(\))1557 706 y Fo(\003)1607 736 y Fj(,)k(de\014ne)l(d)e(in)h(\(157\)) g(has)g(the)g(fol)t(lowing)i(r)l(epr)l(esentation)d(in)h(terms)f(of)118 836 y(inte)l(gr)l(al)c(kernels:)802 1051 y Fk(\000)841 1118 y Ft(L)898 1130 y Fu(S)939 1138 y Fi(\003)977 1150 y(1)1009 1138 y Fy(;)p Fi(\003)1074 1118 y Fs(\036)1123 1051 y Fk(\001)1175 1118 y Fw(\()p Fn(w)1276 1130 y Fo(\003)1321 1138 y Fi(1)1377 1118 y Ft(_)19 b Fn(w)1520 1130 y Fo(\003)1565 1138 y Fi(2)1602 1118 y Fw(\))2075 b(\(159\))886 1303 y(=)1033 1190 y Fk(Z)1079 1379 y Fo(\000)1120 1360 y Fi(\003)1158 1372 y(1)1213 1303 y Fs(d\026)1306 1269 y Fo(\003)1351 1277 y Fi(1)1388 1303 y Fw(\()p Fn(z)1462 1315 y Fo(\003)1507 1323 y Fi(1)1545 1303 y Fw(\))23 b Fs(\036)p Fw(\()p Fn(z)1723 1315 y Fo(\003)1768 1323 y Fi(1)1825 1303 y Ft(_)18 b Fn(w)1967 1315 y Fo(\003)2012 1323 y Fi(2)2049 1303 y Fw(\))2120 1224 y Fk(Y)2095 1402 y Fu(q)r Fm(2)p Fo(\003)2217 1410 y Fi(1)2373 1247 y Fw(\()p Fs(S)2456 1259 y Fo(\003)2501 1267 y Fi(1)2534 1259 y Fu(;)p Fo(\003)2603 1247 y Fw(\()p Fn(z)2677 1259 y Fo(\003)2722 1267 y Fi(1)2779 1247 y Ft(_)g Fn(w)2921 1259 y Fo(\003)2966 1267 y Fi(2)3003 1247 y Fw(\)\))3067 1259 y Fu(q)p 2274 1284 930 4 v 2274 1360 a Fw(\()p Fs(S)2357 1372 y Fo(\003)2402 1380 y Fi(1)2435 1372 y Fu(;)p Fo(\003)2505 1360 y Fw(\()p Fn(z)2579 1372 y Fo(\003)2624 1380 y Fi(1)2680 1360 y Ft(_)h Fn(w)2823 1372 y Fo(\003)2868 1380 y Fi(2)2904 1360 y Fw(\)\))2968 1372 y Fu(q)3024 1360 y Ft(\000)f Fs(w)3166 1372 y Fu(q)118 1571 y Fj(for)31 b Fs(\036)23 b Ft(2)h(H)472 1583 y Fo(\003)521 1571 y Fj(.)118 1753 y Fn(Pro)s(of)59 b Fw(Let)28 b Fs( )e Ft(2)e(C)766 1723 y Fo(0)803 1753 y Fw(\(\()p Fs(S)923 1723 y Fo(1)960 1753 y Fw(\))992 1723 y Fo(\003)1042 1753 y Fw(\).)37 b(W)-7 b(e)28 b(use)g(the)g(notation)f Fs(\036)1945 1765 y Fl(w)1999 1773 y Fi(\003)2037 1785 y(2)2105 1753 y Fw(for)h(the)g(function)g Fn(w)2770 1765 y Fo(\003)2815 1773 y Fi(1)2875 1753 y Ft(7!)23 b Fs(\036)p Fw(\()p Fn(w)3131 1765 y Fo(\003)3176 1773 y Fi(1)3232 1753 y Ft(_)c Fn(w)3375 1765 y Fo(\003)3420 1773 y Fi(2)3457 1753 y Fw(\).)645 1944 y Fk(Z)691 2133 y Fo(\()p Fu(S)761 2116 y Fi(1)793 2133 y Fo(\))819 2116 y Fi(\003)879 2057 y Fs(d\026)972 2023 y Fo(\003)1022 2057 y Fw(\()p Fn(w)1123 2069 y Fo(\003)1173 2057 y Fw(\))k Fs( )f Ft(\016)c Fs(S)1415 2069 y Fo(\003)1460 2077 y Fi(1)1492 2069 y Fu(;)p Fo(\003)1562 2057 y Fw(\()p Fn(w)1663 2069 y Fo(\003)1713 2057 y Fw(\))g Fs(\036)p Fw(\()p Fn(w)1913 2069 y Fo(\003)1964 2057 y Fw(\))1713 b(\(160\))728 2294 y(=)876 2181 y Fk(Z)922 2369 y Fo(\()p Fu(S)992 2353 y Fi(1)1024 2369 y Fo(\))1050 2351 y Fi(\003)1088 2363 y(2)1142 2294 y Fs(d\026)1235 2259 y Fo(\003)1280 2267 y Fi(2)1318 2294 y Fw(\()p Fn(w)1419 2306 y Fo(\003)1464 2314 y Fi(2)1501 2294 y Fw(\))1561 2181 y Fk(Z)1607 2369 y Fo(\()p Fu(S)1677 2353 y Fi(1)1709 2369 y Fo(\))1735 2351 y Fi(\003)1773 2363 y(1)1827 2294 y Fs(d\026)1920 2259 y Fo(\003)1965 2267 y Fi(1)2003 2294 y Fw(\()p Fn(w)2104 2306 y Fo(\003)2149 2314 y Fi(1)2186 2294 y Fw(\))23 b Fs( )2295 2306 y Fl(w)2349 2314 y Fi(\003)2387 2326 y(2)2446 2294 y Ft(\016)18 b Fs(\031)2553 2306 y Fo(\003)2598 2314 y Fi(1)2654 2294 y Ft(\016)g Fs(S)2765 2306 y Fo(\003)2810 2314 y Fi(1)2843 2306 y Fu(;)p Fo(\003)2912 2294 y Fw(\()p Fn(w)3013 2306 y Fo(\003)3058 2314 y Fi(1)3114 2294 y Ft(_)h Fn(w)3257 2306 y Fo(\003)3302 2314 y Fi(2)3339 2294 y Fw(\))2057 2479 y Ft(\002)14 b Fs(\036)2185 2491 y Fl(w)2239 2499 y Fi(\003)2277 2511 y(2)2317 2479 y Fw(\()p Fn(w)2418 2491 y Fo(\003)2463 2499 y Fi(1)2500 2479 y Fw(\))728 2742 y(=)876 2629 y Fk(Z)922 2817 y Fo(\()p Fu(S)992 2801 y Fi(1)1024 2817 y Fo(\))1050 2799 y Fi(\003)1088 2811 y(2)1142 2742 y Fs(d\026)1235 2708 y Fo(\003)1280 2716 y Fi(2)1318 2742 y Fw(\()p Fn(w)1419 2754 y Fo(\003)1464 2762 y Fi(2)1501 2742 y Fw(\))1561 2629 y Fk(Z)1607 2817 y Fo(\()p Fu(S)1677 2801 y Fi(1)1709 2817 y Fo(\))1735 2799 y Fi(\003)1773 2811 y(1)1827 2742 y Fs(d\026)1920 2708 y Fo(\003)1965 2716 y Fi(1)2003 2742 y Fw(\()p Fn(w)2104 2754 y Fo(\003)2149 2762 y Fi(1)2186 2742 y Fw(\))23 b Fs( )2295 2754 y Fl(w)2349 2762 y Fi(\003)2387 2774 y(2)2428 2742 y Fw(\()p Fn(w)2529 2754 y Fo(\003)2574 2762 y Fi(1)2611 2742 y Fw(\))2057 2927 y Ft(\002)p Fw(\()p Ft(L)2211 2942 y Fu(\031)2250 2950 y Fi(\003)2288 2962 y(1)2324 2942 y Fm(\016)p Fu(S)2399 2950 y Fi(\003)2437 2962 y(1)2469 2950 y Fy(;)p Fi(\003)2530 2942 y Fo(\()11 b Fm(\001)g(_)p Fl(w)2697 2950 y Fi(\003)2735 2962 y(2)2772 2942 y Fo(\))2802 2927 y Fs(\036)2851 2939 y Fl(w)2905 2947 y Fi(\003)2943 2959 y(2)2984 2927 y Fw(\)\()p Fn(w)3117 2939 y Fo(\003)3162 2947 y Fi(1)3199 2927 y Fw(\))728 3192 y(=)876 3079 y Fk(Z)922 3268 y Fo(\()p Fu(S)992 3252 y Fi(1)1024 3268 y Fo(\))1050 3252 y Fi(\003)1110 3192 y Fs(d\026)1203 3158 y Fo(\003)1253 3192 y Fw(\()p Fn(w)1354 3204 y Fo(\003)1404 3192 y Fw(\))j Fs( )s Fw(\()p Fn(w)1608 3204 y Fo(\003)1658 3192 y Fw(\))g(\()p Ft(L)1793 3207 y Fu(\031)1832 3215 y Fi(\003)1870 3227 y(1)1906 3207 y Fm(\016)p Fu(S)1981 3215 y Fi(\003)2019 3227 y(1)2051 3215 y Fy(;)p Fi(\003)2112 3207 y Fo(\()e Fm(\001)f(_)p Fl(w)2280 3215 y Fi(\003)2318 3227 y(2)2354 3207 y Fo(\))2384 3192 y Fs(\036)2433 3204 y Fl(w)2487 3212 y Fi(\003)2525 3224 y(2)2566 3192 y Fw(\)\()p Fn(w)2699 3204 y Fo(\003)2744 3212 y Fi(1)2782 3192 y Fw(\))118 3436 y(Using)19 b(the)g(represen)n (tation)e(of)i(the)g(transfer)f(op)r(erator)g(for)g Fs(\031)2022 3448 y Fo(\003)2067 3456 y Fi(1)2105 3436 y Ft(\016)q Fs(S)2199 3448 y Fo(\003)2244 3456 y Fi(1)2277 3448 y Fu(;)p Fo(\003)2346 3436 y Fw(\()c Ft(\001)h(_)q Fn(w)2555 3448 y Fo(\003)2600 3456 y Fi(2)2638 3436 y Fw(\))k(that)g(w)n(e)g (established)f(in)h(Prop)r(osition)118 3536 y(3.3)27 b(of)g([13],)g(w)n(e)h(obtain)f(the)h(rhs)f(of)g(\(159\).)3808 3635 y Fh(2)118 3889 y Fn(Remark)k(3.3)142 b Fw(1.)41 b(W)-7 b(e)41 b(see)e(in)i(particular)e(that)h Ft(L)1909 3901 y Fu(S)1950 3909 y Fi(\003)1988 3921 y(1)2021 3909 y Fy(;)p Fi(\003)2126 3889 y Fw(`acts)g(on)g(the)g(\003)2674 3901 y Fo(1)2711 3889 y Fw(-co)r(ordinates')f(only)-7 b(.)74 b(There)40 b(is)g(no)326 3989 y(in)n(tegration)26 b(wrt)h(the)h(\003)1101 4001 y Fo(2)1138 3989 y Fw(-co)r(ordinates.)326 4121 y(F)-7 b(or)27 b Fs(q)f Ft(2)d Fw(\003)674 4133 y Fo(1)739 4121 y Fw(w)n(e)k(can)g(split)h(the)g(factor)609 4344 y(\()p Fs(S)692 4356 y Fo(\003)737 4364 y Fi(1)770 4356 y Fu(;)p Fo(\003)840 4344 y Fw(\()p Fn(z)914 4356 y Fo(\003)959 4364 y Fi(1)1015 4344 y Ft(_)19 b Fn(w)1158 4356 y Fo(\003)1203 4364 y Fi(2)1239 4344 y Fw(\)\))1303 4356 y Fu(q)p 510 4381 V 510 4457 a Fw(\()p Fs(S)593 4469 y Fo(\003)638 4477 y Fi(1)671 4469 y Fu(;)p Fo(\003)741 4457 y Fw(\()p Fn(z)815 4469 y Fo(\003)860 4477 y Fi(1)916 4457 y Ft(_)g Fn(w)1059 4469 y Fo(\003)1104 4477 y Fi(2)1141 4457 y Fw(\)\))1205 4469 y Fu(q)1260 4457 y Ft(\000)f Fs(w)1402 4469 y Fu(q)1473 4400 y Fw(=)k Fs(h)1608 4412 y Fu(q)1645 4400 y Fw(\()p Fs(w)1736 4412 y Fu(q)1773 4400 y Fs(;)14 b(z)1849 4412 y Fu(q)1885 4400 y Fw(\))19 b(+)f Fs(r)2056 4412 y Fu(q)2093 4400 y Fw(\()p Fs(w)2184 4412 y Fu(q)2221 4400 y Fs(;)c(z)2297 4412 y Fu(q)2334 4400 y Fw(\))k(+)2495 4296 y Fm(1)2468 4321 y Fk(X)2467 4500 y Fu(k)q Fo(=1)2602 4400 y Fs(\014)2649 4412 y Fu(q)r(;k)2742 4400 y Fw(\()p Fs(w)2833 4412 y Fu(q)2870 4400 y Fs(;)c Fn(z)2949 4412 y Fo(\003)2994 4420 y Fi(1)3050 4400 y Ft(_)19 b Fn(w)3193 4412 y Fo(\003)3238 4420 y Fi(2)3293 4400 y Ft(_)g Fs(\030)3403 4416 y Fo(\003)3448 4399 y Fy(C)3501 4400 y Fw(\))176 b(\(161\))326 4661 y(as)31 b(in)i([13)o(].)50 b(The)32 b(in)n(tegral)f(k)n(ernels)g Fs(h)1548 4673 y Fu(q)1617 4661 y Fw(and)h Fs(r)1820 4673 y Fu(q)1889 4661 y Fw(corresp)r(ond)f(to)h(the)g(op)r(erators)e Ft(Q)3010 4673 y Fu(q)3079 4661 y Fw(and)i Ft(R)3315 4673 y Fu(q)3352 4661 y Fw(,)i(in)n(tro)r(duced)d(in)326 4761 y(\(150\))26 b(and)i Fs(\014)751 4773 y Fu(q)r(;k)872 4761 y Fw(to)f Ft(B)1028 4773 y Fu(q)r(;k)1121 4761 y Fw(,)g(sa)n(y)-7 b(.)36 b(In)28 b(addition)f(to)h(\(153\))f(w)n(e)g(ha) n(v)n(e)1870 4994 y Ft(Q)1938 5006 y Fu(q)1993 4994 y Ft(\016)18 b(B)2108 5006 y Fu(q)r(;k)2224 4994 y Fw(=)k(0)1356 b(\(162\))326 5176 y(for)27 b(all)g Fs(k)s Fw(.)37 b(\(Cf.)28 b([13)o(])g(for)f(details.\))1966 5682 y(28)p eop %%Page: 29 29 29 28 bop 220 354 a Fw(2.)41 b(W)-7 b(e)29 b(ha)n(v)n(e)e(established)h (a)g(represen)n(tation)f(of)h(the)h(transfer)f(op)r(erator)f(also)g (for)h(up)r(datings)g(at)h(more)e(than)i(one)326 454 y(p)r(oin)n(t)38 b(at)g(one)g(time.)70 b(Suc)n(h)38 b(sim)n(ultaneous)f (up)r(datings)i(happ)r(en,)i(for)d(example,)i(in)f(certain)e(discrete)h (time)326 554 y(pro)r(cesses)26 b(with)i(p)r(ositiv)n(e)f(probabilit)n (y)g(\(cf.)h([12)o(]\).)326 686 y(As)f(in)h(the)g(systems)f(considered) g(here)g(sim)n(ultaneous)g(up)r(datings)h Fs(P)12 b Fw(-almost)26 b(nev)n(er)h(happ)r(en,)h(w)n(e)f(can)g(restrict)326 786 y(ourself)g(in)g(the)h(follo)n(wing)f(to)g(the)h(case)f(of)h(up)r (datings)f(at)h(single)f(sites,)g(i.e.)h(\003)2815 798 y Fo(1)2875 786 y Fw(=)23 b Ft(f)p Fs(q)s Ft(g)p Fw(.)118 968 y Fn(De\014nition)31 b(3.2)40 b Fw(W)-7 b(e)33 b(de\014ne)g(for)f (\014xed)h(\003)e Ft(2)g(F)f(n)21 b(f;g)p Fw(,)33 b Fs(\030)i Ft(2)d Fs(M)41 b Fw(and)32 b(a)g(\014nite)h(sequence)g Fs(j)j Fw(=)31 b(\()p Fs(q)3290 980 y Fo(1)3327 968 y Fs(;)14 b(:)g(:)g(:)g(;)g(q)3549 980 y Fu(n)3594 968 y Fw(\))31 b Ft(2)h Fs(J)41 b Fw(of)118 1068 y(p)r(oin)n(ts)28 b(in)f(\003)h(the)g(map)1342 1350 y Fs(S)1393 1362 y Fu(j;)p Fo(\003)1512 1350 y Fw(:)23 b(\()p Fs(S)1646 1316 y Fo(1)1684 1350 y Fw(\))1716 1316 y Fo(\003)1853 1350 y Ft(!)87 b Fw(\()p Fs(S)2111 1316 y Fo(1)2148 1350 y Fw(\))2180 1316 y Fo(\003)3709 1350 y Fw(\(163\))1618 1503 y Fs(S)1669 1515 y Fu(j;)p Fo(\003)1848 1456 y(def)1862 1503 y Fw(=)96 b Fs(S)2074 1515 y Fu(q)2104 1523 y Fy(n)2145 1515 y Fu(;)p Fo(\003)2233 1503 y Ft(\016)18 b Fs(:)c(:)g(:)k Ft(\016)g Fs(S)2519 1515 y Fu(q)2549 1523 y Fi(1)2582 1515 y Fu(;)p Fo(\003)2651 1503 y Fs(:)118 1685 y Fw(Here)25 b Fs(S)363 1697 y Fu(q)r(;)p Fo(\003)489 1685 y Fw(is)g(the)h(map)e (for)h(the)g(up)r(dating)g(at)g(site)g Fs(q)s Fw(.)36 b(Recall)25 b(that)g(in)g(De\014nition)h(1.13)d(w)n(e)i(de\014ned)g (the)h(maps)e Fn(j)p Fw(.)36 b(F)-7 b(or)118 1785 y(all)27 b Fs(!)f Ft(2)e Fw(\012)j(there)h(is)f(a)g(\014nite)h(sequence)f Fn(j)p Fw(\()p Fs(!)s Fw(\))d(=)f(\()p Fs(q)1727 1797 y Fo(1)1764 1785 y Fs(;)14 b(:)g(:)g(:)g(;)g(q)1986 1797 y Fu(n)2031 1785 y Fw(\))28 b(and)f(so)1452 1984 y Ft(L)1509 1996 y Fu(S)1550 2007 y Fc(j)p Fi(\()p Fy(!)q Fi(\))p Fy(;)p Fi(\003)1741 1937 y Fo(def)1755 1984 y Fw(=)36 b Ft(L)1913 1996 y Fu(S)1954 2004 y Fy(q)1981 2012 y(n)2022 2004 y(;)p Fi(\003)2106 1984 y Ft(\016)17 b Fs(:)d(:)g(:)19 b Ft(\016)f(L)2398 1996 y Fu(S)2439 2004 y Fy(q)2466 2016 y Fi(1)2499 2004 y Fy(;)p Fi(\003)3709 1984 y Fw(\(164\))118 2133 y(is)28 b(w)n(ell-de\014ned.)118 2316 y(Before)c(establishing)h (particular)f(represen)n(tations)f(of)i(the)g(transfer)f(op)r(erators)g Ft(L)2740 2285 y Fu(T)2740 2338 y Fo(\003)2817 2316 y Fw(and)h Fs(\031)3023 2328 y Fo(\003)3087 2316 y Ft(\016)13 b(L)3199 2285 y Fu(T)3276 2316 y Fw(w)n(e)25 b(consider)f(some)118 2415 y(sp)r(ecial)j(examples.)37 b(F)-7 b(or)27 b(these)g(w)n(e)g(need) h(the)g(follo)n(wing)f(de\014nition.)118 2598 y Fn(De\014nition)k(3.3) 40 b Fw(W)-7 b(e)28 b(de\014ne:)1492 2925 y Ft(R)p Fw(\()p Fs(t)p Fw(\))84 b(=)e Fs(e)1926 2891 y Fm(\000)p Fu(\025t)2088 2822 y Fm(1)2061 2846 y Fk(X)2060 3025 y Fu(k)q Fo(=0)2205 2869 y Fw(\()p Fs(\025t)p Fw(\))2347 2839 y Fu(k)p 2205 2906 184 4 v 2262 2982 a Fs(k)s Fw(!)2413 2925 y Ft(R)2483 2891 y Fu(k)3709 2925 y Fw(\(165\))1740 3170 y(=)g(exp\()p Fs(\025t)14 b Fw(\(id)20 b Ft(\000)e(R)p Fw(\)\))p Fs(:)1210 b Fw(\(166\))118 3353 y(Then)28 b(w)n(e)f(ha)n(v)n(e,)g(using)g (\(152\),)1481 3635 y Ft(R)p Fw(\()p Fs(t)1613 3647 y Fo(1)1651 3635 y Fw(\))14 b Ft(R)p Fw(\()p Fs(t)1829 3647 y Fo(2)1867 3635 y Fw(\))83 b(=)g Ft(R)p Fw(\()p Fs(t)2262 3647 y Fo(1)2318 3635 y Fw(+)18 b Fs(t)2431 3647 y Fo(2)2468 3635 y Fw(\))1209 b(\(167\))1651 3842 y Ft(kR)p Fw(\()p Fs(t)p Fw(\))p Ft(k)83 b(\024)g Fs(c)2166 3854 y Fu(r)2202 3842 y Fs(e)2241 3808 y Fm(\000)p Fo(\(1)p Fm(\000)p Fu(\021)r Fo(\))p Fu(\025t)3709 3842 y Fw(\(168\))118 4025 y Fn(Example)30 b(3.1)41 b Fw(Consider)26 b(a)g(single)h(site)g (system)g(with)g(an)g(up)r(dating)g(map)g Fs(f)32 b Fw(:)23 b Fs(S)2762 3995 y Fo(1)2822 4025 y Ft(7!)g Fs(S)2984 3995 y Fo(1)3048 4025 y Fw(that)k(satis\014es)g Fn(Assump-)118 4124 y(tion)36 b(I)h Fw(.)32 b(W)-7 b(e)32 b(ha)n(v)n(e)e(for)i (\014xed)f(time)i Fs(T)40 b(>)30 b Fw(0)h(and)h(jump)g(rate)f Fs(\025)g(>)e Fw(0)j(a)f(Mark)n(o)n(v)e(k)n(ernel)i Fs(K)3087 4094 y Fu(T)3081 4148 y(f)3139 4124 y Fw(,)i(acting)e(on)g(functions) 118 4235 y Fs( )26 b Ft(2)e(C)326 4205 y Fo(0)362 4235 y Fw(\()p Fs(S)450 4205 y Fo(1)488 4235 y Fw(\))i(as)f(describ)r(ed)h (in)g(Example)g(1.3.)35 b(According)25 b(to)h(\(164\))f(and)h(Example)f (1.3)g(w)n(e)h(get)g(a)f(transfer)g(op)r(erator)118 4335 y Ft(L)175 4305 y Fu(T)175 4358 y(f)228 4335 y Fw(,)i(acting)g(on)h Ft(H)713 4347 y Fu(p)751 4335 y Fw(:)1403 4662 y Ft(L)1460 4628 y Fu(T)1460 4683 y(f)1596 4662 y Fw(=)1773 4559 y Fm(1)1746 4584 y Fk(X)1743 4759 y Fu(n)p Fo(=0)1882 4662 y Fs(e)1921 4628 y Fm(\000)p Fu(\025t)2051 4606 y Fw(\()p Fs(\025t)p Fw(\))2193 4576 y Fu(n)p 2051 4643 189 4 v 2109 4719 a Fs(n)p Fw(!)2263 4662 y Ft(L)2320 4628 y Fu(n)2320 4683 y(f)1596 4938 y Fw(=)1773 4834 y Fm(1)1746 4859 y Fk(X)1743 5035 y Fu(n)p Fo(=0)1882 4938 y Fs(e)1921 4904 y Fm(\000)p Fu(\025t)2051 4882 y Fw(\()p Fs(\025t)p Fw(\))2193 4852 y Fu(n)p 2051 4919 V 2109 4995 a Fs(n)p Fw(!)2263 4938 y(\()p Ft(Q)19 b Fw(+)f Ft(R)p Fw(\))2567 4904 y Fu(n)3709 4938 y Fw(\(169\))1596 5151 y(=)82 b(\(1)18 b Ft(\000)h Fs(e)1958 5117 y Fm(\000)p Fu(\025t)2078 5151 y Fw(\))14 b Ft(Q)k Fw(+)g Ft(R)p Fw(\()p Fs(t)p Fw(\))1252 b(\(170\))118 5333 y(with)31 b Ft(R)p Fw(\()p Fs(t)p Fw(\))h(as)e(in)g(De\014nition)i(3.3.)45 b(Note)30 b(that)h(w)n(e)f(think)h(of)g(the)g(summand)f Fs(e)2672 5303 y Fm(\000)p Fu(\025t)2792 5333 y Fw(id)q(,)h(corresp)r (onding)e(to)h Fs(n)e Fw(=)f(0,)k(as)118 5433 y Fs(e)157 5403 y Fm(\000)p Fu(\025t)277 5433 y Fs(R)341 5403 y Fo(0)378 5433 y Fw(.)37 b(By)27 b(\(153\))g(w)n(e)g(ha)n(v)n(e)g(for)g Fs(n)c Ft(\025)f Fw(1)28 b(that)g(\()p Ft(Q)18 b Fw(+)g Ft(R)p Fw(\))1939 5403 y Fu(n)2008 5433 y Fw(=)23 b Ft(Q)18 b Fw(+)g Ft(R)2335 5403 y Fu(n)2380 5433 y Fw(,)28 b(and)g(so)f(w)n(e)g (get)g(\(170\).)1966 5682 y(29)p eop %%Page: 30 30 30 29 bop 695 1164 333 4 v 695 168 V 860 1162 4 997 v 527 1162 V 496 1164 67 4 v 413 1179 a Fw(0)p 496 168 V 346 183 a Ft(\000)p Fs(T)894 664 y(h)p Fw(-strip)p 1359 1164 333 4 v 1359 168 V 1522 1162 7 997 v 1525 1162 V 1519 1162 V 422 w Fs(r)r Fw(-strip)118 1428 y(Figure)26 b(2:)36 b(single)26 b(site,)h(unp)r(erturb)r(ed)g(case:)35 b(There)26 b(are)g(only)118 1528 y(t)n(w)n(o)h(gum)g(con\014gurations)p 2748 1164 333 4 v 2748 168 V 2910 1162 7 84 v 2914 1162 V 2907 1162 V 2831 1083 167 7 v 2831 1089 V 2831 1076 V 2914 996 a Ff(q)2914 930 y(q)2914 863 y(q)p 2831 751 V 2831 757 V 2831 744 V 2910 747 7 333 v 2914 747 V 2907 747 V 2831 584 167 7 v 2831 591 V 2831 578 V 2831 418 V 2831 425 V 2831 412 V 2912 415 4 250 v 2580 1162 4 997 v 2548 1164 67 4 v 2465 1179 a Fw(0)p 2548 1081 V 2465 1096 a Fs(t)2495 1108 y Fu(K)3013 1096 y Fs(B)p 2548 749 V 2465 764 a(t)2495 776 y Fo(3)3013 764 y Fs(B)p 2548 583 V 2465 598 a(t)2495 610 y Fo(2)3013 598 y Fs(B)p 2548 417 V 2465 432 a(t)2495 444 y Fo(1)3013 432 y Fs(B)p 2548 168 V 2399 183 a Ft(\000)p Fs(T)2947 332 y(h)p Fw(-strip)2947 515 y Fs(r)r Fw(-strip)2947 681 y Fs(r)2947 1146 y(r)2324 1428 y Fw(Figure)k(3:)46 b(single)31 b(site,)j(p)r(erturb)r(ed)e(case:) 45 b(ex-)2324 1528 y(ample)27 b(for)g(a)g(gum)h(con\014guration)118 2064 y Fn(Example)i(3.2)41 b Fw(No)n(w)26 b(w)n(e)g(consider)f(a)h (small)g(p)r(erturbation)g Fs(f)2118 2076 y Fu(\017)2176 2064 y Fw(of)g(the)h(single)f(site)h(system)f Fs(f)3108 2076 y Fo(0)3168 2064 y Fw(=)c Fs(f)36 b Fw(of)26 b(Example)f(3.1,)118 2164 y(that)32 b(dep)r(ends)g(on)f(\014xed)g(b)r(oundary)g(conditions.) 48 b(F)-7 b(or)31 b(simplicit)n(y)g(w)n(e)h(split)f(the)h(transfer)f (op)r(erator)e(for)i(the)h(single)118 2263 y(up)r(dating)f(in)n(to)f Ft(L)702 2275 y Fu(f)734 2283 y Fy(\017)795 2263 y Fw(=)d Ft(Q)20 b Fw(+)g Ft(R)h Fw(+)f Ft(B)32 b Fw(where)e Ft(B)i Fw(is)e(the)h(di\013erence)f(b)r(et)n(w)n(een)g(the)h(op)r(erators)d (for)i(the)g(p)r(erturb)r(ed)h(and)118 2363 y(the)d(unp)r(erturb)r(ed)g (system.)118 2463 y(F)-7 b(or)27 b(a)g(giv)n(en)g(n)n(um)n(b)r(er)g Fs(N)37 b Fw(of)27 b(up)r(datings)h(w)n(e)f(distribute)1616 2662 y Ft(L)1673 2628 y Fu(N)1673 2682 y(f)1705 2690 y Fy(\017)1763 2662 y Fw(=)c(\()p Ft(Q)18 b Fw(+)g Ft(R)h Fw(+)f Ft(B)s Fw(\))2314 2628 y Fu(N)2376 2662 y Fs(;)1310 b Fw(\(171\))118 2811 y(using)29 b(\(153\))f(and)g(\(162\).)40 b(Let)29 b Fs(K)34 b Fw(denote)29 b(the)g(n)n(um)n(b)r(er)g(of)g (factors)f Ft(B)s Fw(,)g Fs(n)2469 2823 y Fo(0)2535 2811 y Fw(the)h(n)n(um)n(b)r(er)g(of)g(factors,)f(either)g(all)h Ft(Q)g Fw(or)118 2911 y(all)h Ft(R)p Fw(,)i(b)r(efore)e(the)h(\014rst)g (factor)e Ft(B)k Fw(and)e Fs(n)1477 2923 y Fu(i)1535 2911 y Fw(the)g(n)n(um)n(b)r(er)f(of)g(factors)g Fs(R)h Fw(after)f(the)h Fs(i)p Fw(th)g Ft(B)s Fw(.)45 b(So)30 b(the)h(total)f(n)n(um)n(b)r(er)h(of)118 3011 y(factors)c(is)g Fs(N)32 b Fw(=)23 b Fs(n)709 3023 y Fo(0)764 3011 y Fw(+)18 b Fs(:)c(:)g(:)k Fw(+)g Fs(n)1095 3023 y Fu(K)1178 3011 y Fw(+)g Fs(K)6 b Fw(.)118 3110 y(W)-7 b(e)28 b(get)403 3438 y Ft(L)460 3404 y Fu(T)460 3459 y(f)492 3467 y Fy(\017)610 3438 y Fw(=)796 3334 y Fm(1)769 3359 y Fk(X)757 3537 y Fu(N)6 b Fo(=0)914 3438 y Fs(e)953 3404 y Fm(\000)p Fu(\025T)1106 3382 y Fw(\()p Fs(\025T)12 b Fw(\))1279 3352 y Fu(N)p 1106 3419 237 4 v 1175 3495 a Fs(N)d Fw(!)1371 3438 y Ft(L)1428 3404 y Fu(N)1428 3459 y(f)1460 3467 y Fy(\017)3709 3438 y Fw(\(172\))610 3716 y(=)796 3612 y Fm(1)769 3637 y Fk(X)757 3815 y Fu(N)d Fo(=0)914 3716 y Fs(e)953 3682 y Fm(\000)p Fu(\025T)1303 3637 y Fk(X)1199 3805 y Fy(n)1236 3817 y Fi(0)1268 3805 y Fy(;:::;n)1400 3819 y(K)1454 3805 y Fe(\025)p Fi(0)1120 3856 y Fy(n)1157 3868 y Fi(0)1189 3856 y(+)p Fy(:::)p Fi(+)p Fy(n)1369 3870 y(K)1422 3856 y Fi(+)p Fy(K)s Fi(=)p Fy(N)1640 3660 y Fw(\()p Fs(\025T)12 b Fw(\))1813 3630 y Fu(N)6 b Fo(+)p Fu(K)p 1640 3697 347 4 v 1643 3773 a Fw(\()p Fs(N)28 b Fw(+)18 b Fs(K)6 b Fw(\)!)2016 3716 y Ft(R)2086 3682 y Fu(n)2127 3690 y Fy(K)2204 3716 y Ft(\016)18 b(B)i(\016)e Fs(:)c(:)g(:)19 b Ft(\016)f(B)i(\016)e(R)2780 3682 y Fu(n)2821 3690 y Fi(1)2877 3716 y Ft(\016)g(B)i(\016)e Fw(\()p Ft(Q)h Fw(+)f Ft(R)p Fw(\))3376 3682 y Fu(n)3417 3690 y Fi(0)3709 3716 y Fw(\(173\))610 4049 y(=)796 3945 y Fm(1)769 3970 y Fk(X)757 4148 y Fu(K)t Fo(=0)1029 3970 y Fk(X)1018 4138 y Fy(n)1055 4150 y Fi(0)1087 4138 y Fe(\025)p Fi(1)925 4189 y Fy(n)962 4201 y Fi(1)994 4189 y Fy(;:::;n)1126 4203 y(K)1180 4189 y Fe(\025)p Fi(0)1277 4049 y Fs(e)1316 4015 y Fm(\000)p Fu(\025T)1525 3993 y Fw(\()p Fs(\025T)12 b Fw(\))1698 3963 y Fu(n)1739 3971 y Fi(0)1771 3963 y Fo(+)p Fu(:::n)1923 3971 y Fy(K)1977 3963 y Fo(+)p Fu(K)p 1469 4030 679 4 v 1469 4106 a Fw(\()p Fs(n)1551 4118 y Fo(0)1607 4106 y Fw(+)18 b Fs(:)c(:)g(:)g(n)1851 4118 y Fu(K)1933 4106 y Fw(+)k Fs(K)6 b Fw(\)!)2176 4049 y Ft(R)2246 4015 y Fu(n)2287 4023 y Fy(K)2364 4049 y Ft(\016)18 b(B)j(\016)d Fs(:)c(:)g(:)k Ft(\016)g(B)j(\016)c(R)2940 4015 y Fu(n)2981 4023 y Fi(1)3037 4049 y Ft(\016)h(B)i(\016)e(Q)409 b Fw(\(174\))771 4382 y(+)888 4278 y Fm(1)861 4303 y Fk(X)850 4481 y Fu(K)t Fo(=0)1123 4303 y Fk(X)1007 4481 y Fu(n)1048 4489 y Fi(0)1081 4481 y Fu(;:::)o(;n)1221 4489 y Fy(K)1275 4481 y Fm(\025)p Fo(0)1373 4382 y Fs(e)1412 4348 y Fm(\000)p Fu(\025T)1621 4326 y Fw(\()p Fs(\025T)12 b Fw(\))1794 4296 y Fu(n)1835 4304 y Fi(0)1868 4296 y Fo(+)p Fu(:::)o(n)2019 4304 y Fy(K)2073 4296 y Fo(+)p Fu(K)p 1566 4363 V 1566 4439 a Fw(\()p Fs(n)1648 4451 y Fo(0)1703 4439 y Fw(+)18 b Fs(:)c(:)g(:)g(n)1947 4451 y Fu(K)2029 4439 y Fw(+)k Fs(K)6 b Fw(\)!)2273 4382 y Ft(R)2343 4348 y Fu(n)2384 4356 y Fy(K)2442 4382 y Ft(B)21 b(\016)d Fs(:)c(:)g(:)k Ft(\016)g(B)j(\016)d(R)2959 4348 y Fu(n)3000 4356 y Fi(1)3055 4382 y Ft(\016)g(B)i(\016)e(R)3320 4348 y Fu(n)3361 4356 y Fi(0)610 4671 y Fw(=)796 4567 y Fm(1)769 4592 y Fk(X)757 4770 y Fu(K)t Fo(=0)915 4558 y Fk(Z)998 4579 y Fu(T)961 4747 y Fo(0)1050 4671 y Fs(\025dt)1171 4683 y Fo(1)1223 4558 y Fk(Z)1306 4579 y Fu(T)1269 4747 y(t)1294 4755 y Fi(1)1358 4671 y Fs(\025dt)1479 4683 y Fo(2)1531 4671 y Ft(\001)c(\001)g(\001)1642 4558 y Fk(Z)1725 4579 y Fu(T)1688 4747 y(t)1713 4755 y Fy(K)1777 4671 y Fs(\025dt)1898 4683 y Fu(K)1976 4671 y Fw(\(1)19 b Ft(\000)f Fs(e)2191 4637 y Fm(\000)p Fu(\025t)2307 4645 y Fi(1)2343 4671 y Fw(\))h Ft(R)p Fw(\()p Fs(t)2526 4683 y Fu(n)2572 4671 y Fw(\))g Ft(\016)f(B)i(\016)e(\001)c(\001)g (\001)k(\016)g(B)j(\016)d(R)p Fw(\()p Fs(t)3261 4683 y Fo(1)3299 4671 y Fw(\))g Ft(\016)g(B)j(\016)d(Q)96 b Fw(\(175\))771 4949 y(+)888 4845 y Fm(1)861 4870 y Fk(X)850 5048 y Fu(K)t Fo(=0)1007 4836 y Fk(Z)1090 4856 y Fu(T)1053 5025 y Fo(0)1143 4949 y Fs(\025dt)1264 4961 y Fo(1)1315 4836 y Fk(Z)1398 4856 y Fu(T)1361 5025 y(t)1386 5033 y Fi(1)1451 4949 y Fs(\025dt)1572 4961 y Fo(2)1623 4949 y Fs(:)14 b(:)g(:)1734 4836 y Fk(Z)1817 4856 y Fu(T)1780 5025 y(t)1805 5033 y Fy(K)1869 4949 y Fs(\025dt)1990 4961 y Fu(K)2069 4949 y Ft(R)p Fw(\()p Fs(t)2201 4961 y Fu(n)2247 4949 y Fw(\))k Ft(\016)g(B)j(\016)d Fs(:)c(:)g(:)k Ft(\016)g(R)p Fw(\()p Fs(t)2800 4961 y Fo(0)2838 4949 y Fw(\))118 5210 y(F)-7 b(or)25 b(the)h(step)g(from)f(\(174\))g(to)h (\(175\))f(w)n(e)g(ha)n(v)n(e)g(used)g(Lemma)h(3.2)f(\(s.)g(b)r(elo)n (w\).)37 b(W)-7 b(e)26 b(in)n(terpret)f(\(175\))g(in)h(the)g(follo)n (wing)118 5310 y(w)n(a)n(y)-7 b(.)52 b(W)-7 b(e)34 b(write)f(the)g(op)r (erator)f Ft(L)1241 5280 y Fu(T)1241 5333 y(f)1273 5341 y Fy(\017)1340 5310 y Fw(as)h(a)g(sum)g(of)g(op)r(erators)e Ft(L)2234 5322 y Fm(C)2270 5330 y Fy(g)2305 5322 y Fu(;T)2377 5310 y Fw(.)54 b(F)-7 b(or)32 b(the)i(time)f(b)r(eing)h(w)n(e)e(think)i (of)f Ft(C)3677 5322 y Fu(g)3749 5310 y Fw(as)f(a)118 5409 y(diagram)22 b(lik)n(e,)h(for)g(example,)g(in)g(Figure)g(3.)34 b(The)24 b(v)n(ertical)d(axis)h(from)h(top)g(to)g(b)r(ottom)g(corresp)r (onds)e(to)i(the)g(p)r(ositiv)n(ely)1966 5682 y(30)p eop %%Page: 31 31 31 30 bop 118 354 a Fw(orien)n(ted)21 b(time)i(line.)35 b(Along)22 b(this)h(axis)e(w)n(e)h(dra)n(w)f Fs(K)28 b Fw(thic)n(k)22 b(horizon)n(tal)e(bars,)j(denoted)f(b)n(y)g Fs(B)t Fw(,)h(that)g(corresp)r(ond)d(to)i(the)118 454 y(op)r(erators)27 b Ft(B)s Fw(.)39 b(Bet)n(w)n(een)29 b(consecutiv)n(e)e Fs(B)t Fw('s)i(or)f(b)r(et)n(w)n(een)h(the)g (\014rst)g Fs(B)k Fw(and)28 b(the)h(top)g(or)f(the)h(last)g Fs(B)k Fw(and)28 b(the)h(b)r(ottom)118 554 y(or,)24 b(in)g(the)h(case)e Fs(K)29 b Fw(=)22 b(0,)j(b)r(et)n(w)n(een)f(the)h(top)f(and)g(the)g(b)r (ottom)h(w)n(e)e(dra)n(w)g(either)h(a)g(thic)n(k)g(\()p Fs(r)r Fw(-strip\))h(or)e(a)h(thin)h(\()p Fs(h)p Fw(-strip\))118 653 y(line,)32 b(represen)n(ting)d(the)h(c)n(hoice)g(of)h(factors)e Ft(R)i Fw(or)e Ft(Q)i Fw(in)g(the)f(pro)r(duct)h(\(171\),)f(resp)r (ectiv)n(ely)-7 b(.)45 b(Note)31 b(that)f(b)r(elo)n(w)g(a)g Fs(B)118 753 y Fw(there)e(m)n(ust)f(b)r(e)h(a)f(thic)n(k)h(v)n(ertical) e(line.)37 b(F)-7 b(or)27 b(a)h(\014xed)f Fs(T)39 b Fw(and)27 b(a)h Fs(K)6 b Fw(-tuple)1150 952 y Fn(t)23 b Fw(=)g(\()p Fs(t)1360 964 y Fo(1)1397 952 y Fs(;)14 b(:)g(:)g(:)g(;)g(t)1612 964 y Fu(K)1676 952 y Fw(\))23 b Ft(2)h(f)p Fn(t)e Fw(:)h Ft(\000)p Fs(T)34 b(<)23 b(t)2223 964 y Fo(1)2283 952 y Fs(<)g(:)14 b(:)g(:)22 b(<)h(t)2608 964 y Fu(K)2695 952 y Fs(<)g Fw(0)p Ft(g)842 b Fw(\(176\))118 1102 y(w)n(e)30 b(think)h(of)f(the)h(top)g(as)e(b)r(eing)i(\014xed)f(at)h(time)f Ft(\000)p Fs(T)12 b Fw(,)30 b(the)h(b)r(ottom)g(at)f(0)g(and)g(the)h Fs(i)p Fw(th)g(sym)n(b)r(ol)f Fs(B)k Fw(at)c Fs(t)3452 1114 y Fu(i)3480 1102 y Fw(.)45 b(That)31 b(also)118 1201 y(\014xes)f(the)g(lengths)g(of)f(the)i(particular)d Fs(h)p Fw(-)i(and)g Fs(r)r Fw(-strips.)43 b(W)-7 b(e)31 b(assign)d(to)i(the)g(triple)g(\()p Ft(C)2914 1213 y Fu(g)2953 1201 y Fs(;)14 b(T)7 b(;)14 b(t)p Fw(\))29 b(an)h(op)r(erator)e Ft(L)3686 1213 y Fm(C)3722 1221 y Fy(g)3757 1213 y Fu(;T)5 b(;)p Fl(t)3875 1201 y Fw(.)118 1301 y(F)-7 b(or)27 b(example)g(with)h Ft(C)825 1313 y Fu(g)864 1301 y Fw(,)f Fs(T)39 b Fw(and)28 b Fn(t)f Fw(as)g(in)h(Figure)f(3,)g(w)n(e)h(get)792 1500 y Ft(L)849 1512 y Fm(C)885 1520 y Fy(g)920 1512 y Fu(;T)5 b(;)p Fl(t)1060 1500 y Fw(=)23 b(\(1)18 b Ft(\000)g Fs(e)1362 1466 y Fm(\000)p Fu(\025)p Fo(\()p Fu(t)1504 1474 y Fi(1)1537 1466 y Fo(+)p Fu(T)9 b Fo(\))1666 1500 y Fw(\))19 b Ft(R)p Fw(\(0)f Ft(\000)g Fs(t)1992 1512 y Fu(K)2056 1500 y Fw(\))h Ft(\016)f(B)e Fs(:)e(:)g(:)k Ft(\016)g(B)i(\016)e(R)p Fw(\()p Fs(t)2680 1512 y Fo(2)2737 1500 y Ft(\000)g Fs(t)2850 1512 y Fo(1)2887 1500 y Fw(\))h Ft(\016)f(B)i(\016)e(Q)p Fs(:)485 b Fw(\(177\))118 1649 y(T)-7 b(o)27 b(get)h Ft(L)436 1619 y Fu(T)436 1672 y Fm(C)472 1680 y Fy(g)539 1649 y Fw(w)n(e)f(in)n(tegrate)f(\(177\))h(o)n(v)n(er)f(the)i(simplex)f (giv)n(en)g(b)n(y)g(\(176\))g(wrt)h(the)g(scaled)f(Leb)r(esgue)g (measure)118 1767 y Fs(\025)166 1737 y Fu(K)231 1767 y Fs(dt)304 1779 y Fo(1)355 1767 y Fs(:)14 b(:)g(:)g(dt)539 1779 y Fu(K)603 1767 y Fw(.)42 b(The)29 b(simplex)h(has)f(dimension)g Fs(K)6 b Fw(,)29 b(so)g(for)g(eac)n(h)f Fs(B)34 b Fw(in)29 b Ft(C)2478 1779 y Fu(g)2546 1767 y Fw(w)n(e)g(get)g(one)g(in)n (tegration.)41 b(Heuristically)-7 b(,)118 1866 y(the)33 b(measure)e(`)p Fs(\025dt)738 1878 y Fo(1)776 1866 y Fw(')h(corresp)r(onds)f(to)h(the)g(probabilit)n(y)f(that)i(a)f(P)n (oisson)e(pro)r(cess)h(with)i(rate)e Fs(\025)i Fw(jumps)g(in)f(a)g (small)118 1966 y(time)i(in)n(terv)-5 b(al.)52 b(The)33 b(approac)n(h)e(of)i(appro)n(ximating)e(the)i(con)n(tin)n(uous)f(time)h (system)g(b)n(y)g(discrete)f(time)h(systems)g(is)118 2066 y(made)24 b(precise)f(in)h([12)o(].)36 b(W)-7 b(e)24 b(also)f(note)h(the)g(sp)r(ecial)g(case)f Fs(K)28 b Fw(=)23 b(0)g(where)g(the)i(simplex)f(degenerates)e(to)i(a)f(single)h(p)r(oin)n (t)118 2165 y(of)k(measure)e(1.)118 2348 y(Ab)r(o)n(v)n(e)h(w)n(e)g(ha) n(v)n(e)g(used)g(the)h(follo)n(wing)f(lemma.)118 2531 y Fn(Lemma)j(3.2)40 b Fw(Let)25 b(\()p Fs(n)848 2543 y Fu(i;j)927 2531 y Fw(\))971 2532 y Fi(0)p Fe(\024)p Fy(i)p Fe(\024)p Fy(K)969 2579 y Fi(1)p Fe(\024)p Fy(j)r Fe(\024)p Fy(N)1202 2531 y Fw(b)r(e)h(a)e(family)h(of)g(non-negativ)n (e)e(in)n(tegers)h(and)h Fs(\025;)14 b(T)34 b(>)23 b Fw(0.)36 b(Then,)25 b(with)h(the)f(notation)118 2692 y Fs(t)148 2704 y Fo(0)208 2645 y(def)222 2692 y Fw(=)36 b Ft(\000)p Fs(T)12 b Fw(,)26 b Fs(t)528 2704 y Fu(K)t Fo(+1)699 2645 y(def)713 2692 y Fw(=)36 b(0:)566 3009 y Fs(e)605 2975 y Fm(\000)p Fu(N)6 b(\025T)899 2953 y Fw(\()p Fs(\025T)12 b Fw(\))1072 2923 y Fu(n)1113 2931 y Fi(0)p Fy(;)p Fi(1)1193 2923 y Fo(+)p Fu(:::)o Fo(+)p Fu(n)1395 2931 y Fy(K;N)1514 2923 y Fo(+)p Fu(K)p 817 2990 895 4 v 817 3066 a Fw(\()p Fs(n)899 3078 y Fo(0)p Fu(;)p Fo(1)1008 3066 y Fw(+)18 b Fs(:)c(:)g(:)k Fw(+)g Fs(n)1339 3078 y Fu(K;N)1496 3066 y Fw(+)g Fs(K)6 b Fw(\)!)1735 2892 y Fk(\022)1796 2953 y Fs(n)1846 2965 y Fo(0)p Fu(;)p Fo(1)1954 2953 y Fw(+)18 b Fs(:)c(:)g(:)19 b Fw(+)f Fs(n)2286 2965 y Fo(0)p Fu(;N)1854 3066 y Fs(n)1904 3078 y Fo(0)p Fu(;)p Fo(1)1994 3066 y Fs(;)c(:)g(:)g(:)f(;)h(n)2228 3078 y Fo(0)p Fu(;N)2401 2892 y Fk(\023)2481 3009 y Ft(\001)k Fs(:)c(:)g(:)19 b Ft(\001)2679 2892 y Fk(\022)2740 2953 y Fs(n)2790 2965 y Fu(K;)p Fo(1)2922 2953 y Fw(+)f Fs(:)c(:)g(:)k Fw(+)g Fs(n)3253 2965 y Fu(K;N)2798 3066 y Fs(n)2848 3078 y Fu(K;)p Fo(1)2961 3066 y Fs(;)c(:)g(:)g(:)g(;)g(n)3196 3078 y Fu(K;N)3392 2892 y Fk(\023)3709 3009 y Fw(\(178\))649 3336 y(=)797 3223 y Fk(Z)880 3244 y Fo(0)843 3412 y Fm(\000)p Fu(T)947 3336 y Fs(\025dt)1068 3348 y Fo(1)1120 3223 y Fk(Z)1203 3244 y Fo(0)1166 3412 y Fm(\000)p Fu(t)1243 3420 y Fi(1)1279 3336 y Fs(\025dt)1400 3348 y Fo(2)1452 3336 y Ft(\001)g(\001)g(\001)1563 3223 y Fk(Z)1646 3244 y Fo(0)1609 3412 y Fm(\000)p Fu(t)1686 3420 y Fy(K)s Fe(\000)p Fi(1)1817 3336 y Fs(\025dt)1938 3348 y Fu(K)2071 3257 y Fk(Y)2029 3426 y Fi(0)p Fe(\024)p Fy(i)p Fe(\024)p Fy(K)2027 3473 y Fi(1)p Fe(\024)p Fy(j)r Fe(\024)p Fy(N)2245 3219 y Fk(\022)2306 3336 y Fs(e)2345 3302 y Fm(\000)p Fu(\025)p Fo(\()p Fu(t)2487 3310 y Fy(i)p Fi(+1)2584 3302 y Fm(\000)p Fu(t)2661 3310 y Fy(i)2687 3302 y Fo(\))2746 3280 y Fw(\()p Fs(\025)p Fw(\()p Fs(t)2888 3292 y Fu(i)p Fo(+1)3019 3280 y Ft(\000)k Fs(t)3132 3292 y Fu(i)3160 3280 y Fw(\)\))3224 3250 y Fu(n)3265 3258 y Fy(i;j)p 2746 3317 596 4 v 2968 3393 a Fs(n)3018 3405 y Fu(i;j)3096 3393 y Fw(!)3352 3219 y Fk(\023)3427 3336 y Fs(:)118 3644 y Fn(Pro)s(of)59 b Fw(W)-7 b(e)28 b(see)g(that)g(the)g(rhs)f(of)g (\(178\))g(is)g(equal)h(to)374 3882 y Fs(e)413 3848 y Fm(\000)p Fu(N)6 b(\025T)615 3882 y Fs(\025)663 3848 y Fu(n)704 3856 y Fi(0)p Fy(;)p Fi(1)784 3848 y Fo(+)p Fu(:::)o Fo(+)p Fu(n)986 3856 y Fy(K;N)1106 3848 y Fo(+)p Fu(K)1288 3803 y Fk(Y)1246 3971 y Fi(0)p Fe(\024)p Fy(i)p Fe(\024)p Fy(K)1244 4019 y Fi(1)p Fe(\024)p Fy(j)r Fe(\024)p Fy(N)1527 3826 y Fw(1)p 1472 3863 152 4 v 1472 3939 a Fs(n)1522 3951 y Fu(i;j)1600 3939 y Fw(!)1647 3769 y Fk(Z)1730 3789 y Fo(0)1693 3958 y Fm(\000)p Fu(T)1811 3882 y Fs(dt)1884 3894 y Fo(1)1936 3882 y Ft(\001)14 b(\001)g(\001)2046 3769 y Fk(Z)2129 3789 y Fo(0)2092 3958 y Fm(\000)p Fu(t)2169 3966 y Fy(K)s Fe(\000)p Fi(1)2301 3882 y Fs(dt)2374 3894 y Fu(K)2509 3803 y Fk(Y)2452 3981 y Fo(0)p Fm(\024)p Fu(i)p Fm(\024)p Fu(K)2672 3882 y Fw(\()p Fs(t)2734 3894 y Fu(i)p Fo(+1)2865 3882 y Ft(\000)k Fs(t)2978 3894 y Fu(i)3005 3882 y Fw(\))3037 3848 y Fu(n)3078 3856 y Fy(i;)p Fi(1)3152 3848 y Fo(+)p Fu(:::)o Fo(+)p Fu(n)3354 3856 y Fy(i;N)3709 3882 y Fw(\(179\))118 4157 y(and)28 b(so)e(\(178\))h(follo)n(ws)g(b)n(y)g Fs(K)33 b Fw(times)28 b(applying)f(the)h(iden)n(tit)n(y)1253 4281 y Fk(Z)1336 4302 y Fo(0)1299 4470 y Fm(\000)p Fu(\034)1407 4394 y Fs(dt)23 b Fw(\()p Ft(\000)p Fs(t)p Fw(\))1662 4360 y Fu(n)1707 4394 y Fw(\()p Fs(t)c Fw(+)f Fs(\034)9 b Fw(\))1948 4360 y Fu(m)2035 4394 y Fw(=)2197 4338 y Fs(n)p Fw(!)14 b Fs(m)p Fw(!)p 2133 4375 312 4 v 2133 4451 a(\()p Fs(n)k Fw(+)g Fs(m)p Fw(\)!)2478 4394 y Fs(\034)2523 4360 y Fu(n)p Fo(+)p Fu(m)p Fo(+1)3709 4394 y Fw(\(180\))118 4593 y(whic)n(h)28 b(can)f(b)r(e)h(easily)f(sho)n(wn)f(b)n(y)i (iterated)f(in)n(tegration)f(b)n(y)i(parts.)1535 b Fh(2)118 4846 y Fn(Example)30 b(3.3)41 b Fw(No)n(w)25 b(w)n(e)h(consider)f(a)g (system)h(with)g(nearest)f(neigh)n(b)r(our)g(coupling)h(where)f(only)g (t)n(w)n(o)h(adjacen)n(t)f(sites)118 4946 y(\(1)i(and)h(2\))f(are)f(up) r(dated)i(and)g(the)g(states)f(at)g(the)h(other)f(sites)g(are)f (\014xed.)37 b(This)28 b(time)f(an)h(expansion)e(as)h(in)h(Example)118 5045 y(3.2)f(giv)n(es)f(rise)h(to)h(summing)f(op)r(erators)f Ft(L)1490 5057 y Fm(C)1526 5065 y Fy(g)1561 5057 y Fu(;T)1661 5045 y Fw(where)h(the)h Ft(C)2088 5057 y Fu(g)2154 5045 y Fw(corresp)r(ond)e(to)h(diagrams)f(lik)n(e)h(in)h(Figure)f(4.)118 5145 y(W)-7 b(e)28 b(ha)n(v)n(e)f(to)g(consider)g(all)g(\014nite)h (ordered)e(sequences)h(of)h(sym)n(b)r(ols)f Fs(B)2357 5157 y Fo(1)2421 5145 y Fw(and)h Fs(B)2646 5157 y Fo(2)2683 5145 y Fw(.)118 5245 y(In)h(the)g(example)g(of)g(Figure)f(4)h(w)n(e)f (ha)n(v)n(e)g(the)h(sequence)g(\()p Fs(B)2025 5257 y Fo(2)2062 5245 y Fs(;)14 b(B)2162 5257 y Fo(1)2199 5245 y Fw(\).)41 b(F)-7 b(or)29 b(eac)n(h)f Fs(B)2697 5257 y Fu(i)2753 5245 y Fw(w)n(e)h(dra)n(w)f(a)g(thic)n(k)h(horizon)n(tal)e (bar,)118 5344 y(cen)n(tered)g(in)g(the)g(column)g(corresp)r(onding)f (to)h(the)g Fs(i)p Fw(-th)g(site)g(and)g(connecting)g(to)f(all)h(sites) g(\(columns\))h(on)e(whic)n(h)h(the)1966 5682 y(31)p eop %%Page: 32 32 32 31 bop 1120 1164 997 4 v 1120 168 V 1451 1162 4 1013 v 1783 1162 V 1120 834 665 7 v 1120 840 V 1120 827 V 1120 797 a Fs(B)1183 809 y Fo(1)p 1452 501 V 1452 508 V 1452 495 V 2150 531 a Fs(B)2213 543 y Fo(2)p 1449 1162 7 333 v 1452 1162 V 1446 1162 V 1781 830 7 665 v 1784 830 V 1778 830 V 953 1162 4 997 v 921 1164 67 4 v 838 1179 a Fw(0)p 921 832 V 838 847 a Fs(t)868 859 y Fo(2)p 921 500 V 838 515 a Fs(t)868 527 y Fo(1)p 921 168 V 772 183 a Ft(\000)p Fs(T)1436 133 y Fw(1)290 b(2)1303 996 y Fs(n)1353 1008 y Fo(2)p Fu(;)p Fo(1)1303 664 y Fs(n)1353 676 y Fo(1)p Fu(;)p Fo(1)1303 332 y Fs(n)1353 344 y Fo(0)p Fu(;)p Fo(1)1818 996 y Fs(n)1868 1008 y Fo(2)p Fu(;)p Fo(2)1818 664 y Fs(n)1868 676 y Fo(1)p Fu(;)p Fo(2)1818 332 y Fs(n)1868 344 y Fo(0)p Fu(;)p Fo(2)564 1428 y Fw(Figure)27 b(4:)37 b(t)n(w)n(o)26 b(sites,)i(example)f(for)g(a)g(gum)h (con\014gurations)118 1948 y(op)r(erator)c Ft(B)506 1960 y Fu(i)559 1948 y Fw(dep)r(ends.)37 b(W)-7 b(e)26 b(dra)n(w)e(the)j (sequence)e(of)g Fs(B)1895 1960 y Fu(i)1923 1948 y Fw('s)h(`do)n(wn)n (w)n(ards'.)34 b(As)26 b(w)n(e)f(consider)g(only)g(nearest)g(neigh)n(b) r(our)118 2047 y(up)r(datings)j(here,)f(the)h Fs(B)913 2059 y Fu(i)968 2047 y Fw(ha)n(v)n(e)f(width)h(2.)118 2147 y(Then)h(w)n(e)g(can)g(c)n(ho)r(ose)f(at)h(sites)g(1)f(and)h(2)g (b)r(et)n(w)n(een)g Fs(h)p Fw(-)g(and)g Fs(r)r Fw(-strips.)41 b(Note)30 b(that)f(in)g(Figure)g(4)g(at)g(site)g(2)g(an)f Fs(h)p Fw(-strip)118 2247 y(follo)n(ws)g(an)h Fs(r)r Fw(-strip.)41 b(This)29 b(is)g(p)r(ossible)f(b)r(ecause)h(they)g(are)f (`separated')g(b)n(y)g Fs(B)2634 2259 y Fo(1)2672 2247 y Fw('.)41 b(No)n(w)28 b(w)n(e)h(consider)f(the)h(particular)118 2346 y Ft(C)162 2358 y Fu(g)238 2346 y Fw(and)38 b Fs(T)49 b Fw(sho)n(wn)37 b(in)g(Figure)h(4.)66 b(Let)38 b Fs(n)1489 2358 y Fo(0)p Fu(;)p Fo(1)1617 2346 y Fw(denote)f(the)i(n)n(um)n(b)r (er)e(of)h(jumps)g(at)g(site)f(1)h(b)r(efore)f(the)h(jumps)g(that)118 2446 y(corresp)r(onds)25 b(to)i(id)743 2458 y Fo(1)797 2446 y Ft(\012)16 b(B)933 2458 y Fo(2)997 2446 y Fw(etc.)37 b(The)27 b(de\014nition)g(of)f Fs(n)1844 2458 y Fu(i;j)1949 2446 y Fw(in)h(general)f(is)g(analogous.)35 b(So)26 b(the)h(total)g(n)n (um)n(b)r(er)f(of)h(jumps)118 2545 y(is)f Fs(N)32 b Fw(=)22 b Fs(n)436 2557 y Fo(0)p Fu(;)p Fo(1)541 2545 y Fw(+)15 b Fs(:)f(:)g(:)h Fw(+)g Fs(n)863 2557 y Fo(2)p Fu(;)p Fo(2)967 2545 y Fw(+)g(2.)36 b(The)26 b(\014rst)g(t)n(w)n(o)f(factors)g (in)h(\(182\))f(giv)n(e)g(the)h(probabilit)n(y)f(that)h(a)g(particular) e(sequence)118 2645 y(of)30 b(exactly)g Fs(N)39 b Fw(jumps)30 b(o)r(ccurs.)44 b(The)30 b(\014rst)g(binomial)g(co)r(e\016cien)n(t)g (coun)n(ts)f(the)i(com)n(binations)e(\(di\013eren)n(t)h(sequences\))118 2745 y(of)e Fs(n)263 2757 y Fo(0)p Fu(;)p Fo(2)380 2745 y Fw(jumps)g(at)g(site)g(1)f(and)g Fs(n)1162 2757 y Fo(0)p Fu(;)p Fo(2)1280 2745 y Fw(jumps)h(at)f(site)h(2,)f(the)h(others)f(are) g(explained)g(analogously)-7 b(.)118 2844 y(W)g(e)28 b(get)f(the)h(op)r(erator:)208 3106 y Ft(L)265 3118 y Fm(C)301 3126 y Fy(g)336 3118 y Fu(;T)3709 3106 y Fw(\(181\))291 3291 y(=)619 3213 y Fk(X)449 3381 y Fy(n)486 3393 y Fi(0)p Fy(;)p Fi(1)565 3381 y Fy(;n)621 3393 y Fi(1)p Fy(;)p Fi(1)700 3381 y Fy(;n)756 3393 y Fi(2)p Fy(;)p Fi(2)836 3381 y Fe(\025)p Fi(1)449 3440 y Fy(n)486 3452 y Fi(0)p Fy(;)p Fi(2)565 3440 y Fy(;n)621 3452 y Fi(1)p Fy(;)p Fi(2)700 3440 y Fy(;n)756 3452 y Fi(0)p Fy(;)p Fi(2)836 3440 y Fe(\025)p Fi(0)933 3291 y Fs(e)972 3257 y Fm(\000)p Fo(2)p Fu(\025T)1187 3235 y Fw(\()p Fs(\025T)12 b Fw(\))1360 3205 y Fu(n)1401 3213 y Fi(0)p Fy(;)p Fi(1)1481 3205 y Fo(+)p Fu(:::)o Fo(+)p Fu(n)1683 3213 y Fi(2)p Fy(;)p Fi(2)1763 3205 y Fo(+2)p 1158 3272 723 4 v 1158 3348 a Fw(\()p Fs(n)1240 3360 y Fo(0)p Fu(;)p Fo(1)1349 3348 y Fw(+)18 b Fs(:)c(:)g(:)f(n)1592 3360 y Fo(2)p Fu(;)p Fo(2)1701 3348 y Fw(+)18 b(2\)!)1890 3174 y Fk(\022)1952 3235 y Fs(n)2002 3247 y Fo(0)p Fu(;)p Fo(1)2110 3235 y Fw(+)g Fs(n)2243 3247 y Fo(0)p Fu(;)p Fo(2)1984 3348 y Fs(n)2034 3360 y Fo(0)p Fu(;)p Fo(1)2124 3348 y Fs(;)c(n)2211 3360 y Fo(0)p Fu(;)p Fo(2)2333 3174 y Fk(\023\022)2455 3235 y Fs(n)2505 3247 y Fo(1)p Fu(;)p Fo(1)2613 3235 y Fw(+)k Fs(n)2746 3247 y Fo(1)p Fu(;)p Fo(2)2487 3348 y Fs(n)2537 3360 y Fo(1)p Fu(;)p Fo(1)2627 3348 y Fs(;)c(n)2714 3360 y Fo(1)p Fu(;)p Fo(2)2836 3174 y Fk(\023\022)2959 3235 y Fs(n)3009 3247 y Fo(2)p Fu(;)p Fo(1)3117 3235 y Fw(+)k Fs(n)3250 3247 y Fo(2)p Fu(;)p Fo(2)2991 3348 y Fs(n)3041 3360 y Fo(2)p Fu(;)p Fo(1)3131 3348 y Fs(;)c(n)3218 3360 y Fo(2)p Fu(;)p Fo(2)3340 3174 y Fk(\023)3709 3291 y Fw(\(182\))457 3655 y Ft(\002R)592 3610 y Fu(n)633 3618 y Fi(2)p Fy(;)p Fi(1)592 3677 y Fo(1)736 3655 y Ft(\012)k(Q)887 3667 y Fo(2)942 3655 y Ft(\016)g(B)1057 3667 y Fo(1)1112 3655 y Ft(\012)g Fw(id)1264 3667 y Fo(2)1320 3655 y Ft(\016)g(Q)1448 3667 y Fo(1)1504 3655 y Ft(\012)g(R)1657 3610 y Fu(n)1698 3618 y Fi(1)p Fy(;)p Fi(2)1657 3677 y Fo(2)1800 3655 y Ft(\016)g Fw(id)1929 3667 y Fo(1)1985 3655 y Ft(\012)g(B)2123 3667 y Fo(2)2178 3655 y Ft(\016)g(Q)2306 3667 y Fo(1)2362 3655 y Ft(\012)g(R)2515 3610 y Fu(n)2556 3618 y Fi(0)p Fy(;)p Fi(2)2515 3677 y Fo(2)291 3963 y Fw(=)439 3850 y Fk(Z)522 3871 y Fo(0)485 4039 y Fm(\000)p Fu(T)603 3963 y Fs(\025dt)724 3975 y Fo(1)776 3850 y Fk(Z)859 3871 y Fo(0)822 4039 y Fu(t)847 4047 y Fi(1)910 3963 y Fs(\025dt)1031 3975 y Fo(2)1083 3796 y Fk(0)1083 3946 y(@)1198 3884 y(X)1155 4063 y Fu(n)1196 4071 y Fi(2)p Fy(;)p Fi(1)1276 4063 y Fm(\025)p Fo(0)1375 3963 y Fs(e)1414 3929 y Fm(\000)p Fu(\025)p Fo(\()p Fm(j)p Fu(t)1576 3937 y Fi(2)1608 3929 y Fm(j)p Fo(\))1668 3907 y Fw(\()p Fs(\025)p Ft(j)p Fs(t)1801 3919 y Fo(2)1839 3907 y Ft(j)p Fw(\))1894 3877 y Fu(n)1935 3885 y Fi(2)p Fy(;)p Fi(1)p 1668 3944 352 4 v 1762 4020 a Fs(n)1812 4032 y Fo(2)p Fu(;)p Fo(1)1902 4020 y Fw(!)2029 3963 y Ft(R)2099 3918 y Fu(n)2140 3926 y Fi(2)p Fy(;)p Fi(1)2099 3985 y Fo(1)2224 3796 y Fk(1)2224 3946 y(A)2315 3963 y Ft(\012)2398 3796 y Fk(0)2398 3946 y(@)2514 3884 y(X)2471 4063 y Fu(n)2512 4071 y Fi(2)p Fy(;)p Fi(2)2591 4063 y Fm(\025)p Fo(1)2690 3963 y Fs(e)2729 3929 y Fm(\000)p Fu(\025)p Fm(j)p Fu(t)2865 3937 y Fi(2)2897 3929 y Fm(j)2931 3907 y Fw(\()p Fs(\025)p Ft(j)p Fs(t)3064 3919 y Fo(2)3102 3907 y Ft(j)p Fw(\))3157 3877 y Fu(n)3198 3885 y Fi(2)p Fy(;)p Fi(2)p 2931 3944 V 3025 4020 a Fs(n)3075 4032 y Fo(2)p Fu(;)p Fo(2)3165 4020 y Fw(!)3292 3963 y Ft(Q)3360 3975 y Fo(2)3397 3796 y Fk(1)3397 3946 y(A)3709 3963 y Fw(\(183\))453 4378 y Ft(\016B)550 4390 y Fo(1)604 4378 y Ft(\012)g Fw(id)757 4390 y Fo(2)812 4378 y Ft(\016)872 4212 y Fk(0)872 4361 y(@)988 4300 y(X)945 4478 y Fu(n)986 4486 y Fi(1)p Fy(;)p Fi(1)1066 4478 y Fm(\025)p Fo(0)1164 4378 y Fs(e)1203 4344 y Fm(\000)p Fu(\025)p Fo(\()p Fu(t)1345 4352 y Fi(2)1378 4344 y Fm(\000)p Fu(t)1455 4352 y Fi(1)1487 4344 y Fo(\))1527 4322 y Fw(\()p Fs(\025)p Fw(\()p Fs(t)1669 4334 y Fo(2)1726 4322 y Ft(\000)g Fs(t)1839 4334 y Fo(1)1876 4322 y Fw(\)\))1940 4292 y Fu(n)1981 4300 y Fi(1)p Fy(;)p Fi(1)p 1527 4359 539 4 v 1715 4435 a Fs(n)1765 4447 y Fo(1)p Fu(;)p Fo(1)1855 4435 y Fw(!)2076 4378 y Ft(Q)2144 4390 y Fo(1)2181 4212 y Fk(1)2181 4361 y(A)2272 4378 y Ft(\012)2355 4212 y Fk(0)2355 4361 y(@)2470 4300 y(X)2427 4478 y Fu(n)2468 4486 y Fi(1)p Fy(;)p Fi(2)2548 4478 y Fm(\025)p Fo(0)2647 4378 y Fs(e)2686 4344 y Fm(\000)p Fu(\025)p Fo(\()p Fu(t)2828 4352 y Fi(2)2860 4344 y Fm(\000)p Fu(t)2937 4352 y Fi(1)2970 4344 y Fo(\))3010 4322 y Fw(\()p Fs(\025)p Fw(\()p Fs(t)3152 4334 y Fo(2)3208 4322 y Ft(\000)g Fs(t)3321 4334 y Fo(1)3359 4322 y Fw(\))3391 4292 y Fu(n)3432 4300 y Fi(1)p Fy(;)p Fi(2)p 3010 4359 507 4 v 3181 4435 a Fs(n)3231 4447 y Fo(1)p Fu(;)p Fo(2)3321 4435 y Fw(!)3526 4378 y Ft(R)3596 4333 y Fu(n)3637 4341 y Fi(1)p Fy(;)p Fi(2)3596 4401 y Fo(2)3721 4212 y Fk(1)3721 4361 y(A)453 4794 y Ft(\016)p Fw(id)563 4806 y Fo(1)619 4794 y Ft(\012)g(B)757 4806 y Fo(2)812 4794 y Ft(\016)872 4627 y Fk(0)872 4776 y(@)988 4715 y(X)945 4893 y Fu(n)986 4901 y Fi(0)p Fy(;)p Fi(1)1066 4893 y Fm(\025)p Fo(0)1164 4794 y Fs(e)1203 4759 y Fm(\000)p Fu(\025)p Fo(\()p Fu(t)1345 4767 y Fi(1)1378 4759 y Fo(+)p Fu(T)9 b Fo(\))1517 4737 y Fw(\()p Fs(\025)p Fw(\()p Fs(t)1659 4749 y Fo(1)1716 4737 y Fw(+)18 b Fs(T)12 b Fw(\)\))1924 4707 y Fu(n)1965 4715 y Fi(0)p Fy(;)p Fi(1)p 1517 4774 532 4 v 1701 4850 a Fs(n)1751 4862 y Fo(0)p Fu(;)p Fo(1)1841 4850 y Fw(!)2058 4794 y Ft(Q)2126 4806 y Fo(1)2163 4627 y Fk(1)2163 4776 y(A)2254 4794 y Ft(\012)2337 4627 y Fk(0)2337 4776 y(@)2453 4715 y(X)2410 4893 y Fu(n)2451 4901 y Fi(0)p Fy(;)p Fi(2)2531 4893 y Fm(\025)p Fo(0)2630 4794 y Fs(e)2669 4759 y Fm(\000)p Fu(\025)p Fo(\()p Fu(t)2811 4767 y Fi(1)2843 4759 y Fo(+)p Fu(T)d Fo(\))2982 4737 y Fw(\()p Fs(\025)p Fw(\()p Fs(t)3124 4749 y Fo(1)3181 4737 y Fw(+)18 b Fs(T)12 b Fw(\))3357 4707 y Fu(n)3398 4715 y Fi(0)p Fy(;)p Fi(2)p 2982 4774 499 4 v 3150 4850 a Fs(n)3200 4862 y Fo(0)p Fu(;)p Fo(2)3290 4850 y Fw(!)3491 4794 y Ft(R)3561 4749 y Fu(n)3602 4757 y Fi(0)p Fy(;)p Fi(2)3561 4816 y Fo(2)3686 4627 y Fk(1)3686 4776 y(A)291 5171 y Fw(=)439 5058 y Fk(Z)522 5078 y Fu(T)485 5246 y Fo(0)588 5171 y Fs(\025dt)709 5183 y Fo(1)761 5058 y Fk(Z)844 5078 y Fu(T)807 5246 y(t)832 5254 y Fi(1)910 5171 y Fs(\025dt)1031 5183 y Fo(2)1101 5078 y Fk(\020)1151 5171 y Fw(1)18 b Ft(\000)g Fs(e)1333 5136 y Fm(\000)p Fu(\025)p Fm(j)p Fu(t)1469 5144 y Fi(2)1501 5136 y Fm(j)1525 5078 y Fk(\021)13 b(\020)1638 5171 y Fw(1)18 b Ft(\000)g Fs(e)1820 5136 y Fm(\000)p Fu(\025)p Fo(\()p Fu(t)1962 5144 y Fi(2)1994 5136 y Fm(\000)p Fu(t)2071 5144 y Fi(1)2103 5136 y Fo(\))2133 5078 y Fk(\021)c(\020)2246 5171 y Fw(1)k Ft(\000)g Fs(e)2428 5136 y Fm(\000)p Fu(\025)p Fo(\()p Fu(t)2570 5144 y Fi(1)2603 5136 y Fm(\000)p Fu(T)9 b Fo(\))2733 5078 y Fk(\021)457 5433 y Ft(\002R)592 5445 y Fo(1)630 5433 y Fw(\()p Ft(j)p Fs(t)715 5445 y Fo(2)752 5433 y Ft(j)p Fw(\))19 b Ft(\012)f(Q)977 5445 y Fo(1)1032 5433 y Ft(\016)g(B)1147 5445 y Fo(1)1203 5433 y Ft(\012)g Fw(id)1355 5445 y Fo(2)1410 5433 y Ft(\016)g(Q)1538 5445 y Fo(1)1594 5433 y Ft(\012)g(R)1747 5445 y Fo(2)1785 5433 y Fw(\()p Fs(t)1847 5445 y Fo(2)1903 5433 y Ft(\000)g Fs(t)2016 5445 y Fo(1)2053 5433 y Fw(\))h Ft(\016)e Fw(id)2233 5445 y Fo(1)2288 5433 y Ft(\012)h(B)2426 5445 y Fo(2)2482 5433 y Ft(\016)g(Q)2610 5445 y Fo(1)2665 5433 y Ft(\012)g(R)2818 5445 y Fo(2)2856 5433 y Fw(\()p Fs(t)2918 5445 y Fo(1)2974 5433 y Fw(+)g Fs(T)12 b Fw(\))1966 5682 y(32)p eop %%Page: 33 33 33 32 bop 118 354 a Fw(Note)32 b(that)g(the)h(op)r(erator)d(`)p Ft(\012)p Fw(')h(here)h(has)f(higher)h(precedence)f(than)h(`)p Ft(\016)p Fw(',)h(so,)f(for)f(example,)i Fs(A)3175 366 y Fo(1)3234 354 y Ft(\012)21 b Fs(A)3382 366 y Fo(2)3441 354 y Ft(\016)g Fs(A)3566 366 y Fo(3)3624 354 y Ft(\012)g Fs(A)3772 366 y Fo(4)3842 354 y Fw(is)118 454 y(understo)r(o)r(d)27 b(as)g(\()p Fs(A)748 466 y Fo(1)805 454 y Ft(\012)18 b Fs(A)950 466 y Fo(2)987 454 y Fw(\))h Ft(\016)f Fw(\()p Fs(A)1192 466 y Fo(3)1248 454 y Ft(\012)g Fs(A)1393 466 y Fo(4)1431 454 y Fw(\).)118 699 y(After)34 b(these)f(in)n(tro)r (ductory)f(examples)h(w)n(e)g(establish)g(a)g(diagrammatic)e(represen)n (tation)h(of)h(op)r(erators)e Fs(\031)3562 711 y Fo(\003)3607 719 y Fi(1)3667 699 y Ft(\016)22 b(L)3788 669 y Fu(T)3788 722 y(S;)p Fo(\003)118 811 y Fw(and)28 b Fs(\031)327 823 y Fo(\003)372 831 y Fi(1)427 811 y Ft(\016)18 b(L)544 781 y Fu(T)544 834 y(S)597 811 y Fw(.)37 b(W)-7 b(e)28 b(b)r(egin)f(with)i(some)e(formal)f(de\014nitions)i(for)f(the)h (diagrams.)118 985 y Fn(De\014nition)j(3.4)40 b Fw(In)35 b(a)e(lab)r(elled)h(tree)g(\(as)f(de\014ned)i(in)f([13)o(]\))g(w)n(e)g (call)f(the)i(v)n(ertices)e(that)h(ha)n(v)n(e)f(no)g(maximal)h(lab)r (el)118 1085 y(\(i.e.)c(they)f(are)f(not)h(lea)n(v)n(es)e(of)i(the)g (tree)g(and)g(there)f(is)h(a)g(v)n(ertex)f(with)h(a)g(greater)e(\(wrt)i (to)g(the)g(partial)f(order\))g(lab)r(el\))118 1184 y(and)k(that)f(are) g(di\013eren)n(t)h(from)f(the)h(ro)r(ot)f(v)n(ertex,)g Fn(branc)m(hings)p Fw(.)50 b(\(Eac)n(h)30 b(branc)n(hing)h(corresp)r (onds)e(to)j(a)f(particular)118 1284 y Fj(star-gr)l(aph)f Fw(\(lik)n(e)23 b(w)n(e)f(used)h(to)g(de\014ne)g(the)g(term)g Fj(lab)l(el)t(le)l(d)k(tr)l(e)l(e)i Fw(in)23 b([13)o(].\))35 b(It)24 b(is)e(called)h Fs(k)s Fn(-branc)m(hing)g Fw(if)g(it)h(corresp) r(onds)118 1384 y(to)g(a)f(star-graph)e(with)j(exactly)g Fs(v)s Fw(\()p Fs(k)s Fw(\))g(v)n(ertices)f(\(and)g(hence)h(to)g(a)f Fs(k)s Fw(-triangle\).)34 b(The)24 b(n)n(um)n(b)r(er)g Fs(k)i Fw(is)e(called)f(the)h Fn(degree)118 1483 y(of)32 b(the)g(branc)m(hing)p Fw(.)118 1583 y(Recall)d(that)h(in)g([13)o(])g (w)n(e)f(ha)n(v)n(e)g(in)n(tro)r(duced)g(a)g(linear)g(order)f Ft(\036)i Fw(on)f(the)h(set)g(of)f(lab)r(els)h(of)f(v)n(ertices.)42 b(In)30 b(the)g(follo)n(wing)118 1683 y(w)n(e)24 b(will)g(use)g(the)h (same)f(sym)n(b)r(ol)f(for)h(linear)g(orders)e(on)i(di\013eren)n(t)g (sets)g(as)g(w)n(ell.)35 b(It)25 b(should)f(b)r(e)g(clear)g(from)g(the) g(con)n(text)118 1782 y(to)k(whic)n(h)f(order)f(w)n(e)i(refer.)118 1882 y(W)-7 b(e)24 b(sa)n(y)f(that)h(a)f(linear)g(order)g Ft(\036)g Fw(on)h(the)g(set)g(of)g(branc)n(hings)e(is)i Fn(compatible)h(with)i(the)h(lab)s(elling)21 b Fw(if)k(the)f(follo)n (wing)118 1981 y(condition)h(is)g(satis\014ed:)35 b(If)26 b Fn(v)1042 1993 y Fl(1)1109 1981 y Fw(and)f Fn(v)1318 1993 y Fl(2)1385 1981 y Fw(are)f(branc)n(hings,)g(lab)r(elled)h(b)n(y)g (lab)r(el\()p Fn(v)2632 1993 y Fl(1)2675 1981 y Fw(\))g(and)g(lab)r (el\()p Fn(v)3146 1993 y Fl(2)3188 1981 y Fw(\),)h(resp)r(ectiv)n(ely) -7 b(,)25 b(then)1316 2181 y(lab)r(el\()p Fn(v)1571 2193 y Fl(1)1614 2181 y Fw(\))e Ft(\037)g Fw(lab)r(el\()p Fn(v)2012 2193 y Fl(2)2054 2181 y Fw(\))106 b Ft(\))g Fn(v)2431 2193 y Fl(1)2497 2181 y Ft(\036)22 b Fn(v)2634 2193 y Fl(2)2676 2181 y Fs(:)1010 b Fw(\(184\))118 2355 y(W)-7 b(e)28 b(in)n(tro)r(duce)f(a)g(linear)g(order)g(\(also)f (denoted)i(b)n(y)f Ft(\036)p Fw(\))h(on)f Fv(Z)2077 2325 y Fu(d)2109 2355 y Fw(:)612 2554 y(\()p Fs(k)687 2566 y Fo(1)725 2554 y Fs(;)14 b(:)g(:)g(:)f(;)h(k)952 2566 y Fu(d)991 2554 y Fw(\))23 b Ft(\036)g Fw(\()1168 2532 y(~)1166 2554 y Fs(k)1209 2566 y Fo(1)1247 2554 y Fs(;)14 b(:)g(:)g(:)f(;)1433 2532 y Fw(~)1431 2554 y Fs(k)1474 2566 y Fu(d)1513 2554 y Fw(\))84 b(if)28 b Fs(k)1748 2566 y Fu(i)1799 2554 y Fs(<)1888 2532 y Fw(~)1886 2554 y Fs(k)1929 2566 y Fu(i)1985 2554 y Fw(for)f(the)h(lo)n(w)n(est)e (index)i Fs(i)f Fw(suc)n(h)h(that)f Fs(k)3194 2566 y Fu(i)3245 2554 y Ft(6)p Fw(=)3335 2532 y(~)3333 2554 y Fs(k)3376 2566 y Fu(i)3709 2554 y Fw(\(185\))118 2728 y Fn(De\014nition)k(3.5)40 b Fw(A)29 b Fn(gum)h(tree)e Fs(\034)1258 2740 y Fu(g)1325 2728 y Fw(with)g(parameters)e Fs(n)1994 2740 y Fu(\014)2062 2728 y Fw(=)d(\()p Fs(n)2232 2740 y Fu(\014)s(;)p Fo(1)2330 2728 y Fs(;)14 b(n)2417 2740 y Fu(\014)s(;)p Fo(2)2514 2728 y Fs(;)g(:)g(:)g(:)p Fw(\))28 b(and)g(\003)2928 2740 y Fo(2)2988 2728 y Ft(2)c(F)j(n)18 b(f;g)26 b Fw(is)i(giv)n(en)f(b)n(y)g(the)118 2828 y(follo)n(wing)g (data:)220 3002 y(1.)41 b(A)28 b(lab)r(elled)f(tree)h Fs(\034)37 b Fw(with)28 b(parameters)e Fs(n)1631 3014 y Fu(\014)1703 3002 y Fw(and)i Ft(j)p Fw(\003)1946 3014 y Fo(2)1983 3002 y Ft(j)f Fw(\(as)h(de\014ned)g(b)r(efore)f(Lemma)g (8.2)g(of)g([13]\).)220 3164 y(2.)41 b(A)20 b(map)h Fn(pin)f Fw(from)g(the)h(set)f(of)h(v)n(ertices)e(\(except)i(the)g(ro)r(ot\))f (of)g Fs(\034)30 b Fw(to)21 b Fv(Z)2501 3134 y Fu(d)2554 3164 y Fw(that)g(satis\014es)f(the)h(follo)n(wing)e(conditions:)425 3342 y Ft(\017)41 b Fw(The)25 b(restriction)e(of)i(pin)g(to)f(the)h (set)g(of)f(v)n(ertices,)h(that)f(are)g(lab)r(elled)h(b)n(y)f(\(0)p Fs(;)14 b Fw(1\))p Fs(;)g(:)g(:)g(:)f(;)h Fw(\(0)p Fs(;)g Ft(j)p Fw(\003)3369 3354 y Fo(2)3406 3342 y Ft(j)p Fw(\))25 b(\(W)-7 b(e)25 b(denote)508 3441 y(the)h(restriction)f(of)g(pin)h(to)g (this)g(set)f(b)n(y)h(pin)1894 3462 y Fo(0)1931 3441 y Fw(.\),)h(is)e(an)h(order-preserving)c(bijection)k(on)n(to)f(\003) 3426 3453 y Fo(2)3462 3441 y Fw(,)i(i.e.)e(for)g(an)n(y)508 3541 y(t)n(w)n(o)i(suc)n(h)g(v)n(ertices)g Fn(v)i Fw(and)1399 3540 y(~)1394 3541 y Fn(v)1412 3755 y Fw(lab)r(el\()p Fn(v)q Fw(\))c Ft(\036)d Fw(lab)r(el\()2022 3754 y(~)2017 3755 y Fn(v)r Fw(\))107 b Ft(\))f Fw(pin\()p Fn(v)q Fw(\))24 b Ft(\036)f Fw(pin\()2891 3754 y(~)2886 3755 y Fn(v)r Fw(\))p Fs(:)716 b Fw(\(186\))425 3914 y Ft(\017)41 b Fw(If)24 b Fn(v)h Fw(with)f(lab)r(el\()p Fn(v)q Fw(\))g(=)f Fs(s)g Fw(=)g(\()p Fs(s)1468 3926 y Fo(1)1505 3914 y Fs(;)14 b(:)g(:)g(:)g(;)g(s)1729 3926 y Fu(m)1791 3914 y Fw(\))24 b(is)f(a)g Fs(k)s Fw(-branc)n(hing)f(and)h(pin\()p Fn(v)q Fw(\))i(=)d Fs(q)k Ft(2)e Fv(Z)3148 3884 y Fu(d)3204 3914 y Fw(then)g(the)g(restriction)508 4014 y(of)i(pin)h(to)f(the)h (set)f(of)g(v)n(ertices)f(with)i(lab)r(els)f(\()p Fs(s;)14 b Fw(1\))p Fs(;)g(:)g(:)g(:)f(;)h Fw(\()p Fs(s;)g(v)s Fw(\()p Fs(k)s Fw(\)\))27 b(\(W)-7 b(e)27 b(denote)f(the)h(restriction) e(of)h(pin)h(to)508 4113 y(this)h(set)g(b)n(y)f(pin)1030 4134 y Fl(v)1075 4113 y Fw(.\))38 b(is)27 b(an)g(order-preserving)d (bijection)k(on)n(to)f Fs(B)2572 4125 y Fu(k)2613 4113 y Fw(\()p Fs(q)s Fw(\))d Ft(\032)e Fv(Z)2890 4083 y Fu(d)2922 4113 y Fw(.)118 4362 y(A)28 b Fn(gum)i(con\014guration)i Ft(C)1043 4374 y Fu(g)1113 4362 y Fn(on)g(\003)g(ending)f(in)g(\003) 1832 4374 y Fl(1)1901 4362 y Fw(is)d(giv)n(en)f(b)n(y)g(the)h(follo)n (wing)e(data:)220 4536 y(1.)41 b(A)19 b(gum)g(tree)f Fs(\034)776 4548 y Fu(g)834 4536 y Fw(with)i(parameters)d Fs(n)1486 4548 y Fu(\014)1549 4536 y Fw(and)i(\003)1760 4548 y Fo(2)1816 4536 y Fw(suc)n(h)g(that)g(\003)2224 4548 y Fo(2)2284 4536 y Ft(\022)j Fw(\003)2429 4548 y Fo(1)2466 4536 y Fw(.)34 b(The)19 b(corresp)r(onding)e(tree)i(has)f (branc)n(hings)326 4635 y Fn(v)376 4647 y Fl(1)441 4635 y Ft(\036)23 b Fs(:)14 b(:)g(:)22 b Ft(\036)h Fn(v)786 4647 y Fl(n)832 4635 y Fw(,)28 b(sa)n(y)-7 b(,)27 b(with)h(branc)n (hing-degrees)c Fs(b)1944 4647 y Fo(1)1981 4635 y Fs(;)14 b(:)g(:)g(:)g(;)g(b)2202 4647 y Fu(n)2247 4635 y Fw(,)27 b(resp)r(ectiv)n(ely)-7 b(.)220 4798 y(2.)41 b(F)-7 b(or)27 b(eac)n(h)f(1)d Ft(\024)g Fs(i)f Ft(\024)h Fs(n)k Fw(there)h(are)e (maps)1506 5103 y Fs(u)1554 5115 y Fu(i)1664 5103 y Fw(:)83 b Fs(B)1833 5115 y Fu(b)1862 5123 y Fy(i)1893 5103 y Fw(\(pin\()p Fn(v)2122 5115 y Fl(i)2148 5103 y Fw(\)\))19 b Ft(\\)g Fw(\003)k Ft(!)g(f)p Fw(0)p Fs(;)14 b Fw(1)p Ft(g)1012 b Fw(\(187\))1510 5228 y Fs(d)1553 5240 y Fu(i)1664 5228 y Fw(:)83 b Fs(B)1833 5240 y Fu(b)1862 5248 y Fy(i)1893 5228 y Fw(\(pin\()p Fn(v)2122 5240 y Fl(i)2148 5228 y Fw(\)\))19 b Ft(\\)g Fw(\003)k Ft(!)g(f)p Fw(0)p Fs(;)14 b Fw(1)p Ft(g)p Fs(:)989 b Fw(\(188\))326 5433 y(suc)n(h)27 b(that)1966 5682 y(33)p eop %%Page: 34 34 34 33 bop 361 354 a Fw(\(a\))41 b(If)27 b Fs(q)f Ft(2)e Fs(B)795 366 y Fu(b)824 374 y Fy(i)854 354 y Fw(\(pin)q(\()p Fn(v)1084 366 y Fl(i)1110 354 y Fw(\)\))16 b Ft(\\)g Fw(\003)27 b(and)f Fs(j)31 b Fw(is)26 b(the)h(smallest)f(n)n(um)n(b)r (er)g(greater)f(than)h Fs(i)g Fw(suc)n(h)g(that)h Fs(q)f Ft(2)d Fs(B)3511 366 y Fu(b)3540 374 y Fy(j)3575 354 y Fw(\(pin)q(\()p Fn(v)3805 366 y Fl(j)3833 354 y Fw(\)\))508 454 y(\(if)29 b(suc)n(h)e(a)g Fs(j)33 b Fw(exists)27 b(at)g(all\))h(then)g Fs(d)1650 466 y Fu(i)1678 454 y Fw(\()p Fs(q)s Fw(\))c(=)e Fs(u)1941 466 y Fu(j)1976 454 y Fw(\()p Fs(q)s Fw(\).)356 583 y(\(b\))42 b(F)-7 b(or)27 b(ev)n(ery)g(1)22 b Ft(\024)h Fs(i)f Ft(\024)h Fs(n)1920 797 y(d)1963 809 y Fu(i)1991 797 y Fw(\(pin\()p Fn(v)2220 809 y Fl(i)2246 797 y Fw(\)\))h(=)f(1)p Fs(:)1222 b Fw(\(189\))365 956 y(\(c\))42 b(If)28 b Fs(q)e Ft(2)e Fs(B)796 968 y Fu(b)825 976 y Fy(i)855 956 y Fw(\(pin)q(\()p Fn(v)1085 968 y Fl(i)1111 956 y Fw(\)\))19 b Ft(\\)g Fw(\(\003)f Ft(n)g Fw(\003)1494 968 y Fo(1)1531 956 y Fw(\))28 b(and)f(there)h(is)f(no)h Fs(j)g(>)22 b(i)28 b Fw(suc)n(h)f(that)h Fs(q)e Ft(2)d Fs(B)2941 968 y Fu(b)2970 976 y Fy(j)3006 956 y Fw(\()p Fs(q)3075 968 y Fu(j)3110 956 y Fw(\))28 b(then)g Fs(d)3402 968 y Fu(i)3430 956 y Fw(\()p Fs(q)3499 968 y Fu(i)3527 956 y Fw(\))23 b(=)g(0.)326 1134 y(\(W)-7 b(e)37 b(will)g(see)f(later)g(that)h(the)g(maps)f Fs(u)1637 1146 y Fu(i)1701 1134 y Fw(de\014ne)g(from)h(a)f(v)n(ertex)f (up)n(w)n(ards)g(going)h Fs(h)p Fj(-strips)43 b Fw(\(if)37 b Fs(u)3545 1146 y Fu(i)3611 1134 y Fw(=)g(0\))g(or)326 1233 y Fs(r)r Fj(-strips)f Fw(\(if)29 b Fs(u)786 1245 y 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b(\(W)-7 b(e)30 b(note)326 3332 y(that)24 b(w)n(e)f(do)g(not)h(distinguish)f (the)h(order)e(of)i(the)g(v)n(ertices)e(in)i(this)g(notation:)34 b(A)24 b(strip)g(from)f Fn(v)i Fw(to)h Fn(~)-50 b(v)25 b Fw(is)e(the)h(same)326 3431 y(as)j(a)g(strip)g(from)i Fn(~)-50 b(v)30 b Fw(to)d Fn(v)q Fw(.\))243 3594 y Ft(\017)41 b Fw(If)31 b Fs(q)h Ft(2)d Fs(B)628 3606 y Fu(b)657 3614 y Fy(i)687 3594 y Fw(\(pin)q(\()p Fn(v)917 3606 y Fl(i)943 3594 y Fw(\)\))21 b Ft(\\)g Fw(\003)31 b(and)g Fn(v)f Fw(=)e(pin)1646 3559 y Fm(\000)p Fo(1)1646 3614 y Fl(v)1686 3623 y Fc(i)1735 3594 y Fw(\()p Fs(q)s Fw(\))k(and)f(there)g(is)f(no)h Fs(j)j(>)28 b(i)i Fw(suc)n(h)h(that)g Fs(q)h Ft(2)d Fs(B)3267 3606 y Fu(b)3296 3614 y Fy(j)3331 3594 y Fw(\(pin)q(\()p Fn(v)3561 3606 y Fl(j)3589 3594 y Fw(\)\))i(and)g(if)326 3694 y Fs(d)369 3706 y Fu(i)397 3694 y Fw(\()p Fs(q)s Fw(\))23 b(=)g(0)k(\()p Fs(d)756 3706 y Fu(i)784 3694 y Fw(\()p Fs(q)s Fw(\))d(=)e(1\))28 b(w)n(e)f(sa)n(y)g(that)g Ft(C)1590 3706 y Fu(g)1656 3694 y Fw(has)h(an)f Fs(h)p Fn(-strip)k(\()p Fs(r)r Fn(-strip\))h(from)f(v)j(to)d(the)h(b)s(ottom)p Fw(.)243 3856 y Ft(\017)41 b Fw(If)29 b Fs(q)f Ft(2)d Fs(B)618 3868 y Fu(b)647 3876 y Fy(i)677 3856 y Fw(\(pin)q(\()p Fn(v)907 3868 y Fl(i)933 3856 y Fw(\)\))20 b Ft(\\)f Fw(\003)28 b(and)h Fn(v)d Fw(=)f(pin)1621 3821 y Fm(\000)p Fo(1)1621 3877 y Fl(v)1661 3886 y Fc(i)1710 3856 y Fw(\()p Fs(q)s Fw(\))k(and)g(there)f(is)h(no)f(n)n(um)n(b)r(er)h Fs(j)g(<)c(i)j Fw(suc)n(h)g(that)h Fs(q)f Ft(2)d Fs(B)3511 3868 y Fu(b)3540 3876 y Fy(j)3575 3856 y Fw(\(pin)q(\()p Fn(v)3805 3868 y Fl(j)3833 3856 y Fw(\)\))326 3956 y(and)i(if)h Fs(u)611 3968 y Fu(i)638 3956 y Fw(\()p Fs(q)s Fw(\))c(=)f(0)k(\()p Fs(u)1003 3968 y Fu(i)1030 3956 y Fw(\()p Fs(q)s Fw(\))d(=)f(1\))k(w)n (e)g(sa)n(y)g(that)h Ft(C)1837 3968 y Fu(g)1903 3956 y Fw(has)f(an)g Fs(h)p Fn(-strip)k(\()p Fs(r)r Fn(-strip\))i(from)d(v)k (to)d(the)h(top)p Fw(.)243 4118 y Ft(\017)41 b Fw(In)27 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y Fu(g)2604 4543 y Fn(reac)m(hes)33 b(the)e(top)p Fw(.)118 4717 y(W)-7 b(e)28 b(denote)g(b)n(y)f Fn(Conf)843 4729 y(g)895 4717 y Fw(\()p Fn(\003)p Fs(;)14 b Fn(\003)1098 4729 y Fl(1)1139 4717 y Fw(\))28 b(the)g(set)g(of)g(all)f(gum)g (con\014gurations)f(o)n(v)n(er)g(\003)h(ending)h(in)g(\003)3088 4729 y Fo(1)3124 4717 y Fw(.)118 4890 y Fn(De\014nition)j(3.6)40 b Fw(Let)32 b Ft(C)924 4902 y Fu(g)994 4890 y Fw(b)r(e)g(a)g(gum)f (con\014guration)f(o)n(v)n(er)g(\003)h(ending)h(in)g(\003)2582 4902 y Fo(1)2650 4890 y Fw(with)g(branc)n(hings)f Fn(v)3316 4902 y Fl(1)3387 4890 y Ft(\036)f Fs(:)14 b(:)g(:)29 b Ft(\036)h Fn(v)3753 4902 y Fl(n)3831 4890 y Fw(of)118 4990 y(branc)n(hing-orders)24 b Fs(b)789 5002 y Fo(1)826 4990 y Fs(;)14 b(:)g(:)g(:)g(;)g(b)1047 5002 y Fu(n)1092 4990 y Fw(,)27 b(resp)r(ectiv)n(ely)-7 b(,)27 b(and)h(let)g Fs(T)34 b Ft(2)23 b Fw(\(0)p Fs(;)14 b Ft(1)p Fw(].)37 b(Then)28 b(w)n(e)f(de\014ne)966 5189 y(Simplex\()p Ft(C)1330 5201 y Fu(g)1368 5189 y Fs(;)14 b(T)e Fw(\))1521 5142 y Fo(def)1534 5189 y Fw(=)36 b Ft(f)p Fw(\()p Fs(t)1739 5201 y Fo(1)1776 5189 y Fs(;)14 b(:)g(:)g(:)g(;)g(t)1991 5201 y Fu(n)2036 5189 y Fw(\))23 b(:)g Ft(\000)p Fs(T)34 b(<)23 b(t)2403 5201 y Fo(1)2463 5189 y Fs(<)g(:)14 b(:)g(:)23 b(<)f(t)2788 5201 y Fu(n)2856 5189 y Fs(<)h Fw(0)p Ft(g)p Fs(:)658 b Fw(\(194\))118 5333 y(Simplex\()p Ft(C)482 5345 y Fu(g)521 5333 y Fs(;)14 b(T)e Fw(\))27 b(is)g(an)h(op)r(en)f (subset)h(of)f Fv(R)1480 5303 y Fu(n)1559 5333 y Fw(and)h(so)f(carries) f(the)h(induced)i(Leb)r(esgue-measure.)118 5433 y(F)-7 b(or)27 b(the)h(sp)r(ecial)f(case)g Fs(n)906 5445 y Fu(\014)951 5433 y Fw(\()p Ft(C)1027 5445 y Fu(g)1065 5433 y Fw(\))d(=)e(0)27 b(w)n(e)h(de\014ne)g(Simplex\()p Ft(C)2004 5445 y Fu(g)2042 5433 y Fs(;)14 b(T)e Fw(\))27 b(to)h(b)r(e)g(a)f(single)g(p)r(oin)n(t)h (ha)n(ving)e(measure)h(1.)1966 5682 y(34)p eop %%Page: 35 35 35 34 bop 118 354 a Fn(De\014nition)31 b(3.7)40 b Fw(F)-7 b(or)29 b Ft(C)922 366 y Fu(g)985 354 y Ft(2)d Fw(Conf)1239 366 y(g)1285 354 y(\(\003)p Fs(;)14 b Fw(\003)1470 366 y Fo(1)1506 354 y Fw(\),)30 b Fs(T)36 b Ft(2)26 b Fw(\(0)p Fs(;)14 b Ft(1)p Fw(])29 b(and)f Fn(t)e Ft(2)f Fw(Simplex)q(\()p Ft(C)2673 366 y Fu(g)2711 354 y Fs(;)14 b(T)e Fw(\))29 b(w)n(e)f(call)h(the)g(triple)g(\()p Ft(C)3590 366 y Fu(g)3628 354 y Fs(;)14 b(T)7 b(;)14 b Fn(t)p Fw(\))29 b(a)118 454 y Fn(sp)s(eci\014c)i(gum)g(con\014guration)p Fw(.)118 554 y(Sp)r(eci\014c)k(gum)f(con\014gurations)f(can)h(b)r(e)h (view)n(ed)f(graphically:)48 b(The)35 b(v)n(ertices)e(are)h(placed)g (in)g Fv(Z)3252 523 y Fu(d)3307 554 y Ft(\002)23 b Fw([0)p Fs(;)14 b(T)e Fw(])33 b(and)h(the)118 653 y(strips)27 b(are)g(`spanned')g(b)r(et)n(w)n(een)h(v)n(ertices,)e(the)i(top)g(\()p Fs(t)23 b Fw(=)g Ft(\000)p Fs(T)12 b Fw(\))27 b(and)g(the)h(b)r(ottom)g (\()p Fs(t)23 b Fw(=)g(0\):)243 836 y Ft(\017)41 b Fw(W)-7 b(e)24 b(assign)f(to)h(eac)n(h)f(v)n(ertex)g Fn(v)j Fw(in)e Fs(\034)9 b Fw(\()p Ft(C)1526 848 y Fu(g)1565 836 y Fw(\))25 b(a)e(p)r(oin)n(t)h(in)h Fv(Z)2055 806 y Fu(d)2099 836 y Ft(\002)11 b Fw([)p Ft(\000)p Fs(T)c(;)14 b Fw(0])22 b(in)j(the)f(follo)n(wing)f(w)n(a)n(y)-7 b(.)34 b(If)25 b Fn(v)3345 848 y Fl(i)3394 836 y Fw(is)f(a)g(branc)n(hing)326 935 y(of)i(degree)g Fs(b)713 947 y Fu(i)740 935 y Fw(,)i Fs(q)e Ft(2)d Fs(B)995 947 y Fu(b)1024 955 y Fy(i)1055 935 y Fw(\(pin\()p Fn(v)1284 947 y Fl(i)1310 935 y Fw(\)\))28 b(and)e Fn(v)f Fw(=)e(pin)1840 901 y Fm(\000)p Fo(1)1840 956 y Fl(v)1880 965 y Fc(i)1929 935 y Fw(\()p Fs(q)s Fw(\))28 b(then)f Fn(v)i Fw(has)d(time-co)r(ordinate)g Fs(t)3102 947 y Fu(i)3129 935 y Fw(.)37 b(In)27 b(particular)f Fn(v)3725 947 y Fl(i)3777 935 y Fw(has)326 1035 y(time-co)r(ordinate)g Fs(t)953 1047 y Fu(i)981 1035 y Fw(.)37 b(As)27 b(further)h(pin\()p Fn(v)q Fw(\))d(=)d Fs(q)31 b Fw(w)n(e)c(assign)g Fn(v)i Fw(to)e(\()p Fs(q)s(;)14 b(t)2540 1047 y Fu(i)2568 1035 y Fw(\).)326 1168 y(Let)27 b(for)g(the)h(follo)n(wing)f(t)n(w)n(o)g(v)n (ertices)f Fn(v)k Fw(and)f Fn(~)-50 b(v)30 b Fw(b)r(e)e(assigned)e(to)h (\()p Fs(q)s(;)14 b(t)p Fw(\))29 b(and)e(\()p Fs(q)s(;)2886 1153 y Fw(~)2885 1168 y Fs(t)p Fw(\),)h(resp)r(ectiv)n(ely)-7 b(.)243 1334 y Ft(\017)41 b Fw(If)27 b Ft(C)452 1346 y Fu(g)517 1334 y Fw(has)f(an)g Fs(h)p Fw(-strip)g(\()p Fs(r)r Fw(-strip\))i(from)e Fn(v)j Fw(to)f Fn(~)-50 b(v)29 b Fw(w)n(e)d(sa)n(y)f(that)i(\()p Ft(C)2344 1346 y Fu(g)2383 1334 y Fs(;)14 b(T)7 b(;)14 b Fn(t)p Fw(\))26 b(has)h(a)f Fn(maximal)i Fs(h)p Fn(-strip)e Fw(\()p Fn(maximal)326 1434 y Fs(r)r Fn(-strip)p Fw(\))i(from)f(\()p Fs(q)s(;)14 b(t)p Fw(\))28 b(to)g(\()p Fs(q)s(;)1258 1418 y Fw(~)1257 1434 y Fs(t)p Fw(\).)37 b(W)-7 b(e)28 b(de\014ne)g(its)g Fn(length)f Fw(to)g(b)r(e)h Ft(j)p Fs(t)19 b Ft(\000)2537 1418 y Fw(~)2536 1434 y Fs(t)p Ft(j)p Fw(.)243 1600 y Ft(\017)41 b Fw(If)20 b Ft(C)445 1612 y Fu(g)504 1600 y Fw(has)g(an)g Fs(h)p Fw(-strip)g(\()p Fs(r)r Fw(-strip\))h(from)f Fn(v)i Fw(to)e(the)h(b)r(ottom)f(\(this)h(has)f(time-co)r(ordinate)f (0.\))35 b(w)n(e)20 b(sa)n(y)f(that)h(\()p Ft(C)3660 1612 y Fu(g)3699 1600 y Fs(;)14 b(T)7 b(;)14 b Fn(t)p Fw(\))326 1699 y(has)27 b(a)g Fn(maximal)i Fs(h)p Fn(-strip)e Fw(\()p Fn(maximal)j Fs(r)r Fn(-strip)p Fw(\))e(from)f(\()p Fs(q)s(;)14 b(t)p Fw(\))28 b(to)g(\()p Fs(q)s(;)14 b Fw(0\).)37 b(Its)27 b(length)h(is)g Ft(j)p Fs(t)p Ft(j)p Fw(.)243 1865 y Ft(\017)41 b Fw(If)24 b Ft(C)449 1877 y Fu(g)511 1865 y Fw(has)g(an)f Fs(h)p Fw(-strip)h(\()p Fs(r)r Fw(-strip\))g(from)g Fn(v)i Fw(to)d(the)i(top)f(\(this)g(has)f (time-co)r(ordinate)g Ft(\000)p Fs(T)12 b Fw(.\))35 b(w)n(e)23 b(sa)n(y)g(that)h(\()p Ft(C)3660 1877 y Fu(g)3699 1865 y Fs(;)14 b(T)7 b(;)14 b Fn(t)p Fw(\))326 1965 y(has)24 b(a)h Fn(maximal)h Fs(h)p Fn(-strip)f Fw(\()p Fn(maximal)i Fs(r)r Fn(-strip)p Fw(\))e(from)g(\()p Fs(q)s(;)14 b(t)p Fw(\))26 b(to)f(\()p Fs(q)s(;)14 b Ft(\000)p Fs(T)e Fw(\).)35 b(Its)26 b(length)f(is)g Fs(T)g Ft(\000)14 b(j)p Fs(t)p Ft(j)p Fw(.)35 b(\(Note)25 b(that)326 2065 y(for)i Fs(T)34 b Fw(=)22 b Ft(1)28 b Fw(this)g(length)g(is)f Ft(1)p Fw(.\))243 2231 y Ft(\017)41 b Fw(If)26 b Ft(C)451 2243 y Fu(g)516 2231 y Fw(has)g(a)g(long)f Fs(h)p Fw(-strip)h(\(long)g Fs(r)r Fw(-strip\))g(at)h Fs(q)i Fw(w)n(e)d(sa)n(y)f(that)i(\()p Ft(C)2366 2243 y Fu(g)2404 2231 y Fs(;)14 b(T)7 b(;)14 b Fn(t)p Fw(\))26 b(has)g(a)g Fn(long)j Fs(h)p Fn(-strip)h(\(long)f Fs(r)r Fn(-strip\))326 2330 y Fw(at)e Fs(q)s Fw(.)37 b(Its)28 b(length)g(is)f Fs(T)12 b Fw(.)36 b(\(Long)27 b Fs(h)p Fw(-strips)g(\(long)g Fs(r)r Fw(-strips\))h(are)e(also)h (considered)f(as)h(maximal)g(strips.\))118 2513 y(If)i(\()p Ft(C)278 2525 y Fu(g)317 2513 y Fs(;)14 b(T)7 b(;)14 b Fn(t)p Fw(\))29 b(has)f(a)h(maximal)f Fs(h)p Fw(-strip)g(\()p Fs(r)r Fw(-strip\))i(from)f(\()p Fs(q)s(;)2012 2498 y Fw(~)2011 2513 y Fs(t)2041 2525 y Fo(1)2078 2513 y Fw(\))h(to)e(\()p Fs(q)s(;)2353 2498 y Fw(~)2351 2513 y Fs(t)2381 2525 y Fo(4)2419 2513 y Fw(\))h(and)2644 2498 y(~)2643 2513 y Fs(t)2673 2525 y Fo(1)2736 2513 y Ft(\024)2827 2498 y Fw(~)2825 2513 y Fs(t)2855 2525 y Fo(2)2918 2513 y Fs(<)3009 2498 y Fw(~)3008 2513 y Fs(t)3038 2525 y Fo(3)3100 2513 y Ft(\024)3191 2498 y Fw(~)3190 2513 y Fs(t)3220 2525 y Fo(4)3287 2513 y Fw(then)g(w)n(e)g(sa)n(y)e(that)118 2613 y(\()p Ft(C)194 2625 y Fu(g)233 2613 y Fs(;)14 b(T)7 b(;)14 b Fn(t)p Fw(\))27 b(has)g(an)g Fs(h)p Fw(-strip)g(\()p Fs(r)r Fw(-strip\))i(from)e(\()p Fs(q)s(;)1625 2597 y Fw(~)1624 2613 y Fs(t)1654 2625 y Fo(2)1691 2613 y Fw(\))h(to)g(\()p Fs(q)s(;)1963 2597 y Fw(~)1962 2613 y Fs(t)1992 2625 y Fo(3)2029 2613 y Fw(\))g(\(or)f(from)h(\()p Fs(q)s(;)2530 2597 y Fw(~)2529 2613 y Fs(t)2559 2625 y Fo(3)2596 2613 y Fw(\))g(to)f(\()p Fs(q)s(;)2868 2597 y Fw(~)2866 2613 y Fs(t)2896 2625 y Fo(2)2934 2613 y Fw(\)\).)118 2712 y(F)-7 b(or)37 b(a)h(branc)n(hing)f Fn(v)803 2724 y Fl(i)866 2712 y Fw(and)h(a)g Fs(q)43 b Ft(2)e Fs(B)1357 2724 y Fu(b)1386 2732 y Fy(i)1416 2712 y Fw(\(pin)q(\()p Fn(v)1646 2724 y Fl(i)1672 2712 y Fw(\)\))d(w)n(e)g(call)f(the)i(maximal)e Fs(h)p Fw(-strip)g(\(if)i(an)n(y\))f(from)f(\()p Fs(q)s(;)14 b(t)3516 2724 y Fu(i)3544 2712 y Fw(\))38 b(to)g(\()p Fs(q)s(;)14 b(t)p Fw(\))118 2812 y(with)32 b Fs(t)341 2824 y Fu(i)398 2812 y Fs(<)d(t)i Fw(\()p Fs(t)615 2824 y Fu(i)673 2812 y Fs(>)e(t)p Fw(\))i(a)g Fn(do)m(wn)m(w)m(ards)38 b(going)d(\(up)m(w)m(ards)i(going\))e Fs(h)p Fn(-strip)h(asso)s(ciated) f(to)i(the)f(branc)m(hing)p Fw(.)118 2911 y(\(Note)31 b(that)f(in)g(our)g(pictures)g(the)g(p)r(ositiv)n(ely)g(orien)n(ted)f (time-axis)h(go)r(es)f(do)n(wn)n(w)n(ards.\))42 b(The)30 b(notation)g(for)g Fs(r)r Fw(-strips)118 3011 y(is)e(analogous.)118 3111 y(\()p Ft(C)194 3123 y Fu(g)233 3111 y Fs(;)14 b(T)7 b(;)14 b Fn(t)p Fw(\))31 b(m)n(ust)h(ha)n(v)n(e)f(a)h(do)n(wn)n(w)n (ards)d(going)i Fs(r)r Fw(-strip)h(at)g(the)g(p)r(oin)n(ts)g(\(pin)q (\()p Fn(v)2603 3123 y Fl(i)2629 3111 y Fw(\))p Fs(;)14 b(t)2728 3123 y Fu(i)2755 3111 y Fw(\))33 b(b)r(ecause)e(of)h (condition)g(2.b.)49 b(W)-7 b(e)118 3210 y(call)27 b(it)h(an)g Fn(ap)s(ex-)p Fs(r)r Fn(-strip)p Fw(.)118 3310 y(An)c Fs(h)p Fw(-strip)f(\()p Fs(r)r Fw(-strip\))i(in)f(\()p Ft(C)1008 3322 y Fu(g)1047 3310 y Fs(;)14 b(T)7 b(;)14 b Fn(t)p Fw(\))23 b Fn(go)s(es)j(to)h(the)h(b)s(ottom)d(\(to)i(the)h (top\))c Fw(if)g(the)g(corresp)r(onding)e Fs(h)p Fw(-strip)h(\()p Fs(r)r Fw(-strip\))118 3410 y(in)28 b Ft(C)259 3422 y Fu(g)325 3410 y Fw(go)r(es)f(to)g(the)h(b)r(ottom)g(\(to)g(the)g (top\).)118 3509 y(W)-7 b(e)28 b(de\014ne)1293 3708 y(~)-44 b Fs(c)p Fw(\()p Ft(C)1403 3720 y Fu(g)1441 3708 y Fs(;)14 b(T)7 b(;)14 b Fn(t)p Fw(\))1663 3661 y Fo(def)1676 3708 y Fw(=)1777 3630 y Fk(Y)1801 3808 y Fu(H)1897 3708 y Fw(\()q(1)k Ft(\000)g Fw(exp\()p Fs(\025)c Fw(length\()p Fs(H)7 b Fw(\)\)\))985 b(\(195\))118 3936 y(where)27 b(the)h(pro)r(duct)g(is)f(o)n(v)n(er)f(all)h(maximal)g Fs(h)p Fw(-strips)g Fs(H)34 b Fw(that)28 b(do)g(not)f(end)h(in)g (\(\003)18 b Ft(n)g Fw(\003)2857 3948 y Fo(1)2894 3936 y Fw(\))h Ft(\002)f(f)p Fw(0)p Ft(g)p Fw(.)118 4119 y(W)-7 b(e)31 b(dra)n(w)e(in)h(the)h(sp)r(eci\014c)f(gum)g(con\014guration)f 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(in)h(the)f(lab)r(elled)h(gum)f(tree)h(\(except)f(the)h(ro)r(ot\))f (are)g(assigned)f(to)i(p)r(oin)n(ts)f(in)h Fv(Z)3562 4487 y Fu(d)3628 4517 y Fw(\(in)g(this)118 4617 y(example)27 b Fs(d)d Fw(=)e(1\))28 b(b)n(y)f(the)h(map)f(pin)q(.)37 b(F)-7 b(or)27 b(example)g(pin\()p Fn(v)1988 4629 y Fl(1)2031 4617 y Fw(\))c(=)g Fs(q)2211 4629 y Fo(2)2248 4617 y Fw(.)118 4800 y Fn(De\014nition)31 b(3.8)40 b Fw(W)-7 b(e)27 b(denote)f(b)n(y)g Ft(R)1319 4812 y Fu(q)1356 4800 y Fw(\()p Fs(t)p Fw(\))g(the)h(op)r(erator)d(as)h(de\014ned)i(in)f (De\014nition)h(3.3,)e(acting)h(on)f(the)i Fs(q)s Fw(-th)f(co)r(ordi-) 118 4899 y(nate.)45 b(F)-7 b(or)30 b Ft(C)539 4911 y Fu(g)605 4899 y Ft(2)e Fw(Conf)861 4911 y(g)907 4899 y(\(\003)p Fs(;)14 b Fw(\003)1092 4911 y Fo(1)1128 4899 y Fw(\))31 b(with)g Fs(n)d Ft(\025)f Fw(1)j(branc)n(hings)f(at)h Fn(v)2200 4911 y Fl(1)2270 4899 y Ft(\036)d Fs(:)14 b(:)g(:)27 b Ft(\036)h Fn(v)2629 4911 y Fl(n)2705 4899 y Fw(of)j(degree)e Fs(b)3100 4911 y Fo(1)3137 4899 y Fs(;)14 b(:)g(:)g(:)g(;)g(b)3358 4911 y Fu(n)3402 4899 y Fw(,)31 b(resp)r(ectiv)n(ely)-7 b(,)118 5032 y(w)n(e)27 b(set)h Fs(t)400 5044 y Fo(0)460 4985 y(def)474 5032 y Fw(=)36 b Ft(\000)p Fs(T)12 b Fw(,)26 b Fs(t)780 5044 y Fu(n)p Fo(+1)932 4985 y(def)946 5032 y Fw(=)36 b(0)27 b(and)g(de\014ne:)1966 5682 y(35)p eop %%Page: 36 36 36 35 bop 480 1162 a Ff(u)480 166 y(u)812 1162 y(u)812 166 y(u)1144 1162 y(u)1144 166 y(u)1476 1162 y(u)1476 166 y(u)1809 1162 y(u)1809 166 y(u)p 812 917 665 7 v 1144 913 a(s)p 480 501 V 812 498 a(s)p 478 1162 4 997 v 811 1162 V 1143 1162 V 1475 1162 V 1807 1162 V 477 498 7 333 v 480 498 V 474 498 V 809 498 V 812 498 V 806 498 V 809 913 7 416 v 812 913 V 806 913 V 1473 1162 7 997 v 1476 1162 V 1470 1162 V 1805 1162 V 1809 1162 V 1802 1162 V 1141 1162 7 250 v 1144 1162 V 1138 1162 V 312 1162 4 997 v 281 1164 67 4 v 198 1179 a Fw(0)p 281 915 V 198 930 a Fs(t)228 942 y Fo(2)p 281 500 V 198 515 a Fs(t)228 527 y Fo(1)p 281 168 V 131 183 a Ft(\000)p Fs(T)447 1295 y(q)484 1307 y Fo(1)779 1295 y Fs(q)816 1307 y Fo(2)1111 1295 y Fs(q)1148 1307 y Fo(3)1443 1295 y Fs(q)1480 1307 y Fo(4)1775 1295 y Fs(q)1812 1307 y Fo(5)3174 1245 y Ff(u)3174 913 y(u)3174 581 y(u)3174 249 y(u)2841 581 y(u)-50 b(u)2509 249 y(u)615 b(u)3506 581 y(u)2841 249 y(u)3207 1262 y Fi(\(0\))3207 946 y(\(0)p Fy(;)11 b Fi(1\))3207 598 y(\(0)p Fy(;)g Fi(1)p Fy(;)f Fi(2\))2592 614 y(\(0)p Fy(;)h Fi(1)p Fy(;)g Fi(1\))3539 598 y(\(0)p Fy(;)g Fi(1)p Fy(;)f Fi(3\))2227 266 y(\(0)p Fy(;)h Fi(1)p Fy(;)g Fi(1)p Fy(;)f Fi(1\))735 b(\(0)p Fy(;)11 b Fi(1)p Fy(;)f Fi(1)p Fy(;)h Fi(3\))2725 199 y(\(0)p Fy(;)g Fi(1)p Fy(;)g Fi(1)p Fy(;)f Fi(2\))3041 946 y Fn(v)3091 958 y Fl(2)2875 614 y Fn(v)2925 626 y Fl(1)p 3172 1245 4 665 v 2840 581 4 333 v 3174 913 a Fb(\000)3257 830 y(\000)3340 747 y(\000)3423 664 y(\000)3091 913 y(@)3008 830 y(@)2924 747 y(@)2841 664 y(@)2758 581 y(@)2675 498 y(@)2592 415 y(@)2509 332 y(@)2841 581 y(\000)2924 498 y(\000)3008 415 y(\000)3091 332 y(\000)970 1511 y Fw(Figure)27 b(5:)37 b(Sp)r(eci\014c)28 b(gum)f(con\014guration)f(and)i (its)f(lab)r(elled)h(tree)779 2230 y(Op)890 2250 y Fo(1)927 2230 y Fw(\()p Fs(i;)14 b Ft(C)1069 2242 y Fu(g)1107 2230 y Fs(;)g(T)7 b(;)14 b Fn(t)p Fw(\))1389 2183 y Fo(def)1402 2230 y Fw(=)1670 2151 y Fk(O)1563 2333 y Fu(q)r Fm(2)p Fo(\003)1685 2341 y Fe(Q)1737 2333 y Fo(\()p Fu(i;)p Fm(C)1842 2341 y Fy(g)1877 2333 y Fo(\))1917 2230 y Ft(Q)1985 2242 y Fu(q)2157 2151 y Fk(O)2049 2333 y Fu(q)r Fm(2)p Fo(\003)2171 2341 y Fe(R)2225 2333 y Fo(\()p Fu(i;)p Fm(C)2330 2341 y Fy(g)2365 2333 y Fo(\))2405 2230 y Ft(R)2475 2242 y Fu(q)2512 2230 y Fw(\()p Fs(t)2574 2242 y Fu(i)p Fo(+1)2704 2230 y Ft(\000)k Fs(t)2817 2242 y Fu(i)2845 2230 y Fw(\))p Fs(;)809 b Fw(\(196\))982 2570 y(Op)1093 2590 y Fo(2)1130 2570 y Fw(\()p Fs(i;)14 b(k)s Fw(\))1389 2523 y Fo(def)1402 2570 y Fw(=)96 b Ft(B)1618 2585 y Fo(pin)o(\()p Fl(v)1776 2594 y Fc(i)1798 2585 y Fo(\))p Fu(;k)1978 2491 y Fk(O)1917 2673 y Fo(~)-38 b Fu(q)r Fm(2)p Fo(\003)p Fm(nf)p Fu(q)r 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Fw(is)g(b)r(ounded)1966 5682 y(37)p eop %%Page: 38 38 38 37 bop 118 354 a Fw(b)n(y)37 b(4)285 324 y Fm(j)p Fo(\003)350 332 y Fi(2)382 324 y Fm(j)420 292 y Fk(Q)498 313 y Fm(1)498 379 y Fu(k)q Fo(=1)623 354 y Fw(\(exp\()r(~)-44 b Fs(c)850 366 y Fu(d)889 354 y Fs(k)935 324 y Fu(d)974 354 y Fw(\)\))1038 324 y Fu(n)1079 333 y Fy(\014)r(;k)1174 354 y Fw(.)66 b(F)-7 b(or)37 b(eac)n(h)f Fs(k)s Fw(-branc)n(hing)g(w)n (e)g(get)i(a)e(factor)h Fs(\025)14 b(\017)g Fw(exp\()p Ft(\000)p Fs(c)3063 366 y Fu(g)3101 354 y Fs(k)3147 324 y Fu(d)3185 354 y Fw(\))38 b(from)f(the)h(uniform)118 454 y(estimates)32 b(for)f(the)i(corresp)r(onding)d(op)r(erator.)48 b(Summing)32 b(o)n(v)n(er)e(all)i(p)r(ossible)g(c)n(hoices)e(of)i(up)n (w)n(ards)f(or)g(do)n(wn)n(w)n(ards)118 554 y(going)c Fs(h)p Fw(-strips)g(or)g Fs(r)r Fw(-strips)h(asso)r(ciated)f(with)h (the)h(branc)n(hing,)e(w)n(e)g(get)h(a)g(factor)f(exp\()p Fs(c)2942 566 y Fo(12)3012 554 y Fs(k)3058 523 y Fu(d)3097 554 y Fw(\).)38 b(There)28 b(are)f(not)h(more)118 653 y(than)23 b Ft(j)p Fw(\003)388 665 y Fo(1)425 653 y Ft(j)9 b(\000)g(j)p Fw(\003)612 665 y Fo(2)650 653 y Ft(j)23 b Fw(sites)g(for)f(whic)n(h)h(w)n(e)g(can)g(c)n(ho)r(ose)f(b)r(et)n(w)n (een)h(long)f Fs(h)p Fw(-strips)g(and)h(long)f Fs(r)r Fw(-strips.)36 b(A)23 b(long)g Fs(r)r Fw(-strip)g(giv)n(es)118 753 y(rise)j(to)h(a)f(factor)g Fs(c)712 765 y Fu(r)763 753 y Fw(exp\()p Ft(\000)p Fw(\(1)16 b Ft(\000)h Fs(\021)s Fw(\))p Fs(\025T)12 b Fw(\),)27 b(and)g(a)f(long)g Fs(h)p Fw(-strip)h(to)f(a)h(factor)f(at)g(most)h Fs(c)2849 765 y Fu(h)2892 753 y Fw(.)37 b(The)27 b(norm)f Ft(k)p Fs(\036)3429 768 y Fo(\003\()p Fm(C)3536 776 y Fy(g)3571 768 y Fo(\))3601 753 y Ft(k)3643 768 y Fo(\003\()p Fm(C)3750 776 y Fy(g)3785 768 y Fo(\))3842 753 y Fw(is)118 852 y(estimated)i(b)n(y)1204 1135 y Ft(k)p Fs(\036)1295 1150 y Fo(\003\()p Fm(C)s Fo(\))1435 1135 y Ft(k)1477 1150 y Fo(\003\()p Fm(C)s Fo(\))1700 1135 y Ft(\024)83 b Fs(#)1897 1100 y Fm(\000j)p Fo(\003)2014 1108 y Fy(r)2047 1100 y Fm(j\000)2119 1057 y Fd(P)2188 1075 y Fe(1)2188 1119 y Fy(k)q Fi(=1)2296 1100 y Fo(\(3)p Fu(k)q Fo(\))2417 1075 y Fy(d)2452 1100 y Fu(n)2493 1109 y Fy(\014)r(;k)2588 1135 y Ft(k)p Fs(\036)p Ft(k)2721 1147 y Fu(#)3709 1135 y Fw(\(208\))1700 1335 y Ft(\024)g Fs(#)1897 1300 y Fm(\000j)p Fo(\003)2014 1308 y Fy(r)2047 1300 y Fm(j)2112 1231 y(1)2092 1256 y Fk(Y)2085 1434 y Fu(k)q Fo(=1)2219 1335 y Fs(#)2268 1300 y Fm(\000)p Fo(\(3)p Fu(k)q Fo(\))2441 1275 y Fy(d)2477 1300 y Fu(n)2518 1309 y Fy(\014)r(;k)2612 1335 y Ft(k)p Fs(\036)p Ft(k)2745 1347 y Fu(#)2789 1335 y Fs(:)118 1596 y Fw(Gum)29 b(con\014gurations)e Ft(C)908 1608 y Fu(g)974 1596 y Fw(without)i(branc)n(hings)e(\(i.e.)i Fs(n)1920 1608 y Fu(\014)1965 1596 y Fw(\()p Ft(C)2041 1608 y Fu(g)2079 1596 y Fw(\))c(=)f(0\))k(can)g(only)g(ha)n(v)n(e)f (long)h(r-c)n(hains)f(\(that)i(m)n(ust)f(end)118 1696 y(in)g(\003)273 1708 y Fo(1)310 1696 y Fw(\))g(or)f(long)f(h-c)n (hains.)36 b(This)28 b(case)f(corresp)r(onds)e(to)j(the)g(summand)f (for)h Fs(K)g Fw(=)23 b(0)k(in)h(\(210\).)118 1796 y(W)-7 b(e)32 b(remark)f(that)h(the)g(sum)g(`)1088 1733 y Fk(P)1176 1754 y Fm(1)1176 1820 y Fu(n)p Fo(=0)1305 1796 y Fw(')g(in)g(\(210\))f (also)g(includes)h(the)g(estimate)g(for)f(the)h(sp)r(ecial)g(case)f Fs(K)36 b Fw(=)29 b(0.)50 b(Then)118 1895 y(the)28 b(gum)g (con\014gurations)e(ha)n(v)n(e)g(no)i(branc)n(hings)e(and)i(so)f Fs(n)2021 1907 y Fu(\014)2089 1895 y Fw(=)c(0.)37 b(The)27 b(sum)h(`)2648 1833 y Fk(P)2736 1853 y Fm(1)2736 1920 y Fu(n)p Fo(=0)2865 1895 y Fw(')g(should)g(then)g(b)r(e)g(replaced)f(b) n(y)118 1995 y(a)g(factor)g(1)g(\(to)h(a)n(v)n(oid)e(confusion\).)37 b(Ho)n(w)n(ev)n(er,)25 b(this)j(sum)g(is)g(at)f(least)g(1)g(and)h(so)f (the)h(estimate)f(is)h(correct.)118 2094 y(W)-7 b(e)28 b(estimate)g(the)g(left-hand)f(side)h(\(lhs\))g(of)g(\(203\):)888 2376 y(~)878 2398 y Fs(#)927 2364 y Fm(j)p Fo(\003)992 2372 y Fi(1)1024 2364 y Fm(j)1245 2320 y Fk(X)1062 2501 y Fm(C)1098 2509 y Fy(g)1133 2501 y Fm(2)p Fo(Conf)1315 2509 y(g)1352 2501 y(\(\003)p Fu(;)p Fo(\003)1488 2509 y Fi(1)1521 2501 y Fo(\))1561 2285 y Fk(Z)1607 2474 y Fo(Simplex)n(\()p Fm(C)1899 2482 y Fy(g)1934 2474 y Fu(;T)9 b Fo(\))2046 2398 y Fs(\025)2094 2364 y Fm(j)p Fu(n)2155 2373 y Fy(\014)2194 2364 y Fo(\()p Fm(C)2256 2372 y Fy(g)2291 2364 y Fo(\))p Fm(j)2340 2398 y Fs(d)p Fn(t)24 b Ft(k)p Fs(\031)2533 2410 y Fo(\003)2578 2418 y Fi(1)2633 2398 y Ft(\016)18 b(L)2750 2364 y Fu(T)2750 2419 y Fm(C)2786 2427 y Fy(g)2821 2419 y Fu(;)p Fl(t)2874 2398 y Fs(\036)2923 2410 y Fo(\003)2973 2398 y Ft(k)694 b Fw(\(209\))961 2725 y Ft(\024)1119 2704 y Fw(~)1108 2725 y Fs(#)1157 2691 y Fm(j)p Fo(\003)1222 2699 y Fi(1)1255 2691 y Fm(j)1306 2614 y(j)p Fo(\003)1371 2622 y Fi(1)1403 2614 y Fm(j)1305 2647 y Fk(X)1293 2825 y Fu(K)t Fo(=0)1450 2608 y Fk(\022)1511 2669 y Ft(j)p Fw(\003)1592 2681 y Fo(1)1629 2669 y Ft(j)1544 2782 y Fs(K)1652 2608 y Fk(\023)1714 2725 y Fw(\()p Fs(c)1782 2737 y Fu(h)1843 2725 y Fw(+)18 b Fs(#)1975 2691 y Fm(\000)p Fo(1)2064 2725 y Fs(c)2100 2737 y Fu(r)2151 2725 y Fw(exp\()p Ft(\000)p Fw(\(1)g Ft(\000)g Fs(\021)s Fw(\))p Fs(\025T)12 b Fw(\)\))2799 2691 y Fm(j)p Fo(\003)2864 2699 y Fi(1)2897 2691 y Fm(j\000)p Fu(K)3033 2725 y Fw(4)3075 2691 y Fu(K)3709 2725 y Fw(\(210\))1122 3011 y Ft(\002)1243 2908 y Fm(1)1216 2933 y Fk(X)1201 3111 y Fu(n)p Fo(=)p Fu(K)1366 2869 y Fk( )1432 3011 y Fs(\017\025)1555 2908 y Fm(1)1528 2933 y Fk(X)1528 3111 y Fu(k)q Fo(=1)1662 3011 y Fw(exp\()p Ft(\000)p Fs(c)1922 3023 y Fu(g)1961 3011 y Fs(k)2007 2977 y Fu(d)2045 3011 y Fw(\))33 b(exp\(\()r(~)-44 b Fs(c)2337 3023 y Fu(d)2394 3011 y Fw(+)18 b Fs(c)2513 3023 y Fo(12)2584 3011 y Fw(\))p Fs(k)2662 2977 y Fu(d)2700 3011 y Fw(\))h Fs(c)2787 2977 y Fu(k)2823 2952 y Fy(d)2863 3011 y Fs(#)2912 2977 y Fm(\000)p Fu(ck)3030 2952 y Fy(d)1122 3303 y Ft(\002)1210 3190 y Fk(Z)1293 3210 y Fu(T)1257 3378 y Fo(0)1360 3303 y Fs(dt)14 b(c)1483 3315 y Fu(r)1533 3303 y Fw(exp\()p Ft(\000)p Fw(\(1)k Ft(\000)g Fs(\021)s Fw(\))p Fs(\025T)12 b Fw(\))2149 3161 y Fk(!)2215 3178 y Fu(n)2274 3303 y Ft(k)p Fs(\036)p Ft(k)2407 3315 y Fu(#)961 3608 y Ft(\024)82 b Fs(c)1144 3620 y Fo(4)1209 3466 y Fk( )1285 3586 y Fw(~)1275 3608 y Fs(#c)1360 3620 y Fu(h)1421 3608 y Fw(+)1525 3530 y(~)1514 3552 y Fs(#)p 1514 3589 50 4 v 1514 3665 a(#)1587 3608 y(c)1623 3620 y Fu(r)1687 3608 y Fw(exp\()p Ft(\000)p Fw(\(1)18 b Ft(\000)g Fs(\021)s Fw(\))p Fs(\025T)12 b Fw(\))19 b(+)2416 3586 y(~)2405 3608 y Fs(#\017)2488 3620 y Fo(1)2525 3466 y Fk(!)2591 3483 y Fm(j)p Fo(\003)2656 3491 y Fi(1)2689 3483 y Fm(j)2726 3608 y Ft(k)p Fs(\036)p Ft(k)2859 3620 y Fu(#)118 3880 y Fw(with)28 b(lim)423 3892 y Fu(\017)p Fm(!)p Fo(0)567 3880 y Fs(\017)601 3892 y Fo(1)661 3880 y Fw(=)23 b(0.)37 b(So)27 b(there)g(are)g(0)22 b Fs(<)1480 3858 y Fw(~)1469 3880 y Fs(#)h(<)g(#)28 b Fw(suc)n(h)f(that)h(for)f (su\016cien)n(tly)g(small)h Fs(\017)1341 4100 y Fw(~)1331 4122 y Fs(#c)1416 4134 y Fu(h)1477 4122 y Fw(+)1581 4044 y(~)1570 4066 y Fs(#)p 1570 4103 V 1570 4179 a(#)1643 4122 y(c)1679 4134 y Fu(r)1743 4122 y Fw(exp\()p Ft(\000)p Fw(\(1)18 b Ft(\000)g Fs(\021)s Fw(\))p Fs(T)12 b Fw(\))19 b(+)2423 4100 y(~)2413 4122 y Fs(#\017)2496 4134 y Fo(1)2556 4122 y Fs(<)j Fw(1)1024 b(\(211\))118 4314 y(and)28 b(so)g(\(203\))g (holds)g(uniformly)g(in)h(\003)1354 4326 y Fo(1)1419 4314 y Fw(and)f(\003.)39 b(F)-7 b(or)28 b(su\016cien)n(tly)g(large)f Fs(T)40 b Fw(w)n(e)28 b(can)g(c)n(ho)r(ose)3111 4292 y(~)3100 4314 y Fs(#)c Fw(=)g Fs(#)29 b Fw(suc)n(h)f(that)h(\(211\))118 4414 y(holds.)37 b(So)27 b(\(203\))g(is)g(pro)n(v)n(ed.)2758 b Fh(2)118 4693 y Fn(Prop)s(osition)30 b(3.2)41 b Ft(L)855 4662 y Fu(T)855 4715 y(S;)p Fo(\003)988 4645 y(def)1001 4693 y Fw(=)36 b Fs(\031)1149 4705 y Fo(\003)1218 4693 y Ft(\016)18 b(L)1335 4662 y Fu(T)1335 4715 y(S;)p Fo(\003)1474 4693 y Fj(is)30 b(the)g(tr)l(ansfer)g(op)l(er)l(ator,)h(r)l(estricte)l (d)f(to)g Ft(H)2896 4705 y Fo(\003)2945 4693 y Fj(,)h(for)f Fs(K)3210 4662 y Fu(T)3204 4715 y(S;)p Fo(\003)3313 4693 y Fj(,)h(i.e.)1152 4800 y Fk(Z)1198 4989 y Fo(\()p Fu(S)1268 4973 y Fi(1)1300 4989 y Fo(\))1326 4973 y Fi(\003)1386 4913 y Fs(d\026)14 b Fw(\()p Fs(K)1602 4879 y Fu(T)1596 4934 y(S;)p Fo(\003)1706 4913 y Fs( )1760 4925 y Fo(\003)1809 4913 y Fw(\))g Fs(\036)1904 4925 y Fo(\003)1978 4913 y Fw(=)2065 4800 y Fk(Z)2111 4989 y Fo(\()p Fu(S)2181 4973 y Fi(1)2214 4989 y Fo(\))2240 4973 y Fi(\003)2300 4913 y Fs(d\026)23 b( )2470 4925 y Fo(\003)2533 4913 y Fw(\()p Ft(L)2622 4879 y Fu(T)2622 4934 y(S;)p Fo(\003)2733 4913 y Fs(\036)2782 4925 y Fo(\003)2832 4913 y Fw(\))845 b(\(212\))118 5132 y Fj(F)-6 b(or)34 b(al)t(l)h Fs( )e Ft(2)d(C)621 5101 y Fo(0)658 5132 y Fw(\(\()p Fs(S)778 5101 y Fo(1)816 5132 y Fw(\))848 5101 y Fo(\003)898 5132 y Fw(\))j Fj(and)i Fs(\036)1178 5144 y Fo(\003)1258 5132 y Ft(2)30 b(H)1413 5144 y Fo(\003)1497 5132 y Fj(the)j(op)l(er)l(ator)i Fs(\031)2016 5144 y Fo(\003)2061 5152 y Fi(1)2120 5132 y Ft(\016)20 b(L)2239 5101 y Fu(T)2239 5154 y(S;)p Fo(\003)2383 5132 y Fj(is)34 b(the)g(c)l(omp)l(osite)h(of)f Fs(\031)3150 5144 y Fo(\003)3195 5152 y Fi(1)3266 5132 y Fj(and)g Ft(L)3488 5101 y Fu(T)3488 5154 y(S;)p Fo(\003)3632 5132 y Fj(for)g(any)118 5231 y Fw(\003)176 5243 y Fo(1)236 5231 y Ft(\032)23 b Fw(\003)p Fj(.)1966 5682 y Fw(38)p eop %%Page: 39 39 39 38 bop 118 354 a Fn(Pro)s(of)50 b Fw(The)24 b(\014rst)f(claim)g(is)h (a)e(sp)r(ecial)i(case)e(of)h(the)h(last)f(statemen)n(t.)36 b(Note)23 b(that)h Fs(\031)2751 366 y Fo(\003)2796 374 y Fi(1)2843 354 y Ft(\016)10 b(L)2952 324 y Fu(T)2952 377 y(S;)p Fo(\003)3085 354 y Fw(is)23 b(a)g(priori)g(the)g(op)r (erator)118 454 y(de\014ned)40 b(in)f(\(202\).)72 b(No)n(w)39 b(w)n(e)f(pro)n(v)n(e)g(that)i(it)g(is)f(actually)f(the)i(comp)r(osite) f(of)g Fs(\031)2798 466 y Fo(\003)2843 474 y Fi(1)2920 454 y Fw(and)g(the)h(transfer)e(op)r(erator,)118 554 y(restricted)27 b(to)g Ft(H)659 566 y Fo(\003)709 554 y Fw(,)h(for)f Fs(K)964 523 y Fu(T)958 576 y(S;)p Fo(\003)1067 554 y Fw(.)37 b(The)28 b(con)n(v)n(ergence)d(of)j(the)f(follo)n(wing)g (expressions)f(follo)n(ws)g(from)i(Prop)r(osition)e(3.1.)118 653 y(W)-7 b(e)25 b(consider)f(all)h(ordered)e(\014nite)j(sequences)e (of)h(jump-sites)g(in)g(\003.)36 b(A)25 b(particular)e(sequence)i(\()p Fs(q)3151 665 y Fo(1)3188 653 y Fs(;)14 b(:)g(:)g(:)g(;)g(q)3410 665 y Fu(k)3451 653 y Fw(\))25 b(o)r(ccurs)f(in)h(a)118 776 y(\014xed)e(time)h(in)n(terv)-5 b(al)22 b(\()p Ft(\000)p Fs(T)7 b(;)14 b Fw(0\))23 b(with)g(probabilit)n(y)f Fs(e)1724 746 y Fm(\000j)p Fo(\003)p Fm(j)p Fu(\025T)1962 736 y Fo(\()p Fu(\025T)9 b Fo(\))2101 711 y Fy(N)p 1962 757 193 4 v 2020 805 a Fu(N)d Fo(!)2188 776 y Fw(b)r(ecause)23 b(this)h(is)f(the)g(probabilit)n(y)f(of)h(ha)n(ving)f(exactly)118 876 y Fs(N)33 b Fw(jumps)26 b(in)e(total,)h(divided)g(b)n(y)f(the)h(n)n (um)n(b)r(er)f Ft(j)p Fw(\003)p Ft(j)1720 845 y Fu(N)1808 876 y Fw(of)g(di\013eren)n(t)h(sequences)f(of)g(length)h Fs(k)i Fw(\(whic)n(h)e(ha)n(v)n(e)e(all)h(the)h(same)118 975 y(probabilit)n(y\).)118 1075 y(The)36 b(sequence)f(corresp)r(onds)e (to)j(a)f(map)g Fs(S)1542 1087 y Fu(q)1572 1095 y Fy(N)1654 1075 y Ft(\016)23 b Fs(:)14 b(:)g(:)23 b Ft(\016)h Fs(S)1956 1087 y Fu(q)1986 1095 y Fi(1)2058 1075 y Fw(\(cf.)36 b(De\014nition)g(3.2\))f(and)h(so)e(b)n(y)i(\(164\),)g(to)g(a)f (transfer)118 1174 y(op)r(erator)1601 1374 y Fs(\031)1648 1386 y Fo(\003)1693 1394 y Fi(1)1748 1374 y Ft(\016)18 b(L)1865 1386 y Fu(S)1906 1394 y Fy(q)1933 1408 y(N)2014 1374 y Ft(\016)g Fs(:)c(:)g(:)k Ft(\016)g(L)2306 1386 y Fu(S)2347 1394 y Fy(q)2374 1406 y Fi(1)3709 1374 y Fw(\(213\))118 1536 y(So)23 b(the)g(comp)r(osite)f(of)h Fs(\031)889 1548 y Fo(\003)934 1556 y Fi(1)994 1536 y Fw(and)f(the)h(transfer)f(op)r(erator)f(for)h Fs(K)2120 1506 y Fu(T)2114 1559 y(S;)p Fo(\003)2246 1536 y Fw(is)h(equal)f(to)h (the)g(follo)n(wing)f(sum)g(o)n(v)n(er)f(all)i(\(p)r(ossibly)118 1636 y(empt)n(y\))28 b(sequences:)1261 1791 y Fk(X)1172 1973 y Fo(\()p Fu(q)1228 1981 y Fi(1)1261 1973 y Fu(;:::)o(;q)1390 1981 y Fy(N)1443 1973 y Fo(\))1483 1870 y Fs(e)1522 1836 y Fm(\000j)p Fo(\003)p Fm(j)p Fu(\025T)1760 1814 y Fw(\()p Fs(\025T)12 b Fw(\))1933 1784 y Fu(N)p 1760 1851 237 4 v 1829 1927 a Fs(N)d Fw(!)2029 1870 y Fs(\031)2076 1882 y Fo(\003)2121 1890 y Fi(1)2177 1870 y Ft(\016)18 b(L)2294 1882 y Fu(S)2335 1890 y Fy(q)2362 1904 y(N)2442 1870 y Ft(\016)g Fs(:)c(:)g(:)19 b Ft(\016)f(L)2735 1882 y Fu(S)2776 1890 y Fy(q)2803 1902 y Fi(1)3709 1870 y Fw(\(214\))118 2116 y(b)r(ecause)33 b(this)h(is)g(equal)f(to)g(the)h (rhs)f(of)h(\(145\).)54 b(The)33 b(probabilit)n(y)g(space)g(\012)g(is)h (partitioned)f(in)n(to)g(coun)n(tably)g(man)n(y)118 2216 y(sets,)f(eac)n(h)f(corresp)r(onding)e(to)j(a)f(particular)f(sequence)h (of)g(jump)i(sites.)48 b(So)31 b(w)n(e)g(can)g(write)g(the)h(in)n (tegral)e(here)h(as)g(a)118 2316 y(w)n(eigh)n(ted)c(sum.)118 2415 y(The)h(factors)e(in)i(\(213\))f(can)g(b)r(e)h(split)1302 2668 y Ft(L)1359 2680 y Fu(S)1400 2688 y Fy(q)1460 2668 y Fw(=)1547 2526 y Fk( )1613 2668 y Ft(Q)1681 2680 y Fu(q)1736 2668 y Fw(+)18 b Ft(R)1889 2680 y Fu(q)1945 2668 y Fw(+)2054 2565 y Fm(1)2028 2590 y Fk(X)2035 2768 y Fu(l)p Fo(=1)2161 2668 y Ft(B)2216 2680 y Fu(q)r(;l)2294 2526 y Fk(!)2378 2668 y Ft(\012)g Fw(id)2530 2683 y Fo(\003)p Fm(nf)p Fu(q)r Fm(g)3709 2668 y Fw(\(215\))118 2897 y(Expanding)30 b(the)g(pro)r(duct)h(in)g(\(213\),)f(w)n(e)g(get)g(a)h(sum)f(of)h(op)r (erators.)43 b(Recall)30 b(the)h(rules)f(\(153\))g(and)g(\(162\),)g (and)h(also)118 2997 y(that)d(w)n(e)f(ha)n(v)n(e)1738 3279 y Fs(\031)1785 3291 y Fu(q)1841 3279 y Ft(\016)18 b(R)1971 3291 y Fu(q)2091 3279 y Fw(=)82 b(0)1429 b(\(216\))1713 3403 y Fs(\031)1760 3415 y Fu(q)1815 3403 y Ft(\016)18 b(B)1930 3415 y Fu(q)r(;l)2091 3403 y Fw(=)82 b(0)p Fs(;)118 3586 y Fw(for)27 b(all)h Fs(q)e Ft(2)d Fw(\003)28 b(and)f Fs(l)d Ft(\025)f Fw(1)k(\(as)h Ft(Q)1158 3598 y Fu(q)1217 3586 y Fw(=)23 b Fs(h)1353 3598 y Fu(q)1408 3586 y Ft(\001)18 b Fs(\031)1496 3598 y Fu(q)1533 3586 y Fw(\).)38 b(So)27 b(some)g(of)h(the)g(summands)f(in)h(the)g(expansion)f(are)f(zero,)h (namely)g(if)118 3686 y Ft(Q)186 3698 y Fu(q)252 3686 y Fw(is)j(follo)n(w)n(ed)f(b)n(y)g Ft(R)848 3698 y Fu(q)885 3686 y Fw(,)h(or)f Ft(R)1112 3698 y Fu(q)1179 3686 y Fw(or)g Ft(B)1338 3698 y Fu(q)1404 3686 y Fw(are)f(follo)n(w)n(ed)h(b)n (y)h Ft(Q)2053 3698 y Fu(q)2119 3686 y Fw(or)f(a)g(pro)5 b(jection)29 b(at)g(site)36 b(~)-48 b Fs(q)39 b(=)-52 b Ft(2)27 b Fw(\003)3155 3698 y Fo(1)3222 3686 y Fw(due)j(to)g(the)g (oprator)118 3785 y Fs(\031)165 3797 y Fo(\003)210 3805 y Fi(1)247 3785 y Fw(.)44 b(`F)-7 b(ollo)n(wing')29 b(here)h(means)f (that)h(there)g(is)g(in)g(b)r(et)n(w)n(een)g(no)g Ft(B)2252 3797 y Fu(q)r(;l)2359 3785 y Fw(with)h Ft(k)6 b Fw(~)-48 b Fs(q)22 b Ft(\000)e Fs(q)s Ft(k)26 b(\024)h Fs(l)r Fw(.)43 b(In)30 b(the)h(follo)n(wing)d(w)n(e)i(rule)118 3885 y(out)e(these)f(com)n(binations.)36 b(\(Compare)27 b(this)h(to)f(the)h(notion)f(of)h(non-zero)e(con\014gurations)g(in)i ([13)o(].\))118 3985 y(No)n(w)f(w)n(e)g(represen)n(t)g(eac)n(h)g (summand)1665 4184 y Fs(\031)1712 4196 y Fo(\003)1757 4204 y Fi(1)1813 4184 y Ft(\016)18 b Fs(A)1935 4196 y Fu(N)2016 4184 y Ft(\016)g Fs(:)c(:)g(:)k Ft(\016)g Fs(A)2313 4196 y Fo(1)3709 4184 y Fw(\(217\))118 4333 y(in)41 b(the)g(expansion)f (of)h(\(213\))f(whic)n(h)h(is)f(the)h(comp)r(osite)g(of)f(op)r(erators) f Fs(\031)2563 4345 y Fo(\003)2608 4353 y Fi(1)2646 4333 y Fw(,)44 b Ft(B)2768 4345 y Fu(q)2831 4333 y Ft(\012)27 b Fw(id)2992 4348 y Fo(\003)p Fm(nf)p Fu(q)r Fm(g)3176 4333 y Fw(,)45 b Ft(R)3314 4345 y Fu(q)3378 4333 y Ft(\012)27 b Fw(id)3539 4348 y Fo(\003)p Fm(nf)p Fu(q)r Fm(g)3764 4333 y Fw(and)118 4433 y Ft(Q)186 4445 y Fu(q)241 4433 y Ft(\012)18 b Fw(id)393 4448 y Fo(\003)p Fm(nf)p Fu(q)r Fm(g)605 4433 y Fw(with)28 b(v)-5 b(ariable)27 b Fs(q)f Ft(2)d Fw(\003,)28 b(b)n(y)f(a)g(gum)h(con\014guration)e(as)h(follo)n (ws:)118 4533 y(W)-7 b(e)19 b(de\014ne)g(the)g(ob)n(vious)f(order)f(of) i(the)g(factors)f(in)h(\(217\))f(suc)n(h)g(that)h Fs(A)2306 4545 y Fo(1)2363 4533 y Fw(comes)f(b)r(efore)g Fs(A)2901 4545 y Fo(2)2958 4533 y Fw(etc.)34 b(Eac)n(h)18 b Ft(B)3374 4545 y Fu(q)r(;l)3470 4533 y Fw(corresp)r(onds)118 4632 y(to)29 b(a)f Fs(l)r Fw(-branc)n(hing)f(\(whic)n(h)i(is)g(assigned)f(b) n(y)g(the)i(map)e(pin)i(to)e Fs(q)s Fw(\))i(and)e(the)i(order)d(of)i (the)g(op)r(erators)e Ft(B)3402 4644 y Fu(q)r(;l)3508 4632 y Fw(de\014nes)i(the)118 4732 y(linear)21 b(order)g(of)h(the)h (branc)n(hings.)33 b(The)22 b(other)g(factors)f Ft(Q)1940 4744 y Fu(q)1999 4732 y Fw(and)g Ft(R)2224 4744 y Fu(q)2284 4732 y Fw(determine)h(the)g Fs(h)p Fw(-)g(and)g Fs(r)r Fw(-strips)g(in)g(the)h(follo)n(wing)118 4831 y(w)n(a)n(y:)220 5014 y(1.)41 b Fj(strips)34 b(b)l(etwe)l(en)f(two)g(vertic)l(es:)51 b Fw(Let)32 b Fn(v)1572 5026 y Fu(i)1630 5014 y Ft(\036)d Fn(v)1774 5026 y Fu(j)1841 5014 y Fw(b)r(e)k(t)n(w)n(o)d(branc)n(hings) h(of)g(degree)g Fs(b)2939 5026 y Fu(i)2998 5014 y Fw(and)g Fs(b)3199 5026 y Fu(j)3234 5014 y Fw(,)i(resp)r(ectiv)n(ely)-7 b(,)32 b(and)326 5114 y Fs(q)26 b Ft(2)d Fs(B)530 5126 y Fu(b)559 5134 y Fy(i)590 5114 y Fw(\(pin\()p Fn(v)819 5126 y Fu(i)848 5114 y Fw(\)\))12 b Ft(\\)g Fs(B)1054 5126 y Fu(b)1083 5134 y Fy(j)1119 5114 y Fw(\(pin)q(\()p Fn(v)1349 5126 y Fu(j)1384 5114 y Fw(\)\))g Ft(\\)g Fw(\003)25 b(suc)n(h)f(that)h(there)f(is)g(no)h(other)e(branc)n(hing)h Fn(v)3019 5126 y Fu(k)3060 5114 y Fw(,)h(of)f(degree)g Fs(b)3491 5126 y Fu(k)3531 5114 y Fw(,)h(sa)n(y)-7 b(,)24 b(with)326 5223 y Fn(v)376 5235 y Fu(i)427 5223 y Ft(\036)e Fn(v)564 5235 y Fu(k)629 5223 y Ft(\036)g Fn(v)766 5235 y Fu(j)826 5223 y Fw(and)i Fs(q)j Ft(2)c Fs(B)1189 5235 y Fu(b)1218 5244 y Fy(k)1259 5223 y Fw(\(pin\()p Fn(v)1488 5235 y Fu(k)1530 5223 y Fw(\)\).)36 b(Then)25 b Ft(C)1911 5235 y Fu(g)1974 5223 y Fw(has)e(an)i Fs(h)p Fw(-strip)e(b)r(et)n(w)n (een)i(pin)2930 5188 y Fm(\000)p Fo(1)2930 5244 y Fl(v)2970 5252 y Fy(i)3019 5223 y Fw(\()p Fs(q)s Fw(\))g(and)f(pin)3422 5188 y Fm(\000)p Fo(1)3422 5244 y Fl(v)3462 5252 y Fy(j)3511 5223 y Fw(\()p Fs(q)s Fw(\))h(if)g(there)326 5323 y(is)g(a)h(factor)f Ft(Q)779 5335 y Fu(q)831 5323 y Ft(\012)14 b Fw(id)979 5338 y Fo(\003)p Fm(nf)p Fu(q)r Fm(g)1190 5323 y Fw(in)26 b(\(217\))f(b)r(et)n(w)n(een)h(the)g(t)n(w)n(o)f(factors)g(corresp)r (onding)f(to)i Fn(v)3066 5335 y Fu(i)3120 5323 y Fw(and)f Fn(v)3329 5335 y Fu(j)3365 5323 y Fw(.)36 b(Otherwise)25 b Ft(C)3859 5335 y Fu(g)326 5422 y Fw(has)i(an)g Fs(r)r Fw(-strip)h(there)f(\(ev)n(en)h(if)g(there)f(is)h(no)f(factor)g Ft(R)2080 5434 y Fu(q)2135 5422 y Ft(\012)18 b Fw(id)2287 5437 y Fo(\003)p Fm(nf)p Fu(q)r Fm(g)2472 5422 y Fw(.\))1966 5682 y(39)p eop %%Page: 40 40 40 39 bop 220 354 a Fw(2.)41 b Fj(strips)30 b(fr)l(om)h(a)g(vertex)e (and)i(the)g(top:)43 b Fw(Let)29 b Fn(v)1750 366 y Fu(j)1813 354 y Fw(b)r(e)g(a)e Fs(b)2032 366 y Fu(j)2067 354 y Fw(-branc)n(hing)g(and)h(assume)f(there)h(is)g(no)g Fs(b)3379 366 y Fu(i)3406 354 y Fw(-branc)n(hing)f Fn(v)3870 366 y Fu(i)326 454 y Fw(with)34 b Fn(v)571 466 y Fu(i)631 454 y Ft(\036)f Fn(v)779 466 y Fu(j)848 454 y Fw(and)g Fs(q)j Ft(2)d Fs(B)1239 466 y Fu(b)1268 474 y Fy(i)1298 454 y Fw(\(pin)q(\()p Fn(v)1528 466 y Fu(i)1556 454 y Fw(\)\).)55 b(Then)34 b Ft(C)1965 466 y Fu(g)2037 454 y Fw(has)f(an)g Fs(h)p Fw(-strip)g(from)g(pin)2905 419 y Fm(\000)p Fo(1)2905 474 y Fl(v)2945 482 y Fy(j)2994 454 y Fw(\()p Fs(q)s Fw(\))h(to)f(the)h(top)g(if)g(there)f(is)326 554 y(a)d(factor)f Ft(Q)706 566 y Fu(q)763 554 y Ft(\012)20 b Fw(id)917 569 y Fo(\003)p Fm(nf)p Fu(q)r Fm(g)1132 554 y Fw(in)31 b(\(217\))f(b)r(efore)g(the)h(factor)f(corresp)r(onding) e(to)j Fn(v)2782 566 y Fu(j)2817 554 y Fw(.)46 b(Otherwise)29 b Ft(C)3325 566 y Fu(g)3394 554 y Fw(has)h(an)g Fs(r)r Fw(-strip)326 653 y(there.)220 817 y(3.)41 b Fj(strips)29 b(fr)l(om)h(a)g(vertex)f(and)h(the)f(b)l(ottom:)42 b Fw(Let)28 b Fn(v)1872 829 y Fu(i)1927 817 y Fw(b)r(e)g(a)e Fs(b)2144 829 y Fu(i)2172 817 y Fw(-branc)n(hing)f(and)i(assume)g (there)g(is)g(no)g Fs(b)3478 829 y Fu(j)3512 817 y Fw(-branc)n(hing)326 917 y Fn(v)376 929 y Fu(j)436 917 y Fw(with)e Fn(v)672 929 y Fu(i)723 917 y Ft(\036)e Fn(v)861 929 y Fu(j)921 917 y Fw(and)i Fs(q)h Ft(2)d Fs(B)1284 929 y Fu(b)1313 937 y Fy(j)1349 917 y Fw(\(pin\()p Fn(v)1578 929 y Fu(j)1614 917 y Fw(\)\).)36 b(Then)25 b Ft(C)1995 929 y Fu(g)2059 917 y Fw(has)f(an)g Fs(h)p Fw(-strip)g(from)h(pin)2892 882 y Fm(\000)p Fo(1)2892 937 y Fl(v)2932 945 y Fy(i)2981 917 y Fw(\()p Fs(q)s Fw(\))h(to)e(the)i(b)r(ottom)f(if)g(there)326 1016 y(is)h(a)g(factor)f Ft(Q)780 1028 y Fu(q)833 1016 y Ft(\012)15 b Fw(id)982 1031 y Fo(\003)p Fm(nf)p Fu(q)r Fm(g)1193 1016 y Fw(in)27 b(\(217\))e(after)h(the)h(factor)e(corresp)r (onding)g(to)h Fn(v)2760 1028 y Fu(j)2822 1016 y Fw(or)f(if)i Fs(q)f Ft(2)d Fw(\003)16 b Ft(n)f Fw(\003)3327 1028 y Fo(1)3364 1016 y Fw(.)37 b(Otherwise)25 b Ft(C)3859 1028 y Fu(g)326 1116 y Fw(has)i(an)g Fs(r)r Fw(-strip)h(there.)220 1280 y(4.)41 b Fj(long)30 b(strips:)42 b Fw(Let)28 b Fs(q)e Ft(2)d Fw(\003)k(and)g(assume)g(that)h(there)f(is)g(no)g Fs(b)2218 1292 y Fu(i)2245 1280 y Fw(-branc)n(hing)f Fn(v)2708 1292 y Fu(i)2763 1280 y Fw(with)i Fs(q)e Ft(2)e Fs(B)3157 1292 y Fu(b)3186 1300 y Fy(i)3216 1280 y Fw(\(pin)q(\()p Fn(v)3446 1292 y Fu(i)3474 1280 y Fw(\)\).)38 b(Then)27 b Ft(C)3859 1292 y Fu(g)326 1380 y Fw(has)g(a)h(long)g Fs(h)p Fw(-strip)f(at)i(site)f Fs(q)j Fw(if)e(there)f(is)g(a)g(factor)g Ft(Q)2070 1392 y Fu(q)2125 1380 y Ft(\012)18 b Fw(id)2278 1395 y Fo(\003)p Fm(nf)p Fu(q)r Fm(g)2490 1380 y Fw(in)29 b(\(217\))e(or)g(if)i Fs(q)e Ft(2)e Fw(\003)19 b Ft(n)f Fw(\003)3323 1392 y Fo(1)3360 1380 y Fw(.)39 b(Otherwise)27 b Ft(C)3859 1392 y Fu(g)326 1479 y Fw(has)g(a)g(long)g Fs(r)r Fw(-strip)h(there.)118 1657 y(The)e(assignmen)n(t)e(of)h(a)g (summand)g(in)h(the)g(sequence)f(\(217\))f(of)h(op)r(erators)f(to)h(a)g (gum)g(con\014guration)f(is)h(not)g(injectiv)n(e,)118 1756 y(as)32 b(w)n(e)f(ha)n(v)n(e)g(already)g(seen)h(in)g(the)h(simple) f(Example)g(3.1.)49 b(No)n(w)32 b(w)n(e)g(consider)f(a)h(\014xed)g(gum) g(con\014guration)e Ft(C)3739 1768 y Fu(g)3810 1756 y Fw(on)118 1856 y(\003)35 b(ending)f(in)h(\003)647 1868 y Fo(1)719 1856 y Fw(and)f(all)h(sequences)f(corresp)r(onding)e(to)j (it.)59 b(W)-7 b(e)35 b(assume)f Ft(C)2668 1868 y Fu(g)2741 1856 y Fw(to)g(ha)n(v)n(e)g(at)g(least)h(one)f(branc)n(hing.)118 1956 y(The)k(case)e(of)i(no)f(branc)n(hing)f(is)h(treated)g(in)h(a)f (similar)g(but)h(easier)e(w)n(a)n(y)-7 b(.)65 b(T)-7 b(o)37 b(k)n(eep)g(the)h(notation)f(simpler,)i(that)118 2055 y(\003)25 b(=)g Ft(f)p Fw(1)p Fs(;)14 b(:)g(:)g(:)f(;)h Ft(j)p Fw(\003)p Ft(jg)p Fw(.)40 b(An)n(y)30 b(suc)n(h)e(sequence)h (has)f(the)i(factors)e Ft(B)2103 2067 y Fu(q)2133 2075 y Fy(i)2159 2067 y Fu(;l)2200 2075 y Fy(i)2259 2055 y Fw(\()p Fs(i)e Fw(=)f(1)p Fs(;)14 b(:)g(:)g(:)f(;)h(K)6 b Fw(\))29 b(that)g(corresp)r(ond)e(1-1)h(and)h(order)118 2155 y(preserving)f(to)h(the)h Fs(K)35 b Fw(branc)n(hings)28 b(of)i Ft(C)1435 2167 y Fu(g)1502 2155 y Fw(as)f(describ)r(ed)h(in)f (ab)r(o)n(v)n(e.)41 b(Bet)n(w)n(een)29 b(t)n(w)n(o)g(consecutiv)n(e)f (factors)h Ft(B)3607 2167 y Fu(q)3637 2175 y Fy(i)3663 2167 y Fu(;l)3704 2175 y Fy(i)3764 2155 y Fw(and)118 2255 y Ft(B)173 2267 y Fu(q)203 2275 y Fy(i)p Fi(+1)300 2267 y Fu(;l)341 2275 y Fy(i)p Fi(+1)466 2255 y Fw(there)22 b(can)h(b)r(e)h(factors)e Ft(Q)1264 2267 y Fu(q)1310 2255 y Ft(\012)9 b Fw(id)1453 2270 y Fo(\003)p Fm(nf)p Fu(q)r Fm(g)1660 2255 y Fw(\(factors)22 b Ft(R)2028 2267 y Fu(q)2075 2255 y Ft(\012)9 b Fw(id)2218 2270 y Fo(\003)p Fm(nf)p Fu(q)r Fm(g)2402 2255 y Fw(\))24 b(if)f(for)g(an)n(y)f Fn(t)h Ft(2)h Fw(Simplex\()p Ft(C)3307 2267 y Fu(g)3346 2255 y Fs(;)14 b(T)e Fw(\))22 b(the)i(sp)r(eci\014c)118 2354 y(gum)32 b(con\014guration)e(\()p Ft(C)890 2366 y Fu(g)928 2354 y Fs(;)14 b(T)7 b(;)14 b Fn(t)p Fw(\))32 b(has)f(an)g Fs(h)p Fw(-strip)h(\(an)f Fs(r)r Fw(-strip\))h(from)g(\()p Fs(q)s(;)14 b(t)2494 2366 y Fu(i)2522 2354 y Fw(\))32 b(to)f(\()p Fs(q)s(;)14 b(t)2830 2366 y Fu(i)p Fo(+1)2942 2354 y Fw(\).)50 b(Similarly)-7 b(,)32 b(the)g(option)g(of)118 2454 y(ha)n(ving)25 b(suc)n(h)g(factors)f(b)r(efore)i Ft(B)1140 2466 y Fu(q)1170 2474 y Fi(1)1202 2466 y Fu(;l)1243 2474 y Fi(1)1305 2454 y Fw(or)f(after)g Ft(B)1654 2466 y Fu(q)1684 2474 y Fy(K)1738 2466 y Fu(;l)1779 2474 y Fy(K)1863 2454 y Fw(dep)r(ends)h(on)f(the)h Fs(h)p Fw(-strips)f(and)g Fs(r)r Fw(-strips)h(of)f Ft(C)3325 2466 y Fu(g)3389 2454 y Fw(in)h(the)g(ob)n(vious)118 2553 y(w)n(a)n(y)-7 b(.)36 b(F)-7 b(urther)27 b(suc)n(h)h(factors)e(b)r(elong)h(to)h(particular)e (maximal)h Fs(r)r Fw(-strips)h(or)e Fs(h)p Fw(-strips)h(in)h(the)g(ob)n (v)n(ous)e(w)n(a)n(y)-7 b(.)118 2653 y(Let)30 b(us)f(denote)h(the)f(n)n (um)n(b)r(er)h(of)f(factors)g Ft(Q)1533 2665 y Fu(q)1589 2653 y Ft(\012)19 b Fw(id)1742 2668 y Fo(\003)p Fm(nf)p Fu(q)r Fm(g)1956 2653 y Fw(or)28 b Ft(R)2129 2665 y Fu(q)2186 2653 y Ft(\012)19 b Fw(id)2340 2668 y Fo(\003)p Fm(nf)p Fu(q)r Fm(g)2553 2653 y Fw(b)r(efore)30 b Ft(B)2860 2665 y Fu(q)2890 2673 y Fi(1)2922 2665 y Fu(;l)2963 2673 y Fi(1)3029 2653 y Fw(b)n(y)f Fs(n)3196 2665 y Fo(0)p Fu(;q)3315 2653 y Fw(and)g(the)h(n)n(um)n(b)r(er)118 2753 y(of)e(suc)n(h)f (factors)f(after)i Ft(B)922 2765 y Fu(q)952 2773 y Fy(i)978 2765 y Fu(;l)1019 2773 y Fy(i)1077 2753 y Fw(b)n(y)f Fs(n)1242 2765 y Fu(i;q)1322 2753 y Fw(.)118 2852 y(Note)34 b(that)f(for)g(ev)n(ery)f(maximal)h Fs(h)p Fw(-strip)g(to)g(the)h(b)r (ottom)g(at)f(a)g(site)g Fs(q)j Ft(2)d Fw(\003)2603 2864 y Fo(1)2674 2852 y Fw(there)g(m)n(ust)g(b)r(e)h(at)g(least)f(one)g (factor)118 2952 y Fs(Q)184 2964 y Fu(q)231 2952 y Ft(\012)10 b Fw(id)375 2967 y Fo(\003)p Fm(nf)p Fu(q)r Fm(g)583 2952 y Fw(b)r(ecause)23 b(of)h(rule)f(3)g(on)g(page)g(40.)35 b(W)-7 b(e)24 b(denote)f(this)h(condition)f(on)h(the)g(family)f(\()p Fs(n)3149 2964 y Fu(i;q)3229 2952 y Fw(\))3271 2953 y Fi(0)p Fe(\024)p Fy(i)p Fe(\024)p Fy(K)3314 3001 y(q)q Fe(2)p Fi(\003)3500 2952 y Fw(of)g(n)n(um)n(b)r(ers)118 3075 y(b)n(y)k(`condition)h(A'.)g(In)g(total)f(there)g(are)1626 3334 y Fs(N)32 b Fw(=)1814 3255 y Fk(X)1812 3434 y Fu(q)r Fm(2)p Fo(\003)1979 3231 y Fu(K)1949 3255 y Fk(X)1955 3432 y Fu(i)p Fo(=0)2082 3334 y Fs(n)2132 3346 y Fu(i;q)2231 3334 y Fw(+)18 b Fs(K)1324 b Fw(\(218\))118 3566 y(factors)25 b(that)h(corresp)r(ond)e(to)h(a)g(particular)g(sequence)g(of)h Fs(N)34 b Fw(jump)27 b(sites)e(and)h(suc)n(h)f(a)g(sequence)g(o)r (ccurs)g(with)h(proba-)118 3684 y(bilit)n(y)h Fs(e)373 3654 y Fm(\000j)p Fo(\003)p Fm(j)p Fu(\025T)611 3644 y Fo(\()p Fu(\025T)9 b Fo(\))750 3619 y Fy(N)p 611 3665 193 4 v 669 3713 a Fu(N)d Fo(!)814 3684 y Fw(.)37 b(The)1044 3622 y Fk(P)1132 3709 y Fu(q)r Fm(2)p Fo(\003)1272 3684 y Fs(n)1322 3696 y Fo(0)p Fu(;q)1439 3684 y Fw(factors)26 b(b)r(efore)h Ft(B)2013 3696 y Fu(q)2043 3704 y Fi(1)2075 3696 y Fu(;l)2116 3704 y Fi(1)2180 3684 y Fw(can)f(o)r(ccur)h(in)g(an)n (y)g(of)g(the)3044 3617 y Fk(\000)3082 3647 y Fu(n)3123 3655 y Fi(0)p Fy(;)p Fi(1)3203 3647 y Fo(+)p Fu(:::)o Fo(+)p Fu(n)3405 3658 y Fi(0)p Fy(;)p Fe(j)p Fi(\003)p Fe(j)3113 3713 y Fu(n)3154 3721 y Fi(0)p Fy(;)p Fi(1)3234 3713 y Fu(;:::)o(;n)3374 3724 y Fi(0)p Fy(;)p Fe(j)p Fi(\003)p Fe(j)3532 3617 y Fk(\001)3598 3684 y Fw(di\013eren)n(t)118 3800 y(orders,)f(that)i(all)f(ha)n(v)n(e)g(the)h(same)f(probabilit)n(y) -7 b(.)118 3900 y(So)29 b(if)h(w)n(e)f(sum)g(o)n(v)n(er)e(all)i(\(pro)r (ducts)g(of)6 b(\))30 b(op)r(erators)d(that)j(corresp)r(ond)d(to)i Ft(C)2545 3912 y Fu(g)2584 3900 y Fw(,)h(w)n(eigh)n(ted)e(with)i(the)f (probabilit)n(y)f(that)118 3999 y(the)34 b(corresp)r(onding)e(sequence) h(of)g(jumps)h(\(in)g(the)g(underlying)f(P)n(oisson)f(pro)r(cess\))g(o) r(ccurs,)j(w)n(e)e(get)g(In)h(\(219\))f(eac)n(h)118 4099 y(op)r(erator)26 b Fs(A)515 4111 y Fu(k)q(;q)636 4099 y Fw(stands)h(for)g(either)h Ft(Q)1326 4111 y Fu(q)1390 4099 y Fw(or)f Ft(R)1562 4111 y Fu(q)1599 4099 y Fw(.)311 4332 y Fk(X)293 4501 y Fi(\()p Fy(n)352 4514 y(i;q)427 4501 y Fi(\))201 4572 y Fo(condition)21 b(A)565 4411 y Fs(e)604 4376 y Fm(\000j)p Fo(\003)p Fm(j)p Fu(\025T)842 4355 y Fw(\()p Fs(\025T)12 b Fw(\))1015 4324 y Fu(N)p 842 4392 237 4 v 911 4468 a Fs(N)d Fw(!)1111 4294 y Fk(\022)1173 4355 y Fs(n)1223 4367 y Fo(0)p Fu(;)p Fo(1)1331 4355 y Fw(+)18 b Fs(:)c(:)g(:)k Fw(+)g Fs(n)1662 4370 y Fo(0)p Fu(;)p Fm(j)p Fo(\003)p Fm(j)1230 4468 y Fs(n)1280 4480 y Fo(0)p Fu(;)p Fo(1)1370 4468 y Fs(;)c(:)g(:)g(:)g(;)g(n)1605 4483 y Fo(0)p Fu(;)p Fm(j)p Fo(\003)p Fm(j)1804 4294 y Fk(\023)1879 4411 y Ft(\001)g(\001)g(\001)1990 4294 y Fk(\022)2051 4355 y Fs(n)2101 4367 y Fu(K;)p Fo(1)2232 4355 y Fw(+)k Fs(:)c(:)g(:)19 b Fw(+)f Fs(n)2564 4370 y Fu(K;)p Fm(j)p Fo(\003)p Fm(j)2108 4468 y Fs(n)2158 4480 y Fu(K;)p Fo(1)2271 4468 y Fs(;)c(:)g(:)g(:)g(;)g(n)2506 4483 y Fu(K;)p Fm(j)p Fo(\003)p Fm(j)2728 4294 y Fk(\023)3709 4411 y Fw(\(219\))422 4703 y Ft(\002)p Fs(A)549 4658 y Fu(n)590 4666 y Fy(K;)p Fi(1)549 4727 y Fu(K;)p Fo(1)711 4703 y Ft(\012)k Fs(:)c(:)g(:)k Ft(\012)g Fs(A)1054 4653 y Fu(K;n)1171 4664 y Fe(j)p Fi(\003)p Fe(j)1054 4731 y Fu(K;)p Fm(j)p Fo(\003)p Fm(j)1274 4703 y Ft(\016)g(B)1389 4715 y Fu(q)1419 4723 y Fy(K)1473 4715 y Fu(;l)1514 4723 y Fy(K)1591 4703 y Ft(\012)g Fw(id)1743 4718 y Fo(\003)p Fm(nf)p Fu(q)1886 4726 y Fy(K)1941 4718 y Fm(g)1998 4703 y Ft(\016)g Fs(:)c(:)g(:)k Ft(\016)g(B)2288 4715 y Fu(q)2318 4723 y Fi(1)2350 4715 y Fu(;l)2391 4723 y Fi(1)2446 4703 y Ft(\012)g Fw(id)2598 4718 y Fo(\003)p Fm(nf)p Fu(q)2741 4726 y Fi(1)2775 4718 y Fm(g)2831 4703 y Ft(\016)g Fs(A)2953 4658 y Fu(n)2994 4666 y Fi(0)p Fy(;)p Fi(1)2953 4725 y Fo(0)p Fu(;)p Fo(1)3097 4703 y Ft(\012)g Fs(:)c(:)g(:)k Ft(\012)g Fs(A)3440 4653 y Fu(n)3481 4664 y Fi(0)p Fy(;)p Fe(j)p Fi(\003)p Fe(j)3440 4731 y Fo(0)p Fu(;)p Fm(j)p Fo(\003)p Fm(j)274 4982 y Fw(=)542 4903 y Fk(X)524 5073 y Fi(\()p Fy(n)583 5086 y(i;q)657 5073 y Fi(\))432 5143 y Fo(condition)i(A)796 4869 y Fk(Z)842 5058 y Fo(Simplex)n(\()p Fm(C)1134 5066 y Fy(g)1169 5058 y Fu(;T)9 b Fo(\))1280 4982 y Fs(\025)1328 4948 y Fu(K)1393 4982 y Fs(d)p Fn(t)1506 4865 y Fk(\022)1567 4982 y Fs(e)1606 4948 y Fm(\000)p Fu(\025)p Fo(\()p Fu(t)1748 4956 y Fi(1)1780 4948 y Fo(+)p Fu(T)g Fo(\))1919 4926 y Fw(\()p Fs(\025)p Fw(\()p Fs(t)2061 4938 y Fo(1)2118 4926 y Fw(+)18 b Fs(T)12 b Fw(\)\))2326 4896 y Fu(n)2367 4904 y Fy(K;)p Fi(1)p 1919 4963 551 4 v 2101 5039 a Fs(n)2151 5051 y Fu(K;)p Fo(1)2264 5039 y Fw(!)2479 4982 y Fs(A)2541 4937 y Fu(n)2582 4945 y Fy(K;)p Fi(1)2541 5007 y Fu(K;)p Fo(1)2685 4865 y Fk(\023)2765 4982 y Ft(\012)18 b Fs(:)c(:)g(:)422 5388 y Ft(\012)501 5271 y Fk(\022)561 5388 y Fs(e)600 5354 y Fm(\000)p Fu(\025)p Fo(\()p Fu(t)742 5362 y Fi(1)774 5354 y Fo(+)p Fu(T)9 b Fo(\))914 5332 y Fw(\()p Fs(\025)p Fw(\()p Fs(t)1056 5344 y Fo(1)1112 5332 y Fw(+)18 b Fs(T)12 b Fw(\)\))1320 5302 y Fu(n)1361 5313 y Fy(K;)p Fe(j)p Fi(\003)p Fe(j)p 914 5369 598 4 v 1094 5445 a Fs(n)1144 5460 y Fu(K;)p Fm(j)p Fo(\003)p Fm(j)1308 5445 y Fw(!)1521 5388 y Fs(A)1583 5338 y Fu(n)1624 5349 y Fy(K;)p Fe(j)p Fi(\003)p Fe(j)1583 5416 y Fu(K;)p Fm(j)p Fo(\003)p Fm(j)1775 5271 y Fk(\023)1855 5388 y Ft(\016)18 b(B)1970 5400 y Fu(q)2000 5408 y Fy(K)2054 5400 y Fu(;l)2095 5408 y Fy(K)2171 5388 y Ft(\012)g Fw(id)2324 5403 y Fo(\003)p Fm(nf)p Fu(q)2467 5411 y Fy(K)2522 5403 y Fm(g)2578 5388 y Ft(\016)g Fs(:)c(:)g(:)19 b Ft(\016)f(B)2869 5400 y Fu(q)2899 5408 y Fi(1)2931 5400 y Fu(;l)2972 5408 y Fi(1)3027 5388 y Ft(\012)g Fw(id)3179 5403 y Fo(\003)p Fm(nf)p Fu(q)3322 5411 y Fi(1)3355 5403 y Fm(g)1966 5682 y Fw(40)p eop %%Page: 41 41 41 40 bop 422 354 a Ft(\016)478 237 y Fk(\022)538 354 y Fs(e)577 320 y Fm(\000)p Fu(\025)p Fo(\()p Fu(t)719 328 y Fi(1)751 320 y Fo(+)p Fu(T)9 b Fo(\))891 298 y Fw(\()p Fs(\025)p Fw(\()p Fs(t)1033 310 y Fo(1)1089 298 y Fw(+)18 b Fs(T)12 b Fw(\)\))1297 268 y Fu(n)1338 276 y Fi(0)p Fy(;)p Fi(1)p 891 335 532 4 v 1075 411 a Fs(n)1125 423 y Fo(0)p Fu(;)p Fo(1)1215 411 y Fw(!)1432 354 y Fs(A)1494 309 y Fu(n)1535 317 y Fi(0)p Fy(;)p Fi(1)1494 376 y Fo(0)p Fu(;)p Fo(1)1619 237 y Fk(\023)1698 354 y Ft(\012)18 b Fs(:)c(:)g(:)19 b Ft(\012)1980 237 y Fk(\022)2041 354 y Fs(e)2080 320 y Fm(\000)p Fu(\025)p Fo(\()p Fu(t)2222 328 y Fi(1)2254 320 y Fo(+)p Fu(T)9 b Fo(\))2393 298 y Fw(\()p Fs(\025)p Fw(\()p Fs(t)2535 310 y Fo(1)2592 298 y Fw(+)18 b Fs(T)12 b Fw(\)\))2800 268 y Fu(n)2841 279 y Fi(0)p Fy(;)p Fe(j)p Fi(\003)p Fe(j)p 2393 335 579 4 v 2575 411 a Fs(n)2625 426 y Fo(0)p Fu(;)p Fm(j)p Fo(\003)p Fm(j)2767 411 y Fw(!)2982 354 y Fs(A)3044 304 y Fu(n)3085 315 y Fi(0)p Fy(;)p Fe(j)p Fi(\003)p Fe(j)3044 383 y Fo(0)p Fu(;)p Fm(j)p Fo(\003)p Fm(j)3217 237 y Fk(\023)274 727 y Fw(=)422 614 y Fk(Z)468 803 y Fo(Simplex)n(\()p Fm(C)760 811 y Fy(g)794 803 y Fu(;T)d Fo(\))906 727 y Fs(\025)954 693 y Fu(k)995 727 y Fs(d)p Fn(t)1108 648 y Fk(Y)1132 826 y Fu(H)1228 635 y Fk(\020)1278 727 y Fw(1)17 b Ft(\000)i Fs(e)1460 693 y Fm(\000)p Fu(\025)p Fo(length)n(\()p Fu(H)t Fo(\))1846 635 y Fk(\021)1909 560 y(0)1909 710 y(@)2034 648 y(X)1982 827 y Fu(n)2023 835 y Fy(K;)p Fi(1)2122 827 y Fm(\025)p Fo(0)2221 727 y Fs(e)2260 693 y Fm(\000)p Fu(\025)p Fo(\()p Fu(t)2402 701 y Fi(1)2434 693 y Fo(+)p Fu(T)9 b Fo(\))2573 671 y Fw(\()p Fs(\025)p Fw(\()p Fs(t)2715 683 y Fo(1)2772 671 y Fw(+)18 b Fs(T)12 b Fw(\)\))2980 641 y Fu(n)3021 649 y Fy(K;)p Fi(1)p 2573 708 551 4 v 2755 784 a Fs(n)2805 796 y Fu(K;)p Fo(1)2918 784 y Fw(!)3133 727 y Fs(A)3195 682 y Fu(n)3236 690 y Fy(K;)p Fi(1)3195 752 y Fu(K;)p Fo(1)3339 560 y Fk(1)3339 710 y(A)3430 727 y Ft(\012)18 b Fs(:)c(:)g(:)422 1142 y Ft(\012)501 975 y Fk(0)501 1125 y(@)649 1063 y(X)573 1242 y Fu(n)614 1253 y Fy(K;)p Fe(j)p Fi(\003)p Fe(j)760 1242 y Fm(\025)p Fo(0)859 1142 y Fs(e)898 1108 y Fm(\000)p Fu(\025)p Fo(\()p Fu(t)1040 1116 y Fi(1)1072 1108 y Fo(+)p Fu(T)9 b Fo(\))1211 1086 y Fw(\()p Fs(\025)p Fw(\()p Fs(t)1353 1098 y Fo(1)1410 1086 y Fw(+)18 b Fs(T)12 b Fw(\)\))1618 1056 y Fu(n)1659 1067 y Fy(K;)p Fe(j)p Fi(\003)p Fe(j)p 1211 1123 598 4 v 1391 1199 a Fs(n)1441 1214 y Fu(K;)p Fm(j)p Fo(\003)p Fm(j)1606 1199 y Fw(!)1819 1142 y Fs(A)1881 1092 y Fu(n)1922 1103 y Fy(K;)p Fe(j)p Fi(\003)p Fe(j)1881 1171 y Fu(K;)p Fm(j)p Fo(\003)p Fm(j)2073 975 y Fk(1)2073 1125 y(A)2164 1142 y Ft(\016)18 b(B)2279 1154 y Fu(q)2309 1162 y Fy(K)2363 1154 y Fu(;l)2404 1162 y Fy(K)2481 1142 y Ft(\012)g Fw(id)2633 1157 y Fo(\003)p Fm(nf)p Fu(q)2776 1165 y Fy(K)2831 1157 y Fm(g)2888 1142 y Ft(\016)g Fs(:)c(:)g(:)k Ft(\016)g(B)3178 1154 y Fu(q)3208 1162 y Fi(1)3240 1154 y Fu(;l)3281 1162 y Fi(1)3336 1142 y Ft(\012)g Fw(id)3488 1157 y Fo(\003)p Fm(nf)p Fu(q)3631 1165 y Fi(1)3664 1157 y Fm(g)440 1558 y Ft(\016)496 1391 y Fk(0)496 1540 y(@)611 1479 y(X)568 1657 y Fu(n)609 1665 y Fi(0)p Fy(;)p Fi(1)689 1657 y Fm(\025)p Fo(0)788 1558 y Fs(e)827 1524 y Fm(\000)p Fu(\025)p Fo(\()p Fu(t)969 1532 y Fi(1)1001 1524 y Fo(+)p Fu(T)9 b Fo(\))1140 1502 y Fw(\()p Fs(\025)p Fw(\()p Fs(t)1282 1514 y Fo(1)1339 1502 y Fw(+)18 b Fs(T)12 b Fw(\)\))1547 1472 y Fu(n)1588 1480 y Fi(0)p Fy(;)p Fi(1)p 1140 1539 532 4 v 1324 1615 a Fs(n)1374 1627 y Fo(0)p Fu(;)p Fo(1)1464 1615 y Fw(!)1681 1558 y Fs(A)1743 1513 y Fu(n)1784 1521 y Fi(0)p Fy(;)p Fi(1)1743 1580 y Fo(0)p Fu(;)p Fo(1)1868 1391 y Fk(1)1868 1540 y(A)1960 1558 y Ft(\012)18 b Fs(:)c(:)g(:)k Ft(\012)2241 1391 y Fk(0)2241 1540 y(@)2380 1479 y(X)2313 1657 y Fu(n)2354 1668 y Fi(0)p Fy(;)p Fe(j)p Fi(\003)p Fe(j)2482 1657 y Fm(\025)p Fo(0)2581 1558 y Fs(e)2620 1524 y Fm(\000)p Fu(\025)p Fo(\()p Fu(t)2762 1532 y Fi(1)2794 1524 y Fo(+)p Fu(T)9 b Fo(\))2933 1502 y Fw(\()p Fs(\025)p Fw(\()p Fs(t)3075 1514 y Fo(1)3132 1502 y Fw(+)18 b Fs(T)12 b Fw(\)\))3340 1471 y Fu(n)3381 1482 y Fi(0)p Fy(;)p Fe(j)p Fi(\003)p Fe(j)p 2933 1539 579 4 v 3115 1615 a Fs(n)3165 1630 y Fo(0)p Fu(;)p Fm(j)p Fo(\003)p Fm(j)3307 1615 y Fw(!)3522 1558 y Fs(A)3584 1508 y Fu(n)3625 1519 y Fi(0)p Fy(;)p Fe(j)p Fi(\003)p Fe(j)3584 1586 y Fo(0)p Fu(;)p Fm(j)p Fo(\003)p Fm(j)3757 1391 y Fk(1)3757 1540 y(A)274 1877 y Fw(=)83 b Ft(L)479 1843 y Fu(T)479 1898 y Fm(C)515 1906 y Fy(g)550 1898 y Fu(;T)3709 1877 y Fw(\(220\))118 2060 y(W)-7 b(e)40 b(ha)n(v)n(e)e(in)n(terc)n(hanged)g(summation)h (with)h(and)f(in)n(tegration)g(and)g(m)n(ultiplication)g(and)h(applied) f(the)h(follo)n(wing)118 2160 y(computations)34 b(to)h(eac)n(h)e(set)i (of)g(factors)e(b)r(elonging)h(to)h(the)g(same)f(maximal)g Fs(h)p Fw(-strip)g(that)g(do)r(es)h(not)f(end)h(in)g(\(\003)23 b Ft(n)118 2259 y Fw(\003)176 2271 y Fo(1)213 2259 y Fw(\))13 b Ft(\002)g(f)p Fw(0)p Ft(g)p Fw(.)35 b(F)-7 b(or)25 b(notational)f(simplicit)n(y)-7 b(,)25 b(let)h(the)f(maximal)g Fs(h)p Fw(-strip)f(under)h(consideration)e(co)n(v)n(er)h(the)h(time)h (in)n(terv)-5 b(als)118 2359 y(\()p Fs(t)180 2371 y Fo(0)218 2359 y Fs(;)14 b(t)285 2371 y Fo(1)322 2359 y Fw(\))p Fs(;)g(:)g(:)g(:)g(;)g Fw(\()p Fs(t)601 2371 y Fu(l)p Fm(\000)p Fo(1)711 2359 y Fs(;)g(t)778 2371 y Fu(l)804 2359 y Fw(\).)37 b(So)27 b(its)h(length)g(is)f Fs(t)1494 2371 y Fu(l)1538 2359 y Ft(\000)18 b Fs(t)1651 2371 y Fo(0)1688 2359 y Fw(.)37 b(And)28 b(let)g(the)g(op)r(erators)e Ft(Q)2629 2371 y Fu(q)2693 2359 y Fw(ha)n(v)n(e)g(exp)r(onen)n(ts)i Fs(n)3325 2371 y Fo(1)3362 2359 y Fs(;)14 b(:)g(:)g(:)f(;)h(n)3596 2371 y Fu(l)3622 2359 y Fw(:)868 2623 y Fk(X)779 2792 y Fy(n)816 2804 y Fi(0)848 2792 y Fy(;:::;n)980 2807 y(l)1004 2792 y Fe(\025)p Fi(0)755 2844 y Fy(n)792 2856 y Fi(0)824 2844 y(+)p Fy(:::)p Fi(+)p Fy(n)1004 2859 y(l)1027 2844 y Fe(\025)p Fi(1)1168 2598 y Fu(l)1125 2623 y Fk(Y)1124 2800 y Fu(i)p Fo(=1)1246 2585 y Fk(\022)1307 2702 y Fs(e)1346 2668 y Fm(\000)p Fu(\025)p Fo(\()p Fu(t)1488 2676 y Fy(i)1514 2668 y Fm(\000)p Fu(t)1591 2676 y Fy(i)p Fe(\000)p Fi(1)1691 2668 y Fo(\))1731 2646 y Fw(\()p Fs(\025)p Fw(\()p Fs(t)1873 2658 y Fu(i)1920 2646 y Ft(\000)k Fs(t)2033 2658 y Fu(i)p Fm(\000)p Fo(1)2145 2646 y Fw(\)\))2209 2616 y Fu(n)2250 2624 y Fy(i)p 1731 2683 551 4 v 1956 2759 a Fs(n)2006 2771 y Fu(i)2033 2759 y Fw(!)2314 2702 y Ft(Q)2382 2714 y Fu(q)2419 2585 y Fk(\023)3709 2702 y Fw(\(221\))828 3053 y(=)1019 2949 y Fu(l)977 2974 y Fk(Y)976 3151 y Fu(i)p Fo(=1)1097 2911 y Fk( )1206 2949 y Fm(1)1179 2974 y Fk(X)1163 3150 y Fu(n)1204 3158 y Fy(i)1230 3150 y Fo(=0)1328 3053 y Fs(e)1367 3019 y Fm(\000)p Fu(\025)p Fo(\()p Fu(t)1509 3027 y Fy(i)1536 3019 y Fm(\000)p Fu(t)1613 3027 y Fy(i)p Fe(\000)p Fi(1)1712 3019 y Fo(\))1752 2997 y Fw(\()p Fs(\025)p Fw(\()p Fs(t)1894 3009 y Fu(i)1941 2997 y Ft(\000)g Fs(t)2054 3009 y Fu(i)p Fm(\000)p Fo(1)2167 2997 y Fw(\)\))2231 2967 y Fu(n)2272 2975 y Fy(i)p 1752 3034 V 1977 3110 a Fs(n)2027 3122 y Fu(i)2055 3110 y Fw(!)2336 3053 y Ft(Q)2404 3065 y Fu(q)2440 2911 y Fk(!)2525 3053 y Ft(\000)2651 2949 y Fu(l)2608 2974 y Fk(Y)2608 3151 y Fu(i)p Fo(=1)2729 3053 y Fs(e)2768 3019 y Fm(\000)p Fu(\025)p Fo(\()p Fu(t)2910 3027 y Fy(i)2936 3019 y Fm(\000)p Fu(t)3013 3027 y Fy(i)p Fe(\000)p Fi(1)3113 3019 y Fo(\))3166 3053 y Ft(Q)3234 3065 y Fu(q)828 3378 y Fw(=)976 3286 y Fk(\020)1026 3378 y Fw(1)g Ft(\000)g Fs(e)1208 3344 y Fm(\000)p Fu(\025)p Fo(\()p Fu(t)1350 3353 y Fy(l)1374 3344 y Fm(\000)p Fu(t)1451 3352 y Fi(0)1483 3344 y Fo(\))1513 3286 y Fk(\021)1576 3378 y Fs(Q)1642 3390 y Fu(q)3709 3378 y Fw(\(222\))118 3586 y(This)27 b(explains)f(the)h(app)r(earance)e (of)i(the)g(factor)f(\(195\))g(in)h(\(220\).)35 b(Recall)27 b(that)g(eac)n(h)e(op)r(erator)g Fs(A)3247 3598 y Fu(i;q)3354 3586 y Fw(in)i(\(220\))f(stands)118 3685 y(for)h(either)h Ft(Q)549 3697 y Fu(q)613 3685 y Fw(or)e Ft(R)784 3697 y Fu(q)822 3685 y Fw(.)36 b(So)28 b(w)n(e)f(can)g(replace)971 3916 y Fk(X)931 4094 y Fu(n)972 4102 y Fy(i;q)1046 4094 y Fm(\025)p Fo(0)1145 3995 y Fs(e)1184 3960 y Fm(\000)p Fu(\025)p Fo(\()p Fu(t)1326 3968 y Fy(i)1352 3960 y Fm(\000)p Fu(t)1429 3968 y Fy(i)p Fe(\000)p Fi(1)1529 3960 y Fo(\))1569 3939 y Fw(\()p Fs(\025)p Fw(\()p Fs(t)1711 3951 y Fu(i)1758 3939 y Ft(\000)18 b Fs(t)1871 3951 y Fu(i)p Fm(\000)p Fo(1)1984 3939 y Fw(\)\))2048 3908 y Fu(n)2089 3916 y Fy(i;q)p 1569 3976 599 4 v 1792 4052 a Fs(n)1842 4064 y Fu(i;q)1922 4052 y Fw(!)2196 3995 y Ft(Q)2264 3960 y Fu(n)2305 3968 y Fy(i;q)2264 4015 y Fu(q)2466 3995 y Fw(=)83 b Ft(Q)2682 4007 y Fu(q)3709 3995 y Fw(\(223\))969 4274 y Fk(X)929 4453 y Fu(n)970 4461 y Fy(i;q)1044 4453 y Fm(\025)p Fo(0)1143 4353 y Fs(e)1182 4319 y Fm(\000)p Fu(\025)p Fo(\()p Fu(t)1324 4327 y Fy(i)1350 4319 y Fm(\000)p Fu(t)1427 4327 y Fy(i)p Fe(\000)p Fi(1)1526 4319 y Fo(\))1566 4297 y Fw(\()p Fs(\025)p Fw(\()p Fs(t)1708 4309 y Fu(i)1756 4297 y Ft(\000)18 b Fs(t)1869 4309 y Fu(i)p Fm(\000)p Fo(1)1981 4297 y Fw(\)\))2045 4267 y Fu(n)2086 4275 y Fy(i;q)p 1566 4334 V 1789 4410 a Fs(n)1839 4422 y Fu(i;q)1919 4410 y Fw(!)2193 4353 y Ft(R)2263 4319 y Fu(n)2304 4327 y Fy(i;q)2263 4374 y Fu(q)2466 4353 y Fw(=)83 b Ft(R)2684 4365 y Fu(q)2721 4353 y Fw(\()p Fs(t)2783 4365 y Fu(i)2829 4353 y Ft(\000)18 b Fs(t)2942 4365 y Fu(i)p Fm(\000)p Fo(1)3055 4353 y Fw(\))622 b(\(224\))118 4631 y(and)28 b(get)f(\(220\).)118 4731 y(So)g(w)n(e)h(ha)n(v)n(e)e(seen)h(that)h (\(214\))f(is)h(equal)f(to)g Fs(\031)1575 4743 y Fo(\003)1620 4751 y Fi(1)1676 4731 y Ft(\016)18 b(L)1793 4700 y Fu(T)1793 4753 y(S;)p Fo(\003)1903 4731 y Fw(,)27 b(as)g(de\014ned)h(in)g (\(202\).)3808 4830 y Fh(2)118 5001 y Fw(F)-7 b(or)27 b(the)h(represen)n(tation)e(of)h(the)h(transfer)f(op)r(erator)f(for)h (the)g(in\014nite)i(dimensional)e(system)g(w)n(e)g(need)h(the)g(follo)n (wing)118 5100 y(de\014nition.)118 5283 y Fn(De\014nition)j(3.9)40 b Fw(Let)g(\003)946 5295 y Fo(1)983 5283 y Fs(;)14 b Fw(\003)1078 5295 y Fo(2)1158 5283 y Ft(\022)43 b Fw(\003)g Ft(2)h(F)34 b(n)27 b(f;g)38 b Fw(and)h Ft(C)2010 5295 y Fu(g)2092 5283 y Ft(2)44 b Fw(Conf)2364 5295 y(g)2409 5283 y(\(\003)p Fs(;)14 b Fw(\003)2594 5295 y Fo(1)2631 5283 y Fw(\).)74 b(W)-7 b(e)40 b(sa)n(y)f(that)h Ft(C)3307 5295 y Fl(g)3395 5283 y Fn(lies)k(in)i(\003)3768 5295 y Fl(2)3849 5283 y Fw(if)118 5401 y(\003\()p Ft(C)252 5413 y Fu(g)290 5401 y Fw(\))26 b Ft([)f Fw(\003)486 5413 y Fo(1)562 5401 y Ft(\022)39 b Fw(\003)724 5413 y Fo(2)761 5401 y Fw(.)67 b(\(Recall)37 b(that)h(\003\()p Ft(C)1467 5413 y Fu(g)1505 5401 y Fw(\))g(w)n(as)e(de\014ned)i(in)g (\(193\).\))66 b(Let)38 b(b)r(oth)f Ft(C)2866 5413 y Fu(g)2944 5401 y Ft(2)j Fw(Conf)3212 5413 y(g)3258 5401 y(\(\003)p Fs(;)14 b Fw(\003)3443 5413 y Fo(1)3479 5401 y Fw(\))38 b(and)3733 5380 y Fk(e)3720 5401 y Ft(C)3764 5413 y Fu(g)3842 5401 y Ft(2)1966 5682 y Fw(41)p eop %%Page: 42 42 42 41 bop 118 354 a Fw(Conf)291 366 y(g)337 354 y(\()375 333 y Fk(e)369 354 y Fw(\003)p Fs(;)14 b Fw(\003)522 366 y Fo(1)558 354 y Fw(\))27 b(lie)f(in)h(\003)15 b Ft(\\)972 333 y Fk(e)967 354 y Fw(\003)o(.)37 b(If)26 b(further)h Ft(C)1488 366 y Fu(g)1552 354 y Fw(and)1725 333 y Fk(e)1712 354 y Ft(C)1756 366 y Fu(g)1821 354 y Fw(ha)n(v)n(e)e(the)i(same)e(gum)i(tree)f(with)g(the)h(same)f(linear)f (order)g(and)h(if)118 472 y(they)c(ha)n(v)n(e)e(the)i(same)g Fs(r)r Fw(-strips)f(then)h(w)n(e)g(sa)n(y)e(that)i Ft(C)1767 484 y Fu(g)1827 472 y Fw(is)g Fn(equiv)-5 b(alen)m(t)21 b Fw(to)2458 451 y Fk(e)2445 472 y Ft(C)2489 484 y Fu(g)2527 472 y Fw(.)35 b(Then)22 b(w)n(e)f(ha)n(v)n(e)g(de\014ned)h(an)f(equiv) -5 b(alence)118 585 y(relation)27 b(and)g(further,)h(for)f Ft(C)1058 597 y Fu(g)1124 585 y Fw(equiv)-5 b(alen)n(t)27 b(to)1632 564 y Fk(e)1620 585 y Ft(C)1664 597 y Fu(g)1702 585 y Fw(,)h(w)n(e)f(ha)n(v)n(e:)960 858 y(Simplex\()p Ft(C)1324 870 y Fu(g)1363 858 y Fs(;)14 b(T)e Fw(\))82 b(=)h(Simplex\()2056 837 y Fk(e)2043 858 y Ft(C)2087 870 y Fu(g)2126 858 y Fs(;)14 b(T)e Fw(\))82 b(for)27 b(all)g Fs(T)34 b Ft(2)24 b Fw(\(0)p Fs(;)14 b Ft(1)p Fw(])p Fs(;)727 b Fw(\(225\))928 982 y Fs(\031)975 994 y Fo(\003)1020 1002 y Fi(1)1076 982 y Ft(\016)18 b(L)1193 948 y Fu(T)1193 1003 y Fm(C)1229 1011 y Fy(g)1264 1003 y Fu(;)p Fl(t)1335 982 y Ft(\016)g Fs(\031)1442 994 y Fo(\003)1575 982 y Fw(=)83 b Fs(\031)1770 994 y Fo(\003)1815 1002 y Fi(1)1871 982 y Ft(\016)18 b(L)1988 948 y Fu(T)1998 1003 y Fd(e)1988 1018 y Fm(C)2024 1026 y Fy(g)2059 1018 y Fu(;)p Fl(t)2130 982 y Ft(\016)g Fs(\031)2241 995 y Fd(e)2237 1010 y Fo(\003)2370 982 y Fw(for)27 b(all)g Fn(t)d Ft(2)f Fw(Simplex\()p Ft(C)3115 994 y Fu(g)3154 982 y Fs(;)14 b(T)e Fw(\))425 b(\(226\))733 1141 y(and)110 b Fs(\031)1024 1153 y Fo(\003)1069 1161 y Fi(1)1125 1141 y Ft(\016)18 b(L)1242 1107 y Fu(T)1242 1162 y Fm(C)1278 1170 y Fy(g)1335 1141 y Ft(\016)g Fs(\031)1442 1153 y Fo(\003)1575 1141 y Fw(=)83 b Fs(\031)1770 1153 y Fo(\003)1815 1161 y Fi(1)1871 1141 y Ft(\016)18 b(L)1988 1107 y Fu(T)1998 1162 y Fd(e)1988 1177 y Fm(C)2024 1185 y Fy(g)2081 1141 y Ft(\016)g Fs(\031)2192 1154 y Fd(e)2188 1169 y Fo(\003)2238 1141 y Fs(:)1448 b Fw(\(227\))118 1346 y(\(226\))24 b(and)g(\(227\))f (sa)n(y)h(that)g(the)h(op)r(erators)d(in)j Fs(L)p Fw(\()p Ft(H)1777 1358 y Fu(#)1821 1346 y Fs(;)14 b Ft(H)1928 1358 y Fo(\003)1973 1366 y Fi(1)2010 1346 y Fw(\))25 b(are)e(the)i(same.)35 b(W)-7 b(e)25 b(de\014ne)f(b)n(y)g Fn(Conf)3268 1358 y(g)3320 1346 y Fw(\()p Fv(Z)3414 1315 y Fl(d)3454 1346 y Fs(;)14 b Fn(\003)3558 1358 y Fl(1)3599 1346 y Fw(\))25 b(the)g(set)118 1445 y(of)30 b(equiv)-5 b(alence)30 b(classes.)43 b(Because)30 b(of)g(\(225\))f(and)h(\(226\))g (the)g(simplices)g(and)g(op)r(erators)f(for)g(eac)n(h)h(equiv)-5 b(alen)n(t)30 b(class)118 1545 y(can)d(b)r(e)h(de\014ned)g(as)f(b)r (eing)h(equal)f(to)g(the)h(corresp)r(onding)e(ob)5 b(ject)27 b(for)h(an)n(y)e(represen)n(tativ)n(e.)118 1644 y(W)-7 b(e)23 b(will)g(write)g Fs(\031)663 1656 y Fo(\003)708 1664 y Fi(1)754 1644 y Ft(\016)9 b(L)862 1614 y Fu(T)862 1667 y Fm(C)898 1675 y Fy(g)932 1667 y Fu(;)p Fl(t)1008 1644 y Fw(instead)23 b(of)f Fs(\031)1426 1656 y Fo(\003)1471 1664 y Fi(1)1517 1644 y Ft(\016)9 b(L)1625 1614 y Fu(T)1625 1667 y Fm(C)1661 1675 y Fy(g)1695 1667 y Fu(;)p Fl(t)1757 1644 y Ft(\016)g Fs(\031)1855 1656 y Fo(\003)1927 1644 y Fw(and)23 b Fs(\031)2131 1656 y Fo(\003)2176 1664 y Fi(1)2222 1644 y Ft(\016)9 b(L)2330 1614 y Fu(T)2330 1667 y Fm(C)2366 1675 y Fy(g)2427 1644 y Fw(instead)23 b(of)f Fs(\031)2845 1656 y Fo(\003)2890 1664 y Fi(1)2936 1644 y Ft(\016)9 b(L)3044 1614 y Fu(T)3044 1667 y Fm(C)3080 1675 y Fy(g)3127 1644 y Ft(\016)g Fs(\031)3225 1656 y Fo(\003)3297 1644 y Fw(for)23 b(the)g(op)r(erators)118 1744 y(from)k Ft(H)384 1756 y Fu(#)456 1744 y Fw(to)h Ft(H)628 1756 y Fo(\003)673 1764 y Fi(1)710 1744 y Fw(.)118 1917 y Fn(Theorem)j(3.1)138 b Fj(1.)43 b(Ther)l(e)c(ar)l(e)g Fw(0)g Fs(<)1487 1895 y Fw(~)1476 1917 y Fs(#)h(<)e(#)h(<)g Fw(1)f Fj(such)h(that)f(for)i(su\016ciently)f(smal)t(l)g Fs(\017)p Fj(,)i(lar)l(ge)f Fs(c)3461 1929 y Fu(g)3538 1917 y Fj(and)f(every)326 2017 y Fs(T)34 b Ft(2)23 b Fw([0)p Fs(;)14 b Ft(1)p Fw(])29 b Fj(we)h(c)l(an)g(de\014ne)g(an)g(op) l(er)l(ator)h Ft(L)1745 1987 y Fu(T)1745 2039 y(S)1827 2017 y Fj(fr)l(om)f Ft(H)2093 2029 y Fu(#)2167 2017 y Fj(to)g Ft(H)2346 2027 y Fo(~)2337 2042 y Fu(#)2411 2017 y Fj(by)1403 2247 y Fs(\031)1450 2259 y Fo(\003)1495 2267 y Fi(1)1551 2247 y Ft(\016)18 b(L)1668 2213 y Fu(T)1668 2268 y(S)1720 2247 y Fs(\036)24 b Fw(=)2077 2168 y Fk(X)1880 2352 y Fm(C)1916 2360 y Fy(g)1951 2352 y Fm(2)p Fo(Conf)2134 2360 y(g)2171 2352 y(\()p Fp(Z)2242 2335 y Fy(d)2271 2352 y Fu(;)p Fo(\003)2336 2360 y Fi(1)2368 2352 y Fo(\))2408 2247 y Fs(\031)2455 2259 y Fo(\003)2500 2267 y Fi(1)2556 2247 y Ft(\016)18 b(L)2673 2213 y Fu(T)2673 2268 y Fm(C)2709 2276 y Fy(g)2748 2247 y Fs(\036:)889 b Fw(\(228\))326 2563 y Fj(Ther)l(e)30 b(is)g(a)h Fs(T)775 2575 y Fo(0)834 2563 y Fs(>)23 b Fw(0)29 b Fj(such)h(that)g(for)g Fs(T)35 b Ft(\025)22 b Fs(T)1704 2575 y Fo(0)1771 2563 y Fj(the)30 b(op)l(er)l(ator)g Ft(L)2292 2533 y Fu(T)2292 2586 y(S)2375 2563 y Fj(maps)g Ft(H)2661 2575 y Fu(#)2735 2563 y Fj(into)g Ft(H)2977 2575 y Fu(#)3021 2563 y Fj(.)326 2694 y Ft(L)383 2664 y Fu(T)383 2717 y(S)465 2694 y Fj(is)g(the)g(tr)l(ansfer)g(op)l (er)l(ator,)h(r)l(estricte)l(d)f(to)f Ft(H)1887 2664 y Fu(bv)1886 2718 y(#)1956 2694 y Fj(,)h(for)h(the)f(kernel)g Fs(K)2606 2664 y Fu(T)2600 2717 y(S)2658 2694 y Fj(,)g(i.e.)1505 2833 y Fk(Z)1551 3022 y Fu(M)1638 2946 y Fs(d\026)14 b Fw(\()p Fs(K)1854 2912 y Fu(T)1848 2967 y(S)1906 2946 y Fs( )s Fw(\))g Fs(\036)24 b Fw(=)2170 2833 y Fk(Z)2216 3022 y Fu(M)2303 2946 y Fs(d\026)g( )d Fw(\()p Ft(L)2584 2912 y Fu(T)2584 2967 y(S)2637 2946 y Fs(\036)p Fw(\))991 b(\(229\))326 3176 y Fj(for)30 b(al)t(l)h Fs( )26 b Ft(2)e(C)785 3146 y Fo(0)821 3176 y Fw(\()p Fs(M)9 b Fw(\))30 b Fj(and)h Fs(\036)23 b Ft(2)g(H)1388 3146 y Fu(bv)1387 3199 y(#)1457 3176 y Fj(.)216 3338 y(2.)43 b(The)30 b(family)i Fw(\()p Ft(L)837 3308 y Fu(T)837 3361 y(S)890 3338 y Fw(\))922 3350 y Fu(T)9 b Fm(\025)p Fo(0)1089 3338 y Fj(in)30 b Fs(L)p Fw(\()p Ft(H)1350 3350 y Fu(#)1394 3338 y Fw(\))g Fj(c)l(onver)l(ges)g(exp)l(onential)t(ly)h(fast)f(to)g Ft(L)2643 3308 y Fm(1)2643 3361 y Fu(S)2714 3338 y Fj(:)1549 3569 y Ft(kL)1648 3534 y Fm(1)1648 3589 y Fu(S)1737 3569 y Ft(\000)18 b(L)1877 3534 y Fu(T)1877 3589 y(S)1929 3569 y Ft(k)1971 3584 y Fu(L)p Fo(\()p Fm(H)2100 3593 y Fy(#)2139 3584 y Fu(;)p Fm(H)2224 3592 y Fi(~)2216 3603 y Fy(#)2255 3584 y Fo(\))2308 3569 y Ft(\024)23 b Fs(c)2432 3581 y Fo(5)2469 3569 y Fs(e)2508 3534 y Fm(\000)p Fu(c)2590 3542 y Fi(6)2622 3534 y Fu(T)3709 3569 y Fw(\(230\))326 3743 y Fj(for)34 b(some)f(p)l(ositive)i(c)l (onstants)e Fs(c)1385 3755 y Fo(5)1422 3743 y Fj(,)h Fs(c)1517 3755 y Fo(6)1554 3743 y Fj(.)50 b(F)-6 b(or)33 b(su\016ciently)h(lar)l(ge)g Fs(T)44 b Fj(estimate)33 b(\(230\))i(holds)g(also)f(in)f(the)h(norm)326 3843 y(of)h Fs(L)p Fw(\()p Ft(H)587 3855 y Fu(#)631 3843 y Fw(\))p Fj(.)53 b(So)34 b(among)h(the)g(pr)l(ob)l(ability)h(me)l(asur)l(es)e(c) l(orr)l(esp)l(onding)i(to)f(elements)f(in)g Ft(H)3199 3855 y Fu(#)3278 3843 y Fj(ther)l(e)g(is)h(a)g(unique)326 3943 y Fs(K)403 3913 y Fu(T)397 3966 y(S)454 3943 y Fj(-invariant)f(pr) l(ob)l(ability)i(me)l(asur)l(e)d Fs(\027)1622 3913 y Fm(\003)1694 3943 y Fj(on)h Fs(M)9 b Fj(,)34 b(say)g(c)l(orr)l(esp)l (onding)h(to)e Fs(\027)i Ft(2)30 b(H)2978 3955 y Fu(#)3023 3943 y Fj(.)50 b(The)34 b(op)l(er)l(ator)g Ft(L)3658 3913 y Fm(1)3658 3966 y Fu(S)3762 3943 y Fj(is)g(a)326 4042 y(pr)l(oje)l(ction)d(onto)f Fw(span)13 b Fs(\027)5 b Fj(:)1846 4273 y Ft(L)1903 4239 y Fm(1)1903 4293 y Fu(S)1974 4273 y Fs(\036)24 b Fw(=)e Fs(\026)p Fw(\()p Fs(\036)p Fw(\))14 b Fs(\027)q(:)1333 b Fw(\(231\))118 4446 y Fn(Pro)s(of)73 b Fw(The)33 b(in\014nite)i(sum)f(on)f(the)h(rhs)f (of)h(\(228\))f(con)n(v)n(erges)e(as)i(the)h(pro)n(v)n(e)e(of)i (estimate)f(\(203\))g(applies)g(literally)118 4545 y(to)e(the)g(case)f (\003)f(=)f Fv(Z)789 4515 y Fu(d)822 4545 y Fw(.)47 b(Next)31 b(w)n(e)g(w)n(an)n(t)f(to)h(sho)n(w)f(that)h Fs(\031)1974 4557 y Fo(\003)2019 4565 y Fi(1)2077 4545 y Ft(\016)20 b(L)2196 4515 y Fu(T)2196 4568 y(S)2280 4545 y Fw(is)31 b(the)g(limit)h(of)f Fs(\031)2860 4557 y Fo(\003)2905 4565 y Fi(1)2963 4545 y Ft(\016)20 b(L)3082 4515 y Fu(T)3082 4568 y(S;)p Fo(\003)3223 4545 y Fw(\(as)30 b(\003)f Ft(!)f Fv(Z)3620 4515 y Fu(d)3652 4545 y Fw(\).)48 b(The)118 4658 y(di\013erence)32 b(b)r(et)n(w)n(een)f(these)h(t)n(w)n(o)f(op)r (erators)f(is)i(due)g(to)f(con\014gurations)f Ft(C)2506 4670 y Fu(g)2576 4658 y Fw(in)i(Conf)2850 4670 y(g)2896 4658 y(\(\003)p Fs(;)14 b Fw(\003)3081 4670 y Fo(1)3118 4658 y Fw(\))32 b(or)f(in)h(Conf)3562 4670 y(g)3607 4658 y(\()p Fv(Z)3701 4628 y Fu(d)3734 4658 y Fs(;)14 b Fw(\003)3829 4670 y Fo(1)3865 4658 y Fw(\))118 4758 y(with)40 b(\003\()p Ft(C)453 4770 y Fu(g)491 4758 y Fw(\))k Ft(6\032)e Fw(\003.)72 b(F)-7 b(or)39 b(these)h(w)n(e)f(can)g(split)h(in)g(estimate)f(\(204\)) g(the)h(factor)e(that)i(arises)e(from)h(the)h(deca)n(y)f(of)118 4858 y(in)n(teraction)27 b(in)h(the)g(follo)n(wing)e(w)n(a)n(y)g (\(whic)n(h)i(is)g(the)g(same)f(as)g(the)h(splitting)g(\(110\))e(in)i ([13)o(]\).)766 5111 y(exp)907 4969 y Fk( )973 5111 y Ft(\000)p Fs(c)1074 5123 y Fu(g)1153 5007 y Fm(1)1126 5032 y Fk(X)1126 5211 y Fu(k)q Fo(=1)1260 5111 y Fs(k)1306 5076 y Fu(d)1345 5111 y Fs(n)1395 5123 y Fu(\014)s(;k)1496 4969 y Fk(!)1585 5111 y Ft(\024)22 b Fw(exp)1813 4969 y Fk( )1879 5111 y Ft(\000)r Fw(~)-44 b Fs(c)1980 5123 y Fu(g)2059 5007 y Fm(1)2032 5032 y Fk(X)2032 5211 y Fu(k)q Fo(=1)2166 5111 y Fs(k)2212 5076 y Fu(d)2251 5111 y Fs(n)2301 5123 y Fu(\014)s(;k)2402 4969 y Fk(!)2495 5111 y Fw(exp\()p Ft(\000)p Fs(\030)18 b Fw(dist\(\003)2997 5123 y Fo(1)3034 5111 y Fs(;)c Fw(\003)3129 5076 y Fu(C)3185 5111 y Fw(\)\))460 b(\(232\))118 5333 y(with)32 b(a)e(suitably)h(c)n (hosen)f Fs(\030)j(>)c Fw(0)i(suc)n(h)f(that)j(~)-43 b Fs(c)1618 5345 y Fu(g)1685 5333 y Fw(=)28 b Fs(c)1814 5345 y Fu(g)1874 5333 y Ft(\000)20 b Fs(\030)33 b(>)c Fw(0.)47 b(\(Note)31 b(that)g(w)n(e)g(can)g(c)n(ho)r(ose)f Fs(\030)35 b Fw(so)30 b(small)h(that)g(the)118 5433 y(estimates,)c (formerly)g(done)g(with)h Fs(c)1262 5445 y Fu(g)1329 5433 y Fw(w)n(ork)e(with)k(~)-44 b Fs(c)1757 5445 y Fu(g)1823 5433 y Fw(instead)28 b(as)f(w)n(ell.\))36 b(So)28 b(w)n(e)f(can)g (estimate)1966 5682 y(42)p eop %%Page: 43 43 43 42 bop 1158 493 a Fw(~)1147 515 y Fs(#)1196 481 y Fm(j)p Fo(\003)1261 489 y Fi(1)1293 481 y Fm(j)1317 515 y Ft(k)p Fs(\031)1406 527 y Fo(\003)1451 535 y Fi(1)1507 515 y Ft(\016)18 b(L)1624 481 y Fu(T)1624 535 y(S;)p Fo(\003)1752 515 y Ft(\000)g Fs(\031)1882 527 y Fo(\003)1927 535 y Fi(1)1983 515 y Ft(\016)g(L)2100 481 y Fu(T)2100 535 y(S)2152 515 y Ft(k)2194 530 y Fu(L)p Fo(\()p Fm(H)2323 539 y Fy(#)2362 530 y Fu(;)p Fm(H)2439 538 y Fi(\003)2477 550 y(1)2513 530 y Fo(\))3709 515 y Fw(\(233\))1230 671 y Ft(\024)82 b Fw(2)1613 592 y Fk(X)1443 771 y Fe(C)1475 779 y Fy(g)1510 771 y Fe(2)p Fo(Conf)p Fi(\()p Fa(Z)1743 759 y Fy(d)1772 771 y(;)p Fi(\003)1829 783 y(1)1861 771 y(\))p Fy(;)1556 827 y Fi(\003\()p Fe(C)1648 835 y Fy(g)1684 827 y Fi(\))p Fe(6\032)p Fi(\003)1937 649 y Fw(~)1926 671 y Fs(#)1975 637 y Fm(j)p Fo(\003)2040 645 y Fi(1)2073 637 y Fm(j)2097 671 y Ft(k)p Fs(\031)2186 683 y Fo(\003)2231 691 y Fi(1)2286 671 y Ft(\016)18 b(L)2403 637 y Fu(T)2403 691 y Fm(C)2439 699 y Fy(g)2478 671 y Ft(k)2520 686 y Fu(L)p Fo(\()p Fm(H)2649 695 y Fy(#)2688 686 y Fu(;)p Fm(H)2765 694 y Fi(\003)2803 706 y(1)2839 686 y Fo(\))1230 959 y Ft(\024)82 b Fs(c)1413 971 y Fo(7)1478 959 y Fw(exp\()p Ft(\000)p Fs(\030)18 b Fw(dist\(\003)1980 971 y Fo(1)2017 959 y Fs(;)c Fw(\003)2112 925 y Fu(C)2168 959 y Fw(\)\))p Fs(:)118 1120 y Fw(Next)28 b(w)n(e)f(sho)n(w)g(\(229\))g(for)g(the)h (sp)r(ecial)f(case)g(that)h Fs( )i Fw(dep)r(ends)e(only)g(on)f(the)h (\003)2666 1132 y Fo(1)2703 1120 y Fw(-co)r(ordinates,)e(using)h (\(212\):)1012 1289 y Fk(Z)1058 1477 y Fu(M)1145 1402 y Fs(d\026)p Fw(\()p Fn(z)p Fw(\))20 b(\()p Fs(K)1473 1367 y Fu(T)1467 1422 y(S)1525 1402 y Fs( )s Fw(\)\()p Fn(z)p Fw(\))14 b Fs(\036)p Fw(\()p Fn(z)p Fw(\))1820 b(\(234\))1095 1623 y(=)117 b(lim)1242 1683 y Fo(\003)p Fm(!)p Fp(Z)1399 1666 y Fy(d)1441 1510 y Fk(Z)1487 1699 y Fu(M)1575 1623 y Fs(d\026)1668 1589 y Fo(\003)1718 1623 y Fw(\()p Fn(z)1792 1635 y Fo(\003)1842 1623 y Fw(\))23 b(\()p Fs(K)2006 1589 y Fu(T)2000 1644 y(S;)p Fo(\003)2110 1623 y Fs( )s Fw(\)\()p Fn(z)2273 1635 y Fo(\003)2324 1623 y Fw(\))18 b Fs(\036)2423 1635 y Fo(\003)2473 1623 y Fw(\()p Fn(z)2547 1635 y Fo(\003)2598 1623 y Fw(\))1095 1845 y(=)117 b(lim)1242 1905 y Fo(\003)p Fm(!)p Fp(Z)1399 1888 y Fy(d)1441 1732 y Fk(Z)1487 1921 y Fo(\()p Fu(S)1557 1904 y Fi(1)1590 1921 y Fo(\))1616 1904 y Fi(\003)1676 1845 y Fs(d\026)1769 1811 y Fo(\003)1819 1845 y Fw(\()p Fn(z)1893 1857 y Fl(\003)1950 1845 y Fw(\))23 b Fs( )s Fw(\()p Fn(z)2136 1857 y Fo(\003)2186 1845 y Fw(\))c(\()p Ft(L)2326 1811 y Fu(T)2326 1866 y(S;)p Fo(\003)2436 1845 y Fs(\036)2485 1857 y Fo(\003)2535 1845 y Fw(\)\()p Fn(z)2641 1857 y Fo(\003)2692 1845 y Fw(\))1095 2082 y(=)117 b(lim)1242 2141 y Fo(\003)p Fm(!)p Fp(Z)1399 2124 y Fy(d)1441 1969 y Fk(Z)1487 2157 y Fo(\()p Fu(S)1557 2141 y Fi(1)1590 2157 y Fo(\))1616 2139 y Fi(\003)1654 2151 y(1)1708 2082 y Fs(d\026)1801 2047 y Fo(\003)1846 2055 y Fi(1)1883 2082 y Fw(\()p Fn(z)1957 2094 y Fl(\003)2014 2082 y Fw(\))24 b Fs( )s Fw(\()p Fn(z)2201 2094 y Fo(\003)2246 2102 y Fi(1)2283 2082 y Fw(\))19 b(\()p Fs(\031)2413 2094 y Fo(\003)2458 2102 y Fi(1)2514 2082 y Ft(\016)f(L)2631 2047 y Fu(T)2631 2102 y(s)2684 2082 y Fs(\036)2733 2094 y Fo(\003)2783 2082 y Fw(\)\()p Fn(z)2889 2094 y Fo(\003)2934 2102 y Fi(1)2972 2082 y Fw(\))1095 2318 y(=)1242 2205 y Fk(Z)1288 2394 y Fu(M)1376 2318 y Fs(d\026)p Fw(\()p Fn(z)p Fw(\))24 b Fs( )s Fw(\()p Fn(z)p Fw(\))c(\()p Ft(L)1871 2284 y Fu(T)1871 2339 y(S)1924 2318 y Fs(\036)p Fw(\))-14 b(\()p Fn(z)p Fw(\))p Fs(:)118 2530 y Fw(W)-7 b(e)27 b(conclude)f(\(229\))f(for)h(general)f Fs( )h Ft(2)d(C)1433 2500 y Fo(0)1470 2530 y Fw(\()p Fs(M)9 b Fw(\))27 b(b)n(y)f(appro)n(ximating)e(it)j(b)n(y)f Fs( )2567 2542 y Fo(\003)2612 2550 y Fi(1)2675 2530 y Fw(\(cf.)h(\(139\)\),)f(dep)r(ending)h(only)e(on)h(the)118 2630 y(\003)176 2642 y Fo(1)213 2630 y Fw(-co)r(ordinates)g(and)h (using)h(con)n(tin)n(uit)n(y)f(wrt)g Fs( )k Fw(of)c(b)r(oth)h(sides)g (of)f(\(229\).)36 b(So)28 b(1.)f(is)g(pro)n(v)n(ed.)118 2730 y(Next)i(w)n(e)f(sho)n(w)g(\(230\).)39 b(W)-7 b(e)29 b(note)f(that)h(for)f(\003)1600 2742 y Fo(1)1662 2730 y Fw(=)c Ft(;)k Fw(the)h(lhs)g(\(235\))e(in)i(the)g(follo)n(wing)e (estimate)i(is)f(equal)g(to)h(zero)e(as)118 2829 y(b)r(oth)f(transfer)e (op)r(erators)g(preserv)n(e)f(the)j(Leb)r(esgue)f(in)n(tegral)f(\()p Fs(\026)h Fw(is)h(a)e(`left)j(eigen)n(v)n(ector')c(with)j(eigen)n(v)-5 b(alue)24 b(1.\))36 b(So)25 b(w)n(e)118 2929 y(only)i(ha)n(v)n(e)g(to)g (consider)g(the)h(case)e Ft(j)p Fw(\003)1316 2941 y Fo(1)1353 2929 y Ft(j)e(\025)e Fw(1.)521 3167 y(~)510 3189 y Fs(#)559 3155 y Fm(j)p Fo(\003)624 3163 y Fi(1)657 3155 y Fm(j)680 3189 y Ft(k)p Fs(\031)769 3201 y Fo(\003)814 3209 y Fi(1)870 3189 y Ft(\016)c(L)987 3155 y Fm(1)987 3210 y Fu(S)1076 3189 y Ft(\000)g Fs(\031)1206 3201 y Fo(\003)1251 3209 y Fi(1)1307 3189 y Ft(\016)f(L)1423 3155 y Fu(T)1423 3210 y(S)1476 3189 y Ft(k)1518 3204 y Fu(L)p Fo(\()p Fm(H)1647 3213 y Fy(#)1686 3204 y Fu(;)p Fm(H)1763 3212 y Fi(\003)1801 3224 y(1)1837 3204 y Fo(\))3709 3189 y Fw(\(235\))593 3345 y Ft(\024)751 3323 y Fw(~)741 3345 y Fs(#)790 3311 y Fm(j)p Fo(\003)855 3319 y Fi(1)887 3311 y Fm(j)1103 3266 y Fk(X)925 3450 y Fm(C)961 3458 y Fy(g)996 3450 y Fm(2)p Fo(Conf)o(\()p Fp(Z)1249 3434 y Fy(d)1278 3450 y Fu(;)p Fo(\003)1343 3458 y Fi(1)1375 3450 y Fo(\))1415 3345 y Ft(k)p Fs(\031)1504 3357 y Fo(\003)1549 3365 y Fi(1)1604 3345 y Ft(\016)h(L)1721 3311 y Fm(1)1721 3366 y(C)1757 3374 y Fy(g)1815 3345 y Ft(\000)g Fs(\031)1945 3357 y Fo(\003)1990 3365 y Fi(1)2046 3345 y Ft(\016)g(L)2163 3311 y Fu(T)2163 3366 y Fm(C)2199 3374 y Fy(g)2238 3345 y Ft(k)2280 3360 y Fu(L)p Fo(\()p Fm(H)2409 3369 y Fy(#)2447 3360 y Fu(;)p Fm(H)2524 3368 y Fi(\003)2562 3380 y(1)2598 3360 y Fo(\))593 3586 y Ft(\024)751 3564 y Fw(~)741 3586 y Fs(#)790 3552 y Fm(j)p Fo(\003)855 3560 y Fi(1)887 3552 y Fm(j)925 3586 y Ft(kQ)1035 3598 y Fo(\003)1080 3606 y Fi(1)1135 3586 y Ft(\016)g Fs(\031)1242 3598 y Fo(\003)1287 3606 y Fi(1)1342 3586 y Ft(\000)g Fw(\(1)h Ft(\000)f Fs(e)1640 3552 y Fm(\000)p Fu(\025T)1783 3586 y Fw(\))1815 3552 y Fm(j)p Fo(\003)1880 3560 y Fi(1)1913 3552 y Fm(j)1937 3586 y Ft(Q)2005 3598 y Fo(\003)2050 3606 y Fi(1)2105 3586 y Ft(\016)g Fs(\031)2212 3598 y Fo(\003)2257 3606 y Fi(1)2294 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5433 y Ft(\016)g(L)1863 5403 y Fu(T)1863 5456 y Fm(C)1899 5464 y Fy(g)1967 5433 y Fw(where)27 b Ft(C)2251 5445 y Fu(g)2318 5433 y Fw(is)h(the)g(gum)g (con\014guration)f(that)h(has)g(only)f(long)1966 5682 y(43)p eop %%Page: 44 44 44 43 bop 118 354 a Fs(h)p Fw(-strips)26 b(\(no)h(branc)n(hings)e(or)h Fs(r)r Fw(-strips\).)37 b(The)27 b(second)f(summand)h(\(237\))f(tak)n (es)g(all)h Ft(C)2846 366 y Fu(g)2911 354 y Fw(in)n(to)g(accoun)n(t)f (that)h(reac)n(h)e(the)118 454 y(top.)46 b(So)31 b(all)f(sp)r (eci\014ed)h(con\014gurations)e(\()p Ft(C)1497 466 y Fu(g)1536 454 y Fs(;)14 b(T)7 b(;)14 b Fn(t)p Fw(\))30 b(ha)n(v)n(e)g(an)g Fs(r)r Fw(-strip)h(ending)g(at)f(time)i Ft(\000)p Fs(T)12 b Fw(.)45 b(All)31 b(\()p Ft(C)3320 466 y Fu(g)3358 454 y Fs(;)14 b Ft(1)p Fs(;)g Fn(t)p Fw(\))31 b(ha)n(v)n(e)f(an)118 554 y(in\014nitely)g(long)e Fs(r)r Fw(-strip)i(and)e(so)h(the)g(corresp)r(onding)e(op)r(erator)h (is)h(zero)f(\(cf.)i(Remark)e(3.4.3\))g(and)h(do)r(es)g(not)g(app)r (ear)118 653 y(in)h(\(237\).)42 b(The)30 b(last)f(three)g(summands,)h (\(238\),)g(\(239\))e(and)i(\(240\),)f(corresp)r(ond)f(to)i Ft(C)2895 665 y Fu(g)2963 653 y Fw(that)g(do)f(not)g(reac)n(h)g(the)h (top)118 753 y(and)23 b(do)g(not)h(consist)f(only)f(of)i Fs(h)p Fw(-strips.)34 b(That)24 b(implies)f(that)h(it)g(has)e(at)i (least)f(one)g(branc)n(hing)f(and)h(the)h(corresp)r(onding)118 852 y(domains)31 b(of)g(in)n(tegration,)f(Simplex)q(\()p Ft(C)1362 864 y Fu(g)1400 852 y Fs(;)14 b Ft(1)p Fw(\))32 b(and)f(Simplex\()p Ft(C)2113 864 y Fu(g)2152 852 y Fs(;)14 b(T)e Fw(\))o(,)32 b(are)e(not)h(degenerated)f(to)i(a)e(p)r(oin)n(t.)48 b(W)-7 b(e)32 b(divide)118 952 y(them)g(in)n(to)e(Simplex)q(\()p Ft(C)870 964 y Fu(g)908 952 y Fs(;)955 919 y Fu(T)p 955 933 49 4 v 963 981 a Fo(2)1013 952 y Fw(\))i(and)e(the)i(particular)e (complemen)n(ts.)46 b(The)31 b(reason)f(for)g(this)h(will)h(b)r(ecome)f (clear)f(when)118 1052 y(w)n(e)25 b(do)h(the)g(estimates.)36 b(In)25 b(\(238\))g(w)n(e)g(in)n(tegrate)g(the)h(norm)f(of)g(the)h(op)r (erator)e(di\013erence)i Fs(\031)3012 1064 y Fo(\003)3057 1072 y Fi(1)3108 1052 y Ft(\016)14 b(L)3221 1022 y Fm(1)3221 1075 y(C)3257 1083 y Fy(g)3311 1052 y Ft(\000)g Fs(\031)3437 1064 y Fo(\003)3482 1072 y Fi(1)3534 1052 y Ft(\016)g(L)3647 1022 y Fu(T)3647 1075 y Fm(C)3683 1083 y Fy(g)3748 1052 y Fw(o)n(v)n(er)118 1172 y(Simplex\()p Ft(C)482 1184 y Fu(g)521 1172 y Fs(;)568 1139 y Fu(T)p 568 1153 V 576 1200 a Fo(2)626 1172 y Fw(\))31 b(and)f(in)h(\(239\))e(and)h(\(240\))g (w)n(e)g(in)n(tegrate)f(the)i(norms)f(of)g(the)h(t)n(w)n(o)e(op)r (erators)g(separately)g(o)n(v)n(er)f(the)118 1271 y(particular)e (complemen)n(t)i(sets.)118 1371 y(No)n(w)f(w)n(e)g(estimate)h(eac)n(h)f (summand:)37 b(The)27 b(\014rst)h(summand)f(\(236\))g(is)h(estimated)f (b)n(y)866 1678 y(~)856 1700 y Fs(#)905 1665 y Fm(j)p Fo(\003)970 1673 y Fi(1)1002 1665 y Fm(j)1026 1700 y Fs(c)1062 1657 y Fm(j)p Fo(\003)1127 1665 y Fi(1)1160 1657 y Fm(j)1062 1725 y Fu(h)1184 1700 y Fw(\(1)18 b Ft(\000)g Fw(\(1)g Ft(\000)g Fs(e)1573 1665 y Fm(\000)p Fu(\025T)1716 1700 y Fw(\))1748 1665 y Fm(j)p Fo(\003)1813 1673 y Fi(1)1846 1665 y Fm(j)1870 1700 y Fw(\))83 b Ft(\024)g Fw(\()2176 1678 y(~)2165 1700 y Fs(#)14 b(c)2264 1712 y Fu(h)2307 1700 y Fw(\))2339 1665 y Fm(j)p Fo(\003)2404 1673 y Fi(1)2437 1665 y Fm(j)2476 1589 y(j)p Fo(\003)2541 1597 y Fi(1)2574 1589 y Fm(j)2475 1621 y Fk(X)2475 1800 y Fu(k)q Fo(=1)2609 1583 y Fk(\022)2670 1644 y Ft(j)p Fw(\003)2751 1656 y Fo(1)2788 1644 y Ft(j)2718 1757 y Fs(k)2811 1583 y Fk(\023)2872 1700 y Fw(\()p Fs(e)2943 1665 y Fm(\000)p Fu(\025T)3087 1700 y Fw(\))3119 1665 y Fu(k)3709 1700 y Fw(\(241\))1985 1924 y Ft(\024)83 b Fw(\()2176 1903 y(~)2165 1924 y Fs(#)14 b(c)2264 1936 y Fu(h)2321 1924 y Fw(\(1)k(+)g Fs(e)2535 1890 y Fm(\000)2597 1868 y Fi(1)p 2597 1877 29 3 v 2597 1910 a(2)2635 1890 y Fu(\025T)2727 1924 y Fw(\)\))2791 1890 y Fm(j)p Fo(\003)2856 1898 y Fi(1)2889 1890 y Fm(j)2913 1924 y Fs(e)2952 1890 y Fm(\000)3013 1868 y Fi(1)p 3013 1877 V 3013 1910 a(2)3051 1890 y Fu(\025T)1985 2062 y Ft(\024)83 b Fs(e)2172 2028 y Fm(\000)2233 2005 y Fi(1)p 2233 2014 V 2233 2048 a(2)2272 2028 y Fu(\025T)118 2220 y Fw(where)27 b(the)h(last)f(inequalit)n(y)h (holds)f(if)1348 2198 y(~)1337 2220 y Fs(#)h Fw(is)f(c)n(hosen)g (su\016cien)n(tly)g(small.)118 2319 y(F)-7 b(or)28 b(estimating)g(the)g (last)g(summand)h(\(240\))e(w)n(e)h(note)g(that)h(for)f Fn(t)c Ft(2)h Fw(Simplex\()p Ft(C)2695 2331 y Fu(g)2734 2319 y Fs(;)14 b(T)e Fw(\))18 b Ft(n)g Fw(Simplex\()p Ft(C)3306 2331 y Fu(g)3345 2319 y Fs(;)3392 2286 y Fu(T)p 3392 2300 49 4 v 3400 2348 a Fo(2)3450 2319 y Fw(\))29 b(the)f(sum)h(of)118 2428 y(the)k(lengths)g(of)f(all)g Fs(r)r Fw(-strips)h(of)g(\()p Ft(C)1253 2440 y Fu(g)1291 2428 y Fs(;)14 b(T)7 b(;)14 b Fn(t)p Fw(\))33 b(is)f(at)h(least)1927 2396 y Fu(T)p 1927 2410 V 1935 2457 a Fo(2)1985 2428 y Fw(.)52 b(\(This)33 b(is)f(b)r(ecause)g(\()p Ft(C)2763 2440 y Fu(g)2802 2428 y Fs(;)14 b(T)7 b(;)14 b 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Fw(-c)n(hain)g Fs(R)q Fw(.)118 3494 y(W)-7 b(e)28 b(get)170 3759 y(~)160 3781 y Fs(#)209 3746 y Fm(j)p Fo(\003)274 3754 y Fi(1)306 3746 y Fm(j)556 3702 y Fk(X)367 3881 y Fe(C)399 3889 y Fy(g)434 3881 y Fe(2)p Fo(Conf)611 3892 y(g)648 3881 y Fi(\()p Fa(Z)705 3869 y Fy(d)733 3881 y(;)p Fi(\003)790 3893 y(1)823 3881 y(\))p Fy(;)246 3952 y Fe(C)278 3960 y Fy(g)313 3952 y Fo(do)r(es)22 b(not)g(reac)n(h)f(the)h(top)o Fy(;)466 4006 y Fe(j)p Fy(n)522 4021 y(\014)561 4006 y Fi(\()p Fe(C)615 4014 y Fy(g)650 4006 y Fi(\))p Fe(j\025)p Fi(1)1019 3668 y Fk(Z)1100 3818 y Fo(Simplex)n Fi(\()p Fe(C)1384 3826 y Fy(g)1420 3818 y(;T)7 b Fi(\))1075 3887 y Fe(n)p Fo(Simplex)n Fi(\()p Fe(C)1390 3895 y Fy(g)1425 3887 y(;)1454 3871 y(T)p 1454 3878 42 3 v 1461 3909 a Fi(2)1506 3887 y(\))1556 3781 y Fs(\025)1604 3746 y Fm(j)p Fu(n)1665 3755 y Fy(\014)1705 3746 y Fo(\()p Fm(C)1767 3754 y Fy(g)1801 3746 y Fo(\))p Fm(j)1851 3781 y Fs(d)p Fn(t)24 b Ft(k)p Fs(\031)2044 3793 y Fo(\003)2089 3801 y Fi(1)2127 3781 y 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b(the)h(top)1902 4422 y Ft(k)p Fs(\031)1991 4434 y Fo(\003)2036 4442 y Fi(1)2091 4422 y Ft(\016)c(L)2208 4388 y Fu(T)2208 4443 y Fm(C)2244 4451 y Fy(g)2284 4422 y Ft(k)2326 4437 y Fu(L)p Fo(\()p Fm(H)2455 4446 y Fy(#)2493 4437 y Fu(;)p Fm(H)2570 4445 y Fi(\003)2608 4457 y(1)2644 4437 y Fo(\))3709 4422 y Fw(\(244\))1764 4772 y Ft(\024)23 b Fs(c)1888 4784 y Fo(9)1953 4772 y Fw(exp)2093 4655 y Fk(\022)2155 4772 y Ft(\000)2243 4716 y Fw(1)18 b Ft(\000)g Fs(\021)p 2243 4753 188 4 v 2316 4829 a Fw(2)2454 4772 y Fs(\025)c(T)2577 4655 y Fk(\023)2651 4772 y Fs(:)706 5087 y Fw(~)695 5109 y Fs(#)744 5075 y Fm(j)p Fo(\003)809 5083 y Fi(1)842 5075 y Fm(j)1076 5030 y Fk(X)880 5214 y Fm(C)916 5222 y Fy(g)950 5214 y Fm(2)p Fo(Conf)1133 5222 y(g)1170 5214 y(\()p Fp(Z)1241 5198 y Fy(d)1270 5214 y Fu(;)p Fo(\003)1335 5222 y Fi(1)1367 5214 y Fo(\))1407 4996 y Fk(Z)1481 5147 y Fo(Simplex)n Fi(\()p Fe(C)1765 5155 y Fy(g)1801 5147 y(;)p Fe(1)p Fi(\))1463 5216 y Fe(n)p Fo(Simplex)n Fi(\()p Fe(C)1778 5224 y Fy(g)1813 5216 y(;)1842 5200 y(T)p 1842 5207 42 3 v 1849 5238 a Fi(2)1894 5216 y(\))1944 5109 y Fs(\025)1992 5075 y Fm(j)p Fu(n)2053 5084 y Fy(\014)2093 5075 y Fo(\()p Fm(C)2155 5083 y Fy(g)2189 5075 y Fo(\))p Fm(j)2239 5109 y Fs(d)p Fn(t)g Ft(k)p Fs(\031)2422 5121 y Fo(\003)2467 5129 y Fi(1)2523 5109 y Ft(\016)k(L)2640 5075 y Fm(1)2640 5130 y(C)2676 5138 y Fy(g)2711 5130 y Fu(;)p Fl(t)2764 5109 y Ft(k)2806 5124 y Fu(L)p Fo(\()p Fm(H)2935 5133 y Fy(#)2973 5124 y Fu(;)p Fm(H)3050 5132 y Fi(\003)3088 5144 y(1)3124 5124 y Fo(\))3709 5109 y Fw(\(245\))2205 5396 y Ft(\024)23 b Fs(c)2329 5408 y Fo(10)2427 5396 y Fw(exp)2568 5279 y Fk(\022)2629 5396 y Ft(\000)2704 5340 y Fw(1)17 b Ft(\000)h Fs(\021)p 2704 5377 188 4 v 2776 5453 a Fw(2)2914 5396 y Fs(\025)2986 5340 y(T)p 2986 5377 61 4 v 2995 5453 a Fw(2)3056 5279 y Fk(\023)3131 5396 y Fs(:)1966 5682 y Fw(44)p eop %%Page: 45 45 45 44 bop 118 354 a Fw(W)-7 b(e)32 b(estimate)g(the)g(third)g(summand)g (\(238\))f(w)n(e)h(use)g(Lemma)f(3.3)g(\(see)h(b)r(elo)n(w\).)50 b(F)-7 b(or)31 b(that)h(w)n(e)g(note)f(that)h(for)g Ft(C)3774 366 y Fu(g)3842 354 y Ft(2)118 454 y Fw(Conf)291 466 y(g)337 454 y(\()p Fv(Z)430 424 y Fu(d)463 454 y Fs(;)14 b Fw(\003)558 466 y Fo(1)595 454 y Fw(\))30 b(and)g Fn(t)d Ft(2)h Fw(Simplex\()p Ft(C)1332 466 y Fu(g)1371 454 y Fs(;)1417 421 y Fu(T)p 1417 435 49 4 v 1425 483 a Fo(2)1476 454 y Fw(\))i(the)g(op)r(erators)e Fs(\031)2100 466 y Fo(\003)2145 474 y Fi(1)2203 454 y Ft(\016)19 b(L)2321 424 y Fm(1)2321 477 y(C)2357 485 y Fy(g)2392 477 y Fu(;)p Fl(t)2476 454 y Fw(and)29 b Fs(\031)2686 466 y Fo(\003)2731 474 y Fi(1)2789 454 y Ft(\016)19 b(L)2907 424 y Fu(T)2907 477 y Fm(C)2943 485 y Fy(g)2978 477 y Fu(;)p Fl(t)3062 454 y Fw(can)29 b(b)r(oth)i(b)r(e)f(written)g(as)118 589 y(a)d(pro)r(duct)g(of)g(n)n(um)n(b)r(ers)g(\(1)17 b Ft(\000)g Fs(e)1139 559 y Fm(\000)p Fw(length)p Fo(\()p Fu(H)t Fo(\))1531 589 y Fw(\))28 b(and)f(op)r(erators)e Ft(Q)2187 601 y Fu(q)2241 589 y Ft(\012)17 b Fw(id)2392 604 y Fo(\003)p Fm(nf)p Fu(q)r Fm(g)2576 589 y Fw(,)28 b(corresp)r(onding)d(to)i(maximal)g Fs(h)p Fw(-strips,)118 689 y Ft(R)188 701 y Fu(q)246 689 y Ft(\012)19 b Fw(id)400 704 y Fo(\003)p Fm(nf)p Fu(q)r Fm(g)584 689 y Fw(,)31 b(corresp)r(onding)e(to)h(maximal)g Fs(r)r Fw(-strips,)h(and)g Ft(B)2162 701 y Fu(q)2218 689 y Ft(\012)20 b Fw(id)2372 704 y Fo(\003)p Fm(nf)p Fu(q)r Fm(g)2556 689 y Fw(,)32 b(corresp)r(onding)c(to)j(branc)n(hings.)44 b(They)118 789 y(ha)n(v)n(e)31 b(the)i(same)e(structure)h(in)g(the)h(sense)f(that) g(these)g(factors)f(are)h(in)g(1-1-corresp)r(ondence)d(and)j(the)g (quan)n(titativ)n(e)118 888 y(di\013erence)25 b(is)f(only)h(due)g(to)f Fs(h)p Fw(-strips)g(going)f(to)i(the)g(top)g(or)f(long)g Fs(h)p Fw(-strips)f(in)i(\003)2628 900 y Fo(1)2690 888 y Fw(as)f(w)n(e)g(can)h(see)f(from)g(represen)n(tation)118 988 y(\(198\))32 b(for)h Ft(L)530 958 y Fm(1)530 1011 y(C)566 1019 y Fy(g)601 1011 y Fu(;)p Fl(t)688 988 y Fw(and)g Ft(L)912 958 y Fu(T)912 1011 y Fm(C)948 1019 y Fy(g)983 1011 y Fu(;)p Fl(t)1069 988 y Fw(and)g(also)f(from)h(Remark) f(3.4.1.)52 b(So)33 b(they)h(di\013er)f(only)g(in)g(the)h(constan)n(ts) g(~)-44 b Fs(c)p Fw(\()p Ft(C)3633 1000 y Fu(g)3671 988 y Fs(;)14 b Ft(1)p Fs(;)g Fn(t)p Fw(\))118 1098 y(and)36 b(~)-43 b Fs(c)p Fw(\()p Ft(C)399 1110 y Fu(g)437 1098 y Fs(;)14 b(T)7 b(;)14 b Fn(t)p Fw(\).)58 b(More)34 b(precisely)-7 b(,)36 b(an)e Fs(h)p Fw(-strip)g(in)h Ft(C)1851 1110 y Fu(g)1924 1098 y Fw(that)g(go)r(es)f(to)g(the)i(top)e(and)h (therefore)f(corresp)r(onds)e(to)j(an)118 1197 y Fs(h)p Fw(-strip)29 b(in)h(\()p Ft(C)565 1209 y Fu(g)604 1197 y Fs(;)14 b(T)7 b(;)14 b Fn(t)p Fw(\),)30 b(sa)n(y)f(from)g(\()p Fs(q)s(;)14 b(t)1339 1209 y Fu(i)1367 1197 y Fw(\))30 b(to)g(\()p Fs(q)s(;)14 b Ft(\000)p Fs(T)e Fw(\),)29 b(and)h(so)f(giv)n(es)g(rise)g(to)g(a)h(factor)f(1)19 b Ft(\000)h Fw(exp)o(\()p Ft(\000)p Fs(\025)p Fw(\()p Fs(T)32 b Ft(\000)19 b(j)p Fs(t)3563 1209 y Fu(i)3591 1197 y Ft(j)p Fw(\)\))31 b(\(note)118 1297 y(that)25 b Ft(j)p Fs(t)348 1309 y Fu(i)376 1297 y Ft(j)e Fs(<)520 1264 y Fu(T)p 520 1278 V 528 1325 a Fo(2)578 1297 y Fw(\))i(whilst)h (the)f(corresp)r(onding)e Fs(h)p Fw(-strip)i(in)g(\()p Ft(C)1987 1309 y Fu(g)2026 1297 y Fs(;)14 b Ft(1)p Fs(;)g Fn(t)p Fw(\))25 b(ends)g(at)g(time)g Ft(\0001)g Fw(and)g(giv)n(es)f (rise)g(to)h(a)g(factor)f(1.)118 1396 y(Similarly)k(a)f(long)g Fs(h)p Fw(-strip)h(of)g Ft(C)1131 1408 y Fu(g)1197 1396 y Fw(in)g(\003)1352 1408 y Fo(1)1417 1396 y Fw(giv)n(es)f(rise)g(to)h (factors)f(1)18 b Ft(\000)g Fw(exp\()p Ft(\000)p Fs(\025T)12 b Fw(\))28 b(and)f(1,)h(resp)r(ectiv)n(ely)-7 b(.)37 b(In)28 b(b)r(oth)h(cases)118 1496 y(the)f(di\013erence)g(b)r(et)n(w)n (een)f(the)h(scalar)e(factors)h(\(for)g(eac)n(h)g Fs(h)p Fw(-strip)g(to)g(the)h(top\))g(is)f(b)r(ounded)h(b)n(y)1661 1724 y Fs(\016)1701 1690 y Fo(2)1761 1724 y Fw(=)23 b(exp)1989 1607 y Fk(\022)2050 1724 y Ft(\000)2125 1668 y Fs(\025)p 2125 1705 V 2128 1781 a Fw(2)2197 1724 y Fs(T)2258 1607 y Fk(\023)2332 1724 y Fs(:)1354 b Fw(\(246\))118 1932 y(The)26 b(n)n(um)n(b)r(er)f(of)h Fs(h)p Fw(-strips)f(to)h(the)g(top)f (is)h(b)r(ounded)g(b)n(y)1900 1870 y Fk(P)1987 1890 y Fm(1)1987 1957 y Fu(k)q Fo(=1)2126 1932 y Fw(3)2168 1902 y Fu(d)2206 1932 y Fs(n)2256 1944 y Fu(\014)s(;k)2357 1932 y Fs(k)2403 1902 y Fu(d)2468 1932 y Fw(and)f(the)h(n)n(um)n(b)r (er)g(of)f(long)g Fs(h)p Fw(-strips)g(at)h(sites)118 2032 y(in)i(\003)273 2044 y Fo(1)338 2032 y Fw(b)n(y)f Ft(j)p Fw(\003)534 2044 y Fo(1)571 2032 y Ft(j)18 b(\000)g Fs(K)33 b Fw(\(where)28 b Fs(n)1122 2044 y Fu(\014)s(;k)1251 2032 y Fw(and)f Fs(K)33 b Fw(are)27 b(the)h(parameters)e(of)h(the)h (lab)r(elled)g(tree)f(of)h Ft(C)3077 2044 y Fu(g)3115 2032 y Fw(.\))118 2131 y(So)f(w)n(e)h(estimate)621 2413 y Ft(k)p Fs(\031)710 2425 y Fo(\003)755 2433 y Fi(1)810 2413 y Ft(\016)18 b(L)927 2379 y Fm(1)927 2434 y(C)963 2442 y Fy(g)998 2434 y Fu(;)p Fl(t)1069 2413 y Ft(\000)g Fs(\031)1199 2425 y Fo(\003)1244 2433 y Fi(1)1300 2413 y Ft(\016)g(L)1417 2379 y Fu(T)1417 2434 y Fm(C)1453 2442 y Fy(g)1488 2434 y Fu(;)p Fl(t)1541 2413 y Ft(k)1583 2428 y Fu(L)p Fo(\()p Fm(H)1712 2437 y Fy(#)1751 2428 y Fu(;)p Fm(H)1828 2436 y Fi(\003)1866 2448 y(1)1902 2428 y Fo(\))3709 2413 y Fw(\(247\))704 2629 y Ft(\024)82 b Fs(\016)946 2525 y Fm(1)926 2550 y Fk(Y)919 2729 y Fu(k)q Fo(=1)1040 2629 y Fw(\(1)18 b(+)g Fs(\016)s Fw(\))1287 2595 y Fo(3)1320 2569 y Fy(d)1355 2595 y Fu(n)1396 2604 y Fy(\014)r(;k)1486 2595 y Fu(k)1522 2569 y Fy(d)1562 2629 y Fw(\(1)g(+)g Fs(\016)s Fw(\))1809 2595 y Fm(j)p Fo(\003)1874 2603 y Fi(1)1907 2595 y Fm(j\000)p Fu(K)2043 2629 y Fw(\()p Fs(c)2111 2641 y Fo(3)2148 2629 y Fs(\017)p Fw(\))2214 2595 y Fm(j)p Fu(n)2275 2604 y Fy(\014)2314 2595 y Fm(j)2352 2629 y Fw(exp)2493 2487 y Fk( )2558 2629 y Ft(\000)p Fs(c)2659 2641 y Fu(g)2739 2525 y Fm(1)2712 2550 y Fk(X)2711 2729 y Fu(k)q Fo(=1)2846 2629 y Fs(k)2892 2595 y Fu(d)2930 2629 y Fs(n)2980 2641 y Fu(\014)s(;k)3081 2487 y Fk(!)3161 2629 y Fs(c)3201 2591 y Fo(~)-37 b Fu(n)3238 2600 y Fy(h)3197 2654 y Fu(h)3281 2629 y Fs(c)3321 2595 y Fo(~)g Fu(n)3358 2603 y Fy(r)3317 2649 y Fu(r)865 2857 y Ft(\002)944 2778 y Fk(Y)971 2957 y Fu(R)1063 2857 y Fw(exp\()p Ft(\000)p Fw(\(1)18 b Ft(\000)g Fs(\021)s Fw(\))p Fs(\025)c Fw(length)q(\()p Fs(R)q Fw(\)\))21 b(~)-44 b Fs(c)p Fw(\()p Ft(C)2118 2869 y Fu(g)2157 2857 y Fs(;)14 b(T)7 b(;)14 b Fn(t)p Fw(\))g Ft(k)p Fs(\036)2461 2872 y Fo(\003\()p Fm(C)2568 2880 y Fy(g)2602 2872 y Fo(\))2632 2857 y Ft(k)2674 2872 y Fo(\003\()p Fm(C)2781 2880 y Fy(g)2816 2872 y Fo(\))2846 2857 y Fs(:)118 3140 y Fw(The)28 b(factors)e(\(1)18 b(+)g Fs(\016)s Fw(\))806 3110 y Fo(3)839 3085 y Fy(d)874 3110 y Fu(n)915 3119 y Fy(\014)r(;k)1005 3110 y Fu(k)1041 3085 y Fy(d)1109 3140 y Fw(are)26 b(comp)r(ensated)h(for)g(b)n(y)g(exp\()p Ft(\000)p Fs(c)2245 3152 y Fu(g)2284 3140 y Fs(k)2330 3110 y Fu(d)2368 3140 y Fs(n)2418 3152 y Fu(\014)s(;k)2519 3140 y Fw(\))h(`in)g(the)g(usual)f(w)n(a)n(y'.)35 b(If)28 b Fs(\017)f Fw(is)h(su\016cien)n(tly)118 3240 y(small)f(and)h Fs(c)533 3252 y Fu(g)599 3240 y Fw(large)e(w)n(e)h(can)h(estimate)402 3522 y(~)391 3544 y Fs(#)440 3510 y Fm(j)p Fo(\003)505 3518 y Fi(1)538 3510 y Fm(j)787 3465 y Fk(X)599 3644 y Fe(C)631 3652 y Fy(g)666 3644 y Fe(2)p Fo(Conf)843 3655 y(g)880 3644 y Fi(\()p Fa(Z)936 3632 y Fy(d)965 3644 y(;)p Fi(\003)1022 3656 y(1)1054 3644 y(\))p Fy(;)477 3715 y Fe(C)509 3723 y Fy(g)545 3715 y Fo(do)r(es)21 b(not)h(reac)n(h)g(the)f(top)p Fy(;)698 3769 y Fe(j)p Fy(n)754 3784 y(\014)792 3769 y Fi(\()p Fe(C)846 3777 y Fy(g)882 3769 y Fi(\))p Fe(j\025)p Fi(1)1251 3431 y Fk(Z)1297 3619 y Fo(Simplex)n(\()p Fm(C)1589 3627 y Fy(g)1623 3619 y Fu(;)1653 3597 y Fy(T)p 1653 3606 42 3 v 1660 3639 a Fi(2)1705 3619 y Fo(\))1749 3544 y Fs(\025)1797 3510 y Fm(j)p Fu(n)1858 3519 y Fy(\014)1897 3510 y Fo(\()p Fm(C)1959 3518 y Fy(g)1994 3510 y Fo(\))p Fm(j)2044 3544 y Fs(d)p Fn(t)i Ft(k)p Fs(\031)2236 3556 y Fo(\003)2281 3564 y Fi(1)2336 3544 y Ft(\016)18 b(L)2453 3510 y Fm(1)2453 3564 y(C)2489 3572 y Fy(g)2524 3564 y Fu(;)p Fl(t)2596 3544 y Ft(\000)g Fs(\031)2726 3556 y Fo(\003)2771 3564 y Fi(1)2827 3544 y Ft(\016)g(L)2944 3510 y Fu(T)2944 3564 y Fm(C)2980 3572 y Fy(g)3015 3564 y Fu(;)p Fl(t)3068 3544 y Ft(k)3110 3559 y Fu(L)p Fo(\()p Fm(H)3239 3568 y Fy(#)3277 3559 y Fu(;)p Fm(H)3354 3567 y Fi(\003)3392 3579 y(1)3428 3559 y Fo(\))3709 3544 y Fw(\(248\))904 3918 y Ft(\024)23 b Fs(c)1028 3930 y Fo(11)1112 3918 y Fs(e)1151 3884 y Fm(\000)1212 3862 y Fy(\025)p 1212 3871 35 3 v 1215 3904 a Fi(4)1257 3884 y Fu(T)1309 3918 y Fs(:)118 4101 y Fw(F)-7 b(rom)27 b(\(241\),)g(\(244\),)g(\(248\),)g (\(245\),)g(and)g(\(243\))g(w)n(e)g(conclude)g(\(230\))g(with)h Fs(c)2579 4113 y Fo(6)2640 4101 y Fw(=)2737 4064 y Fo(1)p Fm(\000)p Fu(\021)p 2737 4082 122 4 v 2781 4130 a Fo(4)2868 4101 y Fs(\025)g Fw(and)g Fs(c)3142 4113 y Fo(5)3207 4101 y Fw(su\016cien)n(tly)f(large.)118 4201 y(F)-7 b(or)27 b(an)n(y)g Fs(\036)c Ft(2)h(H)645 4213 y Fu(#)717 4201 y Fw(and)j(an)n(y)g(\003)c Ft(2)g(F)36 b Fw(w)n(e)27 b(ha)n(v)n(e)977 4483 y Fs(\031)1024 4495 y Fo(\003)1093 4483 y Ft(\016)18 b(L)1210 4449 y Fm(1)1210 4503 y Fu(S)1280 4483 y Fs(\036)84 b Fw(=)1871 4404 y Fk(X)1698 4583 y Fe(C)1730 4591 y Fy(g)1765 4583 y Fe(2)p Fo(Conf)1942 4594 y(g)1979 4583 y Fi(\()p Fa(Z)2036 4571 y Fy(d)2064 4583 y(;)p Fi(\003\))p Fy(;)1570 4654 y Fe(C)1602 4662 y Fy(g)1637 4654 y Fo(do)r(es)22 b(not)g(reac)n(h)g(the)f(top)2315 4483 y Fs(\031)2362 4495 y Fo(\003)2430 4483 y Ft(\016)d(L)2547 4449 y Fm(1)2547 4503 y(C)2583 4511 y Fy(g)2622 4483 y Fs(\036)1038 b Fw(\(249\))1413 4808 y(=)1871 4729 y Fk(X)1698 4908 y Fe(C)1730 4916 y Fy(g)1765 4908 y Fe(2)p Fo(Conf)1942 4919 y(g)1979 4908 y Fi(\()p Fa(Z)2036 4896 y Fy(d)2064 4908 y(;)p Fi(\003\))p Fy(;)1570 4979 y Fe(C)1602 4987 y Fy(g)1637 4979 y Fo(do)r(es)22 b(not)g(reac)n(h)g(the)f(top)2315 4715 y Fk(\020)2364 4808 y Fs(\031)2411 4820 y Fo(\003)2479 4808 y Ft(\016)d(L)2596 4773 y Fm(1)2596 4828 y(C)2632 4836 y Fy(g)2672 4808 y Fs(h)2720 4824 y Fp(Z)2764 4808 y Fy(d)2797 4715 y Fk(\021)2874 4808 y Fs(\026)p Fw(\()p Fs(\036)p Fw(\))118 5165 y(The)27 b(sum)g(in)h(\(249\))e(is)h(a)g (priori)f(o)n(v)n(er)f(all)i Ft(C)1490 5177 y Fu(g)1551 5165 y Ft(2)c Fw(Conf)1802 5177 y Fu(g)1841 5165 y Fw(\()p Fv(Z)1935 5135 y Fu(d)1967 5165 y Fs(;)14 b Fw(\003\))27 b(but,)h(as)f(w)n(e)f(ha)n(v)n(e)g(seen)h(b)r(efore,)g(if)g Ft(C)3281 5177 y Fu(g)3347 5165 y Fw(reac)n(hes)e(the)j(top)118 5265 y(the)f(corresp)r(onding)d(op)r(erator)h Fs(\031)1173 5277 y Fo(\003)1239 5265 y Ft(\016)15 b(L)1353 5235 y Fm(1)1353 5288 y(C)1389 5296 y Fy(g)1455 5265 y Fw(is)26 b(zero.)35 b(If)27 b Ft(C)1870 5277 y Fu(g)1935 5265 y Fw(do)r(es)f(not)g(reac)n(h)f(the)i(top)f(there)g(are)f(only)h Fs(h)p Fw(-strips)g(going)f(to)118 5375 y(the)j(top)g(\()p Ft(\0001)p Fw(\))g(and)f Fs(\031)857 5387 y Fo(\003)925 5375 y Ft(\016)18 b(L)1042 5345 y Fm(1)1042 5398 y(C)1078 5406 y Fy(g)1145 5375 y Fw(is)28 b(a)f(pro)5 b(jection)26 b(on)n(to)h(span\()p Fs(h)2126 5392 y Fp(Z)2171 5375 y Fy(d)r Fw(\).)1966 5682 y(45)p eop %%Page: 46 46 46 45 bop 118 354 a Fw(W)-7 b(e)30 b(set)g Fs(\027)436 366 y Fo(\003)513 307 y(def)526 354 y Fw(=)40 b Fs(\031)678 366 y Fo(\003)748 354 y Ft(\016)19 b(L)866 324 y Fm(1)866 377 y(C)902 385 y Fy(g)941 354 y Fs(h)989 371 y Fp(Z)1034 355 y Fy(d)1097 354 y Fw(and)30 b(this)g(de\014nes)g Fs(\027)i Fw(=)26 b(\()p Fs(\027)1937 366 y Fo(\003)1987 354 y Fw(\))2019 366 y Fo(\003)p Fm(2F)2167 354 y Fw(.)44 b(Note)30 b(that)g(the)g(transfer)f(op)r(erator)f Ft(L)3468 324 y Fm(1)3468 377 y Fu(S)3569 354 y Fw(preserv)n(es)118 454 y(the)g(Leb)r(esgue-in)n(tegral)e(and)h(so)g Fs(\027)1227 469 y Fm(;)1288 454 y Fw(=)c(1,)k(i.e.)h Fs(\027)33 b Fw(corresp)r(onds)25 b(to)j(a)f(probabilit)n(y)g(measure.)3808 554 y Fh(2)118 724 y Fw(In)h(the)g(pro)r(of)f(of)g(Theorem)g(3.1)g(w)n (e)g(ha)n(v)n(e)g(used)g(the)h(follo)n(wing)f(lemma.)118 907 y Fn(Lemma)j(3.3)40 b Fj(L)l(et)22 b Fs(A)818 919 y Fo(1)856 907 y Fs(;)14 b(:)g(:)g(:)f(;)h(A)1102 919 y Fu(n)1148 907 y Fs(;)1206 886 y Fw(~)1185 907 y Fs(A)1247 919 y Fo(1)1284 907 y Fs(;)g(:)g(:)g(:)g(;)1490 886 y Fw(~)1469 907 y Fs(A)1531 919 y Fu(n)1598 907 y Fj(b)l(e)23 b(op)l(er)l(ators)g(on)g(the)f(same)h(Banach)h(sp)l(ac)l(e,)h Fw(0)e Fs(<)f(\016)k(<)d Fw(1)f Fj(and)h Fs(a)3587 919 y Fo(1)3624 907 y Fs(;)14 b(:)g(:)g(:)g(;)g(a)3853 919 y Fu(n)118 1006 y Fj(p)l(ositive)31 b(numb)l(ers)e(such)h(that:)1593 1289 y Ft(k)p Fs(A)1697 1301 y Fu(i)1724 1289 y Ft(k)83 b(\024)g Fs(a)2041 1301 y Fu(i)2238 1289 y Fj(for)30 b(al)t(l)h Fw(1)23 b Ft(\024)f Fs(i)h Ft(\024)g Fs(n)878 b Fw(\(250\))1185 1413 y Fj(and)86 b Ft(k)p Fs(A)1506 1425 y Fu(i)1551 1413 y Ft(\000)1656 1392 y Fw(~)1635 1413 y Fs(A)1697 1425 y Fu(i)1724 1413 y Ft(k)d(\024)g Fs(\016)2037 1379 y Fo(2)2088 1413 y Fs(a)2132 1425 y Fu(i)2159 1413 y Fs(:)1527 b Fw(\(251\))118 1596 y Fj(Then)1014 1795 y Ft(k)p Fs(A)1118 1807 y Fo(1)1174 1795 y Ft(\016)18 b Fs(:)c(:)g(:)k Ft(\016)g Fs(A)1471 1807 y Fu(n)1535 1795 y Ft(\000)1640 1774 y Fw(~)1618 1795 y Fs(A)1680 1807 y Fo(1)1736 1795 y Ft(\016)g Fs(:)c(:)g(:)k Ft(\016)1993 1774 y Fw(~)1971 1795 y Fs(A)2033 1807 y Fu(n)2079 1795 y Ft(k)k(\024)h Fs(\016)17 b Fw(\(1)h(+)g Fs(\016)s Fw(\))2532 1761 y Fu(n)2591 1795 y Fs(a)2635 1807 y Fo(1)2691 1795 y Ft(\001)g Fs(:)c(:)g(:)19 b Ft(\001)f Fs(a)2933 1807 y Fu(n)2978 1795 y Fs(:)708 b Fw(\(252\))118 1978 y Fn(Pro)s(of)59 b Fw(F)-7 b(rom)28 b(\(251\))e(w)n(e)i(get)1669 2177 y Ft(k)1733 2156 y Fw(~)1711 2177 y Fs(A)1773 2189 y Fu(i)1801 2177 y Ft(k)22 b(\024)h Fw(\(1)18 b(+)g Fs(\016)2168 2143 y Fo(2)2206 2177 y Fw(\))c Fs(a)2296 2189 y Fu(i)2323 2177 y Fs(:)1363 b Fw(\(253\))118 2326 y(So)27 b(w)n(e)h(get)f(via)g (`telescop)r(e)g(expansion':)1043 2609 y Ft(k)p Fs(A)1147 2621 y Fo(1)1202 2609 y Ft(\016)18 b Fs(:)c(:)g(:)19 b Ft(\016)f Fs(A)1500 2621 y Fu(n)1564 2609 y Ft(\000)1669 2588 y Fw(~)1647 2609 y Fs(A)1709 2621 y Fo(1)1765 2609 y Ft(\016)g Fs(:)c(:)g(:)k Ft(\016)2022 2588 y Fw(~)2000 2609 y Fs(A)2062 2621 y Fu(n)2107 2609 y Ft(k)1560 b Fw(\(254\))1126 2733 y Ft(\024)83 b(k)p Fs(A)1378 2745 y Fo(1)1433 2733 y Ft(\016)18 b Fs(:)c(:)g(:)k Ft(\016)g Fs(A)1730 2745 y Fu(n)1794 2733 y Ft(\000)1899 2712 y Fw(~)1877 2733 y Fs(A)1939 2745 y Fo(1)1995 2733 y Ft(\016)g Fs(A)2117 2745 y Fo(2)2173 2733 y Ft(\016)g Fs(:)c(:)g(:)k Ft(\016)g Fs(A)2470 2745 y Fu(n)2516 2733 y Ft(k)g Fw(+)g Fs(:)c(:)g(:)1287 2858 y Fw(+)p Ft(k)1415 2837 y Fw(~)1394 2858 y Fs(A)1456 2870 y Fo(1)1511 2858 y Ft(\016)k Fs(:)c(:)g(:)19 b Ft(\016)1769 2837 y Fw(~)1747 2858 y Fs(A)1809 2870 y Fu(n)p Fm(\000)p Fo(1)1958 2858 y Ft(\016)f Fs(A)2080 2870 y Fu(n)2143 2858 y Ft(\000)2248 2837 y Fw(~)2226 2858 y Fs(A)2288 2870 y Fo(1)2344 2858 y Ft(\016)g Fs(:)c(:)g(:)19 b Ft(\016)2602 2837 y Fw(~)2580 2858 y Fs(A)2642 2870 y Fu(n)2687 2858 y Ft(k)1126 2982 y(\024)83 b Fs(\016)1314 2948 y Fo(2)1365 2982 y Fw(\(1)18 b(+)g(\(1)g(+)g Fs(\016)1755 2948 y Fo(2)1793 2982 y Fw(\))g(+)g Fs(:)c(:)g(:)19 b Fw(+)f(\(1)g(+)g Fs(\016)2340 2948 y Fo(2)2377 2982 y Fw(\))2409 2948 y Fu(n)p Fm(\000)p Fo(1)2540 2982 y Fw(\))c Fs(a)2630 2994 y Fo(1)2685 2982 y Ft(\001)19 b Fs(:)14 b(:)g(:)k Ft(\001)h Fs(a)2928 2994 y Fu(n)1126 3107 y Fw(=)83 b(\(\(1)18 b(+)g Fs(\016)1521 3073 y Fo(2)1558 3107 y Fw(\))1590 3073 y Fu(n)1654 3107 y Ft(\000)g Fw(1\))c Fs(a)1869 3119 y Fo(1)1925 3107 y Ft(\001)k Fs(:)c(:)g(:)k Ft(\001)h Fs(a)2167 3119 y Fu(n)1126 3298 y Fw(=)1313 3194 y Fu(n)1274 3219 y Fk(X)1274 3398 y Fu(k)q Fo(=1)1408 3181 y Fk(\022)1469 3242 y Fs(n)1471 3355 y(k)1519 3181 y Fk(\023)1580 3298 y Fs(\016)1620 3264 y Fo(2)p Fu(k)1708 3298 y Fs(a)1752 3310 y Fo(1)1808 3298 y Ft(\001)f Fs(:)c(:)g(:)k Ft(\001)h Fs(a)2050 3310 y Fu(n)1126 3576 y Ft(\024)83 b Fs(\016)1367 3473 y Fu(n)1328 3497 y Fk(X)1327 3676 y Fu(k)q Fo(=1)1462 3459 y Fk(\022)1523 3520 y Fs(n)1525 3633 y(k)1573 3459 y Fk(\023)1634 3576 y Fs(\016)1674 3542 y Fu(k)1729 3576 y Fs(a)1773 3588 y Fo(1)1828 3576 y Ft(\001)19 b Fs(:)14 b(:)g(:)k Ft(\001)h Fs(a)2071 3588 y Fu(n)1126 3780 y Ft(\024)83 b Fs(\016)s Fw(\(1)18 b(+)g Fs(\016)s Fw(\))1561 3746 y Fu(n)1620 3780 y Fs(a)1664 3792 y Fo(1)1720 3780 y Ft(\001)g Fs(:)c(:)g(:)19 b Ft(\001)f Fs(a)1962 3792 y Fu(n)118 3963 y Fw(and)28 b(the)g(lemma)f(is)h(pro)n (v)n(ed.)3808 4062 y Fh(2)118 4316 y Fn(Remark)j(3.5)40 b Fw(Analogously)29 b(to)i(Prop)r(osition)e(6.1.3)g(in)i([13)o(],)g (one)f(can)g(also)g(pro)n(v)n(e)f(a)h(semigroup-lik)n(e)e(prop)r(ert)n (y)i(of)118 4415 y(the)e(family)g(\()p Ft(L)604 4385 y Fu(t)604 4438 y(S)652 4415 y Fw(\))684 4427 y Fu(t)p Fm(\025)p Fo(0)799 4415 y Fw(,)g(using)f(Remark)g(2.5)f(and)i(the)g (diagrammatic)e(represen)n(tation)g(of)h(the)h(op)r(erators.)118 4690 y Fx(4)135 b(Deca)l(y)46 b(of)f(Correlations)118 4872 y Fw(In)38 b(the)f(follo)n(wing)g(theorem)g(whic)n(h)g(is)g (completely)h(analogous)d(to)i(Theorem)f(7.1)h(in)h([13)o(],)i(w)n(e)d (state)g(the)h(mixing)118 4971 y(prop)r(erties)27 b(for)g(the)h(in)n(v) -5 b(arian)n(t)26 b(probabilit)n(y)h(measure)f Fs(\027)1921 4941 y Fm(\003)1988 4971 y Fw(in)h(terms)h(of)f(the)h(w)n(eigh)n(ted)f (norms.)118 5154 y Fn(Theorem)k(4.1)40 b Fj(F)-6 b(or)33 b(su\016ciently)g(smal)t(l)g Fs(#)p Fj(,)1609 5132 y Fw(~)1598 5154 y Fs(#)p Fj(,)h Fs(\017)e Fj(and)i(big)f Fs(c)2107 5166 y Fu(g)2178 5154 y Fj(ther)l(e)g(is)g(a)g Fs(\024)28 b Ft(2)h Fw(\(0)p Fs(;)14 b Fw(1\))32 b Fj(and)h(p)l (ositive)h(c)l(onstants)e Fs(c)3802 5166 y Fo(12)3872 5154 y Fj(,)118 5254 y Fs(c)154 5266 y Fo(13)224 5254 y Fj(,)f Fs(c)316 5266 y Fo(14)416 5254 y Fj(and)f Fs(c)613 5266 y Fo(15)713 5254 y Fj(such)g(that)f(for)i(al)t(l)g(\014nite)e (disjoint)i Fw(\003)1894 5266 y Fo(1)1931 5254 y Fs(;)14 b Fw(\003)2026 5266 y Fo(2)2086 5254 y Ft(\032)23 b Fv(Z)2235 5224 y Fu(d)2297 5254 y Fj(and)31 b Fs( )26 b Ft(2)d(H)2687 5266 y Fo(\003)2732 5274 y Fi(2)2799 5254 y Fj(the)30 b(fol)t(lowing)i(holds:)1966 5682 y Fw(46)p eop %%Page: 47 47 47 46 bop 932 537 a Ft(k)p Fs(\027)1015 549 y Fo(\003)1060 557 y Fi(1)1093 549 y Fm([)p Fo(\003)1183 557 y Fi(2)1238 537 y Ft(\000)18 b Fs(\027)1362 549 y Fo(\003)1407 557 y Fi(1)1444 537 y Fs(\027)1485 549 y Fo(\003)1530 557 y Fi(2)1567 537 y Ft(k)1608 562 y Fo(\003)1653 570 y Fi(1)1686 562 y Fm([)p Fo(\003)1776 570 y Fi(2)1896 537 y Ft(\024)83 b Fs(c)2080 549 y Fo(12)2164 537 y Fs(#)2213 503 y Fm(j)p Fo(\003)2278 511 y Fi(1)2310 503 y Fm([)p Fo(\003)2400 511 y Fi(2)2433 503 y Fm(j)2471 537 y Fs(\024)2519 503 y Fo(dist)o(\(\003)2697 511 y Fi(1)2730 503 y Fu(;)p Fo(\003)2795 511 y Fi(2)2828 503 y Fo(\))2858 537 y Fs(;)828 b Fw(\(255\))945 723 y Ft(k)p Fs(\031)1034 735 y Fo(\003)1079 743 y Fi(1)1115 723 y Fw(\()p Fs( )s(\027)5 b Fw(\))20 b Ft(\000)e Fs(\027)5 b Fw(\()p Fs( )s Fw(\))14 b Fs(\027)1607 735 y Fo(\003)1652 743 y Fi(1)1690 723 y Ft(k)1732 735 y Fo(\003)1777 743 y Fi(1)1896 723 y Ft(\024)83 b Fs(c)2080 735 y Fo(13)2164 723 y Fs(#)2213 688 y Fm(\000j)p Fo(\003)2330 696 y Fi(1)2362 688 y Fm([)p Fo(\003)2452 696 y Fi(2)2485 688 y Fm(j)2509 723 y Ft(k)14 b Fs( )s Ft(k)2664 735 y Fo(\003)2709 743 y Fi(2)2758 723 y Fs(\024)2806 688 y Fo(dist\(\003)2985 696 y Fi(1)3018 688 y Fu(;)p Fo(\003)3083 696 y Fi(2)3115 688 y Fo(\))3145 723 y Fs(;)541 b Fw(\(256\))757 889 y Ft(k)p Fs(\031)846 901 y Fo(\003)891 909 y Fi(1)946 889 y Ft(\016)18 b(L)1063 855 y Fu(T)1063 909 y(S)1115 889 y Fw(\()p Fs( )s(\027)5 b Fw(\))20 b Ft(\000)e Fs(\027)5 b Fw(\()p Fs( )s Fw(\))14 b Fs(\027)1607 901 y Fo(\003)1652 909 y Fi(1)1690 889 y Ft(k)1732 901 y Fo(\003)1777 909 y Fi(1)1896 889 y Ft(\024)83 b Fs(c)2080 901 y Fo(14)2164 889 y Fs(#)2213 855 y Fm(\000j)p Fo(\003)2330 863 y Fi(2)2362 855 y Fm(j)2411 867 y Fw(~)2400 889 y Fs(#)2449 855 y Fm(\000j)p Fo(\003)2566 863 y Fi(1)2599 855 y Fm(j)2622 889 y Ft(k)14 b Fs( )s Ft(k)2777 901 y Fo(\003)2822 909 y Fi(2)2872 889 y Fs(\024)2920 855 y Fo(dist\(\003)3099 863 y Fi(1)3131 855 y Fu(;)p Fo(\003)3196 863 y Fi(2)3229 855 y Fo(\))3709 889 y Fw(\(257\))2062 1013 y Ft(\002)g Fw(exp)f(\()p Ft(\000)p Fs(c)2414 1025 y Fo(15)2484 1013 y Fs(T)f Fw(\))148 1196 y Fj(for)30 b(every)h Fs(T)j(>)23 b Fw(0)p Fj(.)118 1379 y Fn(Pro)s(of)59 b Fw(F)-7 b(or)27 b(a)h(gum)f(con\014guration)f Ft(C)1359 1391 y Fu(g)1425 1379 y Fw(w)n(e)h(de\014ne)h(in)g(analogy)e(to)h(\(109\))g(in)h([13)o (])1590 1624 y Fs(b)p Fw(\()p Ft(C)1702 1636 y Fu(g)1740 1624 y Fw(\))1796 1577 y Fo(def)1809 1624 y Fw(=)1938 1520 y Fm(1)1911 1545 y Fk(X)1910 1724 y Fu(k)q Fo(=1)2045 1624 y Fs(k)17 b(n)2155 1636 y Fu(\014)s(;k)2256 1624 y Fw(\()p Ft(C)2332 1636 y Fu(g)2370 1624 y Fw(\))p Fs(:)1284 b Fw(\(258\))118 1860 y(In)37 b(the)g(follo)n(wing)f(w)n(e)g(split)h (gum)g(con\014gurations)d Ft(C)1850 1872 y Fu(g)1927 1860 y Ft(2)39 b Fw(Conf\()p Fv(Z)2287 1830 y Fu(d)2320 1860 y Fs(;)14 b Fw(\003)2415 1872 y Fo(1)2476 1860 y Ft([)25 b Fw(\003)2614 1872 y Fo(2)2651 1860 y Fw(\))37 b(with)g Fs(b)p Fw(\()p Ft(C)3030 1872 y Fu(g)3068 1860 y Fw(\))i Ft(\024)3252 1828 y Fo(1)p 3252 1842 34 4 v 3252 1889 a(2)3295 1860 y Fw(dist\(\003)3519 1872 y Fo(1)3556 1860 y Fs(;)14 b Fw(\003)3651 1872 y Fo(2)3688 1860 y Fw(\))37 b(in)n(to)118 1960 y Ft(C)162 1972 y Fu(g)224 1960 y Fw(=)22 b Ft(C)360 1930 y Fo(1)355 1981 y Fu(g)415 1960 y Ft([)d(C)538 1930 y Fo(2)533 1981 y Fu(g)603 1960 y Fw(with)28 b Ft(C)841 1930 y Fo(1)836 1981 y Fu(g)901 1960 y Ft(2)23 b Fw(Conf\()p Fv(Z)1246 1930 y Fu(d)1278 1960 y Fs(;)14 b Fw(\003)1373 1972 y Fo(1)1410 1960 y Fw(\),)28 b Ft(C)1542 1930 y Fo(2)1537 1981 y Fu(g)1602 1960 y Ft(2)23 b Fw(Conf\()p Fv(Z)1947 1930 y Fu(d)1980 1960 y Fs(;)14 b Fw(\003)2075 1972 y Fo(2)2112 1960 y Fw(\))28 b(and)f(\003\()p Ft(C)2472 1930 y Fo(1)2467 1981 y Fu(g)2509 1960 y Fw(\))18 b Ft(\\)h Fw(\003\()p Ft(C)2772 1930 y Fo(2)2767 1981 y Fu(g)2809 1960 y Fw(\))k(=)g Ft(;)p Fw(.)118 2060 y(W)-7 b(e)28 b(write,)g(using)f(\(231\))g(and)g (the)h(notation)f(of)h(\(156\):)809 2342 y Fs(\027)850 2354 y Fo(\003)895 2362 y Fi(1)927 2354 y Fm([)p Fo(\003)1017 2362 y Fi(2)1137 2342 y Fw(=)1646 2263 y Fk(X)1295 2443 y Fe(C)1327 2451 y Fy(g)1362 2443 y Fi(=)p Fe(C)1439 2431 y Fi(1)1437 2459 y Fy(g)1472 2443 y Fe([C)1545 2431 y Fi(2)1543 2459 y Fy(g)1578 2443 y Fe(2)p Fo(Conf)1755 2454 y(g)1792 2443 y Fi(\()p Fa(Z)1848 2431 y Fy(d)1877 2443 y(;)p Fi(\003)1934 2455 y(1)1967 2443 y Fe([)p Fi(\003)2044 2455 y(2)2076 2443 y(\))p Fy(;)1434 2518 y(b)p Fi(\()p Fe(C)1514 2526 y Fy(g)1550 2518 y Fi(\))p Fe(\024)1628 2502 y Fi(1)p 1628 2509 29 3 v 1628 2541 a(2)1666 2518 y Fo(dist)p Fi(\(\003)1834 2530 y(1)1866 2518 y Fy(;)p Fi(\003)1923 2530 y(2)1955 2518 y(\))2127 2342 y Fw(\()p Fs(\031)2206 2354 y Fo(\003)2251 2362 y Fi(1)2308 2342 y Ft(\016)17 b(L)2424 2308 y Fu(T)2424 2364 y Fm(C)2463 2348 y Fi(1)2460 2381 y Fy(g)2500 2342 y Fs(h)2548 2359 y Fp(Z)2593 2342 y Fy(d)r Fw(\))i(\()p Fs(\031)2756 2354 y Fo(\003)2801 2362 y Fi(2)2857 2342 y Ft(\016)f(L)2974 2308 y Fu(T)2974 2364 y Fm(C)3013 2348 y Fi(2)3010 2381 y Fy(g)3049 2342 y Fs(h)3097 2359 y Fp(Z)3142 2342 y Fy(d)r Fw(\))1299 2669 y(+)1630 2590 y Fk(X)1387 2770 y Fe(C)1419 2778 y Fy(g)1454 2770 y Fe(2)p Fo(Conf)1631 2781 y(g)1668 2770 y Fi(\()p Fa(Z)1724 2758 y Fy(d)1753 2770 y(;)p Fi(\003)1810 2782 y(1)1843 2770 y Fe([)p Fi(\003)1920 2782 y(2)1952 2770 y(\))p Fy(;)1418 2843 y(b)p Fi(\()p Fe(C)1498 2851 y Fy(g)1534 2843 y Fi(\))p Fy(>)1612 2827 y Fi(1)p 1612 2834 V 1612 2865 a(2)1650 2843 y Fo(dist)p Fi(\(\003)1818 2855 y(1)1850 2843 y Fy(;)p Fi(\003)1907 2855 y(2)1940 2843 y(\))2017 2669 y Fs(\031)2064 2681 y Fo(\003)2109 2689 y Fi(1)2142 2681 y Fm([)p Fo(\003)2232 2689 y Fi(2)2288 2669 y Ft(\016)g(L)2405 2635 y Fu(T)2405 2690 y Fm(C)2441 2698 y Fy(g)2480 2669 y Fs(h)2528 2686 y Fp(Z)2572 2669 y Fy(d)118 3027 y Fw(In)32 b(estimating)g(the)g(norm)f (of)h(the)g(second)g(summand)g(in)g(\(259\))f(w)n(e)g(can)h(tak)n(e)f (out)h(from)g(the)g(estimate)g(for)f Ft(k)p Fs(\031)3753 3039 y Fo(\003)3798 3047 y Fi(1)3856 3027 y Ft(\016)118 3127 y(L)175 3097 y Fm(1)175 3150 y(C)211 3158 y Fy(g)250 3127 y Fs(h)298 3144 y Fp(Z)343 3127 y Fy(d)r Ft(k)c Fw(a)g(factor)1309 3355 y(exp)1463 3238 y Fk(\022)1524 3355 y Ft(\000)p Fs(\030)1657 3299 y Fw(1)p 1657 3336 42 4 v 1657 3412 a(2)1723 3355 y(dist\(\003)1947 3367 y Fo(1)1984 3355 y Fs(;)14 b Fw(\003)2079 3367 y Fo(2)2116 3355 y Fw(\))2148 3238 y Fk(\023)2232 3355 y Fw(=)23 b Fs(\024)2368 3321 y Fo(dist\(\003)2547 3329 y Fi(1)2579 3321 y Fu(;)p Fo(\003)2644 3329 y Fi(2)2677 3321 y Fo(\))3709 3355 y Fw(\(259\))118 3554 y(lik)n(e)k(in)h(\(232\))f(suc)n(h)g(that)h (w)n(e)f(get)869 3754 y Ft(k)1178 3675 y Fk(X)934 3854 y Fe(C)966 3862 y Fy(g)1002 3854 y Fe(2)p Fo(Conf)1178 3865 y(g)1216 3854 y Fi(\()p Fa(Z)1272 3842 y Fy(d)1301 3854 y(;)p Fi(\003)1358 3866 y(1)1390 3854 y Fe([)p Fi(\003)1467 3866 y(2)1499 3854 y(\))p Fy(;)966 3927 y(b)p Fi(\()p Fe(C)1046 3935 y Fy(g)1082 3927 y Fi(\))p Fy(>)1159 3911 y Fi(1)p 1159 3918 29 3 v 1159 3950 a(2)1197 3927 y Fo(dist)p Fi(\(\003)1365 3939 y(1)1398 3927 y Fy(;)p Fi(\003)1455 3939 y(2)1487 3927 y(\))1565 3754 y Fs(\031)1612 3766 y Fo(\003)1657 3774 y Fi(1)1690 3766 y Fm([)p Fo(\003)1780 3774 y Fi(2)1835 3754 y Ft(\016)18 b(L)1952 3719 y Fu(T)1952 3774 y Fm(C)1988 3782 y Fy(g)2027 3754 y Fs(h)2075 3770 y Fp(Z)2120 3754 y Fy(d)r Ft(k)k(\024)h Fs(c)2341 3766 y Fo(16)2425 3754 y Fs(#)2474 3719 y Fm(j)p Fo(\003)2539 3727 y Fi(1)2572 3719 y Fm([)p Fo(\003)2662 3727 y Fi(2)2694 3719 y Fm(j)2737 3754 y Fs(\024)2785 3719 y Fo(dist)o(\(\003)2963 3727 y Fi(1)2996 3719 y Fu(;)p Fo(\003)3061 3727 y Fi(2)3094 3719 y Fo(\))3124 3754 y Fs(:)562 b Fw(\(260\))118 4078 y(W)-7 b(e)28 b(write)f(the)h(\014rst)g(summand)f(in)h(\(259\))f(as) 1119 4282 y Fk(X)768 4461 y Fe(C)800 4469 y Fy(g)835 4461 y Fi(=)p Fe(C)912 4449 y Fi(1)910 4478 y Fy(g)945 4461 y Fe([C)1018 4449 y Fi(2)1016 4478 y Fy(g)1051 4461 y Fe(2)p Fo(Conf)1228 4472 y(g)1265 4461 y Fi(\()p Fa(Z)1321 4449 y Fy(d)1350 4461 y(;)p Fi(\003)1407 4473 y(1)1439 4461 y Fe([)p Fi(\003)1516 4473 y(2)1549 4461 y(\))p Fy(;)907 4537 y(b)p Fi(\()p Fe(C)987 4545 y Fy(g)1023 4537 y Fi(\))p Fe(\024)1100 4521 y Fi(1)p 1100 4528 V 1100 4559 a(2)1139 4537 y Fo(dist)o Fi(\(\003)1306 4549 y(1)1339 4537 y Fy(;)p Fi(\003)1396 4549 y(2)1428 4537 y(\))1600 4360 y Fw(\()p Fs(\031)1679 4372 y Fo(\003)1724 4380 y Fi(1)1780 4360 y Ft(\016)18 b(L)1897 4326 y Fu(T)1897 4383 y Fm(C)1936 4366 y Fi(1)1933 4399 y Fy(g)1973 4360 y Fs(h)2021 4377 y Fp(Z)2066 4361 y Fy(d)2099 4360 y Fw(\))g(\()p Fs(\031)2228 4372 y Fo(\003)2273 4380 y Fi(2)2329 4360 y Ft(\016)g(L)2446 4326 y Fu(T)2446 4383 y Fm(C)2485 4366 y Fi(2)2482 4399 y Fy(g)2522 4360 y Fs(h)2570 4377 y Fp(Z)2615 4361 y Fy(d)2648 4360 y Fw(\))1029 b(\(261\))841 4688 y(=)82 b Fs(\027)1029 4700 y Fo(\003)1074 4708 y Fi(1)1111 4688 y Fs(\027)1152 4700 y Fo(\003)1197 4708 y Fi(2)1253 4688 y Ft(\000)1697 4609 y Fk(X)1346 4788 y Fe(C)1378 4796 y Fy(g)1413 4788 y Fi(=)p Fe(C)1490 4776 y Fi(1)1488 4805 y Fy(g)1523 4788 y Fe([C)1596 4776 y Fi(2)1594 4805 y Fy(g)1629 4788 y Fe(2)p Fo(Conf)1806 4799 y(g)1843 4788 y Fi(\()p Fa(Z)1899 4776 y Fy(d)1928 4788 y(;)p Fi(\003)1985 4800 y(1)2018 4788 y Fe([)p Fi(\003)2095 4800 y(2)2127 4788 y(\))p Fy(;)1485 4864 y(b)p Fi(\()p Fe(C)1565 4872 y Fy(g)1601 4864 y Fi(\))p Fy(>)1679 4848 y Fi(1)p 1679 4855 V 1679 4886 a(2)1717 4864 y Fo(dist)o Fi(\(\003)1884 4876 y(1)1917 4864 y Fy(;)p Fi(\003)1974 4876 y(2)2006 4864 y(\))2178 4688 y Fw(\()p Fs(\031)2257 4700 y Fo(\003)2302 4708 y Fi(1)2358 4688 y Ft(\016)18 b(L)2475 4653 y Fu(T)2475 4710 y Fm(C)2514 4694 y Fi(1)2511 4727 y Fy(g)2551 4688 y Fs(h)2599 4705 y Fp(Z)2644 4688 y Fy(d)r Fw(\))h(\()p Fs(\031)2807 4700 y Fo(\003)2852 4708 y Fi(2)2908 4688 y Ft(\016)f(L)3025 4653 y Fu(T)3025 4710 y Fm(C)3064 4694 y Fi(2)3061 4727 y Fy(g)3100 4688 y Fs(h)3148 4705 y Fp(Z)3193 4688 y Fy(d)r Fw(\))118 5044 y(and)28 b(estimate)522 5243 y Ft(k)939 5164 y Fk(X)587 5343 y Fe(C)619 5351 y Fy(g)655 5343 y Fi(=)p Fe(C)732 5331 y Fi(1)730 5360 y Fy(g)764 5343 y Fe([C)837 5331 y Fi(2)835 5360 y Fy(g)871 5343 y Fe(2)p Fo(Conf)1048 5354 y(g)1085 5343 y Fi(\()p Fa(Z)1141 5331 y Fy(d)1170 5343 y(;)p Fi(\003)1227 5355 y(1)1259 5343 y Fe([)p Fi(\003)1336 5355 y(2)1369 5343 y(\))p Fy(;)727 5419 y(b)p Fi(\()p Fe(C)807 5427 y Fy(g)843 5419 y Fi(\))p Fy(>)920 5403 y Fi(1)p 920 5410 V 920 5442 a(2)958 5419 y Fo(dist)p Fi(\(\003)1126 5431 y(1)1159 5419 y Fy(;)p Fi(\003)1216 5431 y(2)1248 5419 y(\))1420 5243 y Fw(\()p Fs(\031)1499 5255 y Fo(\003)1544 5263 y Fi(1)1600 5243 y Ft(\016)18 b(L)1717 5209 y Fu(T)1717 5265 y Fm(C)1756 5249 y Fi(1)1753 5282 y Fy(g)1793 5243 y Fs(h)1841 5260 y Fp(Z)1885 5243 y Fy(d)r Fw(\))h(\()p Fs(\031)2048 5255 y Fo(\003)2093 5263 y Fi(2)2149 5243 y Ft(\016)f(L)2266 5209 y Fu(T)2266 5265 y Fm(C)2305 5249 y Fi(2)2302 5282 y Fy(g)2342 5243 y Fs(h)2390 5260 y Fp(Z)2435 5243 y Fy(d)r Fw(\))p Ft(k)k(\024)h Fs(c)2688 5255 y Fo(17)2772 5243 y Fs(#)2821 5209 y Fm(j)p Fo(\003)2886 5217 y Fi(1)2919 5209 y Fm([)p Fo(\003)3009 5217 y Fi(2)3041 5209 y Fm(j)3084 5243 y Fs(\024)3132 5209 y Fo(dist)o(\(\003)3310 5217 y Fi(1)3343 5209 y Fu(;)p Fo(\003)3408 5217 y Fi(2)3441 5209 y Fo(\))3471 5243 y Fs(:)215 b Fw(\(262\))1966 5682 y(47)p eop %%Page: 48 48 48 47 bop 118 354 a Fw(F)-7 b(rom)27 b(\(260\),)g(\(261\))g(and)g (\(262\))g(w)n(e)g(conclude)h(\(255\).)36 b(The)27 b(pro)r(of)g(of)h (\(256\),)f(using)g(\(255\),)g(is)g(the)h(same)f(as)g(in)h([13)o(].)118 454 y(T)-7 b(o)27 b(pro)n(v)n(e)f(\(257\))h(w)n(e)g(set)h Fs(\036)23 b Fw(=)g Fs( )s(\027)h Ft(\000)18 b Fs(\027)5 b Fw(\()p Fs( )s Fw(\))14 b Fs(\027)5 b Fw(.)38 b(So)1314 653 y Fs(\031)1361 665 y Fo(\003)1406 673 y Fi(1)1462 653 y Ft(\016)18 b(L)1579 619 y Fu(T)1579 674 y(S)1631 653 y Fs(\036)24 b Fw(=)f Fs(\031)1839 665 y Fo(\003)1884 673 y Fi(1)1939 653 y Ft(\016)18 b(L)2056 619 y Fu(T)2056 674 y(S)2109 653 y Fw(\()p Fs( )f(\027)5 b Fw(\))19 b Ft(\000)f Fs(\027)5 b Fw(\()p Fs( )s Fw(\))20 b Fs(\027)2620 665 y Fo(\003)2665 673 y Fi(1)3709 653 y Fw(\(263\))118 791 y(and)28 b(in)f(particular)1832 990 y Ft(L)1889 956 y Fm(1)1889 1011 y Fu(S)1959 990 y Fs(\036)d Fw(=)e(0)p Fs(:)1525 b Fw(\(264\))118 1128 y(W)-7 b(e)28 b(estimate)g(\(263\),)e (analogously)g(to)h(\(129\))g(in)h([13)o(],)g(using)f(the)h(\014ner)f (estimate)770 1391 y Ft(k)p Fs(\036)861 1406 y Fo(\003\()p Fm(C)s Fo(\))1001 1391 y Ft(k)1043 1406 y Fo(\003\()p Fm(C)s Fo(\))3709 1391 y Fw(\(265\))853 1542 y Ft(\024)82 b Fs(c)1036 1554 y Fo(13)1120 1542 y Fs(#)1169 1508 y Fm(\000j)p Fo(\003)1286 1516 y Fi(2)1319 1508 y Fm(j)1361 1542 y Ft(k)p Fs( )s Ft(k)1502 1554 y Fo(\003)1547 1562 y Fi(2)1602 1542 y Fs(#)1651 1508 y Fm(\000j)p Fo(\003)1768 1516 y Fy(r)1801 1508 y Fo(\()p Fm(C)s Fo(\))p Fm(j\000)1964 1464 y Fd(P)2033 1483 y Fe(1)2033 1527 y Fy(k)q Fi(=1)2141 1508 y Fo(\(3)p Fu(k)q Fo(\))2262 1483 y Fy(d)2297 1508 y Fu(n)2338 1517 y Fy(\014)r(;k)2451 1542 y Fs(\024)2499 1508 y Fo(dist\(\003)2678 1516 y Fi(1)2710 1508 y Fu(;)p Fo(\003)2775 1516 y Fi(2)2808 1508 y Fo(\))p Fm(\000)2886 1464 y Fd(P)2956 1483 y Fe(1)2956 1527 y Fy(k)q Fi(=1)3074 1508 y Fu(k)q(n)3151 1517 y Fy(\014)r(;k)118 1706 y Fw(that)30 b(w)n(e)g(get)f(from)g(\(256\).)43 b(F)-7 b(or)29 b(eac)n(h)g Ft(C)1403 1718 y Fu(g)1471 1706 y Fw(w)n(e)h(get)f(a)g(`go)r(o)r(d')h (factor)e Fs(\024)2346 1675 y Fo(dist\(\003)2525 1683 y Fi(1)2558 1675 y Fu(;)p Fo(\003)2623 1683 y Fi(2)2655 1675 y Fo(\))2715 1706 y Fw(that)i(w)n(e)g(can)f(tak)n(e)g(out)h(of)f (the)i(sum)118 1805 y(\(o)n(v)n(er)e(gum)i(con\014gurations\),)f(and)g (a)h(`bad')f(factor)g Fs(\024)1850 1775 y Fm(\000)1913 1732 y Fd(P)1983 1750 y Fe(1)1983 1794 y Fy(k)q Fi(=1)2101 1775 y Fu(k)q(n)2178 1784 y Fy(\014)r(;k)2274 1805 y Fw(.)46 b(The)31 b(latter)f(is)h(comp)r(ensated)f(for)g(in)h(the)g (usual)118 1905 y(w)n(a)n(y)26 b(b)n(y)i(the)g(factor)f(exp)o(\()p Ft(\000)p Fs(c)1042 1917 y Fu(g)1095 1843 y Fk(P)1182 1863 y Fm(1)1182 1930 y Fu(k)q Fo(=1)1321 1905 y Fs(k)1367 1875 y Fu(d)1406 1905 y Fs(n)1456 1917 y Fu(\014)s(;k)1557 1905 y Fw(\),)h(pro)n(vided)e(that)i Fs(c)2197 1917 y Fu(g)2263 1905 y Fw(is)g(su\016cien)n(tly)f(large.)118 2004 y(Using)g(\(264\))g(and)h(\(265\),)f(w)n(e)g(get)g(with)h(the)g (same)f(argumen)n(t)g(as)g(for)g(the)h(pro)r(of)f(of)g(\(230\):)968 2182 y(~)957 2204 y Fs(#)1006 2169 y Fm(j)p Fo(\003)1071 2177 y Fi(1)1104 2169 y Fm(j)1142 2204 y Ft(k)p Fs(\031)1231 2216 y Fo(\003)1276 2224 y Fi(1)1331 2204 y Ft(\016)18 b(L)1448 2169 y Fu(T)1448 2224 y(S)1500 2204 y Fs(\036)p Ft(k)23 b(\024)g Fs(c)1738 2216 y Fo(18)1822 2204 y Fs(#)1871 2169 y Fm(\000j)p Fo(\003)1988 2177 y Fi(2)2021 2169 y Fm(j)2058 2204 y Ft(k)p Fs( )s Ft(k)18 b Fs(\024)2265 2169 y Fo(dist)o(\(\003)2443 2177 y Fi(1)2476 2169 y Fu(;)p Fo(\003)2541 2177 y Fi(2)2574 2169 y Fo(\))2636 2204 y Fw(exp\()p Ft(\000)p Fs(c)2896 2216 y Fo(15)2966 2204 y Fs(T)12 b Fw(\))650 b(\(266\))118 2342 y(and)28 b(\(257\))e(follo)n(ws.)3808 2441 y Fh(2)118 2612 y Fw(W)-7 b(e)28 b(can)f(state)h(the)g(mixing)f(prop)r(erties)g(of)g Fs(\027)1565 2582 y Fm(\003)1632 2612 y Fw(wrt)g(spatio-temp)r(oral)f 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4549 y(\014)3709 4520 y Fw(\(268\))1096 4719 y Ft(\024)82 b Fs(c)p Fw(\(\003)1369 4731 y Fo(1)1407 4719 y Fs(;)14 b Fw(\003)1502 4731 y Fo(2)1538 4719 y Fs(;)g(\024)p Fw(\))19 b Fs(c)1710 4676 y Fm(j)p Fo(\003)1775 4684 y Fi(1)1807 4676 y Fm(j)p Fo(+)p Fm(j)p Fo(\003)1943 4684 y Fi(2)1976 4676 y Fm(j)1710 4741 y Fo(20)2018 4719 y Ft(k)p Fs( )2114 4731 y Fo(1)2151 4719 y Ft(k)2193 4731 y Fm(1)2276 4719 y Ft(k)p Fs( )2372 4731 y Fo(2)2409 4719 y Ft(k)2451 4731 y Fo(\003)2496 4739 y Fi(2)2551 4719 y Fs(\024)2599 4684 y Fu(m)p Fo(\()p Fu(\034)7 b Fo(\))2765 4719 y Fw(exp\()p Ft(\000)p Fs(c)3025 4731 y Fo(21)3095 4719 y Fs(T)12 b Fw(\))p Fs(:)216 4911 y Fj(3.)43 b(If)30 b(the)g(system)f(is)h(tr)l(anslation-invariant)h(and)f Fs( )1903 4923 y Fo(1)1941 4911 y Fs(;)14 b( )2032 4923 y Fo(2)2092 4911 y Ft(2)23 b(C)5 b Fw(\()p Fs(M)k Fw(\))30 b Fj(then)991 5169 y Fw(lim)780 5227 y Fo(max)o Fm(f)p Fu(m)p Fo(\()p Fu(\034)7 b Fo(\))p Fu(;T)i Fm(g!1)1332 5056 y Fk(Z)1378 5245 y Fu(M)1466 5169 y Fs(d\027)1555 5135 y Fm(\003)1617 5169 y Fs(K)1694 5135 y Fu(T)1688 5190 y(S)1745 5169 y Fw(\()p Fs( )1831 5181 y Fo(1)1887 5169 y Ft(\016)18 b Fs(\034)9 b Fw(\))20 b Fs( )2098 5181 y Fo(2)2243 5169 y Fw(=)2416 5052 y Fk(\022)2477 5056 y(Z)2523 5245 y Fu(M)2610 5169 y Fs(d\027)2699 5135 y Fm(\003)2752 5169 y Fs( )2806 5181 y Fo(1)2843 5052 y Fk(\023)14 b(\022)2979 5056 y(Z)3025 5245 y Fu(M)3113 5169 y Fs(d\027)3202 5135 y Fm(\003)3254 5169 y Fs( )3308 5181 y Fo(2)3346 5052 y Fk(\023)3421 5169 y Fs(:)265 b Fw(\(269\))3808 5362 y Fh(2)1966 5682 y Fw(48)p eop %%Page: 49 49 49 48 bop 118 354 a Fx(Ac)l(kno)l(wledgemen)l(ts)118 536 y Fw(This)29 b(pap)r(er)f(is)g(based)g(on)g(a)g(part)g(of)g(m)n(y)h (thesis)f([12)o(].)40 b(I)28 b(w)n(ould)g(lik)n(e)g(to)h(thank)f(m)n(y) g(sup)r(ervisor)f(Hans)i(Henrik)f(Rugh)118 636 y(for)23 b(his)h(great)f(supp)r(ort,)h(and)g(also)f(Roger)f(T)-7 b(rib)r(e)24 b(for)f(man)n(y)g(helpful)i(discussions)e(and)g (suggestions.)34 b(F)-7 b(urther)24 b(I)g(thank)118 735 y(Viviane)35 b(Baladi)f(and)g(V)n(\023)-39 b(eronique)33 b(Maume-Desc)n(hamps)h(for)g(helpful)i(discussions.)57 b(Finally)34 b(I)h(thank)g(the)g(EC)f(for)118 835 y(\014nancial)27 b(supp)r(ort)h(through)e(the)i(TMR-F)-7 b(ello)n(wship)28 b(ERBFMBICT-961157.)118 1110 y Fx(References)160 1291 y Fw([1])41 b(L.)27 b(Arnold:)37 b(Random)27 b(Dynamical)g(Systems,)h Fj(Springer)p Fw(,)g(1998)160 1457 y([2])41 b(V.)26 b(Baladi,)g(M.)g (Degli)h(Esp)r(osti,)e(S.)i(Isola,)e(E.)h(J\177)-42 b(arv)n(enp\177)g (a\177)g(a,)24 b(A.)i(Kupiainen:)36 b(The)27 b(sp)r(ectrum)f(of)g(w)n (eakly)f(coupled)289 1557 y(map)i(lattices,)h Fj(J.)h(Math.)j(Pur)l(es) d(et)h(Appl.)p Fw(,)e(77,)f(pp.)h(539-584,)d(1998)160 1723 y([3])41 b(H.)28 b(Bauer:)36 b(W)-7 b(ahrsc)n(heinlic)n(hk)n (eitstheorie,)25 b Fj(de)30 b(Gruyter)p Fw(,)e(4th)g(ed.,)f(1991)160 1889 y([4])41 b(J.)27 b(Bricmon)n(t)g(and)g(A.)h(Kupiainen:)37 b(Coupled)27 b(Analytic)h(Maps,)f Fj(Nonline)l(arity)36 b Fw(8,)28 b(pp.379-396,)c(1995)160 2055 y([5])41 b(J.)c(Bricmon)n(t)g (and)h(A.)g(Kupiainen:)57 b(High)37 b(T)-7 b(emp)r(erature)37 b(Expansions)f(and)i(Dynamical)f(Systems,)k Fj(Comm.)289 2155 y(Math.)31 b(Phys.)e Fw(178,)d(1996,)g(pp.73-732)160 2321 y([6])41 b(J.)26 b(Bricmon)n(t)f(and)h(A.)h(Kupiainen:)35 b(In\014nite)27 b(dimensional)f(SRB-measures.)f(Lattice)h(dynamics)f (\(P)n(aris,)g(1995\),)289 2421 y Fj(Physic)l(a)31 b(D)p Fw(,)d(103)e(\(1997\),)g(no.)i(1-4,)e(18{33.)160 2587 y([7])41 b(L.A.)26 b(Bunimo)n(vic)n(h:)35 b(Coupled)26 b(map)g(lattices:)36 b(One)26 b(step)g(forw)n(ard)e(and)i(t)n(w)n(o)g (steps)g(bac)n(k,)f Fj(Physic)l(a)30 b(D)e(86)p Fw(,)g(1995,)289 2686 y(pp.)g(248-255)160 2852 y([8])41 b(L.A.)23 b(Bunimo)n(vic)n(h,)g (Y.G.)g(Sinai:)34 b(Space-time)22 b(c)n(haos)g(in)g(coupled)h(map)f (lattices,)i Fj(Nonline)l(arity)i(1)p Fw(,)e(pp.)f(491-516,)289 2952 y(1988)160 3118 y([9])41 b(R.L.)25 b(Dobrushin:)35 b(Mark)n(o)n(v)23 b(pro)r(cesses)g(with)i(a)f(large)g(n)n(um)n(b)r(er)g (of)h(lo)r(cally)f(in)n(teracting)g(comp)r(onen)n(ts:)35 b(existence)289 3218 y(of)27 b(a)h(limit)g(pro)r(cess)e(and)i(its)f (ergo)r(dicit)n(y)-7 b(,)27 b Fj(Pr)l(oblems)k(Inform.)f(T)-6 b(r)l(ansmission)p Fw(,)29 b(7,)e(pp.)h(149-164,)c(1971)118 3384 y([10])41 b(R.L.)34 b(Dobrushin:)51 b(Mark)n(o)n(v)32 b(pro)r(cesses)h(with)i(man)n(y)e(lo)r(cally)h(in)n(teracting)g(comp)r (onen)n(ts-)f(the)i(rev)n(ersible)e(case)289 3483 y(and)27 b(some)g(generalizations,)f Fj(Pr)l(oblems)k(Inform.)h(T)-6 b(r)l(ansmission)p Fw(,)28 b(7,)g(pp.)g(235-241,)c(1971)118 3649 y([11])41 b(R.L.)32 b(Dobrushin)g(and)g(Y)-7 b(a.G.)32 b(Sinai)g(\(editors\):)45 b(Multicomp)r(onen)n(t)32 b(Random)g (Systems,)h Fj(A)l(dvanc)l(es)h(in)g(pr)l(ob)l(a-)289 3749 y(bility)d(and)f(r)l(elate)l(d)g(topics)p Fw(,)f(v)n(ol.)e(6)g (\(originally)f(published)i(in)g(Russian\),)g(1980)118 3915 y([12])41 b(T.)26 b(Fisc)n(her:)36 b(T)-7 b(ransfer)25 b(Op)r(erators)g(for)h(Deterministic)h(and)g(Sto)r(c)n(hastic)f (Coupled)g(Map)h(Lattices,)f Fj(thesis)p Fw(,)i(Uni-)289 4015 y(v)n(ersit)n(y)e(of)i(W)-7 b(arwic)n(k,)26 b(1998)118 4181 y([13])41 b(T.)24 b(Fisc)n(her,)g(H.H.)g(Rugh:)35 b(T)-7 b(ransfer)23 b(Op)r(erators)f(for)h(Coupled)h(Analytic)f(Maps,)i (to)e(app)r(ear)g(in)h Fj(Er)l(go)l(dic)k(The)l(ory)289 4280 y(&)h(Dynamic)l(al)i(Systems)118 4446 y Fw([14])41 b(R.J.)27 b(Glaub)r(er:)37 b(Time-dep)r(enden)n(t)28 b(statistics)f(of)h(the)g(Ising)f(mo)r(del.)h Fj(J.)i(Math.)h(Physics)p Fw(,)e(4.)e(pp.)h(294-307,)d(1963)118 4612 y([15])41 b(T.E.)19 b(Harris:)31 b(Nearest-Neigh)n(b)r(or)18 b(Mark)n(o)n(v)f(In) n(teraction)h(Pro)r(cesses)g(on)h(Multidimensional)h(Lattices,)g Fj(A)l(dvanc)l(es)289 4712 y(in)30 b(Mathematics)h Fn(9)p Fw(,)c(pp.)h(66-89,)e(1972)118 4878 y([16])41 b(R.)28 b(Holley:)36 b(A)28 b(class)f(of)g(in)n(teractions)g(in)h(an)f (in\014nite)h(particle)f(system,)h Fj(A)l(dv.)i(Math.)p Fw(,)f(5,)e(pp.)h(291-309,)d(1970)118 5044 y([17])41 b(M.)24 b(Jiang:Equilibrium)g(states)f(for)h(lattice)h(mo)r(dels)f(of)h (h)n(yp)r(erb)r(olic)f(t)n(yp)r(e)g Fj(Nonline)l(arity)k(8)p Fw(,)d(no.5,)g(1994,)e(pp.631-)289 5144 y(659)118 5310 y([18])41 b(M.)25 b(Jiang:)35 b(Ergo)r(dic)24 b(Prop)r(erties)f(of)i (Coupled)g(Map)h(Lattices)e(of)i(Hyp)r(erb)r(olic)f(T)n(yp)r(e,)g Fj(Penns.)j(State)f(University)289 5409 y(Dissertation)p Fw(,)h(1995)1966 5682 y(49)p eop %%Page: 50 50 50 49 bop 118 354 a Fw([19])41 b(M.)28 b(Jiang)g(and)g(A.)h(Mazel:)38 b(uniqueness)28 b(of)g(Gibbs)h(States)f(and)g(exp)r(onen)n(tial)g(Deca) n(y)g(of)g(Correlation)e(for)i(some)289 454 y(Lattice)f(mo)r(dels,)h Fj(Journal)h(of)i(Statistic)l(al)f(Physics)36 b Fw(82,)27 b(no.)g(3-4,)g(1995)118 620 y([20])41 b(M.)27 b(Jiang)f(and)h(Y)-7 b(a.B.)27 b(P)n(esin:)36 b(Equilibrium)27 b(Measures)f(for)h(Coupled)g (Map)g(Lattices:)36 b(Existence,)27 b(Uniqueness)289 720 y(and)g(Finite-Dimensional)h(Appro)n(ximations,)e(to)i(app)r(ear)e (in)i(CMP)118 886 y([21])41 b(G.)19 b(Keller)f(and)g(M.)h(K)r(\177)-44 b(unzle:)32 b(T)-7 b(ransfer)18 b(Op)r(erators)f(for)h(Coupled)g(Map)h (Lattices,)h Fj(Er)l(go)l(dic)j(The)l(ory)g(&)f(Dynamic)l(al)289 985 y(Systems)34 b Fw(V)-7 b(ol.)27 b(12,)g(pp.297-318,)e(1992)118 1151 y([22])41 b(S.)28 b(Lang:)36 b(Real)27 b(and)g(F)-7 b(unctional)28 b(Analysis,)f(3rd)g(edition,)h Fj(Springer)p Fw(,)g(1993)118 1317 y([23])41 b(T.M.)32 b(Liggett:)45 b(Existence)31 b(theorems)g(for)g(in\014nite)i(particle)e(systems,)i Fj(T)-6 b(r)l(ans.)34 b(A)n(mer.)f(Math.)i(So)l(c.)p Fw(,)f(165,)d(pp.)289 1417 y(471-481,)24 b(1972)118 1583 y([24])41 b(T.M.)28 b(Liggett:)38 b(In)n(teracting)27 b(P)n(article)g(Systems,)i Fj(Grund)t(lehr)l(en)i(der)g(mathematischen) g(Wissenschaften)p Fw(,)f(276,)289 1683 y Fj(Springer)p Fw(,)e(1985)118 1849 y([25])41 b(C.)29 b(Maes)f(and)g(A.)h(V)-7 b(an)29 b(Mo\013aert:)39 b(Sto)r(c)n(hastic)28 b(Stabilit)n(y)h(of)f(W) -7 b(eakly)29 b(Coupled)f(Lattice)h(Maps,)g Fj(Nonline)l(arity)p Fw(,)289 1948 y(10,)e(1997,)e(pp.715-730)118 2114 y([26])41 b(Y)-7 b(a.B.)24 b(P)n(esin)g(and)g(Y)-7 b(a.G.)25 b(Sinai:)35 b(Space-time)25 b(c)n(haos)e(in)i(c)n(hains)f(of)g(w)n(eakly)f(in)n (teracting)h(h)n(yp)r(erb)r(olic)g(mappings,)289 2214 y Fj(A)l(dvanc)l(es)30 b(in)g(Soviet)g(Mathematics)36 b Fw(V)-7 b(ol.3,)27 b(1991,)f(pp.165-198)118 2380 y([27])41 b(F.)j(Spitzer:)70 b(Random)43 b(Pro)r(cesses)f(De\014ned)j(through)e (the)h(In)n(teraction)f(of)h(an)f(In\014nite)i(P)n(article)e(System,) 289 2480 y Fj(Springer)30 b(L)l(e)l(ctur)l(e)f(Notes)g(in)h (Mathematics)p Fw(,)g(V)-7 b(ol.)27 b(89,)g(pp.)h(201-223,)c(1969)118 2646 y([28])41 b(F.)28 b(Spitzer:)37 b(In)n(teraction)26 b(of)i(Mark)n(o)n(v)d(pro)r(cesses,)h Fj(A)l(dv.)k(Math.)p Fw(,)f(5,)f(pp.)g(246-290,)c(1970)118 2812 y([29])41 b(D.L.)31 b(V)-7 b(olevic)n(h:)44 b(The)31 b(Sinai-Bo)n(w)n(en-Ruelle)f (Measure)g(for)g(a)h(Multidimensional)g(Lattice)h(of)f(in)n(teracting)f (Hy-)289 2911 y(p)r(erb)r(olic)d(Mappings,)g Fj(R)n(uss.)i(A)n(kad.)i (Dokl.)g(Math.)e Fw(47,)e(1993,)e(pp.117-121)118 3077 y([30])41 b(D.L.)34 b(V)-7 b(olevic)n(h:)49 b(Construction)33 b(of)g(an)h(analogue)e(of)i(Bo)n(w)n(en-Ruell-Sinai)e(measure)g(for)h (a)h(m)n(ultidimensional)289 3177 y(lattice)27 b(of)h(in)n(teracting)f (h)n(yp)r(erb)r(olic)g(mappings)g Fj(R)n(uss.)i(A)n(kad.)i(Math.)g(Sb)l (ornik)37 b Fw(V)-7 b(ol.79,)26 b(1994,)g(pp.347-363)118 3443 y(T)-7 b(orsten)27 b(Fisc)n(her)118 3542 y(Niels)h(Bohr)e (Institute)118 3642 y(Blegdamsv)n(ej)g(17)118 3742 y(DK-2100)f(Cop)r (enhagen)i(\037)118 3841 y(email:)37 b(\014sc)n(her@n)n(bi.dk)1966 5682 y(50)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------9906100918258--