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b(is)g(w)m(e)h(construct)g(a)g (Mark)m(o)m(v)h(c)m(hain)e Fr(\033)s Fv(\()p Fr(t)p Fv(\))h(with)50 4239 y(state)40 b(space)f Fk(S)587 4253 y Fj(N)695 4239 y Fv(and)f(discrete)h(time)f Fr(t)h Fk(2)g Fh(N)51 b Fv(b)m(y)39 b(prescribing)d(transition)h(probabilities)f Fr(p)3322 4253 y Fj(N)3390 4239 y Fv(\()p Fr(\033)n(;)15 b(\021)s Fv(\))41 b(=)50 4385 y Fh(P)p Fv([)p Fr(\033)s Fv(\()p Fr(t)21 b Fv(+)f(1\))26 b(=)f Fr(\021)s Fk(j)p Fr(\033)s Fv(\()p Fr(t)p Fv(\))i(=)e Fr(\033)s Fv(])31 b(b)m(y)932 4719 y Fr(p)978 4733 y Fj(N)1046 4719 y Fv(\()p Fr(\033)n(;)15 b(\021)s Fv(\))27 b(=)1377 4506 y Ff(8)1377 4588 y(>)1377 4615 y(<)1377 4779 y(>)1377 4806 y(:)1513 4543 y Ft(1)p 1499 4558 65 4 v 1499 4610 a Fj(N)1574 4579 y Fr(e)1616 4546 y Fi(\000)p Fj(\014)1717 4491 y Fi(p)p 1778 4491 V 1778 4546 a Fj(N)6 b(E)1895 4555 y Fm(\033)1943 4579 y Fr(;)162 b Fv(if)29 b Fk(k)p Fr(\033)24 b Fk(\000)c Fr(\021)s Fk(k)2518 4593 y Ft(2)2585 4579 y Fv(=)2681 4504 y Fk(p)p 2757 4504 46 4 v 75 x Fv(2)1488 4740 y(1)h Fk(\000)f Fr(e)1687 4707 y Fi(\000)p Fj(\014)1788 4653 y Fi(p)p 1848 4653 65 4 v 54 x Fj(N)7 b(E)1966 4716 y Fm(\033)2014 4740 y Fr(;)91 b Fv(if)29 b Fr(\033)g Fv(=)c Fr(\021)1488 4876 y Fv(0)p Fr(;)572 b Fv(otherwise)3523 4719 y(\(1)p Fr(:)p Fv(3\))50 5049 y(Note)29 b(that)g(the)g(dynamics)e (is)g(also)i(random,)f(i.e.)40 b(the)28 b(la)m(w)g(of)h(the)f(Mark)m(o) m(v)i(c)m(hain)e(is)g(a)g(measure)h(v)-5 b(alued)50 5194 y(random)31 b(v)-5 b(ariable)31 b(on)g(\012)h(that)g(tak)m(es)i(v)-5 b(alues)31 b(in)g(the)h(space)g(of)g(Mark)m(o)m(v)i(measures)e(on)f (the)i(path)e(space)p 50 5284 546 4 v 159 5345 a Fp(5)197 5371 y Fo(The)23 b(standard)g(mo)r(del)g(has)g Fe(X)1099 5380 y Fd(\033)1167 5371 y Fo(instead)h(of)e Fe(E)1573 5380 y Fd(\033)1619 5371 y Fo(.)33 b(This)24 b(mo)r(di\014cation)f(has) g(no)g(e\013ect)h(on)e(the)i(equilibrium)g(prop)r(erties)50 5462 y(of)h(the)h(mo)r(del,)g(and)f(will)i(b)r(e)f(helpful)h(for)e (setting)i(up)e(the)h(dynamics.)p eop %%Page: 4 4 4 3 bop 50 295 a Fo(4)3309 b Fl(Se)l(ction)27 b Fo(1)50 513 y Fk(S)112 480 y Fg(N)105 538 y Fj(N)174 513 y Fv(.)71 b(W)-8 b(e)42 b(will)c(mostly)i(tak)m(e)i(a)g(p)s(oin)m(t)m(wise)d(p)s (oin)m(t)g(of)i(view,)i(i.e.)71 b(w)m(e)41 b(consider)f(the)h(dynamics) e(for)h(a)50 658 y(giv)m(en)26 b(\014xed)f(realization)g(of)h(the)g (disorder)e(parameter)i Fr(!)i Fk(2)d Fv(\012)h(\(dep)s(endence)f(on)h (whic)m(h)e(w)m(e)i(p)s(ersisten)m(tly)50 804 y(suppress)i(in)h(the)i (notation\).)159 1022 y(It)g(is)f(easy)i(to)g(see)f(that)h(this)e (dynamics)g(is)g(rev)m(ersible)f(with)h(resp)s(ect)h(to)h(the)f(Gibbs)f (measure)h Fr(\026)3550 1036 y Fj(\014)s(;N)3685 1022 y Fv(.)50 1167 y(On)22 b(also)h(sees)g(that)h(it)e(represen)m(ts)h(a)h (nearest)f(neigh)m(b)s(or)f(random)g(w)m(alk)h(on)g(the)g(h)m(yp)s (ercub)s(e)e(with)h(traps)h(of)50 1313 y(random)d(depths)h(\(i.e.)38 b(the)21 b(probabilit)m(y)e(to)j(mak)m(e)g(a)g(zero)g(step)f(is)g (rather)g(large)g(when)g Fr(E)3070 1327 y Fj(\033)3140 1313 y Fv(is)f(large\))3449 1280 y Ft(6)3491 1313 y Fv(.)37 b(The)50 1458 y(idea)31 b(suggested)h(b)m(y)g(the)f(kno)m(wn)g(b)s(eha) m(vior)g(of)h(the)f(equilibrium)c(distribution)h(is)i(that)i(this)f (dynamics,)50 1604 y(for)37 b Fr(\014)42 b(>)37 b(\014)448 1618 y Fj(c)486 1604 y Fv(,)i(will)c(sp)s(end)h(long)h(p)s(erio)s(ds)e (of)j(time)f(in)f(the)i(states)g Fr(\033)2449 1571 y Ft(\(1\))2547 1604 y Fr(;)15 b(\033)2642 1571 y Ft(\(2\))2740 1604 y Fr(;)g(:)g(:)g(:)54 b Fv(etc.)64 b(and)36 b(will)f(mo)m(v)m(e)50 1749 y(\\quic)m(kly")28 b(from)g(one)h(of)g(these)g(con\014gurations)e (to)j(the)e(next.)41 b(Based)29 b(on)f(this)f(in)m(tuition,)g(Bouc)m (haud)i(et)50 1895 y(al.)38 b(prop)s(osed)21 b(the)i(\\REM-lik)m(e")g (trap)g(mo)s(del:)36 b(the)23 b(state)h(space)f(is)f(reduced)g(to)h Fr(M)33 b Fv(p)s(oin)m(ts,)23 b(represen)m(ting)50 2040 y(the)30 b Fr(M)40 b Fv(\\deep)s(est")31 b(traps.)40 b(Eac)m(h)31 b(of)f(the)g(states)h(is)e(assigned)g(a)i(p)s(ositiv)m(e)e (random)g(energy)h Fr(E)3314 2054 y Fj(k)3389 2040 y Fv(whic)m(h)f(is)50 2185 y(tak)m(en)h(to)f(b)s(e)f(exp)s(onen)m(tially) f(distributed)f(with)h(rate)j(one.)40 b(The)29 b(dynamics)e(is)h(no)m (w)h(a)g(con)m(tin)m(uous)f(time)50 2331 y(Mark)m(o)m(v)h(c)m(hain)e Fr(Y)20 b Fv(\()p Fr(t)p Fv(\))27 b(taking)g(v)-5 b(alues)27 b(in)f Fr(S)1522 2345 y Fj(M)1628 2331 y Fk(\021)f(f)p Fv(1)p Fr(;)15 b(:)g(:)g(:)i(;)e(M)10 b Fk(g)p Fv(.)41 b(If)26 b(the)i(pro)s(cess)f(is)f(in)g(state)j Fr(k)s Fv(,)f(it)f(w)m(aits)g(an)50 2476 y(exp)s(onen)m(tially)k(distributed)f (time)i(with)f(mean)i(prop)s(ortional)e(to)i Fr(e)2425 2443 y Fj(E)2478 2453 y Fm(k)2519 2443 y Fj(\013)2571 2476 y Fv(,)g(and)f(then)h(jumps)e(with)g(equal)50 2622 y(probabilit)m(y)23 b(in)h(one)i(of)f(the)h(other)g(states)h Fr(k)1560 2589 y Fi(0)1611 2622 y Fk(2)d Fr(S)1752 2636 y Fj(M)1833 2622 y Fv(.)39 b(This)24 b(pro)s(cess)h(is)f(then)h (analyzed)h(using)e(essen)m(tially)50 2767 y(tec)m(hniques)j(from)h Fq(r)-5 b(enewal)29 b Fv(theory)-8 b(.)41 b(The)27 b(essen)m(tial)h(p)s (oin)m(t)f(is)g(that)h(if)f(one)h(starts)h(the)f(pro)s(cess)f(from)h (the)50 2913 y(uniform)g(distribution,)e(it)k(is)f(p)s(ossible)f(to)j (sho)m(w)f(that)h(if)e(one)h(only)f(considers)g(the)h(times,)g Fr(T)3283 2927 y Fj(i)3314 2913 y Fv(,)g(at)h(whic)m(h)50 3058 y(the)25 b(pro)s(cess)g(c)m(hanges)h(its)e(state,)k(then)d(the)g (coun)m(ting)g(pro)s(cess,)h Fr(c)p Fv(\()p Fr(t)p Fv(\),)h(that)f (coun)m(ts)f(the)g(n)m(um)m(b)s(er)f(of)h(these)50 3204 y(jumps)31 b(in)g(the)i(time)g(in)m(terv)-5 b(al)32 b(\(0)p Fr(;)15 b(t)p Fv(])34 b(is)d(a)j(classical)d(renew)m(al)i(\(coun)m (ting\))g(pro)s(cess)g([KT];)f(moreo)m(v)m(er,)j(as)50 3349 y Fr(n)28 b Fk(")h(1)p Fv(,)k(this)f(renew)m(al)g(pro)s(cess)g (con)m(v)m(erges)i(to)f(a)g(renew)m(al)f(pro)s(cess)g(with)f(a)i Fq(deterministic)g Fv(la)m(w)f(for)h(the)50 3495 y(renew)m(al)h(time)h (with)e(a)i(hea)m(vy-tailed)g(distribution)c(\(in)j(the)h(sense)g(that) g(that)g(the)g(mean)g(is)f(in\014nite)3636 3462 y Ft(7)3674 3495 y Fv(\))50 3640 y(whose)e(densit)m(y)g(is)f(prop)s(ortional)g(to)i Fr(t)1401 3607 y Fi(\000)p Ft(1)p Fi(\000)p Ft(1)p Fj(=\013)1708 3640 y Fv(where)f Fr(\013)d Fv(=)g Fr(\014)5 b(=\014)2312 3654 y Fj(c)2350 3640 y Fv(.)47 b(It)32 b(is)g(the)g(emergence)i(of)e (suc)m(h)h Fq(non-)50 3785 y(Markovian)24 b Fv(limit)d(pro)s(cesses)i (that)h(is)e(ultimately)g(resp)s(onsible)f(for)i(all)f(the)h(aging)h (phenomena)f(observ)m(ed)50 3931 y(in)32 b(the)i(abundan)m(t)e (literature)h(on)h(this)e(and)h(related)g(mo)s(dels.)49 b(Mathematically)-8 b(,)35 b(the)f(analysis)e(of)i(this)50 4076 y(trap)k(mo)s(del)g(presen)m(ts)h(no)f(particular)f(c)m(hallenge)i (and)f(the)h(analysis)f(presen)m(ted)g(e.g.)67 b(in)38 b(the)h(review)50 4222 y([BCKM])31 b(is)e(essen)m(tially)g(rigorous,)h (or)h(can)f(b)s(e)g(made)g(so)h(with)e(minor)g(e\013orts.)159 4440 y(Our)34 b(purp)s(ose)g(is)h(to)i(sho)m(w,)g(in)d(a)i (mathematically)g(rigorous)f(w)m(a)m(y)-8 b(,)38 b(ho)m(w)e(and)f(to)i (what)e(exten)m(t)j(the)50 4585 y(REM-lik)m(e)26 b(trap)h(mo)s(del)e (can)i(b)s(e)f(view)m(ed)g(as)h(an)f(appro)m(ximation)g(of)h(what)f (happ)s(ens)f(in)g(the)i(REM)g(itself.)50 4731 y(Clearly)22 b(the)i(main)e(di\016cult)m(y)g(in)h(doing)f(this)h(will)e(b)s(e)i(to)h (explain)e(wh)m(y)h(the)h(rather)f(complicated)h(random)50 4876 y(w)m(alk)i(on)g(the)h(h)m(yp)s(ercub)s(e)e(b)s(et)m(w)m(een)i (the)g(most)g(profound)d(traps)j(can)f(b)s(e)g(in)m(terpreted)g(as)h(a) g(simple)d(jump)50 5022 y(pro)s(cess.)40 b(This)29 b(question)g(has)h (t)m(w)m(o)i(asp)s(ects:)p 50 5193 546 4 v 159 5254 a Fp(6)197 5280 y Fo(W)-6 b(e)28 b(ha)n(v)n(e)f(c)n(hosen)i(this)g (particular)f(dynamics)g(for)f(tec)n(hnical)j(reasons.)42 b(T)-6 b(o)26 b(study)j(e.g.)41 b(the)29 b(Metrop)r(olis)f(algorithm)50 5371 y(w)n(ould)d(require)i(some)f(extra)f(w)n(ork,)h(but)f(w)n(e)h (exp)r(ect)h(essen)n(tially)h(the)e(same)g(results)h(to)f(hold.)159 5436 y Fp(7)197 5462 y Fo(This)g(is)h(clearly)g(due)f(to)f(the)h(fact)g (that)f(the)h(a)n(v)n(erage)f(of)g(the)h(witing)g(time)g Fe(e)2470 5436 y Fd(\013)2519 5462 y Fe(E)2576 5473 y Fd(i)2630 5462 y Fo(o)n(v)n(er)f(the)h(disorder)h(is)g(in\014nite.)p eop %%Page: 5 5 5 4 bop 50 295 a Fl(A)l(ging)27 b(in)h(the)g(REM.)f(Part)h(1.)2699 b Fo(5)33 513 y Fv(1\))46 b(Wh)m(y)31 b(do)s(es)f(the)g(pro)s(cess)g (jump)f(with)g(the)i(uniform)d(distribution)e(on)31 b(the)f(extremal)h (states?)33 731 y(2\))46 b(Wh)m(y)23 b(can)f(this)f(pro)s(cess)h(b)s(e) g(seen)g(as)h(a)f(Mark)m(o)m(v)i(pro)s(cess,)g(in)d(particular,)h(wh)m (y)g(are)h(the)f(times)g(b)s(et)m(w)m(een)159 876 y(visits)29 b(of)h(t)m(w)m(o)i(extreme)f(p)s(oin)m(ts)e(asymptotically)h(exp)s (onen)m(tially)f(distributed?)159 1095 y(While)i(these)i(facts)h(ma)m (y)f(app)s(ear)f(\\ob)m(vious")g(to)i(most)f(ph)m(ysicists,)e(the)i (reason)g(wh)m(y)f(they)h(are)f(not)50 1240 y(addressed)38 b(in)g(an)m(y)i(serious)e(w)m(a)m(y)j(in)d(the)i(literature)e(is)h (that)h(a\))g(they)f(are)h(not)g(at)g(all)e(easy)i(to)g(solv)m(e)50 1385 y(and)c(b\))h(they)g(are,)i(strictly)d(sp)s(eaking,)i(not)f(ev)m (en)h(true.)60 b(In)36 b(fact,)k(w)m(e)e(will)c(see)j(in)f(the)h (course)g(of)h(the)50 1531 y(analysis)30 b(\(including)e(the)k(follo)m (w-up)d(pap)s(er)i([BBG]\))i(that)f(suc)m(h)f(prop)s(erties)f(can)h(b)s (e)g(only)f(established)50 1676 y(in)25 b(a)i(v)m(ery)g(w)m(eak)g (asymptotic)g(form,)g(whic)m(h)f(is,)g(ho)m(w)m(ev)m(er,)j(just)d (enough)g(to)i(imply)c(that)j(the)g(predictions)50 1822 y(of)j(Bouc)m(haud's)h(mo)s(del)e(apply)g(to)i(the)f(long)g(time)g (asymptotics)h(of)f(the)h(pro)s(cess.)40 b(While)29 b(this)g(fact)i (will)50 1967 y(emerge)f(here)f(only)f(through)h(some)g(v)m(ery)h (careful)e(and)h(tedious)f(computations,)h(it)g(is)f(clearly)h (desirable)50 2113 y(to)i(dev)m(elop)f(a)h(more)f(profound)f (understanding)f(of)i(the)h(phenomenon.)159 2331 y(In)38 b(this)f(\014rst)h(pap)s(er)g(w)m(e)h(will)d(essen)m(tially)i(address)g (the)h(question)e(1\).)67 b(W)-8 b(e)40 b(will)c(sho)m(w)i(that)i(if)d (w)m(e)50 2476 y(lo)s(ok)f(at)g(the)h(sequence)f(of)g(visits)f(of)h (the)h(pro)s(cess)e(on)h(a)h(selected)f(set)h(of)f(set)h(of)f(the)g (states)i(of)e(lo)m(w)m(est)50 2622 y(energy)-8 b(,)27 b(disregarding)d(the)i(times)f(of)h(these)g(visits,)f(the)h(la)m(w)f (of)h(the)g(sequence)g(can)g(indeed)e(b)s(e)h(describ)s(ed)50 2767 y(asymptotically)36 b(b)m(y)h(a)h(simple)d(discrete)i(time)g(Mark) m(o)m(v)i(c)m(hain)e(on)g(this)f(set,)k(whic)m(h)c(jumps)g(from)g(one) 50 2913 y(p)s(oin)m(t)d(to)i(the)f(next)g(with)f(the)h(uniform)e (distribution.)48 b(W)-8 b(e)35 b(will)c(also)j(consider)f(t)m(w)m(o)i (more)f(questions.)50 3058 y(First)41 b(w)m(e)h(will)e(compute)i(the)g (mean)g(en)m(trance)h(time)f(and)f(the)h(en)m(trance)h(la)m(w)f(on)g (this)e(set)j(starting)50 3204 y(from)e(an)h(arbitrary)e(p)s(oin)m(t)h (on)h(the)g(h)m(yp)s(ercub)s(e.)73 b(Second)41 b(w)m(e)h(will)d (compute)j(the)g(mean)g(transition)50 3349 y(times)29 b(b)s(et)m(w)m(een)h(p)s(oin)m(ts)e(in)g(this)g(set.)41 b(It)30 b(will)c(turn)i(out)i(that)g(these)g(mean)f(transition)f(times) h(do)g(indeed)50 3495 y(dep)s(end,)d(asymptotically)-8 b(,)27 b(only)e(on)i(the)g(starting)f(p)s(oin)m(t.)38 b(Th)m(us,)27 b(mo)s(dulo)e(the)h(Mark)m(o)m(vian)i(h)m(yp)s(othesis,) 50 3640 y(w)m(e)36 b(come)g(v)m(ery)f(close)h(to)g(the)f(heuristic)f (picture)g(outlined)f(ab)s(o)m(v)m(e.)57 b(Moreo)m(v)m(er,)39 b(w)m(e)c(will)e(see)j(that)g(the)50 3785 y(mean)29 b(time)f(to)i(reac) m(h)g(a)f(set)g(of)h(extremes)f(is)f(prop)s(ortional)f(to)j(the)f (smallest)f(\\w)m(aiting)g(times")h(on)g(that)50 3931 y(set)23 b(\(if)g Fr(\014)30 b(>)473 3853 y Fk(p)p 549 3853 197 4 v 78 x Fv(2)15 b(ln)f(2)q(\),)25 b(whic)m(h)d(will)e(b)s(e)i (in)m(terpreted)g(as)i(a)f(\014rst)f(sign)g(of)h(the)h(o)s(ccurrence)f (of)g(aging.)38 b(W)-8 b(e)24 b(will)50 4076 y(also)g(sho)m(w)g(that)i (in)d(con)m(trast,)k(if)c Fr(\014)31 b(<)1364 3999 y Fk(p)p 1439 3999 V 1439 4076 a Fv(2)15 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y(information,)g(w)m(e)h(will)d(discuss)g(in)h (subsection)h(1.3)h(some)g(asp)s(ects)f(of)h(the)g(metastable)g(b)s (eha)m(viour)e(of)50 3131 y(the)30 b(mo)s(del,)g(and)g(state)h (precisely)e(the)i(results)e(w)m(e)i(alluded)d(to)j(ab)s(o)m(v)m(e.)50 3334 y Fs(1.2.)47 b(Equilibrium)34 b(results)h(for)g(the)g(REM.)159 3609 y Fv(In)c(this)g(sub-section)h(w)m(e)g(giv)m(e)h(the)f(necessary)g (bac)m(kground)g(on)g(the)g(\(mostly)g(w)m(ell)g(kno)m(wn,)g(see)g (e.g.)50 3755 y([Ei,GOP)-8 b(,OP)g(,Ru]\))30 b(static)h(asp)s(ects)f (of)g(the)g(REM,)h(i.e.)40 b(w)m(e)31 b(giv)m(e)f(a)g(precise)g (description)e(of)i(the)g(in\014nite)50 3900 y(v)m(olume)f(asymptotics) g(of)h(the)f(Gibbs)f(measures)h(that)h(will)d(help)h(to)i(understand)d (the)j(heuristics)d(of)j(the)50 4045 y(mo)s(del.)38 b(A)26 b(complete)g(exp)s(osition)e(can)i(b)s(e)f(found)g(in)f([Bo].)40 b(The)26 b(basic)f(result)f(is)h(the)h(follo)m(wing)e(theorem)50 4191 y(that)31 b(c)m(haracterizes)h(the)e(precise)g(b)s(eha)m(vior)f 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b(measure)g(is)f(pure)g(p)s(oin)m(t.)47 b(This)32 b(is)g(quite)g(easy)h(to)h(see)50 5462 y(and)c(the)g(details)g(of)g (the)h(argumen)m(t)f(can)h(b)s(e)f(found)f(in)g([Bo].)p eop %%Page: 9 9 9 8 bop 50 295 a Fl(A)l(ging)27 b(in)h(the)g(REM.)f(Part)h(1.)2699 b Fo(9)50 513 y Fs(1.3.)47 b(Metastabilit)m(y)34 b(and)h(statemen)m(t)e (of)i(the)g(main)f(results.)159 757 y Fv(The)26 b(prop)s(erties)e(of)i (the)h(in)m(v)-5 b(arian)m(t)25 b(distribution)d(explained)j(in)f(the)j (previous)d(section)j(clearly)e(imply)50 902 y(that)i(at)g(temp)s (eratures)f(b)s(elo)m(w)g(the)h(critical)e(one)i(the)f(dynamical)f(pro) s(cess)h(will)e(sp)s(end)h(most)i(of)f(its)g(time)50 1047 y(on)34 b(the)h(extreme)h(states.)54 b(This)33 b(suggests)j(that)f (the)g(long)f(time)g(b)s(eha)m(viour)f(of)i(the)g(dynamics)e(can)i(b)s (e)50 1193 y(read)d(of)g(from)g(observ)-5 b(ations)31 b(of)i(the)f(pro)s(cess)g(on)g(visits)e(to)j(these)f(states.)48 b(More)32 b(precisely)-8 b(,)32 b(de\014ne)g(the)50 1338 y(sets,)f(for)f Fr(E)h Fk(2)24 b Fh(R)s Fv(,)1223 1484 y Fr(T)1276 1498 y Fj(N)1345 1484 y Fv(\()p Fr(E)5 b Fv(\))26 b Fk(\021)1609 1410 y Ff(\010)1662 1484 y Fr(\033)i Fk(2)d(S)1883 1498 y Fj(N)1952 1407 y Ff(\014)1952 1461 y(\014)1983 1484 y Fr(E)2050 1498 y Fj(\033)2124 1484 y Fk(\025)g Fr(u)2272 1498 y Fj(N)2341 1484 y Fv(\()p Fr(E)5 b Fv(\))2483 1410 y Ff(\011)3477 1484 y Fv(\(1)p Fr(:)p Fv(18\))50 1687 y(where)24 b Fr(u)359 1701 y Fj(N)428 1687 y Fv(\()p Fr(E)5 b Fv(\))26 b(is)e(de\014ned)f(in)h(\(1.6\).)40 b(W)-8 b(e)26 b(will)c(call)j(the)g(set)g Fr(T)2159 1701 y Fj(N)2228 1687 y Fv(\()p Fr(E)5 b Fv(\))26 b('the)f(top',)i(and)d (frequen)m(tly)g(suppress)50 1832 y(indices,)33 b(writing)e Fr(T)44 b Fv(=)30 b Fr(T)13 b Fv(\()p Fr(E)5 b Fv(\))32 b(=)e Fr(T)1286 1846 y Fj(N)1356 1832 y Fv(\()p Fr(E)5 b Fv(\))34 b(whenev)m(er)g(no)f(confusion)g(is)f(lik)m(ely)-8 b(.)50 b(Moreo)m(v)m(er,)37 b(w)m(e)d(will)d(use)50 1977 y(the)36 b(con)m(v)m(en)m(tion)g(that)g Fr(M)44 b Fk(\021)34 b(j)p Fr(T)1188 1991 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Fv([)p Fr(\034)1884 2857 y Fj(\033)1874 2917 y(I)1959 2894 y Fv(=)g Fr(k)s Fk(j)p Fr(\033)s Fv(\(0\))i(=)e Fr(\033)s Fv(])974 b(\(1)p Fr(:)p Fv(20\))50 3097 y(Let)30 b(us)f(recall)g(that)i(in)d([BEGK1,BEGK2])k(a)f(v)m(ery)f(similar)d (program)j(w)m(as)g(carried)f(out)h(in)f(a)h(situation)50 3242 y(that)k(w)m(e)h(consider)e(generic)h(for)g(systems)g(ha)m(ving)g (\\metastable)h(states".)53 b(A)34 b(k)m(ey)h(c)m(haracterization)h(of) 50 3388 y(the)29 b(e\013ectiv)m(e)h(dynamics)e(on)g(suc)m(h)h(a)g(set)g Fk(M)g Fv(in)m(v)m(olv)m(es)g(the)g(quan)m(tities)f Fh(P)p Fv([)p Fr(\034)2678 3355 y Fj(x)2668 3413 y(I)2750 3388 y Fr(<)d(\034)2896 3355 y Fj(x)2886 3410 y(x)2942 3388 y Fv(])k(\(that,)h(in)d(p)s(oten)m(tial)50 3533 y(theoretic)g (language,)g(are)g(closely)f(related)g(to)h(Newtonian)f(capacities\).) 40 b(There,)26 b(as)h(here,)g(w)m(e)g(iden)m(ti\014ed)50 3679 y(certain)k(subsets)g Fk(M)h Fv(of)g(the)g(state)h(space,)g(\000.) 44 b(They)31 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b(and)d(an)m(y)i(set)g Fr(I)j Fk(\032)28 b(Mnf)p Fr(x;)15 b(y)s Fk(g)34 b Fv(either)e Fh(P)p Fv([)p Fr(\034)2558 4702 y Fj(x)2548 4760 y(I)2633 4735 y Fr(<)c(\034)2782 4702 y Fj(x)2772 4757 y(x)2828 4735 y Fv(])h Fk(\034)g Fr(\026)p Fv(\()p Fr(y)s Fv(\))p Fh(P)p Fv([)p Fr(\034)3305 4691 y Fj(y)3295 4761 y(I)3378 4735 y Fr(<)f(\034)3527 4702 y Fj(y)3517 4757 y(y)3571 4735 y Fv(])33 b(or)159 4880 y Fh(P)p Fv([)p Fr(\034)289 4847 y Fj(x)279 4905 y(I)361 4880 y Fr(<)25 b(\034)507 4847 y Fj(x)497 4902 y(x)553 4880 y Fv(])g Fk(\035)g Fr(\026)p Fv(\()p Fr(y)s Fv(\))p Fh(P)p Fv([)p Fr(\034)1022 4836 y Fj(y)1012 4907 y(I)1092 4880 y Fr(<)g(\034)1238 4847 y Fj(y)1228 4902 y(y)1282 4880 y Fv(],)31 b(and)-8 5098 y(\(ii\))45 b(there)30 b(exists)g Fr(m)722 5112 y Ft(1)788 5098 y Fk(2)25 b(M)p Fv(,)31 b(s.t.)41 b(for)30 b(all)f Fr(x)d Fk(2)f(Mn)p Fr(m)1864 5112 y Ft(1)1905 5098 y Fv(,)30 b Fr(\026)p Fv(\()p Fr(x)p Fv(\))c Fk(\034)f Fr(\026)p Fv(\()p Fr(m)2449 5112 y Ft(1)2490 5098 y Fv(\),)50 5316 y(it)39 b(w)m(as)i(sho)m(wn)e (in)g([BEGK2])j(that)e(the)h(motion)e(on)h(the)h(set)f Fk(M)h Fv(can)f(b)s(e)f(describ)s(ed)f(as)j(a)f(sequence)50 5462 y(of)31 b(exits)g(with)f(asymptotically)g(exp)s(onen)m(tially)g (distributed)e(times)j(\(on)g(distinct)f(scales\))i(to)m(w)m(ards)g (the)p eop %%Page: 10 10 10 9 bop 50 295 a Fo(10)3270 b Fl(Se)l(ction)27 b Fo(1)50 513 y Fv(more)32 b(stable)g(states,)i(i.e.)46 b(the)32 b(equilibrium.)41 b(It)33 b(w)m(as)f(also)g(sho)m(wn)g(that)h(the)f(in) m(v)m(erse)g(mean)g(exit)g(times)50 658 y(from)f(an)m(y)g(p)s(oin)m(t)g Fr(x)26 b Fk(2)h(M)k Fv(are)h(asymptotically)f(equal)f(to)j(the)e (small)f(eigen)m(v)-5 b(alues)31 b(of)h(the)f(generator)i(of)50 804 y(the)d(Mark)m(o)m(v)i(c)m(hain.)159 1022 y(In)k(the)h(random)f (energy)h(mo)s(del)e(w)m(e)i(will)d(\014nd)h(ourselv)m(es)i(in)e(a)i (situation)f(where)g(all)g(of)g(these)i(h)m(y-)50 1167 y(p)s(othesis)30 b(are)i(not)g(satis\014ed.)43 b(When)31 b(c)m(hec)m(king)i(condition)d(\(1.21\))j(with)e Fk(M)c(\021)g Fr(T)13 b Fv(\()p Fr(E)5 b Fv(\))32 b(w)m(e)g(will)d(see)j(that)50 1313 y(this)d(is)h(not)g(satis\014ed,)g(and)g(that,)h(rather,)1025 1517 y(sup)1162 1539 y Fj(\033)r Fi(2S)1300 1548 y Fm(N)1378 1517 y Fh(P)p Fv([)p Fr(\034)1508 1484 y Fj(\033)1498 1546 y(T)10 b Ft(\()p Fj(E)s Ft(\))1694 1517 y Fr(<)25 b(\034)1840 1484 y Fj(\033)1830 1539 y(\033)1889 1517 y Fv(])p 1006 1576 928 4 v 1006 1660 a(inf)1116 1674 y Fj(\033)r Fi(2M)1315 1660 y Fh(P)p Fv([)p Fr(\034)1445 1628 y Fj(\033)1435 1691 y(T)10 b Ft(\()p Fj(E)s Ft(\))p Fi(n)p Fj(\033)1713 1660 y Fr(<)25 b(\034)1859 1633 y Fj(\033)1849 1682 y(\033)1908 1660 y Fv(])1970 1597 y Fk(!)g Fv(1)p Fr(;)107 b Fv(as)91 b Fr(E)30 b Fk(#)c(\0001)712 b Fv(\(1)p Fr(:)p Fv(22\))50 1893 y(Moreo)m(v)m(er,)39 b(all)34 b(the)h(quan)m(tities)g Fh(P)p Fv([)p Fr(\034)1328 1860 y Fj(\033)1318 1922 y(T)10 b Ft(\()p Fj(E)s Ft(\))p Fi(n)p Fj(\033)1604 1893 y Fr(<)33 b(\034)1758 1860 y Fj(\033)1748 1915 y(\033)1807 1893 y Fv(])j(for)f Fr(x)e Fk(2)g Fr(T)13 b Fv(\()p Fr(E)5 b Fv(\))36 b(will)d(turn)h(out)h(to)h (b)s(e)f(comparable.)50 2038 y(Th)m(us)d(the)h(situation)f(is)g (completely)h(di\013eren)m(t)g(than)f(in)g([BEGK2],)j(and)e(w)m(e)g(ha) m(v)m(e)i(to)f(exp)s(ect)f(a)h(m)m(uc)m(h)50 2184 y(more)c(complicated) g(b)s(eha)m(viour)f(of)i(the)g(pro)s(cess)f(on)g Fr(T)13 b Fv(\()p Fr(E)5 b Fv(\).)41 b(Moreo)m(v)m(er,)33 b(there)e(is)e(no)i (natural)e(criterion)50 2329 y(for)j(the)h(c)m(hoice)g(of)g(a)g (particular)e(v)-5 b(alue)32 b(of)g Fr(E)5 b Fv(,)34 b(and)e(w)m(e)h(will,)d(in)i(fact,)i(see)f(later)g(\(in)e([BBG]\))k (that)e(it)f(is)50 2475 y(someho)m(w)i(natural)f(to)h(consider)f (limits)e(as)j Fr(E)i Fk(#)c(\0001)p Fv(.)50 b(In)33 b(an)m(y)h(case)h(our)e(purp)s(ose)f(is)h(the)h(description)50 2620 y(of)c(the)h(pro)s(cess)f(observ)m(ed)g(on)h Fr(T)13 b Fv(\()p Fr(E)5 b Fv(\).)159 2838 y(Our)25 b(\014rst)g(result)g (concerns)h(just)f(the)i(\\motion")f(of)h(the)f(pro)s(cess)f (disregarding)f(time.)39 b(T)-8 b(o)27 b(that)f(e\013ect)50 2984 y(w)m(e)31 b(consider)e(the)i(random)e(times)1049 3179 y Fr(\022)1092 3193 y Ft(0)1158 3179 y Fk(\021)c Fv(min)n Fk(f)p Fr(n)h(>)f Fv(0)p Fk(j)p Fr(\033)s Fv(\()p Fr(n)p Fv(\))h Fk(2)f Fr(T)13 b Fv(\()p Fr(E)5 b Fv(\))p Fk(g)1055 3352 y Fr(\022)1098 3366 y Fj(`)1158 3352 y Fk(\021)25 b Fv(min)n Fk(f)p Fr(n)h(>)f(\022)1670 3366 y Fj(`)p Fi(\000)p Ft(1)1797 3352 y Fk(j)p Fr(\033)s Fv(\()p Fr(n)p Fv(\))h Fk(2)f Fr(T)13 b Fv(\()p Fr(E)5 b Fv(\))p Fk(n)p Fr(\033)s Fv(\()p Fr(\022)2500 3366 y Fj(`)p Fi(\000)p Ft(1)2630 3352 y Fv(\))p Fk(g)3477 3266 y Fv(\(1)p Fr(:)p Fv(23\))50 3583 y(Let)32 b Fr(\030)258 3550 y Ft(1)298 3583 y Fr(;)15 b(:)g(:)g(:)i(;)e(\030)544 3550 y Fi(j)p Fj(T)10 b Ft(\()p Fj(E)s Ft(\))p Fi(j)790 3583 y Fv(b)s(e)30 b(an)i(en)m(umeration)e(of)i(the)f(elemen)m(ts)h(of) f Fr(T)13 b Fv(\()p Fr(E)5 b Fv(\).)45 b(No)m(w)32 b(de\014ne)e(\(for)i (\014xed)e Fr(N)41 b Fv(and)50 3728 y Fr(E)5 b Fv(\),)31 b(the)g(sto)s(c)m(hastic)g(pro)s(cess)f Fr(Y)1160 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Fm(j)2414 4560 y Fi(g)2456 4528 y Fv(])996 b(\(1)p Fr(:)p Fv(25\))50 4737 y(Note)36 b(that)f(this)f(Mark)m(o)m(v)j(c)m (hain)d(has)h(a)g(state)h(space)g(whose)e(size)h Fk(j)p Fr(T)13 b Fv(\()p Fr(E)5 b Fv(\))p Fk(j)36 b Fv(is)e(a)h(random)f(v)-5 b(ariable.)53 b(T)-8 b(o)50 4883 y(form)m(ulate)37 b(our)g(\014rst)g (theorem)g(it)g(will)e(b)s(e)i(con)m(v)m(enien)m(t)h(to)g(\014x)f(the)h (size)f(b)m(y)g(conditioning.)60 b(Th)m(us)36 b(set)50 5028 y Fr(P)108 5042 y Fj(M)189 5028 y Fv(\()p Fk(\001)p Fv(\))26 b Fk(\021)f Fr(P)13 b Fv(\()p Fk(\001j)i(j)p Fr(T)e Fv(\()p Fr(E)5 b Fv(\))p Fk(j)28 b Fv(=)d Fr(M)10 b Fv(\).)p eop %%Page: 11 11 11 10 bop 50 295 a Fl(A)l(ging)27 b(in)h(the)g(REM.)f(Part)h(1.)2661 b Fo(11)50 513 y Fc(Theorem)37 b(1.3:)91 b Fq(L)-5 b(et)33 b Fr(\033)s Fv(\()p Fr(n)p Fv(\))g Fq(denote)h(the)f(Markov)g(chain)g (with)h(tr)-5 b(ansition)34 b(matrix)g(de\014ne)-5 b(d)34 b(in)f(\(1.3\))50 658 y(and)38 b(whose)g(initial)g(distribution)g(is)f 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Fr(\013)h Fk(\000)f Fv(1)3023 4990 y Ff(\023)3477 5118 y Fv(\(1)p Fr(:)p Fv(31\))p 50 5284 546 4 v 159 5345 a Fp(8)197 5371 y Fo(In)e(fact)g(it)h(is)g (enough,)h(for)d(the)i(result)h(to)e(hold,)i(that)e(the)h(initial)h (distribution)f(giv)n(es)g(zero)g(mass)f(to)g(an)g Fe(\017)p Fo(-neigh)n(b)r(orho)r(o)r(d)50 5462 y(of)25 b Fe(T)11 b Fo(\()p Fe(E)t Fo(\).)p eop %%Page: 12 12 12 11 bop 50 295 a Fo(12)3270 b Fl(Se)l(ction)27 b Fo(1)50 513 y Fq(and)33 b Fk(V)282 527 y Fj(N)s(;E)458 513 y Fq(is)g(a)g(r)-5 b(andom)35 b(variable)e(of)g(me)-5 b(an)34 b(zer)-5 b(o)33 b(and)h(varianc)-5 b(e)33 b(one.)159 731 y Fv(Theorem)d(1.4)h(is)f(complemen)m(ted)g(b)m(y)h(a)f(somewhat)h (con)m(v)m(erse)h(result)d(in)g(the)i(case)g Fr(\013)26 b(<)f Fv(1:)50 949 y Fc(Theorem)41 b(1.5:)106 b Fq(Assume)35 b(that)i Fr(\013)31 b(<)g Fv(1)p Fq(.)51 b(Then,)36 b(with)h(pr)-5 b(ob)g(ability)38 b(one,)e(for)g(al)5 b(l)36 b Fr(N)46 b Fq(lar)-5 b(ge)36 b(enough,)50 1095 y(for)d(al)5 b(l)33 b Fr(\033)c Fk(2)24 b(S)546 1109 y Fj(N)615 1095 y Fq(,)716 1350 y Fh(E)11 b Fr(\034)821 1313 y Fj(\033)811 1375 y(T)f Ft(\()p Fj(E)s Ft(\))1012 1350 y Fv(=)1224 1289 y(1)p 1119 1330 255 4 v 1119 1413 a Fr(M)31 b Fk(\000)20 b Fv(1)1385 1350 y Fr(e)1427 1313 y Fj(N)7 b Ft(\()p Fj(\014)1564 1285 y Fn(2)1600 1313 y Fj(=)p Ft(2+ln)13 b(2\))1872 1350 y Fv(\(1)21 b(+)f Fr(O)s Fv(\(1)p Fr(=)-5 b(N)10 b Fv(\)\))27 b Fk(\035)52 b Fv(sup)2552 1427 y Fj(\021)r Fi(2S)2685 1436 y Fm(N)2759 1350 y Fh(E)10 b Fr(\034)2863 1307 y Fj(\021)2853 1381 y Fi(S)2898 1390 y Fm(N)2962 1381 y Fi(n)p Fj(\021)3477 1350 y Fv(\(1)p Fr(:)p Fv(32\))50 1823 y Fs(Remark:)59 b Fv(Since)35 b(as)h Fr(N)45 b Fk(")35 b(1)p Fv(,)j Fh(E)10 b Fk(j)q Fr(T)j Fv(\()p Fr(E)5 b Fv(\))p Fk(j)41 b(!)35 b Fr(e)1735 1790 y Fi(\000)p Fj(E)1854 1823 y Fv(,)i(w)m(e)g(see)g(that)f(for)g Fk(\000)p Fr(E)41 b Fv(v)m(ery)c(large,)g Fk(W)3285 1840 y Fj(\014)s(;N)s(;T)10 b Ft(\()p Fj(E)s Ft(\))3639 1823 y Fk(\030)50 1969 y Fr(e)92 1936 y Fj(\013)p Ft(\()p Fj(E)s Ft(+)p Fj(u)322 1945 y Fm(N)382 1936 y Ft(\(0\)\))508 1969 y Fv(.)65 b(Th)m(us)37 b(\(ii\))g(of)i(Theorem)f(1.4)i(implies)35 b(that)k(if)f Fr(\013)h(>)f Fv(1,)j(for)d(all)g Fr(\033)j Fk(62)e Fr(T)13 b Fv(\()p Fr(E)5 b Fv(\),)41 b(the)e(mean)50 2114 y(time)f(of)g(arriv)-5 b(al)37 b(in)g(the)i(top)g(is)e(prop)s (ortional)f(to)j Fr(e)1910 2081 y Fj(\013)p Ft(\()p Fj(E)s Ft(+)p Fj(u)2140 2090 y Fm(N)2200 2081 y Ft(\(0\)\))2326 2114 y Fv(.)65 b(On)38 b(the)g(other)h(hand,)h(there)e(exists)50 2260 y Fr(\021)c Fk(2)d Fr(T)13 b Fv(\()p Fr(E)5 b Fv(\))34 b(suc)m(h)g(that)872 2183 y Fk(p)p 948 2183 83 4 v 77 x Fr(N)10 b(E)1098 2274 y Fj(\021)1173 2260 y Fk(\030)31 b Fr(E)d Fv(+)22 b Fr(u)1515 2274 y Fj(N)1584 2260 y Fv(\(0\))i(+)e Fr(O)s Fv(\()p Fr(e)1965 2227 y Fj(E)2027 2260 y Fv(\),)35 b(so)f(that)g(the)g(slo)m(w)m(est)h(times)e(of)h(exit) g(from)f(a)50 2405 y(state,)27 b Fh(E)11 b Fr(\034)398 2361 y Fj(\021)388 2436 y Fi(S)433 2445 y Fm(N)497 2436 y Fi(n)p Fj(\021)604 2405 y Fv(=)25 b Fr(e)742 2372 y Fj(\014)786 2318 y Fi(p)p 846 2318 65 4 v 54 x Fj(N)7 b(E)964 2381 y Fm(\021)1008 2405 y Fv(,)27 b(in)c Fr(T)13 b Fv(\()p Fr(E)5 b Fv(\))26 b(are)f(just)f(of)h(the)h(same)f(order.)38 b(This)23 b(can)i(b)s(e)g(expressed)f(b)m(y)h(sa)m(ying)50 2550 y(that)31 b(on)f(the)g(a)m(v)m(erage)j(the)d(pro)s(cess)g(tak)m (es)i(a)e(time)g Fr(t)g Fv(to)h(reac)m(h)g(states)h(that)f(ha)m(v)m(e)g (an)f(exit)g(time)g Fr(t)p Fv(.)41 b(This)50 2696 y(is)28 b(a)h(\014rst,)f(and)g(w)m(eak,)j(manifestation)d(of)g(the)h(aging)g (phenomenon)f(that)h(w)m(e)g(will)d(in)m(v)m(estigate)k(in)e(m)m(uc)m (h)50 2841 y(greater)34 b(detail)f(in)f([BBG].)j(In)d(con)m(trast,)k (if)c Fr(\013)f(<)e Fv(1,)35 b(Theorem)e(1.5)h Fh(E)11 b Fr(\034)2553 2808 y Fj(\033)2543 2871 y(T)f Ft(\()p Fj(E)t Ft(\))2750 2841 y Fk(\035)30 b Fv(sup)3008 2864 y Fj(\021)r Fi(2S)3141 2873 y Fm(N)3219 2841 y Fh(E)11 b Fr(\034)3324 2798 y Fj(\021)3314 2872 y Fi(S)3359 2881 y Fm(N)3423 2872 y Fi(n)p Fj(\021)3504 2841 y Fv(,)34 b(and)50 2987 y(th)m(us)27 b(the)i(time)e(sp)s(en)m(t)h(in)f(top)h (states)h(is)e(irrelev)-5 b(an)m(t)27 b(compared)i(to)f(the)h(time)e(b) s(et)m(w)m(een)i(successiv)m(e)f(visits)50 3132 y(of)37 b(suc)m(h)f(states.)61 b(Th)m(us)36 b(w)m(e)h(see)g(a)h(clear)e (distinction)f(b)s(et)m(w)m(een)i(the)g(high)e(and)h(the)h(lo)m(w)g (temp)s(erature)50 3278 y(phase)30 b(of)g(the)h(REM)f(on)h(the)f (dynamical)f(lev)m(el.)50 3496 y Fs(Remark:)52 b Fv(Statemen)m(t)34 b(iii\))d(of)i(Theorem)g(1.4)h(expresses)f(the)g(fact)h(that)f(the)g (mean)g(times)g(of)g(passage)50 3641 y(from)39 b(a)h(state)h Fr(\021)j Fk(2)d Fr(T)13 b Fv(\()p Fr(E)5 b Fv(\))40 b(to)g(another)g(state)48 b(\026)-52 b Fr(\021)45 b Fk(2)40 b Fr(T)13 b Fv(\()p Fr(E)5 b Fv(\))40 b(are)g(asymptotically)f(indep)s (enden)m(t)f(of)i(the)50 3787 y(terminal)32 b(state)42 b(\026)-51 b Fr(\021)s Fv(.)52 b(This)33 b(con\014rms)g(to)i(some)f (extend)g(the)h(heuristic)d(picture)h(of)h(Bouc)m(haud.)52 b(Indeed,)50 3932 y(if)29 b(w)m(e)i(added)f(the)h(h)m(yp)s(othesis)e (that)i(the)g(pro)s(cess)f(observ)m(ed)g(on)h(the)f(top)h(is)f(Mark)m (o)m(vian,)h(than)g(the)f(t)m(w)m(o)50 4078 y(preceding)i(theorems)h(w) m(ould)e(immediately)g(imply)g(that)i(the)g(w)m(aiting)f(times)h(m)m (ust)f(b)s(e)h(exp)s(onen)m(tially)50 4223 y(distributed)28 b(with)i(rates)i(indep)s(enden)m(t)d(of)i(the)g(terminal)f(state)i(and) f(giv)m(en)g(b)m(y)g(\(1.30\).)45 b(W)-8 b(e)33 b(will)28 b(see)k(in)50 4369 y([BBG])g(that)f(this,)e(ho)m(w)m(ev)m(er,)j(cannot) f(b)s(e)f(justi\014ed.)159 4587 y(The)i(remainder)f(of)i(this)f(pap)s (er)g(is)g(dev)m(oted)i(to)f(pro)m(ving)f(Theorems)h(1.3,)h(1.4,)h(and) e(1.5.)49 b(Section)32 b(2)50 4732 y(will)j(in)h(fact)j(pro)m(v)m(e)g (a)f(n)m(um)m(b)s(er)e(of)i(results)f(that)h(will)d(not)j(only)f(imply) e(Theorem)j(1.3,)j(but)c(will)e(also)50 4878 y(furnish)27 b(basic)j(input)e(to)j(b)s(oth)f(Section)g(3)g(and)g(the)g(follo)m (w-up)f(pap)s(er)g([BBG].)j(Section)e(3)h(con)m(tains)g(the)50 5023 y(pro)s(of)e(of)i(Theorems1.4)g(and)f(1.5.)p eop %%Page: 13 13 13 12 bop 50 295 a Fl(A)l(ging)27 b(in)h(the)g(REM.)f(Part)h(1.)2661 b Fo(13)50 513 y Fu(2.)55 b(Probabilit)m(y)40 b(estimates)159 731 y Fv(In)27 b(this)h(section)g(w)m(e)h(pro)m(vide)e(estimates)i (that)g(will)d(immediately)g(allo)m(w)i(to)h(pro)m(v)m(e)g(Theorem)f (1.3.)41 b(In)50 876 y(fact)30 b(w)m(e)f(will)d(pro)s(of)i(m)m(uc)m(h)h (more,)h(an)m(ticipating)e(what)h(will)d(b)s(e)j(needed)f(in)g(Section) g(3)i(as)f(w)m(ell)f(as)h(in)f(the)50 1022 y(follo)m(w-up)h(pap)s(er)g ([BBG].)j(These)e(results)g(are)g(collected)h(in)e(the)i(follo)m(wing)d (prop)s(osition.)50 1240 y Fc(Prop)s(osition)47 b(2.1:)56 b Fq(Set)39 b Fr(M)48 b Fv(=)38 b Fk(j)p Fr(T)13 b Fv(\()p Fr(E)5 b Fv(\))p Fk(j)p Fq(,)42 b Fr(d)d Fv(=)e(2)1886 1207 y Fj(M)2007 1240 y Fq(and)j Fr(\016)s Fv(\()p Fr(N)10 b Fv(\))40 b Fk(\021)2534 1166 y Ff(\000)2600 1204 y Fj(d)p 2587 1219 65 4 v 2587 1271 a(N)2662 1166 y Ff(\001)2704 1182 y Ft(1)p Fj(=)p Ft(2)2833 1240 y Fv(log)17 b Fr(N)10 b Fq(.)62 b(Ther)-5 b(e)40 b(exists)g(a)50 1385 y(subset)c Fk(E)j(\032)31 b Fv(\012)k Fq(with)i Fr(P)13 b Fv(\()p Fk(E)8 b Fv(\))32 b(=)e(1)p Fq(,)37 b(such)f(that)h(for)g(al)5 b(l)36 b Fr(!)e Fk(2)d(E)8 b Fq(,)36 b(for)h(al)5 b(l)36 b Fr(N)46 b Fq(lar)-5 b(ge)37 b(enough,)f(the)h(fol)5 b(lowing)50 1531 y(holds:)159 1749 y(F)-7 b(or)34 b Fr(")25 b(>)g Fv(0)33 b Fq(a)g(c)-5 b(onstant,)34 b(de\014ne)f(the)g(sets)857 2004 y Fk(B)917 1978 y Fi(p)p 977 1978 99 4 v 55 x Fj("N)1080 2004 y Fv(\()p Fr(\033)s Fv(\))26 b(=)f Fk(f)p Fr(\033)1427 1966 y Fi(0)1479 2004 y Fk(2)g(S)1620 2018 y Fj(N)1714 2004 y Fk(j)g(k)p Fr(\033)1864 1966 y Fi(0)1911 2004 y Fk(\000)20 b Fr(\033)s Fk(k)2102 2018 y Ft(2)2168 2004 y Fk(\024)2264 1922 y(p)p 2340 1922 126 4 v 82 x Fr("N)10 b Fk(g)p Fr(;)77 b(\033)28 b Fk(2)d(S)2833 2018 y Fj(N)3523 2004 y Fv(\(2)p Fr(:)p Fv(1\))50 2258 y Fq(and)1254 2404 y Fr(W)1340 2418 y Fj(")1379 2404 y Fv(\()p Fr(I)7 b Fv(\))26 b Fk(\021)1633 2317 y Ff(\\)1618 2512 y Fj(\033)r Fi(2)p Fj(I)1764 2404 y Fk(B)1827 2366 y Fj(c)1824 2387 y Fi(p)p 1883 2387 99 4 v 1883 2441 a Fj("N)1986 2404 y Fv(\()p Fr(\033)s Fv(\))p Fr(;)77 b(I)33 b Fk(\022)25 b(S)2437 2418 y Fj(N)3523 2404 y Fv(\(2)p Fr(:)p Fv(2\))50 2662 y Fq(Then,)47 2880 y(i\))47 b(F)-7 b(or)34 b(al)5 b(l)33 b Fr(")25 b(>)g Fv(0)33 b Fq(ther)-5 b(e)34 b(exists)f(a)g(c)-5 b(onstant)34 b Fr(c)26 b(>)f Fv(0)33 b Fq(such)f(that,)i(for)f(al)5 b(l)33 b Fr(\021)c Fk(2)c Fr(T)45 b Fq(and)34 b(al)5 b(l)33 b Fr(\033)28 b Fk(2)d Fr(W)3368 2894 y Fj(")3407 2880 y Fv(\()p Fr(T)13 b Fv(\))p Fq(,)1139 3030 y Ff(\014)1139 3085 y(\014)1139 3139 y(\014)1169 3135 y Fh(P)1239 3034 y Ff(\020)1294 3135 y Fr(\034)1344 3097 y Fj(\033)1334 3157 y(\021)1418 3135 y Fr(<)25 b(\034)1564 3097 y Fj(\033)1554 3159 y(T)10 b Fi(n)p Fj(\021)1688 3034 y Ff(\021)1763 3135 y Fk(\000)1885 3099 y Ft(1)p 1865 3114 77 4 v 1865 3166 a Fj(M)1952 3030 y Ff(\014)1952 3085 y(\014)1952 3139 y(\014)2007 3135 y Fk(\024)2166 3099 y Fj(d)p 2114 3114 141 4 v 2114 3166 a(N)d(M)2266 3135 y Fv(\(1)21 b(+)f Fr(c\016)s Fv(\()p Fr(N)10 b Fv(\)\))795 b(\(2)p Fr(:)p Fv(3\))20 3462 y Fq(ii\))46 b(Ther)-5 b(e)33 b(exists)g(a)g(c)-5 b(onstant)35 b Fr(c)25 b(>)g Fv(0)33 b Fq(such)g(that,)h(for)f(al)5 b(l)33 b Fr(\021)c Fk(2)24 b Fr(T)46 b Fq(and)40 b Fv(\026)-52 b Fr(\021)29 b Fk(2)c Fr(T)45 b Fq(with)34 b Fr(\021)28 b Fk(6)p Fv(=)k(\026)-52 b Fr(\021)t Fq(,)979 3612 y Ff(\014)979 3666 y(\014)979 3721 y(\014)1010 3716 y Fr(e)1052 3679 y Fj(\014)1096 3625 y Fi(p)p 1156 3625 65 4 v 54 x Fj(N)7 b(E)1277 3649 y Fn(+)1279 3698 y(\026)-36 b Fm(\021)1333 3716 y Fh(P)1403 3616 y Ff(\020)1458 3716 y Fr(\034)1514 3679 y Ft(\026)-42 b Fj(\021)1498 3739 y(\021)1578 3716 y Fr(<)25 b(\034)1730 3673 y Ft(\026)-42 b Fj(\021)1714 3747 y(T)10 b Fi(n)p Fj(\021)1848 3616 y Ff(\021)1922 3716 y Fk(\000)2044 3681 y Ft(1)p 2024 3696 77 4 v 2024 3748 a Fj(M)2111 3612 y Ff(\014)2111 3666 y(\014)2111 3721 y(\014)2167 3716 y Fk(\024)2325 3681 y Fj(d)p 2274 3696 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Fv(\(1)g(+)e Fr(c\016)s Fv(\()p Fr(N)10 b Fv(\)\))378 b(\(2)p Fr(:)p Fv(6\))33 5207 y Fq(v\))47 b(Ther)-5 b(e)33 b(exists)g(a)g(c)-5 b(onstant)35 b Fr(c)25 b(>)g Fv(0)33 b Fq(such)g(that,)h(for)f(al)5 b(l)33 b Fr(\033)39 b(=)-56 b Fk(2)25 b Fr(T)13 b Fq(,)827 5388 y Ff(\000)868 5462 y Fv(1)21 b Fk(\000)1056 5426 y Ft(1)p 1036 5441 77 4 v 1036 5493 a Fj(M)1123 5388 y Ff(\001)15 b(\000)1221 5462 y Fv(1)21 b Fk(\000)1402 5426 y Fj(d)p 1389 5441 65 4 v 1389 5493 a(N)1464 5462 y Fv(\(1)g(+)f Fr(c\016)s Fv(\()p Fr(N)10 b Fv(\)\))1926 5388 y Ff(\001)1995 5462 y Fk(\024)25 b Fr(e)2133 5424 y Fj(\014)2177 5370 y Fi(p)p 2238 5370 V 2238 5424 a Fj(N)6 b(E)2358 5397 y Fn(+)2355 5442 y Fm(\033)2415 5462 y Fh(P)15 b Fv(\()q Fr(\034)2571 5424 y Fj(\033)2561 5484 y(T)2645 5462 y Fr(<)25 b(\034)2791 5424 y Fj(\033)2781 5484 y(\033)2840 5462 y Fv(\))h Fk(\024)f Fv(1)481 b(\(2)p Fr(:)p Fv(7\))p eop %%Page: 14 14 14 13 bop 50 295 a Fo(14)3270 b Fl(Se)l(ction)27 b Fo(2)6 513 y Fq(vi\))46 b(F)-7 b(or)32 b(al)5 b(l)32 b Fr(")26 b(>)f Fv(0)31 b Fq(ther)-5 b(e)33 b(exists)e(a)h(c)-5 b(onstant)33 b Fr(c)26 b(>)f Fv(0)31 b Fq(such)h(that,)h(for)f(al)5 b(l)31 b Fr(\033)39 b(=)-55 b Fk(2)25 b Fr(T)44 b Fq(and)32 b(al)5 b(l)37 b Fv(\026)-50 b Fr(\033)28 b Fk(2)d Fr(W)3356 527 y Fj(")3395 513 y Fv(\()p Fr(T)30 b Fk([)17 b Fr(\033)s Fv(\))p Fq(,)772 694 y Ft(1)p 706 709 169 4 v 706 761 a Fj(M)7 b Ft(+1)905 729 y Fv(+)1058 694 y Fj(d)p 1007 709 141 4 v 1007 761 a(N)g(M)1158 729 y Fv(\(1)21 b Fk(\000)f Fr(c\016)s Fv(\()p Fr(N)10 b Fv(\)\))28 b Fk(\024)d Fh(P)1814 656 y Ff(\000)1856 729 y Fr(\034)1910 692 y Ft(\026)-40 b Fj(\033)1896 752 y(\033)1980 729 y Fk(\024)25 b Fr(\034)2130 692 y Ft(\026)-40 b Fj(\033)2116 752 y(T)2175 656 y Ff(\001)2242 729 y Fk(\024)2369 694 y Ft(1)p 2349 709 77 4 v 2349 761 a Fj(M)2456 729 y Fv(+)2610 694 y Fj(d)p 2558 709 141 4 v 2558 761 a(N)7 b(M)2710 729 y Fv(\(1)21 b(+)f Fr(c\016)s Fv(\()p Fr(N)10 b Fv(\)\))351 b(\(2)p Fr(:)p Fv(8\))50 1164 y Fa(Pro)s(of)45 b(of)87 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b(e)g(d)36 b(chain)e Fv(through)e(a)h Fq(lumping)i(pr)-5 b(o)g(c)g(e)g(dur)g(e)36 b Fv(that)d(allo)m(ws)f(to) i(reduce)e(the)h(high)f(dimensional)d(state)50 2110 y(space)23 b Fk(S)340 2124 y Fj(N)432 2110 y Fv(to)h(a)g(m)m(uc)m(h)f(smaller)f (one.)38 b(In)22 b(sub-section)h(2.2)h(w)m(e)g(analyse)f(the)g(lump)s (ed)d(c)m(hain)j(and)f(establish)50 2255 y(the)28 b(probabilit)m(y)d (estimates)j(whic)m(h)e(will)f(serv)m(e)j(as)g(basic)f(input)f(to)i (the)g(pro)s(of)f(of)h(Prop)s(osition)d(2.1.)41 b(The)50 2401 y(pro)s(of)29 b(of)i(the)g(prop)s(osition)d(is)h(then)h(carried)g (out)g(in)f(sub-section)h(2.3.)50 2592 y Fs(2.1.)47 b(Lump)s(ed)34 b(c)m(hains:)47 b(de\014nition)35 b(and)g(prop)s(erties)50 2784 y(Lumping)f(pro)s(cedure)159 3002 y Fv(W)-8 b(e)27 b(b)s(egin)e(with)g(some)h(preparatory)g(notation)h(and)e (de\014nitions.)37 b(F)-8 b(or)27 b Fr(M)36 b Fv(an)27 b(in)m(teger,)g(let)f Fk(S)3388 3016 y Fj(M)7 b Fi(\002)p Fj(N)3616 3002 y Fv(b)s(e)50 3148 y(the)33 b(set)g(of)g(all)f Fr(M)g Fk(\002)21 b Fr(N)43 b Fv(matrices)33 b(whose)f(elemen)m(ts)h(b) s(elong)f(to)i Fk(S)i Fv(=)29 b Fk(f\000)p Fv(1)p Fr(;)15 b Fv(1)p Fk(g)p Fv(.)50 b(A)33 b(matrix)f Fr(\030)h Fk(2)c(S)3508 3162 y Fj(M)7 b Fi(\002)p Fj(N)50 3293 y Fv(will)21 b(b)s(e)i(written)g (either)h(in)f(terms)g(of)i(its)e(matrix)g(elemen)m(ts,)j(ro)m(w)e(v)m (ectors)i(or)e(column)e(v)m(ectors)k(according)50 3438 y(to)38 b(the)f(follo)m(wing)f(notation.)62 b(In)37 b(terms)g(of)h(its) e(matrix)h(elemen)m(ts)h(w)m(e)g(will)c(write)j Fr(\030)k Fv(=)36 b(\()p Fr(\030)3275 3395 y Fj(\026)3271 3464 y(i)3324 3438 y Fv(\))3359 3395 y Fj(\026)p Ft(=1)p Fj(;:::)o(;M)3359 3465 y(i)p Ft(=1)p Fj(;:::)o(;N)3685 3438 y Fv(,)50 3584 y(where)31 b Fr(\030)358 3540 y Fj(\026)354 3609 y(i)435 3584 y Fk(2)d(S)39 b Fv(is)31 b(the)h(elemen)m(t)h(lying)d(at)j(the)f (in)m(tersection)g(of)g(the)h Fr(\026)p Fv(-th)e(ro)m(w)i(and)e Fr(i)p Fv(-th)i(column.)44 b(The)50 3729 y(ro)m(w)28 b(and)g(column)g(v)m(ectors)h(of)g Fr(\030)j Fv(will)26 b(b)s(e)i(denoted)h(resp)s(ectiv)m(ely)e(b)m(y)i Fr(\030)2490 3696 y Fj(\026)2567 3729 y Fv(and)f Fr(\030)2782 3743 y Fj(i)2812 3729 y Fv(,)h(and)f(written,)g(in)f(terms)50 3875 y(of)j(their)g(elemen)m(ts,)h(as:)1073 3979 y Fr(\030)1117 3941 y Fj(\026)1191 3979 y Fv(=)25 b(\()p Fr(\030)1366 3935 y Fj(\026)1362 4004 y(i)1415 3979 y Fv(\))1450 3993 y Fj(i)p Ft(=1)p Fj(;:::)n(;N)1770 3979 y Fk(2)g(S)1911 3993 y Fj(N)1980 3979 y Fr(;)76 b(\026)25 b Fk(2)g(f)p Fv(1)p Fr(;)15 b(:)g(:)g(:)i(;)e(M)10 b Fk(g)1096 4152 y Fr(\030)1136 4166 y Fj(i)1191 4152 y Fv(=)25 b(\()p Fr(\030)1366 4108 y Fj(\026)1362 4177 y(i)1415 4152 y Fv(\))1450 4114 y Fj(\026)p Ft(=1)p Fj(;:::)o(;M)1800 4152 y Fk(2)g(S)1941 4166 y Fj(M)2022 4152 y Fr(;)76 b(i)26 b Fk(2)e(f)p Fv(1)p Fr(;)15 b(:)g(:)g(:)j(;)d(N)10 b Fk(g)3523 4065 y Fv(\(2)p Fr(:)p Fv(9\))50 4302 y(Observ)m(e)23 b(that,)j(when)d(carrying)g(an)h(index)e(placed)h(as)h(a)g(sup)s (erscript,)f(the)h(letter)g Fr(\030)k Fv(refers)23 b(to)i(an)e(elemen)m (t)50 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5424 y Fj(M)2331 5462 y Fv(\))1111 b(\(2)p Fr(:)p Fv(11\))p eop %%Page: 15 15 15 14 bop 50 295 a Fl(A)l(ging)27 b(in)h(the)g(REM.)f(Part)h(1.)2661 b Fo(15)159 513 y Fv(Giv)m(en)31 b(a)g(subset)f Fr(I)j Fk(\032)26 b(S)1003 527 y Fj(N)1103 513 y Fv(w)m(e)31 b(de\014ne)g(a)g(partition)e(of)i(the)h(index)d(set)i(\003)c Fk(\021)e(f)p Fv(1)p Fr(;)15 b(:)g(:)g(:)j(;)d(i;)g(:)g(:)g(:)i(;)e(N) 10 b Fk(g)32 b Fv(in)e(the)50 658 y(follo)m(wing)f(w)m(a)m(y)-8 b(.)42 b(Let)31 b Fr(\030)e Fv(=)c(\()p Fr(\030)1049 672 y Ft(1)1090 658 y Fr(;)15 b(:)g(:)g(:)h(;)f(\030)1331 672 y Fj(i)1362 658 y Fr(;)g(:)g(:)g(:)h(;)f(\030)1603 672 y Fj(N)1673 658 y Fv(\))25 b Fk(2)g(S)1874 675 y Fi(j)p Fj(I)5 b Fi(j\002)p Fj(N)2110 658 y Fv(b)s(e)30 b(an)m(y)h(matrix)f(ha)m(ving)g(the)g(prop)s(ert)m(y)g(that)1380 864 y Fr(I)i Fv(=)1548 763 y Ff(n)1609 864 y Fr(\030)1653 826 y Ft(1)1694 864 y Fr(;)15 b(:)g(:)g(:)h(;)f(\030)1939 826 y Fj(\026)1988 864 y Fr(;)g(:)g(:)g(:)i(;)e(\030)2234 826 y Fi(j)p Fj(I)5 b Fi(j)2319 763 y Ff(o)3477 864 y Fv(\(2)p Fr(:)p Fv(12\))50 1069 y(in)20 b(other)h(w)m(ords,)i(an)m(y)e (matrix)g(ha)m(ving)f(the)i(set)f Fr(I)29 b Fv(for)20 b(set)i(of)f(ro)m(w)h(v)m(ectors.)39 b(Next,)24 b(let)d Fk(f)p Fr(e)3047 1083 y Ft(1)3089 1069 y Fr(;)15 b(:)g(:)g(:)h(;)f(e) 3332 1083 y Fj(k)3378 1069 y Fr(;)g(:)g(:)g(:)h(;)f(e)3621 1083 y Fj(d)3664 1069 y Fk(g)50 1214 y Fv(b)s(e)36 b(an)g(arbitrarily)e (c)m(hosen)k(lab)s(eling)c(of)j(all)e Fr(d)h Fv(=)g(2)1878 1181 y Fi(j)p Fj(I)5 b Fi(j)2000 1214 y Fv(elemen)m(ts)37 b(of)f Fk(S)2542 1231 y Fi(j)p Fj(I)5 b Fi(j)2664 1214 y Fv(\(this)36 b(lab)s(eling)e(will)g(b)s(e)i(k)m(ept)50 1360 y(\014xed)44 b(throughout,)k(whatev)m(er)e(the)f(c)m(hoice)g(of)g Fr(I)52 b Fv(is\).)83 b(Then)44 b Fr(\030)k Fv(induces)c(a)h(partition) e(of)i(\003)g(in)m(to)f Fr(d)50 1505 y Fv(disjoin)m(t)31 b(\(p)s(ossibly)f(empt)m(y\))k(subsets,)f(\003)1485 1519 y Fj(k)1530 1505 y Fv(\()p Fr(I)7 b Fv(\),)34 b(obtained)e(b)m(y)h (grouping)e(together)k(all)c(indices)g Fr(i)i 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Fv(\000)1457 5394 y Fj(N)s(;d)1581 5380 y Fv(\()p Fr(I)7 b Fv(\))26 b Fk(\()-15 b(\))25 b Fr(x)h Fv(=)2137 5266 y Fj(d)2090 5293 y Ff(X)2090 5489 y Fj(k)r Ft(=1)2336 5318 y Fr(n)2391 5332 y Fj(k)p 2248 5359 V 2248 5442 a Fk(j)p Fv(\003)2336 5456 y Fj(k)2381 5442 y Fv(\()p Fr(I)7 b Fv(\))p Fk(j)2535 5380 y Fr(u)2587 5394 y Fj(k)3477 5380 y Fv(\(2)p Fr(:)p Fv(17\))p eop %%Page: 16 16 16 15 bop 50 295 a Fo(16)3270 b Fl(Se)l(ction)27 b Fo(2)159 513 y Fq(wher)-5 b(e,)33 b(for)h(al)5 b(l)33 b Fv(1)26 b Fk(\024)e Fr(k)29 b Fk(\024)c Fr(d)p Fq(,)33 b Fk(j)p Fr(n)1245 527 y Fj(k)1289 513 y Fk(j)26 b(\024)f(j)p Fv(\003)1524 527 y Fj(k)1569 513 y Fv(\()p Fr(I)7 b Fv(\))p Fk(j)33 b Fq(has)h(the)f(same)g(p)-5 b(arity)35 b(as)e Fk(j)p Fv(\003)2756 527 y Fj(k)2801 513 y Fv(\()p Fr(I)7 b Fv(\))p Fk(j)p Fq(.)20 731 y(ii\))743 930 y Fk(jf)p Fr(\033)29 b Fk(2)c(S)1035 944 y Fj(N)1129 930 y Fk(j)h Fr(\015)1227 944 y Fj(I)1269 930 y Fv(\()p Fr(\033)s Fv(\))g(=)f Fr(x)p Fk(gj)h Fv(=)1807 816 y Fj(d)1768 843 y Ff(Y)1760 1039 y Fj(k)r Ft(=1)1907 801 y Ff(\022)2071 867 y Fk(j)p Fv(\003)2159 881 y Fj(k)2205 867 y Fv(\()p Fr(I)7 b Fv(\))p Fk(j)1974 994 y(j)p Fv(\003)2062 1008 y Fj(k)2107 994 y Fv(\()p Fr(I)g Fv(\))p Fk(j)2260 956 y Ft(1+)p Fj(x)2393 966 y Fm(k)p 2261 973 174 4 v 2329 1025 a Ft(2)2445 801 y Ff(\023)2512 930 y Fr(;)76 b Fk(8)p Fr(x)25 b Fk(2)f Fv(\000)2883 944 y Fj(N)s(;d)3008 930 y Fv(\()p Fr(I)7 b Fv(\))352 b(\(2)p Fr(:)p Fv(18\))159 1181 y Fq(In)33 b(p)-5 b(articular,)34 b(the)f(r)-5 b(estriction)34 b(of)f Fr(\015)1465 1195 y Fj(I)1540 1181 y Fq(to)g Fr(I)39 b Fq(is)33 b(a)g(one-to-one)g(mapping)h(fr)-5 b(om)34 b Fr(I)40 b Fq(onto)33 b Fr(\015)3273 1195 y Fj(I)3315 1181 y Fv(\()p Fr(I)7 b Fv(\))p Fq(.)-8 1400 y(iii\))46 b(The)33 b(elements)g(of)g Fr(I)39 b Fq(ar)-5 b(e)34 b(mapp)-5 b(e)g(d)35 b(onto)f(c)-5 b(orners)34 b(of)f Fv([)p Fk(\000)p Fv(1)p Fr(;)15 b Fv(1])2273 1367 y Fj(d)2317 1400 y Fq(:)41 b(for)33 b(al)5 b(l)34 b Fr(\033)28 b Fk(2)d Fr(I)411 1637 y(\015)458 1651 y 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Fv(\()p Fr(x;)15 b(x)554 3513 y Fi(0)580 3551 y Fv(\))26 b(=)737 3395 y Ff(\()851 3440 y Fi(j)p Ft(\003)923 3450 y Fm(k)962 3440 y Ft(\()p Fj(I)5 b Ft(\))p Fi(j)p 851 3464 227 4 v 933 3516 a Fj(N)1100 3447 y Ft(1)p Fi(\000)p Fj(sx)1268 3457 y Fm(k)p 1100 3464 209 4 v 1186 3516 a Ft(2)1520 3485 y Fq(if)93 b Fv(\()p Fr(x;)15 b(x)1848 3452 y Fi(0)1874 3485 y Fv(\))26 b Fk(2)e Fr(E)5 b Fv(\(\000)2184 3499 y Fj(N)s(;d)2309 3485 y Fv(\()p Fr(I)i Fv(\)\)\))95 b Fq(and)f Fr(x)2880 3452 y Fi(0)2925 3485 y Fk(\000)20 b Fr(x)26 b Fv(=)f Fr(s)3338 3449 y Ft(2)p 3243 3464 227 4 v 3243 3516 a Fi(j)p Ft(\003)3315 3526 y Fm(k)3354 3516 y Ft(\()p Fj(I)5 b Ft(\))p Fi(j)3481 3485 y Fr(u)3533 3499 y Fj(k)840 3631 y Fv(0)p Fr(;)610 b Fq(otherwise)3477 3742 y Fv(\(2)p Fr(:)p Fv(28\))50 4105 y Fa(Pro)s(of:)44 b Fv(The)24 b(pro)s(of)g(of)h(this)f(lemma)g(is)g(a)h(direct)g (application)e(of)i(the)g(results)e(of)i(Burk)m(e)g(and)f(Rosen)m (blatt)50 4251 y([BR])31 b(on)f(Mark)m(o)m(vian)h(functions)e(of)i (Mark)m(o)m(v)h(Chains.)39 b Fk(})50 4443 y Fs(Comparison)34 b(lemmata)159 4661 y Fv(In)29 b(order)h(to)h(mak)m(e)g(use)f(of)g(the)h (ab)s(o)m(v)m(e)g(set-up)f(w)m(e)h(\014rst)e(need)h(to)h(establish)e (ho)m(w)h(the)g(Mark)m(o)m(v)i(c)m(hain)50 4806 y Fr(\033)s Fv(\()p Fr(t)p Fv(\))f(relates)f(to)i(the)e(OR)-10 b(W.)31 b(This)d(is)i(done)g(in)f(the)h(next)h(lemma.)50 5025 y Fc(Lemma)38 b(2.4:)91 b Fq(L)-5 b(et)33 b Fr(I)f Fk(\032)25 b(S)1105 5039 y Fj(N)1174 5025 y Fq(.)42 b(Then,)47 5243 y(i\))47 b(for)33 b(al)5 b(l)33 b Fr(\033)39 b(=)-55 b Fk(2)25 b Fr(I)39 b Fq(and)34 b Fr(\021)k(=)-55 b Fk(2)25 b Fr(I)j Fk([)19 b Fr(\033)1380 5462 y Fh(P)1450 5388 y Ff(\000)1492 5462 y Fr(\034)1542 5424 y Fj(\033)1532 5484 y(\021)1616 5462 y Fr(<)25 b(\034)1762 5424 y Fj(\033)1752 5484 y(I)1811 5388 y Ff(\001)1878 5462 y Fv(=)g Fh(P)2029 5424 y Fi(\016)2087 5388 y Ff(\000)2128 5462 y Fr(\034)2178 5424 y Fj(\033)2168 5484 y(\021)2252 5462 y Fr(<)g(\034)2398 5424 y Fj(\033)2388 5484 y(I)2447 5388 y Ff(\001)3477 5462 y Fv(\(2)p Fr(:)p Fv(29\))p eop %%Page: 18 18 18 17 bop 50 295 a Fo(18)3270 b Fl(Se)l(ction)27 b Fo(2)20 513 y Fq(ii\))46 b(for)33 b(al)5 b(l)33 b Fr(\033)c Fk(2)c Fr(I)39 b Fq(and)34 b Fr(\021)k(=)-55 b Fk(2)25 b Fr(I)1189 756 y Fh(P)1259 682 y Ff(\000)1302 756 y Fr(\034)1352 718 y Fj(\033)1342 778 y(\021)1426 756 y Fr(<)g(\034)1572 718 y Fj(\033)1562 778 y(I)1621 682 y Ff(\001)1688 756 y Fv(=)g Fr(e)1826 718 y Fi(\000)p Fj(\014)1927 664 y Fi(p)p 1987 664 65 4 v 54 x Fj(N)7 b(E)2108 691 y Fn(+)2105 736 y Fm(\033)2164 756 y Fh(P)2219 718 y Fi(\016)2277 682 y Ff(\000)2318 756 y Fr(\034)2368 718 y Fj(\033)2358 778 y(\021)2443 756 y Fr(<)25 b(\034)2589 718 y Fj(\033)2579 778 y(I)2638 682 y Ff(\001)3477 756 y Fv(\(2)p Fr(:)p Fv(30\))159 1217 y(It)31 b(\014nally)f(remains)g(to)i(establish)e(ho)m (w)h(the)h(quan)m(tities)f Fh(P)2187 1184 y Fi(\016)2244 1143 y Ff(\000)2286 1217 y Fr(\034)2336 1184 y Fj(\033)2326 1239 y(\021)2410 1217 y Fr(<)25 b(\034)2556 1184 y Fj(\033)2546 1242 y(I)2605 1143 y Ff(\001)2678 1217 y Fv(can)32 b(b)s(e)f(expressed) g(in)f(terms)50 1362 y(of)g(a)h(lump)s(ed)d(c)m(hain.)50 1580 y Fc(Lemma)36 b(2.5:)90 b Fq(L)-5 b(et)32 b Fr(I)7 b(;)15 b(J)o(;)g(K)33 b Fk(\032)25 b(S)1315 1594 y Fj(N)1415 1580 y Fq(b)-5 b(e)32 b(such)g(that)g Fr(I)25 b Fk(\\)18 b Fr(J)34 b Fv(=)25 b Fk(;)32 b Fq(and)h Fr(I)24 b Fk([)18 b Fr(J)34 b Fk(\022)25 b Fr(K)7 b Fq(.)41 b(Then,)32 b(Denoting)g(by)50 1726 y Fh(R)109 1693 y Fi(\016)190 1726 y Fq(the)h(law)g(of)g(the)g Fr(K)7 b Fq(-lump)-5 b(e)g(d)34 b(chain,)744 1969 y Fh(P)799 1931 y Fi(\016)857 1969 y Fv(\()p Fr(\034)942 1931 y Fj(\033)932 1991 y(I)1016 1969 y Fk(\024)25 b Fr(\034)1162 1931 y Fj(\033)1152 1991 y(J)1211 1969 y Fv(\))h(=)f Fh(R)1427 1931 y Fi(\016)1490 1868 y Ff(\020)1545 1969 y Fr(\034)1595 1921 y Fj(\015)1633 1930 y Fm(K)1692 1921 y Ft(\()p Fj(\033)r Ft(\))1585 2000 y Fj(\015)1623 2009 y Fm(K)1682 2000 y Ft(\()p Fj(I)5 b Ft(\))1823 1969 y Fk(\024)25 b Fr(\034)1969 1921 y Fj(\015)2007 1930 y Fm(K)2067 1921 y Ft(\()p Fj(\033)r Ft(\))1959 2000 y Fj(\015)1997 2009 y Fm(K)2057 2000 y Ft(\()p Fj(J)6 b Ft(\))2173 1868 y Ff(\021)2242 1969 y Fr(;)199 b Fq(for)34 b(al)5 b(l)93 b Fr(\033)39 b(=)-55 b Fk(2)24 b Fr(I)469 b Fv(\(2)p Fr(:)p Fv(31\))50 2430 y Fs(Remark:)46 b Fv(Note)32 b(that)e Fr(K)37 b Fv(in)30 b(the)g(ab)s(o)m(v)m(e)i(lemma)e(do)s(es)g(not)g(necessarily)g(con)m (tain)g Fr(\033)k Fv(if)29 b Fr(\033)39 b(=)-55 b Fk(2)24 b Fr(J)9 b Fv(.)159 2648 y(W)-8 b(e)31 b(skip)e(the)i(pro)s(ofs)e(of)i (Lemma)f(2.4)i(and)e(2.5)h(as)g(they)f(are)h(nothing)e(but)h(elemen)m (tary)h(exercises.)50 2845 y Fs(2.2.)47 b(Main)35 b(ingredien)m(ts)g (of)g(the)g(pro)s(of)g(of)g(Prop)s(osition)h(2.1)159 3115 y Fv(Observ)m(e)i(that)h(the)g(en)m(trop)m(y)g(pro)s(duced)d(b)m (y)j(the)f(lumping)e(pro)s(cedure)h(giv)m(es)h(rise)g(through)f (\(2.27\))50 3261 y(to)k(a)g(p)s(oten)m(tial,)h Fr(F)736 3275 y Fj(N)805 3261 y Fv(\()p Fr(x)p Fv(\))h Fk(\021)e(\000)1178 3225 y Ft(1)p 1164 3240 V 1164 3292 a Fj(N)1254 3261 y Fv(ln)14 b Fh(Q)1410 3228 y Fi(\016)1410 3286 y Fj(N)1485 3261 y Fv(\()p Fr(x)p Fv(\).)71 b(It)41 b(moreo)m(v)m(er)g(follo)m(ws)f (from)g(assertions)f(ii\))h(and)f(iii\))g(of)50 3406 y(Lemma)32 b(2.2)i(that)f(this)e(p)s(oten)m(tial)h(is)f(con)m(v)m(ex)j (and)e(tak)m(es)i(on)e(its)g(global)g(maxim)m(um)f(at)i(the)f(corners)h (of)50 3552 y(the)j(cub)s(e)f([)p Fk(\000)p Fv(1)p Fr(;)15 b Fv(1])683 3519 y Fj(d)727 3552 y Fv(.)58 b(This)34 b(allo)m(ws)i(us)f(to)i(dra)m(w)e(on)h(the)g(results)f(of)h([BEGK1])i (where)d(suc)m(h)h(pro)s(cesses)50 3697 y(w)m(ere)31 b(in)m(v)m(estigated.)159 3915 y(Throughout)e(this)g(section)h Fr(I)37 b Fv(denotes)30 b(an)g(arbitrary)f(\(non)h(empt)m(y\))g(subset) g(of)g Fk(S)3023 3929 y Fj(N)3122 3915 y Fv(whose)g(size,)g Fk(j)p Fr(I)7 b Fk(j)p Fv(,)50 4061 y(do)s(es)30 b(not)g(dep)s(end)f (on)h Fr(N)10 b Fv(.)41 b(Giv)m(en)30 b(0)c Fr(<)f(\017)g(<)g Fv(1)31 b(let)f Fk(K)q Fv(\()p Fr(I)7 b Fv(\))32 b(and)e Fk(K)q Fv(\()p Fr(I)7 b Fv(\))2385 4028 y Fj(c)2453 4061 y Fv(b)s(e)30 b(the)h(sets)f(de\014ned)g(through:)947 4269 y Fk(K)q Fv(\()p Fr(I)7 b Fv(\))27 b Fk(\021)e(K)1326 4283 y Fj(\017)1361 4269 y Fv(\()p Fr(I)7 b Fv(\))26 b Fk(\021)1600 4196 y Ff(\010)1653 4269 y Fr(k)j Fk(2)24 b(f)p Fv(1)p Fr(;)15 b(:)g(:)g(:)j(;)d(d)p Fk(g)26 b(j)g(j)p Fv(\003)2364 4283 y Fj(k)2409 4269 y Fv(\()p Fr(I)7 b Fv(\))p Fk(j)26 b(\025)f Fr(\017)2721 4234 y Fj(N)p 2721 4249 V 2734 4301 a(d)2796 4196 y Ff(\011)910 4442 y Fk(K)q Fv(\()p Fr(I)7 b Fv(\))1097 4405 y Fj(c)1161 4442 y Fk(\021)25 b(K)1326 4456 y Fj(\017)1361 4442 y Fv(\()p Fr(I)7 b Fv(\))1478 4405 y Fj(c)1541 4442 y Fk(\021)25 b(f)p Fv(1)p Fr(;)15 b(:)g(:)g(:)j(;)d(d)p Fk(g)21 b(n)g(K)2178 4456 y Fj(\017)2213 4442 y Fv(\()p Fr(I)7 b Fv(\))3477 4355 y(\(2)p Fr(:)p Fv(32\))50 4670 y(Set)33 b Fr(\024)c Fv(=)g Fk(jK)q Fv(\()p Fr(I)7 b Fv(\))p Fk(j)p Fv(.)50 b(Of)32 b(course)h Fr(\024)d Fk(\025)f Fv(1)k(since)f(supp)s(osing)f Fr(\024)e Fv(=)g(0,)34 b(\(2.32\))h(implies)30 b(that)3086 4602 y Ff(P)3182 4625 y Fj(d)3182 4698 y(k)r Ft(=1)3334 4670 y Fk(j)p Fv(\003)3422 4684 y Fj(k)3467 4670 y Fv(\()p Fr(I)7 b Fv(\))p Fk(j)30 b Fr(<)50 4816 y(\017N)35 b(<)25 b(N)10 b Fv(,)27 b(con)m(tradicting)f(\(2.13\).)42 b(Let)27 b Fr(\031)h Fv(:)e Fh(R)1621 4783 y Fj(d)1695 4816 y Fk(!)f Fh(R)1871 4783 y Fj(\024)1950 4816 y Fv(b)s(e)h(the)g(pro)5 b(jection)26 b(that)h(maps)f Fr(x)f Fv(=)g(\()p Fr(x)3337 4830 y Ft(1)3378 4816 y Fr(;)15 b(:)g(:)g(:)i(;)e(x)3632 4830 y Fj(d)3674 4816 y Fv(\))50 4961 y(in)m(to)30 b Fr(\031)s(x)25 b Fv(=)g(\()p Fr(x)549 4975 y Fj(i)575 4984 y Fn(1)615 4961 y Fr(;)15 b(:)g(:)g(:)i(;)e(x)869 4975 y Fj(i)895 4984 y Fm(\024)942 4961 y Fv(\))31 b(where,)f(for)g (all)f(1)d Fk(\024)f Fr(j)31 b Fk(\024)25 b Fr(\024)p Fv(,)31 b Fr(i)2031 4975 y Fj(j)2094 4961 y Fk(2)25 b(K)q Fv(\()p Fr(I)7 b Fv(\).)42 b(Finally)-8 b(,)29 b(set)1556 5204 y Fr(N)1629 5218 y Fi(\003)1696 5204 y Fv(=)68 b(min)1792 5267 y Fj(k)r Fi(2K)p Ft(\()p Fj(I)5 b Ft(\))2046 5204 y Fk(j)p Fv(\003)2134 5218 y Fj(k)2179 5204 y Fk(j)1273 b Fv(\(2)p Fr(:)p Fv(33\))50 5462 y(With)30 b(this)f(notation)i(w)m(e)f (ha)m(v)m(e:)p eop %%Page: 19 19 19 18 bop 50 295 a Fl(A)l(ging)27 b(in)h(the)g(REM.)f(Part)h(1.)2661 b Fo(19)50 513 y Fc(Lemma)38 b(2.6:)91 b Fq(Ther)-5 b(e)33 b(exists)g(a)g(c)-5 b(onstant)35 b Fr(c)25 b(>)g Fv(0)33 b Fq(such)g(that,)h(for)f(al)5 b(l)33 b Fr(N)43 b Fq(lar)-5 b(ge)33 b(enough,)191 798 y Fh(R)250 761 y Fi(\016)313 798 y Fv(\()q Fr(\034)399 761 y Fj(x)389 821 y Ft(0)470 798 y Fr(<)25 b(\034)616 761 y Fj(x)606 821 y(x)662 798 y Fv(\))g Fk(\025)818 643 y Ff( )890 798 y Fv(1)c Fk(\000)1076 737 y Fv(1)p 1058 777 83 4 v 1058 861 a Fr(N)1239 712 y Ff(X)1167 911 y Fj(\026)p Fi(2K)p Ft(\()p Fj(I)5 b Ft(\))1408 893 y Fm(c)1459 798 y Fk(j)p Fv(\003)1547 812 y Fj(\026)1596 798 y Fk(j)1621 643 y Ff(!)1708 670 y(\022)1775 798 y Fv(1)21 b Fk(\000)1977 737 y Fv(1)p 1942 777 115 4 v 1942 861 a Fr(N)2015 875 y Fi(\003)2088 798 y Fk(\000)2232 737 y Fr(c)p 2190 777 124 4 v 2190 861 a(N)2273 834 y Ft(2)2263 883 y Fi(\003)2325 670 y Ff(\023)2407 798 y Fr(;)199 b Fq(for)33 b(al)5 b(l)94 b Fr(x)25 b Fk(2)g Fr(\015)3177 812 y Fj(I)3219 798 y Fv(\()p Fr(I)7 b Fv(\))141 b(\(2)p Fr(:)p Fv(34\))50 1263 y Fc(Lemma)31 b(2.7:)85 b Fq(L)-5 b(et)27 b Fr(x)f Fk(2)e Fr(\015)1073 1277 y Fj(I)1115 1263 y Fv(\()p Fr(I)7 b Fv(\))28 b Fq(and)h Fr(y)f Fk(2)d Fr(\015)1638 1277 y Fj(I)1679 1263 y Fv(\()p Fk(S)1769 1277 y Fj(N)1839 1263 y Fv(\))j Fq(b)-5 b(e)27 b(such)g(that)i Fk(k)p Fr(\031)s(x)9 b Fk(\000)g Fr(\031)s(y)s Fk(k)2779 1277 y Ft(2)2846 1263 y Fk(\025)25 b Fr(\016)31 b Fq(for)d(some)g(c)-5 b(onstant)61 1373 y Ft(1)p 61 1388 37 4 v 61 1440 a(2)133 1409 y Fr(>)25 b(\016)k(>)c Fv(0)p Fq(.)42 b(Then)33 b(ther)-5 b(e)34 b(exist)f(a)g(c)-5 b(onstant)34 b Fr(h)p Fv(\()p Fr(\016)n(;)15 b(\024)p Fv(\))28 b Fr(>)d Fv(0)33 b Fq(such)g(that,)g(for)g(al)5 b(l)34 b Fr(N)42 b Fq(lar)-5 b(ge)34 b(enough,)1371 1654 y Fh(R)1431 1616 y Fi(\016)1494 1654 y Fv(\()p Fr(\034)1579 1616 y Fj(y)1569 1676 y(x)1648 1654 y Fr(<)25 b(\034)1794 1610 y Fj(y)1784 1678 y Ft(0)1838 1654 y Fv(\))g Fk(\024)g Fr(e)2036 1616 y Fi(\000)p Fj(N)2151 1625 y Fb(\003)2189 1616 y Fj(h)p Ft(\()p Fj(\016)n(;\024)p Ft(\))3477 1654 y Fv(\(2)p Fr(:)p Fv(35\))159 2117 y(As)30 b(an)h(imp)s(ortan)m(t)e(consequence)i(of)g(the)f(previous)f(t)m(w)m(o) j(lemmata)f(w)m(e)f(ma)m(y)h(immediately)e(state:)50 2335 y Fc(Lemma)g(2.8:)84 b Fq(L)-5 b(et)26 b Fr(x)f Fk(2)g Fr(\015)1069 2349 y Fj(I)1111 2335 y Fv(\()p Fr(I)7 b Fv(\))27 b Fq(and)g Fr(J)35 b Fk(\022)24 b Fr(\015)1652 2349 y Fj(I)1694 2335 y Fv(\()p Fr(I)7 b Fv(\))27 b Fq(b)-5 b(e)26 b(such)h(that)g(for)g(al)5 b(l)27 b Fr(y)s(;)15 b(y)2722 2302 y Fi(0)2773 2335 y Fk(2)25 b Fr(J)15 b Fk([)6 b Fr(x)p Fq(,)27 b Fk(k)p Fr(\031)s(y)3246 2302 y Fi(0)3279 2335 y Fk(\000)6 b Fr(\031)s(y)s Fk(k)3504 2349 y Ft(2)3570 2335 y Fk(\025)25 b Fr(\016)50 2481 y Fq(for)33 b(some)g Fr(\016)c(>)c Fv(0)p Fq(.)43 b(Then,)32 b(for)i(al)5 b(l)33 b Fr(N)42 b Fq(lar)-5 b(ge)34 b(enough,)691 2670 y Fr(#)p 663 2711 110 4 v 663 2794 a Fk(j)p Fr(J)9 b Fk(j)809 2732 y(\024)25 b Fh(R)965 2694 y Fi(\016)1028 2658 y Ff(\000)1069 2732 y Fr(\034)1119 2694 y Ft(0)1109 2754 y Fj(x)1185 2732 y Fk(\024)g Fr(\034)1331 2694 y Ft(0)1321 2754 y Fj(J)1372 2658 y Ff(\001)1439 2732 y Fk(\024)1605 2670 y Fv(1)p 1546 2711 164 4 v 1546 2794 a Fr(#)p Fk(j)p Fr(J)9 b Fk(j)1720 2732 y Fr(;)199 b Fq(for)34 b(al)5 b(l)93 b Fr(J)34 b Fk(\022)25 b Fr(\015)2507 2746 y Fj(I)2549 2732 y Fv(\()p Fr(I)7 b Fv(\))p Fr(;)47 b(x)25 b Fk(2)g Fr(\015)2948 2746 y Fj(I)2990 2732 y Fv(\()p Fr(I)7 b Fv(\))370 b(\(2)p Fr(:)p Fv(36\))50 2989 y Fq(wher)-5 b(e)1006 3157 y Fr(#)25 b Fv(=)1180 3002 y Ff( )1252 3157 y Fv(1)c Fk(\000)1439 3096 y Fv(1)p 1420 3136 83 4 v 1420 3219 a Fr(N)1602 3071 y Ff(X)1529 3270 y Fj(\026)p Fi(2K)p Ft(\()p Fj(I)5 b Ft(\))1770 3252 y Fm(c)1821 3157 y Fk(j)p Fv(\003)1909 3171 y Fj(\026)1958 3157 y Fk(j)1983 3002 y Ff(!)2070 3029 y(\022)2137 3157 y Fv(1)21 b Fk(\000)2339 3096 y Fv(1)p 2305 3136 115 4 v 2305 3219 a Fr(N)2378 3233 y Fi(\003)2451 3157 y Fk(\000)2595 3096 y Fr(c)p 2553 3136 124 4 v 2553 3219 a(N)2636 3193 y Ft(2)2626 3242 y Fi(\003)2687 3029 y Ff(\023)3477 3157 y Fv(\(2)p Fr(:)p Fv(37\))50 3432 y Fq(In)32 b(p)-5 b(articular,)35 b(if)e Fk(K)q Fv(\()p Fr(I)7 b Fv(\))888 3399 y Fj(c)951 3432 y Fv(=)25 b Fk(;)p Fq(,)1069 3563 y Ff(\014)1069 3618 y(\014)1069 3672 y(\014)1069 3727 y(\014)1099 3695 y Fh(R)1159 3657 y Fi(\016)1222 3621 y Ff(\000)1264 3695 y Fr(\034)1314 3657 y Ft(0)1304 3717 y Fj(x)1380 3695 y Fk(\024)g Fr(\034)1526 3657 y Ft(0)1516 3717 y Fj(J)1566 3621 y Ff(\001)1628 3695 y Fk(\000)1762 3633 y Fv(1)p 1730 3674 110 4 v 1730 3757 a Fk(j)p Fr(J)9 b Fk(j)1851 3563 y Ff(\014)1851 3618 y(\014)1851 3672 y(\014)1851 3727 y(\014)1906 3695 y Fk(\024)2103 3633 y Fv(1)p 2013 3674 225 4 v 2013 3757 a Fk(j)p Fr(J)g Fk(j)p Fr(N)2195 3771 y Fi(\003)2264 3567 y Ff(\022)2331 3695 y Fv(1)20 b(+)2536 3633 y Fr(c)p 2498 3674 115 4 v 2498 3757 a(N)2571 3771 y Fi(\003)2624 3567 y Ff(\023)3477 3695 y Fv(\(2)p Fr(:)p Fv(38\))50 3953 y Fq(for)33 b(some)g(c)-5 b(onstant)35 b Fr(c)25 b(>)g Fv(0)p Fq(.)50 4171 y Fa(Pro)s(of)49 b(of)f(Lemma)e(2.6:)137 b Fv(An)42 b Fr(L)p Fv(-steps)g(path)f Fr(!)k Fv(on)d(\000)2180 4185 y Fj(N)s(;d)2304 4171 y Fv(\()p Fr(I)7 b Fv(\),)46 b(b)s(eginning)39 b(at)j Fr(x)g Fv(and)g(ending)e(at)50 4317 y Fr(y)h Fv(is)d(de\014ned)f(as)i(sequence)g(of)g Fr(L)f Fv(sites)g Fr(!)k Fv(=)d(\()p Fr(!)1786 4331 y Ft(0)1826 4317 y Fr(;)15 b(!)1923 4331 y Ft(1)1964 4317 y Fr(;)g(:)g(:)g(:)i(;)e(!)2223 4331 y Fj(L)2277 4317 y Fv(\),)41 b(with)d Fr(!)2651 4331 y Ft(0)2730 4317 y Fv(=)h Fr(x)p Fv(,)h Fr(!)3014 4331 y Fj(L)3107 4317 y Fv(=)f Fr(y)s Fv(,)i(and)d Fr(!)3573 4331 y Fj(l)3639 4317 y Fv(=)50 4462 y(\()p Fr(!)145 4429 y Fj(k)142 4488 y(l)190 4462 y Fv(\))225 4476 y Fj(k)r Ft(=1)p Fj(;:::)n(;d)533 4462 y Fk(2)25 b Fr(V)20 b Fv(\(\000)784 4476 y Fj(N)s(;d)909 4462 y Fv(\()p Fr(I)7 b Fv(\)\))31 b(for)f(all)g(1)25 b Fk(\024)g Fr(l)i Fk(\024)e Fr(L)p Fv(,)31 b(that)g(satis\014es:)905 4708 y(\()p Fr(!)997 4722 y Fj(l)1025 4708 y Fr(;)15 b(!)1122 4722 y Fj(l)p Fi(\000)p Ft(1)1243 4708 y Fv(\))26 b Fk(2)e Fr(E)5 b Fv(\(\000)1553 4722 y Fj(N)s(;d)1678 4708 y Fv(\()p Fr(I)i Fv(\)\))p Fr(;)199 b Fv(for)30 b(all)105 b Fr(l)27 b Fv(=)e(1)p Fr(;)15 b(:)g(:)g(:)i(;)e(L)623 b Fv(\(2)p Fr(:)p Fv(39\))50 4953 y(\(W)-8 b(e)32 b(ma)m(y)f(also)f (write)g Fk(j)p Fr(!)s Fk(j)25 b Fv(=)g Fr(L)30 b Fv(to)h(denote)g(the) g(length)f(of)g Fr(!)s Fv(.\))159 5171 y(Recall)35 b(from)h(Lemma)g (2.2)h(that)f(if)f Fr(x)f Fk(2)g Fr(\015)1660 5185 y Fj(I)1702 5171 y Fv(\()p Fr(I)7 b Fv(\),)38 b(then)e(a)g(fortiori)f Fr(x)f Fk(2)g(f\000)p Fv(1)p Fr(;)15 b Fv(1)p Fk(g)2960 5138 y Fj(d)3005 5171 y Fv(.)57 b(Without)35 b(loss)h(of)50 5316 y(generalit)m(y)28 b(w)m(e)g(ma)m(y)g(th)m(us)g(c)m(ho)s(ose)h Fr(x)e Fv(in)g(\(2.34\))j(as)e(the)g(p)s(oin)m(t)f Fr(x)e Fv(=)g(\()p Fr(x)2477 5330 y Fj(k)2522 5316 y Fv(\))2557 5283 y Fj(d)2557 5342 y(k)r Ft(=1)2694 5316 y Fv(,)k Fr(x)2800 5330 y Fj(k)2870 5316 y Fv(=)c(1)j(for)f(all)g(1)f Fk(\024)f Fr(k)j Fk(\024)d Fr(d)p Fv(.)50 5462 y(There)31 b(is)f(no)h(loss)g(of)g(generalit)m(y)h(either)f(in)f(taking)h Fk(K)q Fv(\()p Fr(I)7 b Fv(\))33 b(in)d(\(2.32\))j(to)f(b)s(e)f(the)g (set)h Fk(K)q Fv(\()p Fr(I)7 b Fv(\))28 b(=)f Fk(f)p Fv(1)p Fr(;)15 b(:)g(:)g(:)j(;)d(\024)p Fk(g)p eop %%Page: 20 20 20 19 bop 50 295 a Fo(20)3270 b Fl(Se)l(ction)27 b Fo(2)50 513 y Fv(and)21 b(in)g(assuming)g Fk(j)p Fv(\003)791 527 y Fj(k)836 513 y Fv(\()p Fr(I)7 b Fv(\))p Fk(j)23 b Fv(to)g(b)s(e)f(ev)m(en)h(for)f(all)f Fr(k)29 b Fk(2)24 b(K)q Fv(\()p Fr(I)7 b Fv(\).)40 b(With)21 b(this)h(w)m(e)g(in)m(tro)s (duce)f Fr(\024)i Fv(one-dimensional)50 658 y(paths)g(in)g(\000)448 672 y Fj(N)s(;d)572 658 y Fv(\()p Fr(I)7 b Fv(\),)26 b(eac)m(h)f(b)s(eing)d(of)i(length)f Fr(L)i Fk(\021)g Fr(\024N)1852 672 y Fi(\003)1894 658 y Fr(=)p Fv(2,)h(and)e(connecting) f Fr(x)h Fv(to)g(the)g(endp)s(oin)m(t)f Fr(y)j Fv(de\014ned)50 804 y(b)m(y)894 989 y Fr(y)i Fv(=)d(\()p Fr(y)1143 1003 y Fj(k)1188 989 y Fv(\))1223 952 y Fj(d)1223 1012 y(k)r Ft(=1)1360 989 y Fr(;)91 b(y)1521 1003 y Fj(k)1591 989 y Fv(=)1687 834 y Ff(\()1790 915 y Fv(1)21 b Fk(\000)1976 878 y Fj(N)2034 887 y Fb(\003)p 1958 894 133 4 v 1958 947 a Fi(j)p Ft(\003)2030 957 y Fm(k)2069 947 y Fi(j)2101 915 y Fr(;)183 b Fv(if)89 b Fr(k)29 b Fk(2)c(K)q Fv(\()p Fr(I)7 b Fv(\))1790 1067 y(1)p Fr(;)449 b Fv(if)89 b Fr(k)29 b Fk(2)c(K)q Fv(\()p Fr(I)7 b Fv(\))2801 1034 y Fj(c)3477 989 y Fv(\(2)p Fr(:)p Fv(40\))50 1296 y Fc(De\014nition)38 b(2.9:)99 b Fq(F)-7 b(or)35 b(e)-5 b(ach)35 b Fv(1)29 b Fk(\024)e Fr(\026)h Fk(\024)g Fr(\024)p Fq(,)35 b(let)f Fr(!)s Fv(\()p Fr(\026)p Fv(\))29 b(=)e(\()p Fr(!)2243 1310 y Ft(0)2284 1296 y Fv(\()p Fr(\026)p Fv(\))p Fr(;)15 b(:)g(:)g(:)i(;)e(!)2668 1310 y Fj(n)2718 1296 y Fv(\()p Fr(\026)p Fv(\))p Fr(;)g(:)g(:)g(:)i(;)e(!)3102 1310 y Fj(L)3156 1296 y Fv(\()p Fr(\026)p Fv(\)\))p Fq(,)35 b Fr(!)3436 1310 y Fj(n)3486 1296 y Fv(\()p Fr(\026)p Fv(\))28 b(=)50 1441 y(\()p Fr(!)145 1408 y Fj(k)142 1464 y(n)191 1441 y Fv(\()p Fr(\026)p Fv(\)\))351 1408 y Fj(d)351 1467 y(k)r Ft(=1)489 1441 y Fq(,)k(b)-5 b(e)33 b(the)g(p)-5 b(ath)34 b(in)f Fv(\000)1178 1455 y Fj(N)s(;d)1302 1441 y Fv(\()p Fr(I)7 b Fv(\))33 b Fq(de\014ne)-5 b(d)34 b(thr)-5 b(ough)946 1696 y Fr(!)1006 1658 y Fj(k)1003 1718 y(n)1052 1696 y Fv(\()p Fr(\026)p Fv(\))25 b(=)1298 1540 y Ff(\()1402 1636 y Fr(!)1462 1589 y Ft(\()p Fj(k)r Ft(+)p Fj(\026)p Fi(\000)p Ft(2\))12 b(mo)r(d)p Fj(\024)p Ft(+1)1459 1647 y Fj(n)2041 1636 y Fr(;)187 b Fq(if)92 b Fr(k)29 b Fk(2)c(K)q Fv(\()p Fr(I)7 b Fv(\))1402 1773 y(1)806 b Fq(if)92 b Fr(k)29 b Fk(2)c(K)q Fv(\()p Fr(I)7 b Fv(\))2750 1740 y Fj(c)3477 1696 y Fv(\(2)p Fr(:)p Fv(41\))50 1956 y Fq(wher)-5 b(e)34 b Fr(!)28 b Fv(=)d(\()p Fr(!)580 1970 y Ft(0)620 1956 y Fr(;)15 b(:)g(:)g(:)i(;)e(!)879 1970 y Fj(n)928 1956 y Fr(;)g(:)g(:)g(:)i(;)e(!)1187 1970 y Fj(L)1241 1956 y Fv(\))p Fq(,)33 b Fr(!)1394 1970 y Fj(n)1469 1956 y Fv(=)24 b(\()p Fr(!)1659 1923 y Fj(k)1656 1978 y(n)1706 1956 y Fv(\))1741 1923 y Fj(d)1741 1981 y(k)r Ft(=1)1878 1956 y Fq(,)32 b(is)h(de\014ne)-5 b(d)34 b(by)1745 2161 y Fr(!)1802 2175 y Ft(0)1867 2161 y Fv(=)25 b Fr(x)1462 b Fv(\(2)p Fr(:)p Fv(42\))50 2367 y Fq(and,)33 b(for)g Fv(1)26 b Fk(\024)f Fr(n)g Fk(\024)g Fr(L)p Fq(,)234 2678 y Fr(!)294 2640 y Fj(k)291 2700 y(n)365 2678 y Fv(=)461 2464 y Ff(8)461 2546 y(>)461 2573 y(<)461 2737 y(>)461 2764 y(:)572 2522 y Fv(1)c Fk(\000)835 2486 y Ft(2)p 740 2501 227 4 v 740 2553 a Fi(j)p Ft(\003)812 2563 y Fm(k)851 2553 y Ft(\()p Fj(I)5 b Ft(\))p Fi(j)993 2448 y Ff(\002)1041 2486 y Fj(n)p Fi(\000)p Ft(1)p 1041 2501 138 4 v 1089 2553 a Fj(\024)1190 2448 y Ff(\003)1248 2522 y Fk(\000)1445 2486 y Ft(2)p 1350 2501 227 4 v 1350 2553 a Fi(j)p Ft(\003)1422 2563 y Fm(k)1461 2553 y Ft(\()p Fj(I)g Ft(\))p Fi(j)1588 2522 y Fr(;)186 b Fq(if)93 b Fr(k)28 b Fk(2)d(K)q Fv(\()p Fr(I)7 b Fv(\))94 b Fq(and)g Fr(k)29 b Fk(\024)24 b Fr(n)c Fk(\000)g Fr(\024)3031 2448 y Ff(\002)3080 2486 y Fj(n)p Fi(\000)p Ft(1)p 3080 2501 138 4 v 3128 2553 a Fj(\024)3229 2448 y Ff(\003)572 2682 y Fv(1)h Fk(\000)835 2647 y Ft(2)p 740 2662 227 4 v 740 2714 a Fi(j)p Ft(\003)812 2724 y Fm(k)851 2714 y Ft(\()p Fj(I)5 b Ft(\))p Fi(j)993 2609 y Ff(\002)1041 2647 y Fj(n)p Fi(\000)p Ft(1)p 1041 2662 138 4 v 1089 2714 a Fj(\024)1190 2609 y Ff(\003)1243 2682 y Fr(;)531 b Fq(if)93 b Fr(k)28 b Fk(2)d(K)q Fv(\()p Fr(I)7 b Fv(\))94 b Fq(and)g Fr(k)29 b(>)24 b(n)c Fk(\000)g Fr(\024)3031 2609 y Ff(\002)3080 2647 y Fj(n)p Fi(\000)p Ft(1)p 3080 2662 V 3128 2714 a Fj(\024)3229 2609 y Ff(\003)572 2834 y Fv(1)1182 b Fq(if)93 b Fr(k)28 b Fk(2)d(K)q Fv(\()p Fr(I)7 b Fv(\))2296 2801 y Fj(c)3477 2678 y Fv(\(2)p Fr(:)p Fv(43\))50 2994 y Fq(Her)-5 b(e)40 b Fv([)p Fr(x)p Fv(])p Fq(,)k Fr(x)c Fk(2)f Fh(R)s Fq(,)49 b(denotes)41 b(the)h(inte)-5 b(ger)41 b(p)-5 b(art)42 b(of)f Fr(x)p Fq(.)67 b(\(The)41 b(p)-5 b(aths)43 b Fr(!)s Fv(\()p Fr(\026)p Fv(\))e Fq(ar)-5 b(e)41 b(in)g(fact)g(p)-5 b(aths)43 b(on)e(the)50 3140 y(sub)-5 b(gr)g(aph)34 b Fk(f)p Fr(z)c Fk(2)25 b Fv(\000)680 3154 y Fj(N)s(;d)804 3140 y Fv(\()p Fr(I)7 b Fv(\))27 b Fk(j)e Fr(z)1040 3154 y Fj(k)1110 3140 y Fv(=)g(1)15 b Fk(8)p Fr(k)29 b Fk(2)c(K)q Fv(\()p Fr(I)7 b Fv(\))1666 3107 y Fj(c)1704 3140 y Fk(g)p Fq(.\))159 3358 y Fv(Let)31 b Fr(D)i Fv(b)s(e)d(the)h(subgraph)e(of)h Fk(G)5 b Fv(\(\000)1360 3372 y Fj(N)s(;d)1485 3358 y Fv(\()p Fr(I)i Fv(\)\))32 b(with)d(set)i(of)g(v)m(ertices)g Fr(V)20 b Fv(\()p Fr(D)s Fv(\))26 b(=)g Fk(f)p Fr(x)2892 3325 y Fi(0)2943 3358 y Fk(2)f Fv(\000)3086 3372 y Fj(N)s(;d)3210 3358 y Fv(\()p Fr(I)7 b Fv(\))27 b Fk(j)e(k)p Fr(x)3501 3325 y Fi(0)3527 3358 y Fk(k)3572 3372 y Ft(2)3639 3358 y Fk(\024)50 3503 y(k)p Fr(y)s Fk(k)188 3517 y Ft(2)229 3503 y Fk(g)37 b Fv(and)e(set)i(of)f(edges)h Fr(E)5 b Fv(\()p Fr(D)s Fv(\))35 b(=)g Fk(f)p Fv(\()p Fr(x)1492 3470 y Fi(0)1518 3503 y Fr(;)15 b(x)1610 3470 y Fi(00)1657 3503 y Fv(\))35 b Fk(2)f Fr(E)5 b Fv(\(\000)1986 3517 y Fj(N)s(;d)2111 3503 y Fv(\()p Fr(I)i Fv(\)\))36 b Fk(j)f Fr(x)2411 3470 y Fi(0)2436 3503 y Fr(;)15 b(x)2528 3470 y Fi(0)q(0)2610 3503 y Fk(2)34 b Fr(V)20 b Fv(\()p Fr(D)s Fv(\))p Fk(g)p Fv(.)59 b(Denoting)36 b(b)m(y)g(\001)3661 3517 y Fj(\026)50 3649 y Fv(the)29 b(subgraph)f(of)i Fk(G)5 b Fv(\(\000)853 3663 y Fj(N)s(;d)978 3649 y Fv(\()p Fr(I)i Fv(\)\))30 b(\\generated")i(b)m(y)d(the)h(path)f Fr(!)s Fv(\()p Fr(\026)p Fv(\),)h(i.e.,)g(with)e(set)i(of)f(v)m (ertices)h Fr(V)21 b Fv(\(\001)3530 3663 y Fj(\026)3578 3649 y Fv(\))26 b(=)50 3794 y Fk(f)p Fr(x)147 3761 y Fi(0)198 3794 y Fk(2)f Fv(\000)341 3808 y Fj(N)s(;d)465 3794 y Fv(\()p Fr(I)7 b Fv(\))26 b Fk(j)g(9)710 3808 y Ft(0)p Fi(\024)p Fj(n)p Fi(\024)p Fj(L)999 3794 y Fv(:)40 b Fr(x)1116 3761 y Fi(0)1167 3794 y Fv(=)25 b Fr(!)1320 3808 y Fj(n)1369 3794 y Fv(\()p Fr(\026)p Fv(\))p Fk(g)p Fv(,)32 b(w)m(e)f(set)1554 4040 y(\001)25 b(=)g Fr(D)e Fk([)1977 3926 y Fj(\024)1947 3953 y Ff([)1930 4146 y Fj(\026)p Ft(=1)2081 4040 y Fv(\001)2157 4054 y Fj(\026)3477 4040 y Fv(\(2)p Fr(:)p Fv(44\))50 4305 y(Since)29 b(b)s(oth)h Fr(x)g Fv(and)g(0)h(b)s(elong)e(to)i(\001)f(it)g(follo)m(ws)g(from)f (Lemma)i(\(6.1\))h(that)1026 4486 y Fh(R)1085 4448 y Fi(\016)1148 4486 y Fv(\()q Fr(\034)1234 4448 y Fj(x)1224 4508 y Ft(0)1305 4486 y Fr(<)25 b(\034)1451 4448 y Fj(x)1441 4508 y(x)1496 4486 y Fv(\))h Fk(\025)1661 4462 y Ff(e)1653 4486 y Fh(R)1713 4448 y Fi(\016)1713 4508 y Ft(\001)1798 4486 y Fv(\()p Fr(\034)1883 4448 y Fj(x)1873 4508 y Ft(0)1955 4486 y Fr(<)f(\034)2101 4448 y Fj(x)2091 4508 y(x)2146 4486 y Fv(\))1557 4658 y(=)1661 4635 y Ff(e)1653 4658 y Fh(R)1713 4621 y Fi(\016)1713 4681 y Ft(\001)1798 4585 y Ff(\000)1840 4658 y Fr(\034)1890 4621 y Fj(x)1880 4681 y(y)1961 4658 y Fr(<)g(\034)2107 4621 y Fj(x)2097 4681 y(x)2153 4585 y Ff(\001)2217 4635 y(e)2209 4658 y Fh(R)2269 4621 y Fi(\016)2269 4681 y Ft(\001)2355 4658 y Fv(\()p Fr(\034)2440 4615 y Fj(y)2430 4683 y Ft(0)2509 4658 y Fr(<)g(\034)2655 4621 y Fj(y)2645 4681 y(x)2699 4658 y Fv(\))3477 4568 y(\(2)p Fr(:)p Fv(45\))50 4825 y(where)34 b(the)g(last)g(equalit)m(y)g(is)f(nothing)h(but)f(the)i(Mark)m(o)m(v)h (prop)s(ert)m(y)-8 b(.)52 b(Again,)35 b(the)g(collection)f(\001)3453 4839 y Fj(\026)3501 4825 y Fv(,)i(1)c Fk(\024)50 4971 y Fr(\026)c Fk(\024)h Fr(\024)p Fv(,)34 b(b)s(eing)d(easily)g(seen)i (to)g(v)m(erify)f(conditions)f(\(6.2\))k(and)d(\(6.3\))i(of)f(Lemma)f (6.1)i(\(w.r.t.)47 b(the)33 b(ev)m(en)m(t)50 5043 y Ff(\010)103 5116 y Fr(\034)153 5083 y Fj(x)143 5139 y(y)224 5116 y Fr(<)25 b(\034)370 5083 y Fj(x)360 5139 y(x)416 5043 y Ff(\011)469 5116 y Fv(\),)31 b(w)m(e)f(ha)m(v)m(e,)i(applying)c(the)j (latter)g(lemma)f(t)m(wice)h(in)e(a)h(ro)m(w,)596 5347 y Ff(e)588 5371 y Fh(R)648 5333 y Fi(\016)648 5393 y Ft(\001)733 5297 y Ff(\000)775 5371 y Fr(\034)825 5333 y Fj(x)815 5393 y(y)896 5371 y Fr(<)25 b(\034)1042 5333 y Fj(x)1032 5393 y(x)1088 5297 y Ff(\001)1155 5371 y Fk(\025)1258 5347 y Ff(e)1251 5371 y Fh(R)1310 5333 y Fi(\016)1310 5393 y([)1359 5371 y Fm(\024)1359 5415 y(\026)p Fn(=1)1486 5393 y Ft(\001)1546 5402 y Fm(\026)1609 5297 y Ff(\000)1650 5371 y Fr(\034)1700 5333 y Fj(x)1690 5393 y(y)1771 5371 y Fr(<)g(\034)1917 5333 y Fj(x)1907 5393 y(x)1963 5297 y Ff(\001)2030 5371 y Fk(\025)2173 5257 y Fj(\024)2129 5284 y Ff(X)2126 5477 y Fj(\026)p Ft(=1)2285 5347 y Ff(e)2277 5371 y Fh(R)2337 5333 y Fi(\016)2337 5393 y Ft(\001)2397 5402 y Fm(\026)2466 5270 y Ff(\020)2520 5371 y Fr(\034)2570 5323 y Fj(!)2616 5332 y Fn(0)2651 5323 y Ft(\()p Fj(\026)p Ft(\))2560 5401 y Fj(!)2606 5410 y Fm(L)2653 5401 y Ft(\()p Fj(\026)p Ft(\))2784 5371 y Fr(<)g(\034)2930 5323 y Fj(!)2976 5332 y Fn(0)3012 5323 y Ft(\()p Fj(\026)p Ft(\))2920 5401 y Fj(!)2966 5410 y Fn(0)3001 5401 y Ft(\()p Fj(\026)p Ft(\))3117 5270 y Ff(\021)3477 5371 y Fv(\(2)p Fr(:)p Fv(46\))p eop %%Page: 21 21 21 20 bop 50 295 a Fl(A)l(ging)27 b(in)h(the)g(REM.)f(Part)h(1.)2661 b Fo(21)50 513 y Fv(and)30 b(com)m(bining)e(\(2.45\))33 b(and)d(\(2.46\),)j(w)m(e)e(ha)m(v)m(e,)774 783 y Fh(R)833 745 y Fi(\016)896 783 y Fv(\()q Fr(\034)982 745 y Fj(x)972 805 y Ft(0)1053 783 y Fr(<)25 b(\034)1199 745 y Fj(x)1189 805 y(x)1244 783 y Fv(\))h Fk(\025)1409 759 y Ff(e)1401 783 y Fh(R)1461 745 y Fi(\016)1461 805 y Ft(\001)1546 783 y Fv(\()p Fr(\034)1631 739 y Fj(y)1621 807 y Ft(0)1700 783 y Fr(<)f(\034)1846 745 y Fj(y)1836 805 y(x)1890 783 y Fv(\))1987 669 y Fj(\024)1943 696 y Ff(X)1941 889 y Fj(\026)p Ft(=1)2100 759 y Ff(e)2092 783 y Fh(R)2152 745 y Fi(\016)2152 805 y Ft(\001)2212 814 y Fm(\026)2280 682 y Ff(\020)2335 783 y Fr(\034)2385 735 y Fj(!)2431 744 y Fn(0)2466 735 y Ft(\()p Fj(\026)p Ft(\))2375 814 y Fj(!)2421 823 y Fm(L)2468 814 y Ft(\()p Fj(\026)p Ft(\))2599 783 y Fr(<)g(\034)2745 735 y Fj(!)2791 744 y Fn(0)2826 735 y Ft(\()p Fj(\026)p Ft(\))2735 814 y Fj(!)2781 823 y Fn(0)2816 814 y Ft(\()p Fj(\026)p Ft(\))2932 682 y Ff(\021)3477 783 y Fv(\(2)p Fr(:)p Fv(47\))50 1076 y(The)i(b)s(ound)e (\(2.45\))30 b(is)d(of)g(course)h(meaningless)e(if)h(it)g(so)g(happ)s (ens)f(that)i Fr(y)g Fv(=)d(0.)40 b(In)27 b(this)g(sp)s(ecial)f(case)i (w)m(e)50 1221 y(only)h(use)h(\(2.46\))j(to)e(write)1043 1491 y Fh(R)1103 1454 y Fi(\016)1166 1491 y Fv(\()p Fr(\034)1251 1454 y Fj(x)1241 1514 y Ft(0)1322 1491 y Fr(<)25 b(\034)1468 1454 y Fj(x)1458 1514 y(x)1514 1491 y Fv(\))h Fk(\025)1718 1377 y Fj(\024)1673 1405 y Ff(X)1671 1597 y Fj(\026)p Ft(=1)1830 1468 y Ff(e)1822 1491 y Fh(R)1882 1454 y Fi(\016)1882 1514 y Ft(\001)1942 1523 y Fm(\026)2010 1390 y Ff(\020)2065 1491 y Fr(\034)2115 1444 y Fj(!)2161 1453 y Fn(0)2196 1444 y Ft(\()p Fj(\026)p Ft(\))2105 1522 y Fj(!)2151 1531 y Fm(L)2198 1522 y Ft(\()p Fj(\026)p Ft(\))2329 1491 y Fr(<)f(\034)2475 1444 y Fj(!)2521 1453 y Fn(0)2556 1444 y Ft(\()p Fj(\026)p Ft(\))2465 1522 y Fj(!)2511 1531 y Fn(0)2546 1522 y Ft(\()p Fj(\026)p Ft(\))2662 1390 y Ff(\021)3477 1491 y Fv(\(2)p Fr(:)p Fv(48\))50 1784 y(Th)m(us,)k(in)g(view)h(of)h(\(2.47\))h(and)e(\(2.48\),)j(Lemma)d (2.6)i(will)c(b)s(e)h(pro)m(v)m(en)i(if)e(w)m(e)i(can)g(establish)e (that:)50 2002 y Fc(Lemma)38 b(2.10:)91 b Fq(Under)33 b(the)g(assumptions)i(of)e(L)-5 b(emma)34 b(2.6:)47 2220 y(i\))47 b(Ther)-5 b(e)35 b(exists)g(a)g(c)-5 b(onstant)37 b Fr(c)29 b(>)f Fv(0)35 b Fq(such)g(that,)h(for)f(lar)-5 b(ge)36 b(enough)f Fr(N)10 b Fq(,)35 b(for)g(e)-5 b(ach)35 b Fr(\026)29 b Fk(2)f(K)q Fv(\()p Fr(I)7 b Fv(\))36 b Fq(and)g(with)159 2366 y Fr(N)232 2380 y Fi(\003)306 2366 y Fq(de\014ne)-5 b(d)34 b(as)f(in)g(\(2.33\),)993 2594 y Ff(e)985 2618 y Fh(R)1045 2580 y Fi(\016)1045 2640 y Ft(\001)1105 2649 y Fm(\026)1173 2517 y Ff(\020)1228 2618 y Fr(\034)1278 2570 y Fj(!)1324 2579 y Fn(0)1359 2570 y Ft(\()p Fj(\026)p Ft(\))1268 2649 y Fj(!)1314 2658 y Fm(L)1361 2649 y Ft(\()p Fj(\026)p Ft(\))1492 2618 y Fr(<)25 b(\034)1638 2570 y Fj(!)1684 2579 y Fn(0)1719 2570 y Ft(\()p Fj(\026)p Ft(\))1628 2649 y Fj(!)1674 2658 y Fn(0)1709 2649 y Ft(\()p Fj(\026)p Ft(\))1825 2517 y Ff(\021)1904 2618 y Fk(\025)2011 2556 y(j)p Fv(\003)2099 2570 y Fj(\026)2148 2556 y Fk(j)p 2011 2597 163 4 v 2051 2680 a Fr(N)2200 2489 y Ff(\022)2267 2618 y Fv(1)20 b Fk(\000)2469 2556 y Fv(1)p 2434 2597 115 4 v 2434 2680 a Fr(N)2507 2694 y Fi(\003)2580 2618 y Fk(\000)2724 2556 y Fr(c)p 2682 2597 124 4 v 2682 2680 a(N)2765 2654 y Ft(2)2755 2702 y Fi(\003)2817 2489 y Ff(\023)3477 2618 y Fv(\(2)p Fr(:)p Fv(49\))20 2942 y Fq(ii\))46 b(Assume)32 b(that)i Fr(y)28 b Fk(6)p Fv(=)d(0)p Fq(.)42 b(Ther)-5 b(e)34 b(exists)f(a)g(c)-5 b(onstant)34 b Fr(c)26 b(>)f Fv(0)33 b Fq(such)f(that,)i(for)f(al)5 b(l)33 b Fr(N)43 b Fq(lar)-5 b(ge)33 b(enough,)1264 3152 y Ff(e)1256 3175 y Fh(R)1316 3138 y Fi(\016)1316 3198 y Ft(\001)1401 3175 y Fv(\()p Fr(\034)1486 3132 y Fj(y)1476 3200 y Ft(0)1555 3175 y Fr(<)25 b(\034)1701 3138 y Fj(y)1691 3198 y(x)1745 3175 y Fv(\))h Fk(\025)f Fv(1)20 b Fk(\000)g Fr(cdN)2227 3138 y Ft(3)p Fj(=)p Ft(2)2342 3175 y Fv(2)2387 3138 y Fi(\000)p Fj(\017N)q(=d)3477 3175 y Fv(\(2)p Fr(:)p Fv(50\))50 3627 y Fa(Pro)s(of)36 b(of)f(Lemma)d(2.10)k(i\):)86 b Fv(T)-8 b(o)31 b(simplify)c(the)k(presen)m(tation)f(w)m(e)h(will)c (only)j(treat)h(the)g(case)g Fr(\026)25 b Fv(=)g(1,)50 3773 y(that)31 b(is,)e(with)h(the)g(notation)h(of)f(De\014nition)f (2.9,)j(establish)d(that)1056 3996 y Ff(e)1048 4019 y Fh(R)1108 3982 y Fi(\016)1108 4042 y Ft(\001)1168 4051 y Fm(\026)1237 3946 y Ff(\000)1278 4019 y Fr(\034)1328 3982 y Fj(!)1374 3991 y Fn(0)1318 4042 y Fj(!)1364 4051 y Fm(L)1442 4019 y Fr(<)24 b(\034)1587 3982 y Fj(!)1633 3991 y Fn(0)1577 4042 y Fj(!)1623 4051 y Fn(0)1673 3946 y Ff(\001)1740 4019 y Fk(\025)1847 3958 y(j)p Fv(\003)1935 3972 y Ft(1)1976 3958 y Fk(j)p 1847 3998 155 4 v 1883 4082 a Fr(N)2028 3891 y Ff(\022)2095 4019 y Fv(1)c Fk(\000)2297 3958 y Fv(1)p 2262 3998 115 4 v 2262 4082 a Fr(N)2335 4096 y Fi(\003)2408 4019 y Fk(\000)2552 3958 y Fr(c)p 2510 3998 124 4 v 2510 4082 a(N)2593 4055 y Ft(2)2583 4104 y Fi(\003)2644 3891 y Ff(\023)3477 4019 y Fv(\(2)p Fr(:)p Fv(51\))50 4271 y(As)30 b(is)g(w)m(ell)f(kno)m(wn)h(that)h (\(see)g(e.g.)42 b([Sp])30 b(or)g([BEGK1],)i(Lemma)e(2.5\))834 4553 y Ff(e)826 4577 y Fh(R)886 4539 y Fi(\016)886 4599 y Ft(\001)946 4608 y Fm(\026)1015 4503 y Ff(\000)1056 4577 y Fr(\034)1106 4539 y Fj(!)1152 4548 y Fn(0)1096 4599 y Fj(!)1142 4608 y Fm(L)1220 4577 y Fr(<)24 b(\034)1365 4539 y Fj(!)1411 4548 y Fn(0)1355 4599 y Fj(!)1401 4608 y Fn(0)1451 4503 y Ff(\001)1518 4577 y Fv(=)1614 4421 y Ff(")1711 4463 y Fj(L)1670 4491 y Ff(X)1667 4683 y Fj(n)p Ft(=1)1845 4475 y Ff(e)1835 4499 y Fh(Q)1899 4466 y Fi(\016)1899 4524 y Ft(\001)1959 4533 y Fm(\026)2013 4499 y Fv(\()p Fr(!)2105 4513 y Ft(0)2146 4499 y Fv(\))p 1830 4556 355 4 v 1840 4633 a Ff(e)1830 4656 y Fh(Q)1895 4625 y Fi(\016)1895 4683 y Ft(\001)1955 4692 y Fm(\026)2008 4656 y Fv(\()p Fr(!)2100 4670 y Fj(n)2150 4656 y Fv(\))2467 4515 y(1)p 2207 4556 565 4 v 2209 4639 a Ff(e)-53 b Fr(r)2251 4608 y Fi(\016)2248 4666 y Ft(\001)2308 4675 y Fm(\026)2355 4639 y Fv(\()p Fr(!)2447 4653 y Fj(n)2497 4639 y Fr(;)15 b(!)2594 4653 y Fj(n)p Fi(\000)p Ft(1)2736 4639 y Fv(\))2783 4421 y Ff(#)2836 4432 y Fi(\000)p Ft(1)3477 4577 y Fv(\(2)p Fr(:)p Fv(52\))50 4887 y(whic)m(h)36 b(w)m(e)h(ma)m(y)h(also)f(write,)h (using)e(rev)m(ersibilit)m(y)e(together)39 b(with)d(the)h (de\014nitions)d(of)39 b Ff(e)-52 b Fr(r)3207 4854 y Fi(\016)3204 4912 y Ft(\001)3264 4921 y Fm(\026)3348 4887 y Fv(and)3542 4864 y Ff(e)3532 4887 y Fh(Q)3596 4854 y Fi(\016)3596 4912 y Ft(\001)3656 4921 y Fm(\026)50 5032 y Fv(\(see)31 b(App)s(endix)d(A\),)281 5334 y Ff(e)273 5357 y Fh(R)333 5320 y Fi(\016)333 5380 y Ft(\001)393 5389 y Fm(\026)461 5284 y Ff(\000)503 5357 y Fr(\034)553 5320 y Fj(!)599 5329 y Fn(0)543 5380 y Fj(!)589 5389 y Fm(L)666 5357 y Fr(<)d(\034)812 5320 y Fj(!)858 5329 y Fn(0)802 5380 y Fj(!)848 5389 y Fn(0)898 5284 y Ff(\001)965 5357 y Fv(=)1061 5202 y Ff(")1114 5244 y Fj(L)p Fi(\000)p Ft(1)1120 5271 y Ff(X)1117 5463 y Fj(n)p Ft(=0)1287 5296 y Fh(Q)1352 5310 y Fj(N)1427 5296 y Fv(\()p Fr(!)1519 5310 y Ft(0)1560 5296 y Fv(\))p 1283 5336 317 4 v 1283 5420 a Fh(Q)1348 5434 y Fj(N)1423 5420 y Fv(\()p Fr(!)1515 5434 y Fj(n)1564 5420 y Fv(\))1861 5296 y(1)p 1621 5336 526 4 v 1621 5420 a Fr(r)1662 5434 y Fj(N)1731 5420 y Fv(\()p Fr(!)1823 5434 y Fj(n)1873 5420 y Fr(;)15 b(!)1970 5434 y Fj(n)p Ft(+1)2111 5420 y Fv(\))2158 5202 y Ff(#)2211 5213 y Fi(\000)p Ft(1)2333 5357 y Fv(=)2429 5175 y Ff(2)2429 5338 y(4)2490 5236 y Fj(N)2548 5245 y Fb(\003)2586 5236 y Fj(=)p Ft(2)p Fi(\000)p Ft(1)2555 5271 y Ff(X)2543 5463 y Fj(m)p Ft(=0)2812 5244 y Fj(\024)2767 5271 y Ff(X)2775 5467 y Fj(l)p Ft(=1)2914 5357 y Fr(A)2982 5371 y Fj(m;l)3096 5175 y Ff(3)3096 5338 y(5)3157 5193 y Fi(\000)p Ft(1)3477 5348 y Fv(\(2)p Fr(:)p Fv(53\))p eop %%Page: 22 22 22 21 bop 50 295 a Fo(22)3270 b Fl(Se)l(ction)27 b Fo(2)50 513 y Fv(where)1026 683 y Fr(A)1094 697 y Fj(m;l)1234 683 y Fk(\021)1462 622 y Fh(Q)1527 636 y Fj(N)1602 622 y Fv(\()p Fr(!)1694 636 y Ft(0)1735 622 y Fv(\))p 1341 662 551 4 v 1341 745 a Fh(Q)1405 759 y Fj(N)1480 745 y Fv(\()p Fr(!)1572 759 y Fj(m\024)p Ft(+)p Fj(l)p Fi(\000)p Ft(1)1856 745 y Fv(\))2295 622 y(1)p 1913 662 809 4 v 1913 745 a Fr(r)1954 759 y Fj(N)2023 745 y Fv(\()p Fr(!)2115 759 y Fj(m\024)p Ft(+)p Fj(l)p Fi(\000)p Ft(1)2399 745 y Fr(;)15 b(!)2496 759 y Fj(m\024)p Ft(+)p Fj(l)2687 745 y Fv(\))3477 683 y(\(2)p Fr(:)p Fv(54\))50 934 y(By)30 b(\(2.27\))j(and)d(\(2.18\),)1120 1141 y Fr(A)1188 1102 y Fi(\000)p Ft(1)1188 1168 y Fj(m;l)1327 1141 y Fv(=)1434 1079 y Fk(j)p Fv(\003)1522 1093 y Fj(l)1551 1079 y Fk(j)20 b(\000)g Fr(m)p 1434 1120 333 4 v 1559 1203 a(N)1838 1027 y Fj(\024)1801 1054 y Ff(Y)1793 1250 y Fj(k)r Ft(=1)1940 1013 y Ff(\022)2211 1060 y Fk(j)p Fv(\003)2299 1074 y Fj(k)2344 1060 y Fk(j)2007 1222 y(j)p Fv(\003)2095 1236 y Fj(k)2140 1222 y Fk(j)2176 1162 y Ft(1)p Fi(\000)p Fj(!)2317 1140 y Fm(k)2315 1186 y(m\024)p Fn(+)p Fm(l)p Fb(\000)p Fn(1)p 2176 1201 386 4 v 2351 1253 a Ft(2)2573 1013 y Ff(\023)3477 1141 y Fv(\(2)p Fr(:)p Fv(55\))50 1425 y(and)30 b(b)m(y)g(\(2.43\))976 1613 y Fr(A)1044 1575 y Fi(\000)p Ft(1)1044 1641 y Fj(m;l)1183 1613 y Fv(=)1290 1552 y Fk(j)p Fv(\003)1378 1566 y Fj(l)1407 1552 y Fk(j)20 b(\000)g Fr(m)p 1290 1593 333 4 v 1415 1676 a(N)1698 1527 y Ff(Y)1649 1723 y Fj(k)r(>l)p Fi(\000)p Ft(1)1877 1485 y Ff(\022)1944 1552 y Fk(j)p Fv(\003)2032 1566 y Fj(k)2077 1552 y Fk(j)1984 1676 y Fr(m)2103 1485 y Ff(\023)2233 1527 y(Y)2185 1723 y Fj(k)r Fi(\024)p Fj(l)p Fi(\000)p Ft(1)2413 1485 y Ff(\022)2519 1552 y Fk(j)p Fv(\003)2607 1566 y Fj(k)2652 1552 y Fk(j)2480 1676 y Fr(m)g Fv(+)g(1)2717 1485 y Ff(\023)1183 1930 y Fv(=)1290 1869 y Fk(j)p Fv(\003)1378 1883 y Fj(l)1407 1869 y Fk(j)g(\000)g Fr(m)p 1290 1909 V 1415 1993 a(N)1706 1817 y Fj(\024)1669 1844 y Ff(Y)1649 2040 y Fj(k)1690 2021 y Fb(0)1714 2040 y Ft(=1)1821 1802 y Ff(\022)1888 1869 y Fk(j)p Fv(\003)1976 1883 y Fj(k)2017 1864 y Fb(0)2045 1869 y Fk(j)1939 1993 y Fr(m)2071 1802 y Ff(\023)2201 1844 y(Y)2153 2040 y Fj(k)r Fi(\024)p Fj(l)p Fi(\000)p Ft(1)2392 1869 y Fk(j)p Fv(\003)2480 1883 y Fj(k)2525 1869 y Fk(j)h(\000)f Fr(m)p 2392 1909 350 4 v 2449 1993 a(m)g Fv(+)g(1)3477 1794 y(\(2)p Fr(:)p Fv(56\))50 2221 y(where)35 b(w)m(e)h(use)f(the)h(con)m(v)m(en)m(tion)h(that)f(the)g (second)g(pro)s(duct)e(ab)s(o)m(v)m(e)j(is)e(one)h(whenev)m(er)f(the)h (index)e(set)50 2367 y Fr(k)28 b Fk(\024)d Fr(l)d Fk(\000)e Fv(1)31 b(is)e(empt)m(y)-8 b(.)42 b(F)-8 b(rom)31 b(no)m(w)f(on)h(w)m (e)f(distinguish)d(t)m(w)m(o)32 b(cases.)50 2585 y Fs(1\))j(the)f(case) h Fr(\024)26 b Fv(=)f(1)p Fs(:)41 b Fv(Here)31 b Fr(N)1157 2599 y Fi(\003)1224 2585 y Fv(=)25 b Fk(j)p Fv(\003)1408 2599 y Ft(1)1449 2585 y Fk(j)p Fv(.)41 b(Inserting)29 b(\(2.56\))k(in)c(\(2.53\))j(yields)1083 2932 y Ff(e)1075 2956 y Fh(R)1135 2918 y Fi(\016)1135 2978 y Ft(\001)1195 2987 y Fm(\026)1263 2882 y Ff(\000)1305 2956 y Fr(\034)1355 2918 y Fj(!)1401 2927 y Fn(0)1345 2978 y Fj(!)1391 2987 y Fm(L)1468 2956 y Fr(<)25 b(\034)1614 2918 y Fj(!)1660 2927 y Fn(0)1604 2978 y Fj(!)1650 2987 y Fn(0)1700 2882 y Ff(\001)1767 2956 y Fv(=)1874 2894 y Fk(j)p Fv(\003)1962 2908 y Ft(1)2003 2894 y Fk(j)p 1874 2935 155 4 v 1910 3018 a Fr(N)2054 2773 y Ff(2)2054 2936 y(4)2115 2834 y Fj(N)2173 2843 y Fb(\003)2211 2834 y Fj(=)p Ft(2)p Fi(\000)p Ft(1)2180 2869 y Ff(X)2168 3062 y Fj(m)p Ft(=0)2392 2956 y Fr(C)2457 2970 y Fj(m)2526 2773 y Ff(3)2526 2936 y(5)2587 2792 y Fi(\000)p Ft(1)3477 2956 y Fv(\(2)p Fr(:)p Fv(57\))50 3303 y(where)1044 3487 y Fr(C)1109 3501 y Fj(m)1203 3487 y Fk(\021)1299 3359 y Ff(\022)1366 3426 y Fk(j)p Fv(\003)1454 3440 y Ft(1)1495 3426 y Fk(j)1403 3549 y Fr(m)1520 3359 y Ff(\023)1587 3378 y Fi(\000)p Ft(1)1791 3426 y Fk(j)p Fv(\003)1879 3440 y Ft(1)1921 3426 y Fk(j)p 1696 3466 346 4 v 1696 3549 a(j)p Fv(\003)1784 3563 y Ft(1)1825 3549 y Fk(j)20 b(\000)g Fr(m)2077 3487 y Fv(=)2173 3359 y Ff(\022)2240 3426 y Fk(j)p Fv(\003)2328 3440 y Ft(1)2369 3426 y Fk(j)h(\000)f Fv(1)2356 3549 y Fr(m)2551 3359 y Ff(\023)2618 3378 y Fi(\000)p Ft(1)3477 3487 y Fv(\(2)p Fr(:)p Fv(58\))50 3734 y(Then,)29 b(using)g(the)i(b)s (ound)973 3917 y Ff(\022)1040 3984 y Fk(j)p Fv(\003)1128 3998 y Ft(1)1170 3984 y Fk(j)1078 4108 y Fr(m)1195 3917 y Ff(\023)1262 3936 y Fi(\000)p Ft(1)1385 4045 y Fk(\024)1680 3984 y Fv(6)p 1491 4024 423 4 v 1491 4108 a(\()p Fk(j)p Fv(\003)1614 4122 y Ft(1)1656 4108 y Fk(j)20 b(\000)g Fv(1\))1872 4081 y Ft(3)1940 4045 y Fr(;)91 b Fv(3)26 b Fk(\024)f Fr(m)g Fk(\024)g Fr(N)2497 4059 y Fi(\003)2539 4045 y Fr(=)p Fv(2)c Fk(\000)f Fv(1)691 b(\(2)p Fr(:)p Fv(59\))50 4338 y(easily)29 b(yields)1056 4498 y Ff(e)1048 4521 y Fh(R)1108 4484 y Fi(\016)1108 4544 y Ft(\001)1168 4553 y Fm(\026)1237 4448 y Ff(\000)1278 4521 y Fr(\034)1328 4484 y Fj(!)1374 4493 y Fn(0)1318 4544 y Fj(!)1364 4553 y Fm(L)1442 4521 y Fr(<)24 b(\034)1587 4484 y Fj(!)1633 4493 y Fn(0)1577 4544 y Fj(!)1623 4553 y Fn(0)1673 4448 y Ff(\001)1740 4521 y Fk(\025)1847 4460 y(j)p Fv(\003)1935 4474 y Ft(1)1976 4460 y Fk(j)p 1847 4500 155 4 v 1883 4584 a Fr(N)2028 4393 y Ff(\022)2095 4521 y Fv(1)c Fk(\000)2297 4460 y Fv(1)p 2262 4500 115 4 v 2262 4584 a Fr(N)2335 4598 y Fi(\003)2408 4521 y Fk(\000)2552 4460 y Fr(c)p 2510 4500 124 4 v 2510 4584 a(N)2593 4557 y Ft(2)2583 4606 y Fi(\003)2644 4393 y Ff(\023)3477 4521 y Fv(\(2)p Fr(:)p Fv(60\))50 4768 y(for)30 b(some)h(constan)m(t)g Fr(c)26 b(>)f Fv(0.)50 4987 y Fs(2\))35 b(the)f(case)h Fr(\024)26 b(>)f Fv(1)p Fs(:)41 b Fv(Inserting)29 b(\(2.56\))k(in)c (\(2.53\))j(yields)823 5334 y Ff(e)815 5357 y Fh(R)875 5320 y Fi(\016)875 5380 y Ft(\001)935 5389 y Fm(\026)1003 5284 y Ff(\000)1045 5357 y Fr(\034)1095 5320 y Fj(!)1141 5329 y Fn(0)1085 5380 y Fj(!)1131 5389 y Fm(L)1208 5357 y Fr(<)25 b(\034)1354 5320 y Fj(!)1400 5329 y Fn(0)1344 5380 y Fj(!)1390 5389 y Fn(0)1440 5284 y Ff(\001)1507 5357 y Fv(=)1603 5175 y Ff(2)1603 5338 y(4)1664 5236 y Fj(N)1722 5245 y Fb(\003)1759 5236 y Fj(=)p Ft(2)p Fi(\000)p Ft(1)1729 5271 y Ff(X)1716 5463 y Fj(m)p Ft(=0)1941 5202 y Ff(")2051 5244 y Fj(\024)2014 5271 y Ff(Y)1994 5467 y Fj(k)2035 5448 y Fb(0)2059 5467 y Ft(=1)2166 5229 y Ff(\022)2233 5296 y Fk(j)p Fv(\003)2321 5310 y Fj(k)2362 5291 y Fb(0)2390 5296 y Fk(j)2284 5420 y Fr(m)2416 5229 y Ff(\023)2483 5202 y(#)2536 5221 y Fi(\000)p Ft(1)2648 5357 y Fr(B)2717 5371 y Fj(m)2786 5175 y Ff(3)2786 5338 y(5)2847 5193 y Fi(\000)p Ft(1)3477 5357 y Fv(\(2)p Fr(:)p Fv(61\))p eop %%Page: 23 23 23 22 bop 50 295 a Fl(A)l(ging)27 b(in)h(the)g(REM.)f(Part)h(1.)2661 b Fo(23)50 513 y Fv(where)1063 672 y Fr(B)1132 686 y Fj(m)1227 672 y Fk(\021)1367 558 y Fj(\024)1323 586 y Ff(X)1330 781 y Fj(l)p Ft(=1)1605 610 y Fr(N)p 1480 651 333 4 v 1480 734 a Fk(j)p Fv(\003)1568 748 y Fj(l)1596 734 y Fk(j)21 b(\000)f Fr(m)1887 586 y Ff(Y)1838 781 y Fj(k)r Fi(\024)p Fj(l)p Fi(\000)p Ft(1)2134 610 y Fr(m)g Fv(+)g(1)p 2078 651 350 4 v 2078 734 a Fk(j)p Fv(\003)2166 748 y Fj(k)2211 734 y Fk(j)g(\000)g Fr(m)1227 1005 y Fv(=)1465 944 y Fr(N)p 1333 985 346 4 v 1333 1068 a Fk(j)p Fv(\003)1421 1082 y Ft(1)1463 1068 y Fk(j)g(\000)g Fr(m)1705 850 y Ff(")1758 1005 y Fv(1)h(+)1959 892 y Fj(\024)1914 919 y Ff(X)1922 1115 y Fj(l)p Ft(=2)2119 892 y Fj(l)2073 919 y Ff(Y)2061 1115 y Fj(l)2084 1097 y Fb(0)2109 1115 y Ft(=2)2287 944 y Fr(m)f Fv(+)g(1)p 2227 985 358 4 v 2227 1068 a Fk(j)p Fv(\003)2315 1082 y Fj(l)2338 1063 y Fb(0)2368 1068 y Fk(j)g(\000)g Fr(m)2595 850 y Ff(#)1227 1319 y Fk(\024)1465 1258 y Fr(N)p 1333 1299 346 4 v 1333 1382 a Fk(j)p Fv(\003)1421 1396 y Ft(1)1463 1382 y Fk(j)g(\000)g Fr(m)1705 1164 y Ff(")1758 1319 y Fv(1)h(+)1959 1206 y Fj(\024)1914 1233 y Ff(X)1922 1429 y Fj(l)p Ft(=2)2061 1191 y Ff(\022)2173 1258 y Fr(m)f Fv(+)g(1)p 2139 1299 306 4 v 2139 1382 a Fr(N)2212 1396 y Fi(\003)2274 1382 y Fk(\000)g Fr(m)2455 1191 y Ff(\023)2522 1210 y Fj(l)p Fi(\000)p Ft(1)2643 1164 y Ff(#)3477 1003 y Fv(\(2)p Fr(:)p Fv(62\))50 1567 y(Since)29 b(\()p Fr(m)21 b Fv(+)e(1\))p Fr(=)p Fv(\()p Fr(N)746 1581 y Fi(\003)810 1567 y Fk(\000)h Fr(m)p Fv(\))25 b Fr(<)g Fv(1)31 b(for)f(all)g(0)25 b Fk(\024)g Fr(m)g Fk(\024)g Fr(N)1919 1581 y Fi(\003)1961 1567 y Fr(=)p Fv(2)c Fk(\000)f Fv(1,)1242 1810 y Fr(B)1311 1824 y Fj(m)1405 1810 y Fk(\024)1780 1748 y Fr(N)10 b Fv(\()p Fr(N)1971 1762 y Fi(\003)2033 1748 y Fk(\000)20 b Fr(m)p Fv(\))p 1512 1789 995 4 v 1512 1872 a(\()p Fk(j)p Fv(\003)1635 1886 y Ft(1)1677 1872 y Fk(j)g(\000)g Fr(m)p Fv(\)\()p Fr(N)2036 1886 y Fi(\003)2099 1872 y Fk(\000)g Fv(2)p Fr(m)g Fk(\000)g Fv(1\))3477 1810 y(\(2)p Fr(:)p Fv(63\))50 2054 y(Inserting)29 b(\(2.63\))k(in)c(\(2.61\),)1083 2349 y Ff(e)1075 2373 y Fh(R)1135 2335 y Fi(\016)1135 2395 y Ft(\001)1195 2404 y Fm(\026)1263 2299 y Ff(\000)1305 2373 y Fr(\034)1355 2335 y Fj(!)1401 2344 y Fn(0)1345 2395 y Fj(!)1391 2404 y Fm(L)1468 2373 y Fr(<)c(\034)1614 2335 y Fj(!)1660 2344 y Fn(0)1604 2395 y Fj(!)1650 2404 y Fn(0)1700 2299 y Ff(\001)1767 2373 y Fk(\025)1874 2311 y(j)p Fv(\003)1962 2325 y Ft(1)2003 2311 y Fk(j)p 1874 2352 155 4 v 1910 2435 a Fr(N)2054 2190 y Ff(2)2054 2353 y(4)2115 2251 y Fj(N)2173 2260 y Fb(\003)2211 2251 y Fj(=)p Ft(2)p Fi(\000)p Ft(1)2180 2286 y Ff(X)2168 2479 y Fj(m)p Ft(=0)2392 2373 y Fr(C)2457 2387 y Fj(m)2526 2190 y Ff(3)2526 2353 y(5)2587 2209 y Fi(\000)p Ft(1)3477 2373 y Fv(\(2)p Fr(:)p Fv(64\))50 2667 y(where)890 2854 y Fr(C)955 2868 y Fj(m)1050 2854 y Fv(=)1145 2699 y Ff(")1256 2741 y Fj(\024)1219 2768 y Ff(Y)1198 2964 y Fj(k)1239 2945 y Fb(0)1263 2964 y Ft(=1)1370 2726 y Ff(\022)1437 2793 y Fk(j)p Fv(\003)1525 2807 y Fj(k)1566 2788 y Fb(0)1595 2793 y Fk(j)1489 2917 y Fr(m)1620 2726 y Ff(\023)1687 2699 y(#)1740 2718 y Fi(\000)p Ft(1)2096 2793 y Fv(\()p Fr(N)2204 2807 y Fi(\003)2266 2793 y Fk(\000)20 b Fr(m)p Fv(\))p Fk(j)p Fv(\003)2560 2807 y Ft(1)2601 2793 y Fk(j)p 1864 2833 995 4 v 1864 2917 a Fv(\()p Fr(N)1972 2931 y Fi(\003)2034 2917 y Fk(\000)g Fv(2)p Fr(m)h Fk(\000)f Fv(1\)\()p Fk(j)p Fv(\003)2565 2931 y Ft(1)2607 2917 y Fk(j)g(\000)g Fr(m)p Fv(\))3477 2854 y(\(2)p Fr(:)p Fv(65\))50 3096 y(Finally)-8 b(,)29 b(a)i(few)f(simple)e(computations)i (yield)f(the)h(b)s(ounds)1099 3282 y Fr(C)1164 3296 y Ft(0)1230 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Fj(y)2414 5482 y(y)2468 5471 y Fv(\))3477 5392 y(\(2)p Fr(:)p Fv(69\))p eop %%Page: 24 24 24 23 bop 50 295 a Fo(24)3270 b Fl(Se)l(ction)27 b Fo(2)50 513 y Fv(By)j(rev)m(ersibilit)m(y)e(the)j(n)m(umerator)f(of)h(\(2.69\)) i(ma)m(y)e(b)s(e)e(rewritten)h(as)1073 788 y Ff(e)1065 811 y Fh(R)1125 774 y Fi(\016)1125 834 y Ft(\001)1210 738 y Ff(\000)1252 811 y Fr(\034)1302 774 y Fj(y)1292 834 y(x)1371 811 y Fr(<)25 b(\034)1517 768 y Fj(y)1507 836 y(y)r Fi([)p Ft(0)1636 738 y Ff(\001)1703 811 y Fv(=)1820 726 y Ff(e)1810 750 y Fh(Q)1874 717 y Fi(\016)1874 775 y Ft(\001)1944 750 y Fv(\()p Fr(x)p Fv(\))p 1810 791 258 4 v 1822 867 a Ff(e)1812 891 y Fh(Q)1876 859 y Fi(\016)1876 917 y Ft(\001)1947 891 y Fv(\()p Fr(y)s Fv(\))2086 788 y Ff(e)2078 811 y Fh(R)2138 774 y Fi(\016)2138 834 y Ft(\001)2223 738 y Ff(\000)2265 811 y Fr(\034)2315 774 y Fj(x)2305 834 y(y)2386 811 y Fr(<)g(\034)2532 774 y Fj(x)2522 834 y(x)p Fi([)p Ft(0)2653 738 y Ff(\001)3477 811 y Fv(\(2)p Fr(:)p Fv(70\))50 1132 y(Th)m(us,)k(remem)m(b)s(ering)g (that)1058 1108 y Ff(e)1048 1132 y Fh(Q)1113 1099 y Fi(\016)1113 1157 y Ft(\001)1183 1132 y Fv(\()p Fr(x)1270 1099 y Fi(0)1296 1132 y Fv(\))d(=)f Fh(Q)1517 1099 y Fi(\016)1517 1157 y Fj(N)1592 1132 y Fv(\()p Fr(x)1679 1099 y Fi(0)1705 1132 y Fv(\))p Fr(=)p Fh(Q)1851 1099 y Fi(\016)1851 1157 y Fj(N)1926 1132 y Fv(\(\001\))30 b(w)m(e)h(ha)m(v)m(e,)h(b)m(y)e (\(2.27\),)614 1388 y Ff(e)607 1412 y Fh(R)666 1374 y Fi(\016)666 1434 y Ft(\001)752 1338 y Ff(\000)793 1412 y Fr(\034)843 1374 y Fj(y)833 1434 y(x)912 1412 y Fr(<)25 b(\034)1058 1368 y Fj(y)1048 1436 y(y)r Fi([)p Ft(0)1177 1338 y Ff(\001)1244 1412 y Fk(\024)1351 1350 y Fh(Q)1415 1317 y Fi(\016)1415 1375 y Fj(N)1490 1350 y Fv(\()p Fr(x)p Fv(\))p 1351 1391 263 4 v 1353 1474 a Fh(Q)1417 1443 y Fi(\016)1417 1501 y Fj(N)1492 1474 y Fv(\()p Fr(y)s Fv(\))1649 1412 y(=)1756 1350 y Fk(jf)p Fr(\033)k Fk(j)d Fr(\015)5 b Fv(\()p Fr(\033)s Fv(\))26 b(=)f Fr(x)p Fk(gj)p 1756 1391 624 4 v 1758 1474 a(jf)p Fr(\033)k Fk(j)d Fr(\015)5 b Fv(\()p Fr(\033)s Fv(\))26 b(=)f Fr(y)s Fk(gj)2416 1412 y Fv(=)2810 1350 y(1)p 2523 1391 620 4 v 2523 1474 a Fk(jf)p Fr(\033)k Fk(j)c Fr(\015)5 b Fv(\()p Fr(\033)s Fv(\))27 b(=)e Fr(y)s Fk(gj)3477 1412 y Fv(\(2)p Fr(:)p Fv(71\))50 1694 y(whic)m(h)k(b)m(y)h(\(2.18\),)j(for)d Fr(y)j Fv(de\014ned)c(in)h(\(2.40\),)i(giv)m(es:)835 2024 y Ff(e)828 2047 y Fh(R)888 2010 y Fi(\016)888 2070 y Ft(\001)973 1974 y Ff(\000)1015 2047 y Fr(\034)1065 2010 y Fj(y)1055 2070 y(x)1134 2047 y Fr(<)25 b(\034)1280 2004 y Fj(y)1270 2072 y(y)r Fi([)p Ft(0)1398 1974 y Ff(\001)1465 2047 y Fk(\024)1561 1865 y Ff(2)1561 2028 y(4)1683 1961 y(Y)1622 2160 y Fj(k)r Fi(2K)p Ft(\()p Fj(I)5 b Ft(\))1876 1919 y Ff(\022)1966 1986 y Fk(j)p Fv(\003)2054 2000 y Fj(k)2099 1986 y Fk(j)1942 2110 y Fr(N)2015 2124 y Fi(\003)2057 2110 y Fr(=)p Fv(2)2148 1919 y Ff(\023)2215 1865 y(3)2215 2028 y(5)2276 1884 y Fi(\000)p Ft(1)2399 2047 y Fk(\024)2495 1919 y Ff(\022)2607 1986 y Fr(N)2680 2000 y Fi(\003)2561 2110 y Fr(N)2634 2124 y Fi(\003)2676 2110 y Fr(=)p Fv(2)2767 1919 y Ff(\023)2834 1938 y Fi(\000)p Ft(1)3477 2047 y Fv(\(2)p Fr(:)p Fv(72\))50 2382 y(where)24 b(w)m(e)g(used)g(that)g (there)h(exists)f(at)h(least)f(one)h(index)e Fr(k)28 b Fk(2)d(K)q Fv(\()p Fr(I)7 b Fv(\))25 b(with)e(the)i(prop)s(ert)m(y)e (that)i Fk(j)p Fv(\003)3378 2396 y Fj(k)3423 2382 y Fk(j)h Fv(=)f Fr(N)3643 2396 y Fi(\003)3685 2382 y Fv(.)50 2527 y(Since)d(b)m(y)h(\(2.32\))j Fr(N)35 b Fk(\025)25 b Fr(N)932 2541 y Fi(\003)999 2527 y Fk(\025)g Fr(\017)1143 2492 y Fj(N)p 1143 2507 65 4 v 1156 2559 a(d)1218 2527 y Fv(,)g(Stirling's) 20 b(form)m(ula)j(enables)f(us)h(to)h(conclude)e(that,)k(for)d(large)g (enough)50 2673 y Fr(N)10 b Fv(,)1002 2802 y Ff(e)994 2825 y Fh(R)1054 2788 y Fi(\016)1054 2848 y Ft(\001)1140 2752 y Ff(\000)1181 2825 y Fr(\034)1231 2788 y Fj(y)1221 2848 y(x)1300 2825 y Fr(<)25 b(\034)1446 2782 y Fj(y)1436 2850 y(y)r Fi([)p Ft(0)1565 2752 y Ff(\001)1632 2825 y Fk(\024)g Fr(c)1767 2742 y Ff(p)p 1858 2742 115 4 v 83 x Fr(N)1931 2839 y Fi(\003)1973 2825 y Fv(2)2018 2788 y Fi(\000)p Fj(N)2133 2797 y Fb(\003)2200 2825 y Fk(\024)g Fr(c)2335 2744 y Fk(p)p 2411 2744 83 4 v 81 x Fr(N)10 b Fv(2)2539 2788 y Fi(\000)p Fj(\017N)q(=d)3477 2825 y Fv(\(2)p Fr(:)p Fv(73\))50 3042 y(for)30 b(some)h(constan)m(t)g Fr(c)26 b(>)f Fv(0.)159 3260 y(T)-8 b(o)35 b(b)s(ound)e(the)i (probabilit)m(y)d(app)s(earing)i(in)f(the)i(denominator)f(of)h (\(2.69\))i(w)m(e)e(again)g(resort)g(to)h(the)50 3405 y(path)28 b(tec)m(hnique)h(emplo)m(y)m(ed)g(in)f(the)h(pro)s(of)f(of)h (assertion)f(i\).)40 b(As)29 b(w)m(e)g(need)g(only)f(a)h(rough)g (estimate,)h(this)50 3551 y(probabilit)m(y)e(will)f(b)s(e)j(estimated)h (b)m(y)f(means)g(of)h(a)g(single)e(path,)35 b Ff(e)-56 b Fr(!)s Fv(.)41 b(Setting)1058 3771 y Fr(L)25 b Fv(=)g Fr(L)1303 3785 y Ft(0)1364 3771 y Fv(+)20 b Fk(\001)15 b(\001)g(\001)21 b Fv(+)f Fr(L)1734 3785 y Fj(d)1013 4056 y Fr(L)1075 4070 y Fj(k)1145 4056 y Fv(=)1241 3842 y Ff(8)1241 3924 y(>)1241 3951 y(<)1241 4115 y(>)1241 4142 y(:)1352 3908 y Fv(0)p Fr(;)768 b Fv(if)90 b Fr(k)28 b Fv(=)d(0)1363 4008 y Ft(1)p 1363 4023 37 4 v 1363 4075 a(2)1410 4044 y Fv(\()p Fk(j)p Fv(\003)1533 4058 y Fj(k)1578 4044 y Fv(\()p Fr(I)7 b Fv(\))p Fk(j)22 b(\000)e Fr(N)1906 4058 y Fi(\003)1947 4044 y Fv(\))p Fr(;)183 b Fv(if)90 b Fr(k)28 b Fk(2)d(K)q Fv(\()p Fr(I)7 b Fv(\))1363 4159 y Fi(j)p Ft(\003)1435 4169 y Fm(k)1474 4159 y Ft(\()p Fj(I)e Ft(\))p Fi(j)p 1363 4183 227 4 v 1458 4235 a Ft(2)1600 4204 y Fr(;)565 b Fv(if)90 b Fr(k)28 b Fk(2)d(K)q Fv(\()p Fr(I)7 b Fv(\))2682 4171 y Fj(c)3477 3983 y Fv(\(2)p Fr(:)p Fv(74\))54 4420 y Ff(e)-55 b Fr(!)28 b Fv(=)d(\()5 b Ff(e)-56 b Fr(!)323 4434 y Ft(0)364 4420 y Fr(;)15 b(:)g(:)g(:)h(;)k Ff(e)-56 b Fr(!)622 4434 y Fj(L)677 4420 y Fv(\))30 b(is)g(de\014ned)f(as)h(follo)m(ws)365 4727 y Ff(e)-55 b Fr(!)418 4741 y Fj(n)492 4727 y Fv(=)588 4572 y Ff(\()691 4647 y Fr(y)s(;)790 b Fv(if)90 b Fr(n)25 b Fv(=)g(0)696 4791 y Ff(e)-56 b Fr(!)748 4805 y Fj(n)p Fi(\000)p Ft(1)911 4791 y 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Fv(\000)1672 5214 y Fj(N)s(;d)1797 5200 y Fv(\()p Fr(I)7 b Fv(\))26 b Fk(j)f(9)2041 5214 y Ft(0)p Fi(\024)p Fj(n)p Fi(\024)p Fj(L)2330 5200 y Fv(:)41 b Fr(x)2448 5167 y Fi(0)2499 5200 y Fv(=)29 b Ff(e)-55 b Fr(!)2652 5214 y Fj(n)2701 5200 y Fk(g)p Fv(\),)31 b(w)m(e)g(ha)m(v)m(e)1662 5439 y Ff(e)1643 5462 y Fr(D)d Fk(\032)d Fr(D)j Fk(\032)d Fv(\001)1360 b(\(2)p Fr(:)p Fv(76\))p eop %%Page: 25 25 25 24 bop 50 295 a Fl(A)l(ging)27 b(in)h(the)g(REM.)f(Part)h(1.)2661 b Fo(25)50 513 y Fv(and)30 b(th)m(us,)g(b)m(y)g(Lemma)h(6.1,)921 727 y Ff(e)914 751 y Fh(R)973 713 y Fi(\016)973 773 y Ft(\001)1059 677 y Ff(\000)1100 751 y Fr(\034)1150 707 y Fj(y)1140 775 y(x)p Fi([)p Ft(0)1296 751 y Fr(<)25 b(\034)1442 713 y Fj(y)1432 773 y(y)1486 677 y Ff(\001)1553 751 y Fk(\025)1657 727 y Ff(e)1649 751 y Fh(R)1709 713 y Fi(\016)1709 773 y Ft(\001)1794 677 y Ff(\000)1836 751 y Fr(\034)1886 707 y Fj(y)1876 775 y Ft(0)1955 751 y Fr(<)g(\034)2101 713 y Fj(y)2091 773 y(y)2144 677 y Ff(\001)2211 751 y 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y Fr(\034)629 2618 y Ff(e)-50 b Fj(!)676 2627 y Fn(0)619 2707 y Ff(e)g Fj(!)666 2716 y Fm(L)744 2659 y Fr(<)25 b(\034)889 2618 y Ff(e)-50 b Fj(!)936 2627 y Fn(0)879 2707 y Ff(e)g Fj(!)926 2716 y Fn(0)975 2558 y Ff(\021)1055 2659 y Fk(\025)1151 2503 y Ff(")1251 2545 y Fj(d)1204 2572 y Ff(X)1204 2768 y Fj(k)r Ft(=1)1351 2543 y Fj(L)1401 2553 y Fm(k)1441 2543 y Fi(\000)p Ft(1)1377 2572 y Ff(X)1364 2765 y Fj(m)p Ft(=0)1560 2597 y Fk(j)p Fv(\003)1648 2611 y Fj(k)1693 2597 y Fv(\()p Fr(I)7 b Fv(\))p Fk(j)p 1560 2638 V 1634 2721 a Fv(2)p Fr(N)1847 2503 y Ff(#)1900 2513 y Fi(\000)p Ft(1)2023 2659 y Fv(=)2119 2503 y Ff(")2219 2545 y Fj(d)2173 2572 y Ff(X)2172 2768 y Fj(k)r Ft(=1)2319 2659 y Fr(L)2381 2673 y Fj(k)2437 2597 y Fk(j)p Fv(\003)2525 2611 y Fj(k)2570 2597 y Fv(\()p Fr(I)g Fv(\))p Fk(j)p 2437 2638 V 2511 2721 a Fv(2)p Fr(N)2724 2503 y Ff(#)2777 2514 y Fi(\000)p Ft(1)2900 2659 y Fk(\025)3049 2597 y Fv(1)p 3007 2638 131 4 v 3007 2721 a Fr(N)j(d)3477 2659 y Fv(\(2)p Fr(:)p Fv(80\))50 2947 y(where)30 b(the)g(last)g(inequalit)m(y)f(follo)m(ws)h (from)g(\(2.74\).)43 b(Putting)29 b(\(2.80\))k(bac)m(k)e(in)e(\(2.77\)) k(\014nally)28 b(giv)m(es)1370 3161 y Ff(e)1363 3185 y Fh(R)1423 3147 y Fi(\016)1423 3207 y Ft(\001)1508 3111 y Ff(\000)1550 3185 y Fr(\034)1600 3141 y Fj(y)1590 3209 y(x)p Fi([)p Ft(0)1746 3185 y Fr(<)d(\034)1892 3147 y Fj(y)1882 3207 y(y)1935 3111 y Ff(\001)2002 3185 y Fk(\025)g Fv(\()p Fr(N)10 b(d)p Fv(\))2298 3147 y Fi(\000)p Ft(1)3477 3185 y Fv(\(2)p Fr(:)p Fv(81\))50 3423 y(Inserting)27 b(\(2.81\))j(and)e(\(2.73\))i(in)d(\(2.69\))j(and)e(plugging)e(the)i (resulting)f(b)s(ound)f(in)h(\(2.68\))j(yields)c(\(2.50\).)50 3568 y(The)31 b(second)h(assertion)f(of)h(Lemma)f(2.10)j(b)s(eing)c (pro)m(v)m(en,)i(this)f(concludes)g(the)h(pro)s(of)e(of)i(Lemma)g (2.10.)50 3714 y Fk(})159 3932 y Fv(Inserting)d(the)i(b)s(ounds)d(of)i (Lemma)h(2.10)h(in)d(\(2.47\))j(w)m(e)f(obtain)309 4215 y Fh(R)368 4178 y Fi(\016)431 4215 y Fv(\()q Fr(\034)517 4178 y Fj(x)507 4238 y Ft(0)588 4215 y Fr(<)25 b(\034)734 4178 y Fj(x)724 4238 y(x)779 4215 y Fv(\))h Fk(\025)911 4060 y Ff( )983 4215 y Fv(1)20 b Fk(\000)1169 4154 y Fv(1)p 1150 4195 83 4 v 1150 4278 a Fr(N)1332 4129 y Ff(X)1259 4328 y Fj(\026)p Fi(2K)p Ft(\()p Fj(I)5 b Ft(\))1500 4310 y Fm(c)1551 4215 y Fk(j)p Fv(\003)1639 4229 y Fj(\026)1688 4215 y Fk(j)1713 4060 y Ff(!)1801 4087 y(\022)1868 4215 y Fv(1)20 b Fk(\000)2070 4154 y Fv(1)p 2035 4195 115 4 v 2035 4278 a Fr(N)2108 4292 y Fi(\003)2181 4215 y Fk(\000)2325 4154 y Fr(c)p 2283 4195 124 4 v 2283 4278 a(N)2366 4252 y Ft(2)2356 4300 y Fi(\003)2417 4087 y Ff(\023)2500 4115 y(\020)2554 4215 y Fv(1)h Fk(\000)e Fr(c)2749 4178 y Fi(0)2775 4215 y Fr(N)10 b Fv(2)2903 4178 y Fi(\000)p Fj(\017N)q(=d)3129 4215 y Fv(\))3164 4115 y Ff(\021)840 4549 y Fk(\025)911 4393 y Ff( )983 4549 y Fv(1)20 b Fk(\000)1169 4487 y Fv(1)p 1150 4528 83 4 v 1150 4611 a Fr(N)1332 4462 y Ff(X)1259 4662 y Fj(\026)p Fi(2K)p Ft(\()p Fj(I)5 b Ft(\))1500 4643 y Fm(c)1551 4549 y Fk(j)p 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Fv(\000)1729 5472 y Fj(N)s(;d)1853 5458 y Fv(\()p Fr(I)7 b Fv(\))26 b Fk(j)f(k)p Fr(\031)s(x)c Fk(\000)f Fr(\031)s(y)2413 5421 y Fi(0)2438 5458 y Fk(k)2483 5472 y Ft(2)2550 5458 y Fr(>)25 b(\032)p Fk(g)739 b Fv(\(2)p Fr(:)p Fv(83\))p eop %%Page: 26 26 26 25 bop 50 295 a Fo(26)3270 b Fl(Se)l(ction)27 b Fo(2)50 513 y Fv(By)j(h)m(yp)s(othesis,)1651 658 y Fr(y)e Fk(2)d Fv(\000)1867 614 y Fj(x;\016)1867 686 y(N)s(;d)1991 658 y Fv(\()p Fr(I)7 b Fv(\))1369 b(\(2)p Fr(:)p Fv(84\))50 860 y(Observ)m(e)30 b(moreo)m(v)m(er)i(that)f(either)f Fr(y)j Fv(satis\014es)53 1078 y(i\))45 b Fk(8)p Fr(z)29 b Fk(2)c(f\000)p Fv(1)p Fr(;)15 b Fv(1)p Fk(g)658 1045 y Fj(d)722 1078 y Fk(\\)20 b Fv(\000)860 1031 y Fj(x;\016)r(=)p Ft(2)860 1106 y Fj(N)s(;d)1036 1078 y Fv(\()p Fr(I)7 b Fv(\),)31 b Fk(k)p Fr(\031)s(z)26 b Fk(\000)19 b Fr(\031)s(y)s Fk(k)1615 1092 y Ft(2)1682 1078 y Fr(>)25 b(\016)s(=)p Fv(2)159 1297 y(or)30 b(else)28 1515 y(ii\))44 b Fk(9)p Fr(z)29 b Fk(2)c(f\000)p Fv(1)p Fr(;)15 b Fv(1)p Fk(g)658 1482 y Fj(d)722 1515 y Fk(\\)20 b Fv(\000)860 1471 y Fj(x;\016)860 1542 y(N)s(;d)984 1515 y Fv(\()p Fr(I)7 b Fv(\))32 b(suc)m(h)e(that)h Fk(k)p Fr(\031)s(z)25 b Fk(\000)19 b Fr(\031)s(y)s Fk(k)1940 1529 y Ft(2)2007 1515 y Fk(\024)25 b Fr(\016)s(=)p Fv(2.)159 1733 y(W)-8 b(e)29 b(will)d(\014rst)h(sho)m(w)i(that)f(in)f(case)j(i\),)e(\(2.35\)) i(is)e(a)g(direct)g(consequence)h(of)f(rev)m(ersibilit)m(y)-8 b(.)38 b(Indeed,)28 b(as)50 1878 y(in)h(\(2.69\),)1263 2060 y Fh(R)1322 2022 y Fi(\016)1385 2060 y Fv(\()p Fr(\034)1470 2022 y Fj(y)1460 2082 y(x)1539 2060 y Fr(<)c(\034)1685 2016 y Fj(y)1675 2084 y Ft(0)1729 2060 y Fv(\))h(=)1897 1987 y Fh(R)1956 1954 y Fi(\016)2019 1914 y Ff(\000)2061 1987 y Fr(\034)2111 1954 y Fj(y)2101 2010 y(x)2180 1987 y Fr(<)f(\034)2326 1944 y Fj(y)2316 2012 y(y)r Fi([)p Ft(0)2445 1914 y Ff(\001)p 1897 2039 590 4 v 1902 2122 a Fh(R)1962 2096 y Fi(\016)2025 2122 y Fv(\()p Fr(\034)2110 2078 y Fj(y)2100 2146 y(x)p Fi([)p Ft(0)2256 2122 y Fr(<)g(\034)2402 2078 y Fj(y)2392 2133 y(y)2446 2122 y Fv(\))3477 2060 y(\(2)p Fr(:)p Fv(85\))50 2280 y(A)h(straigh)m(tforw)m(ard)g (adaptation)h(of)f(the)h(pro)s(of)e(of)i(the)f(b)s(ound)e(\(2.81\))29 b(to)e(the)g(case)g(at)g(hand)e(sho)m(ws)h(that)50 2425 y(the)k(denominator)g(of)g(\(2.85\))j(ob)s(eys)d(the)h(b)s(ound)1439 2664 y Fh(R)1499 2627 y Fi(\016)1562 2590 y Ff(\000)1603 2664 y Fr(\034)1653 2620 y Fj(y)1643 2688 y(x)p Fi([)p Ft(0)1799 2664 y Fr(<)25 b(\034)1945 2627 y Fj(y)1935 2687 y(y)1989 2590 y Ff(\001)2056 2664 y Fk(\025)g Fv(1)p Fr(=)-5 b(N)1167 b Fv(\(2)p Fr(:)p Fv(86\))50 2903 y(while)28 b(b)m(y)i(rev)m(ersibilit)m(y)e(its)i(n)m(umerator)h(ma)m(y)g(b)s(e)e (rewritten)h(as)1099 3152 y Fh(R)1159 3115 y Fi(\016)1222 3079 y Ff(\000)1263 3152 y Fr(\034)1313 3115 y Fj(y)1303 3175 y(x)1382 3152 y Fr(<)25 b(\034)1528 3109 y Fj(y)1518 3177 y(y)r Fi([)p Ft(0)1647 3079 y Ff(\001)1714 3152 y Fv(=)1821 3091 y Fh(Q)1885 3058 y Fi(\016)1933 3091 y Fv(\()p Fr(x)p Fv(\))p 1821 3131 236 4 v 1823 3215 a Fh(Q)1888 3188 y Fi(\016)1935 3215 y Fv(\()p Fr(y)s Fv(\))2067 3152 y Fh(R)2126 3115 y Fi(\016)2189 3079 y Ff(\000)2231 3152 y Fr(\034)2281 3115 y Fj(x)2271 3175 y(y)2352 3152 y Fr(<)g(\034)2498 3115 y Fj(x)2488 3175 y(x)p Fi([)p Ft(0)2619 3079 y Ff(\001)3477 3152 y Fv(\(2)p Fr(:)p Fv(87\))50 3408 y(Th)m(us,)k(b)m(y)i(\(2.27\),)614 3639 y Ff(e)607 3663 y Fh(R)666 3625 y Fi(\016)666 3685 y Ft(\001)752 3589 y Ff(\000)793 3663 y Fr(\034)843 3625 y Fj(y)833 3685 y(x)912 3663 y Fr(<)25 b(\034)1058 3619 y Fj(y)1048 3687 y(y)r Fi([)p Ft(0)1177 3589 y Ff(\001)1244 3663 y Fk(\024)1351 3601 y Fh(Q)1415 3568 y Fi(\016)1415 3626 y Fj(N)1490 3601 y Fv(\()p Fr(x)p Fv(\))p 1351 3642 263 4 v 1353 3725 a Fh(Q)1417 3694 y Fi(\016)1417 3752 y Fj(N)1492 3725 y Fv(\()p Fr(y)s Fv(\))1649 3663 y(=)1756 3601 y Fk(jf)p Fr(\033)k Fk(j)d Fr(\015)5 b Fv(\()p Fr(\033)s Fv(\))26 b(=)f Fr(x)p Fk(gj)p 1756 3642 624 4 v 1758 3725 a(jf)p Fr(\033)k Fk(j)d Fr(\015)5 b Fv(\()p Fr(\033)s Fv(\))26 b(=)f Fr(y)s Fk(gj)2416 3663 y Fv(=)2810 3601 y(1)p 2523 3642 620 4 v 2523 3725 a Fk(jf)p Fr(\033)k Fk(j)c Fr(\015)5 b Fv(\()p Fr(\033)s Fv(\))27 b(=)e Fr(y)s Fk(gj)3477 3663 y Fv(\(2)p Fr(:)p Fv(88\))50 3922 y(where)34 b(the)h(last)g(equalit)m(y)f(follo)m(ws)g(from)g(the)h(fact)h(that)f Fr(x)e Fk(2)f Fr(\015)2298 3936 y Fj(I)2339 3922 y Fv(\()p Fr(I)7 b Fv(\))36 b(\(see)g(Lemma)e(2.2\).)56 b(T)-8 b(o)35 b(estimate)50 4068 y(the)30 b(last)h(ratio)f(note)h(that)g (condition)e(i\))h(com)m(bined)f(with)h(\(2.84\))i(implies)c(that)1480 4306 y(inf)1346 4369 y Fj(z)1384 4351 y Fb(0)1408 4369 y Fi(2f\000)p Ft(1)p Fj(;)p Ft(1)p Fi(g)1682 4351 y Fm(\024)1740 4306 y Fk(k)p Fr(z)1831 4269 y Fi(0)1877 4306 y Fk(\000)20 b Fr(\031)s(y)s Fk(k)2116 4320 y Ft(2)2183 4306 y Fr(>)25 b(\016)s(=)p Fv(2)1065 b(\(2)p Fr(:)p Fv(89\))50 4569 y(whic)m(h)31 b(in)g(turn)g(implies)e(that)k(there)f(exists)g Fr(k)1670 4536 y Fi(0)1724 4569 y Fk(2)c(K)q Fv(\()p Fr(I)7 b Fv(\))33 b(suc)m(h)f(that)h(inf)2550 4583 y Fj(s)p Ft(=)p Fi(\006)p Ft(1)2752 4569 y Fk(j)p Fr(s)22 b Fk(\000)f Fr(y)2979 4583 y Fj(k)3020 4565 y Fb(0)3048 4569 y Fk(j)28 b Fr(>)g(\016)s(=)p Fv(2)3333 4504 y Fk(p)p 3411 4504 53 4 v 3411 4569 a Fr(\024)p Fv(,)33 b(or)f(in)50 4715 y(other)d(w)m(ords,)f(suc)m(h)h(that)g Fk(j)p Fr(y)1039 4729 y Fj(k)1080 4710 y Fb(0)1108 4715 y Fk(j)d Fr(<)f Fv(1)17 b Fk(\000)f Fr(\016)s(=)p Fv(2)1537 4649 y Fk(p)p 1615 4649 V 1615 4715 a Fr(\024)p Fv(.)40 b(Th)m(us,)28 b(making)g(use)h(of)f(\(2.18\))j(and)d(Stirling's)e(form)m(ula,)867 4993 y Fk(jf)p Fr(\033)j Fk(j)d Fr(\015)5 b Fv(\()p Fr(\033)s Fv(\))26 b(=)f Fr(y)s Fk(gj)1486 4955 y Fi(\000)p Ft(1)1610 4993 y Fk(\024)1706 4865 y Ff(\022)1880 4926 y Fk(j)p Fv(\003)1968 4940 y Fj(k)2009 4922 y Fb(0)2038 4926 y Fv(\()p Fr(I)7 b Fv(\))p Fk(j)1772 5061 y(j)p Fv(\003)1860 5075 y Fj(k)1901 5056 y Fb(0)1930 5061 y Fv(\()p Fr(I)g Fv(\))p Fk(j)2083 5016 y Ft(1+)p Fj(y)2212 5032 y Fm(k)2247 5019 y Fb(0)p 2084 5040 194 4 v 2163 5092 a Ft(2)2288 4865 y Ff(\023)2355 4881 y Fi(\000)p Ft(1)1610 5206 y Fk(\024)25 b Fr(c)15 b Fv(exp)g Fk(f)q(\000j)p Fv(\003)2119 5220 y Fj(k)2160 5201 y Fb(0)2188 5206 y Fv(\()p Fr(I)7 b Fv(\))p Fk(jI)g Fv(\()p Fr(y)2467 5220 y Fj(k)2508 5201 y Fb(0)2536 5206 y Fv(\))p Fk(g)1610 5406 y(\024)25 b Fr(c)15 b Fv(exp)1914 5277 y Ff(\032)1982 5406 y Fk(\000)p Fr(N)2126 5420 y Fi(\003)2351 5406 y Fv(inf)2183 5473 y Fi(j)p Fj(u)p Fi(j)p Fj(<)p Ft(1)p Fi(\000)p Fj(\016)r(=)p Ft(2)2528 5427 y Fi(p)p 2588 5427 43 4 v 46 x Fj(\024)2645 5406 y Fk(I)7 b Fv(\()p Fr(u)p Fv(\))2824 5277 y Ff(\033)3477 5192 y Fv(\(2)p Fr(:)p Fv(90\))p eop %%Page: 27 27 27 26 bop 50 295 a Fl(A)l(ging)27 b(in)h(the)g(REM.)f(Part)h(1.)2661 b Fo(27)50 513 y Fv(for)30 b(some)h(constan)m(t)g Fr(c)26 b(>)f Fv(0,)31 b(with)e Fr(N)1325 527 y Fi(\003)1397 513 y Fv(de\014ned)g(as)i(in)e(\(2.33\),)k(and)d(where)612 769 y Fk(\000I)7 b Fv(\()p Fr(u)p Fv(\))24 b(=)982 640 y Ff(\022)1060 707 y Fv(1)d Fk(\000)f Fr(u)p 1060 748 209 4 v 1142 831 a Fv(2)1280 640 y Ff(\023)1362 769 y Fv(ln)1453 640 y Ff(\022)1531 707 y Fv(1)g Fk(\000)g Fr(u)p 1531 748 V 1612 831 a Fv(2)1750 640 y Ff(\023)1837 769 y Fv(+)1928 640 y Ff(\022)2006 707 y Fv(1)h(+)f Fr(u)p 2006 748 V 2088 831 a Fv(2)2225 640 y Ff(\023)2308 769 y Fv(ln)2398 640 y Ff(\022)2476 707 y Fv(1)h(+)f Fr(u)p 2476 748 V 2558 831 a Fv(2)2696 640 y Ff(\023)2778 769 y Fr(;)76 b Fk(j)p Fr(u)p Fk(j)26 b(\024)f Fv(1)329 b(\(2)p Fr(:)p Fv(91\))50 1020 y(Collecting)29 b(all)g(our)h(b)s(ounds)e(w)m(e) j(arriv)m(e)f(at)933 1271 y Fh(R)993 1233 y Fi(\016)1056 1271 y Fv(\()p Fr(\034)1141 1233 y Fj(y)1131 1293 y(x)1210 1271 y Fr(<)25 b(\034)1356 1227 y Fj(y)1346 1295 y Ft(0)1400 1271 y Fv(\))h Fk(\024)e Fr(cN)i Fv(exp)1848 1142 y Ff(\032)1916 1271 y Fk(\000)p Fr(N)2060 1285 y Fi(\003)2285 1271 y Fv(inf)2117 1338 y Fi(j)p Fj(u)p Fi(j)p Fj(<)p Ft(1)p Fi(\000)p Fj(\016)r(=)p Ft(2)2462 1292 y Fi(p)p 2521 1292 43 4 v 2521 1338 a Fj(\024)2579 1271 y Fk(I)7 b Fv(\()p Fr(u)p Fv(\))2758 1142 y Ff(\033)3477 1271 y Fv(\(2)p Fr(:)p Fv(92\))50 1533 y(As)32 b Fr(\016)f(>)d Fv(0,)33 b(inf)569 1554 y Fi(j)p Fj(u)p Fi(j)p Fj(<)p Ft(1)p Fi(\000)p Fj(\016)r(=)p Ft(2)914 1508 y Fi(p)p 974 1508 V 46 x Fj(\024)1036 1533 y Fk(I)7 b Fv(\()p Fr(u)p Fv(\))28 b Fr(>)f Fv(0.)46 b(Cho)s(osing)30 b Fr(h)p Fv(\()p Fr(\016)n(;)15 b(\024)p Fv(\))31 b(=)c(inf)2347 1554 y Fi(j)p Fj(u)p Fi(j)p Fj(<)p Ft(1)p Fi(\000)p Fj(\016)r(=)p Ft(2)2692 1508 y Fi(p)p 2751 1508 V 2751 1554 a Fj(\024)2813 1533 y Fk(I)7 b Fv(\()p Fr(u)p Fv(\))p Fr(=)p Fv(2,)34 b(it)d(then)h(follo)m(ws)50 1678 y(from)e(\(2.92\))i(that,)g(for)e(all) f Fr(N)40 b Fv(large)31 b(enough,)1226 1916 y Fh(R)1285 1878 y Fi(\016)1348 1916 y Fv(\()q Fr(\034)1434 1878 y Fj(y)1424 1938 y(x)1503 1916 y Fr(<)25 b(\034)1649 1872 y Fj(y)1639 1940 y Ft(0)1692 1916 y Fv(\))h Fk(\024)f Fv(exp)15 b Fk(f\000)p Fr(N)2192 1930 y Fi(\003)2234 1916 y Fr(h)p Fv(\()p Fr(\016)n(;)g(\024)p Fv(\))p Fk(g)946 b Fv(\(2)p Fr(:)p Fv(93\))50 2154 y(Th)m(us,)27 b(under)e(the)i (assumption)e(made)i(in)f(i\),)h(\(2.35\))j(is)25 b(pro)m(v)m(en.)40 b(Let)28 b(us)e(turn)g(to)h(the)h(case)g(ii\).)38 b(Observ)m(e)50 2299 y(that)k(for)f Fr(\016)47 b(<)c Fv(1)p Fr(=)p Fv(2)f(there)g (exists)f(a)g(unique)e(p)s(oin)m(t)i Fr(z)47 b Fk(2)c(f\000)p Fv(1)p Fr(;)15 b Fv(1)p Fk(g)2417 2266 y Fj(\024)2507 2299 y Fv(suc)m(h)41 b(that)h Fk(k)p Fr(\031)s(z)32 b Fk(\000)27 b Fr(\031)s(y)s Fk(k)3351 2313 y Ft(2)3436 2299 y Fk(\024)43 b Fr(\016)s(=)p Fv(2.)50 2445 y(Calling)31 b Fr(z)412 2412 y Fi(\003)487 2445 y Fv(this)i(p)s(oin)m(t)f(and)h(in)m (tro)s(ducing)e(the)j(discrete)f(h)m(yp)s(er-surface)g Fr(H)2714 2461 y Fj(\016)r(=)p Ft(2)2826 2445 y Fv(\()p Fr(z)2907 2412 y Fi(\003)2950 2445 y Fv(\))e(=)f Fk(f)p Fr(z)3208 2412 y Fi(0)3264 2445 y Fk(2)g Fv(\000)3412 2459 y Fj(N)s(;d)3536 2445 y Fv(\()p Fr(I)7 b Fv(\))32 b Fk(j)50 2590 y(k)p Fr(z)141 2557 y Fi(\003)204 2590 y Fk(\000)19 b Fr(\031)s(z)395 2557 y Fi(0)421 2590 y Fk(k)466 2604 y Ft(2)533 2590 y Fv(=)25 b Fr(\016)s(=)p Fv(2)p Fk(g)p Fv(,)33 b(w)m(e)d(ha)m(v)m(e)662 2828 y Fh(R)721 2790 y Fi(\016)784 2828 y Fv(\()q Fr(\034)870 2790 y Fj(y)860 2850 y(x)939 2828 y Fr(<)24 b(\034)1084 2784 y Fj(y)1074 2852 y Ft(0)1128 2828 y Fv(\))i(=)1421 2742 y Ff(X)1285 2941 y Fj(z)1323 2923 y Fb(0)1347 2941 y Fi(2)p Fj(H)1456 2953 y Fm(\016)q(=)p Fn(2)1555 2941 y Ft(\()p Fj(z)1621 2923 y Fb(\003)1659 2941 y Ft(\))1703 2828 y Fh(R)1762 2790 y Fi(\016)1825 2727 y Ff(\020)1880 2828 y Fr(\034)1930 2784 y Fj(y)1920 2855 y(z)1958 2836 y Fb(0)2011 2828 y Fr(<)f(\034)2157 2784 y Fj(y)2147 2859 y(H)2207 2871 y Fm(\016)q(=)p Fn(2)2306 2859 y Ft(\()p Fj(z)2372 2841 y Fb(\003)2410 2859 y Ft(\))2443 2727 y Ff(\021)2513 2828 y Fh(R)2572 2790 y Fi(\016)2635 2727 y Ff(\020)2690 2828 y Fr(\034)2740 2790 y Fj(z)2778 2763 y Fb(0)2730 2850 y Fj(x)2831 2828 y Fr(<)g(\034)2977 2790 y Fj(z)3015 2763 y Fb(0)2967 2850 y Ft(0)3044 2727 y Ff(\021)3477 2828 y Fv(\(2)p Fr(:)p Fv(94\))50 3143 y(No)m(w)38 b(all)f(p)s(oin)m(ts)f Fr(z)724 3110 y Fi(0)787 3143 y Fk(2)h Fr(H)961 3160 y Fj(\016)r(=)p Ft(2)1074 3143 y Fv(\()p Fr(z)1155 3110 y Fi(\003)1198 3143 y Fv(\))h(ha)m(v)m(e) g(the)g(follo)m(wing)e(prop)s(erties:)54 b(\014rstly)-8 b(,)39 b(as)f(is)e(ob)m(vious)h(from)h(the)50 3289 y(de\014nition)j(of) j Fr(H)660 3305 y Fj(\016)r(=)p Ft(2)773 3289 y Fv(\()p Fr(z)854 3256 y Fi(\003)896 3289 y Fv(\),)k Fk(k)p Fr(\031)s(z)33 b Fk(\000)c Fr(\031)s(z)1380 3256 y Fi(0)1406 3289 y Fk(k)1451 3303 y Ft(2)1539 3289 y Fr(>)47 b(\016)s(=)p Fv(4)f(for)d(all)f Fr(z)52 b Fk(2)47 b(f\000)p Fv(1)p Fr(;)15 b Fv(1)p Fk(g)2620 3256 y Fj(d)2693 3289 y Fk(\\)29 b Fv(\000)2840 3245 y Fj(x;\016)2840 3316 y(N)s(;d)2964 3289 y Fv(\()p Fr(I)7 b Fv(\),)48 b(implying)40 b(that)50 3434 y(assumption)25 b(i\))h(is)g(satis\014ed)f(with)h Fr(\016)k Fv(replaced)c(b)m(y)h Fr(\016)s(=)p Fv(2;)i(secondly)-8 b(,)28 b(since)e Fr(z)2633 3401 y Fi(\003)2700 3434 y Fk(2)f Fv(\000)2843 3390 y Fj(x;\016)2843 3462 y(N)s(;d)2967 3434 y Fv(\()p Fr(I)7 b Fv(\))28 b(b)m(y)e(assumption,)50 3580 y(then)518 3725 y Fr(\016)j Fk(\024)c(k)p Fr(\031)s(x)c Fk(\000)f Fr(\031)s(z)1048 3688 y Fi(\003)1090 3725 y Fk(k)1135 3739 y Ft(2)1201 3725 y Fk(\024)25 b(k)p Fr(\031)s(x)c Fk(\000)f Fr(\031)s(z)1662 3688 y Fi(0)1688 3725 y Fk(k)1733 3739 y Ft(2)1794 3725 y Fv(+)g Fk(k)p Fr(\031)s(z)2031 3688 y Fi(0)2078 3725 y Fk(\000)f Fr(\031)s(z)2269 3688 y Fi(\003)2312 3725 y Fk(k)2357 3739 y Ft(2)2423 3725 y Fk(\024)25 b(k)p Fr(\031)s(x)c Fk(\000)f Fr(\031)s(z)2884 3688 y Fi(0)2910 3725 y Fk(k)2955 3739 y Ft(2)3016 3725 y Fv(+)g Fr(\016)s(=)p Fv(2)237 b(\(2)p Fr(:)p Fv(95\))50 3927 y(implying)27 b(that)k Fk(k)p Fr(\031)s(x)21 b Fk(\000)f Fr(\031)s(z)988 3894 y Fi(0)1014 3927 y Fk(k)1059 3941 y Ft(2)1125 3927 y Fk(\025)25 b Fr(\016)s(=)p Fv(2,)32 b(i.e.,)f(that)g Fr(z)1826 3894 y Fi(0)1877 3927 y Fk(2)25 b Fv(\000)2020 3879 y Fj(x;\016)r(=)p Ft(2)2020 3954 y Fj(N)s(;d)2196 3927 y Fv(\()p Fr(I)7 b Fv(\).)159 4153 y(As)28 b(a)g(result,)g(for)g(eac)m(h)h Fr(z)1026 4120 y Fi(0)1077 4153 y Fk(2)c Fr(H)1239 4169 y Fj(\016)r(=)p Ft(2)1352 4153 y Fv(\()p Fr(z)1433 4120 y Fi(\003)1475 4153 y Fv(\),)k(the)g(probabilit)m(y)c Fh(R)2241 4120 y Fi(\016)2304 4052 y Ff(\020)2358 4153 y Fr(\034)2408 4120 y Fj(z)2446 4092 y Fb(0)2398 4175 y Fj(x)2500 4153 y Fr(<)g(\034)2646 4120 y Fj(z)2684 4092 y Fb(0)2636 4175 y Ft(0)2712 4052 y Ff(\021)2795 4153 y Fv(ob)s(eys)i(the)i(b)s (ound)c(\(2.92\))50 4298 y(with)k Fr(\016)34 b Fv(replaced)c(b)m(y)g Fr(\016)s(=)p Fv(2.)43 b(It)30 b(therefore)h(follo)m(ws)e(from)h (\(2.94\))j(that)81 4536 y Fh(R)141 4499 y Fi(\016)204 4536 y Fv(\()p Fr(\034)289 4499 y Fj(y)279 4559 y(x)358 4536 y Fr(<)25 b(\034)504 4492 y Fj(y)494 4560 y Ft(0)548 4536 y Fv(\))h Fk(\024)f Fv(exp)15 b Fk(f\000)p Fr(N)1048 4550 y Fi(\003)1090 4536 y Fr(h)p Fv(\()p Fr(\016)s(=)p Fv(2)p Fr(;)g(\024)p Fv(\))p Fk(g)1652 4450 y Ff(X)1501 4649 y Fj(z)1539 4631 y Fb(0)1563 4649 y Fi(2)p Fj(@)t Fi(B)1703 4661 y Fm(\016)q(=)p Fn(2)1803 4649 y Ft(\()p Fj(z)1869 4631 y Fb(\003)1906 4649 y Ft(\))1950 4536 y Fh(R)2010 4499 y Fi(\016)2073 4435 y Ff(\020)2127 4536 y Fr(\034)2177 4492 y Fj(y)2167 4563 y(z)2205 4545 y Fb(0)2258 4536 y Fr(<)25 b(\034)2404 4492 y Fj(y)2394 4567 y(@)t Fi(B)2485 4579 y Fm(\016)q(=)p Fn(2)2585 4567 y Ft(\()p Fj(z)2651 4549 y Fb(\003)2689 4567 y Ft(\))2722 4435 y Ff(\021)2801 4536 y Fk(\024)g Fv(exp)15 b Fk(f)q(\000)p Fr(N)3241 4550 y Fi(\003)3282 4536 y Fr(h)p Fv(\()p Fr(\016)s(=)p Fv(2)p Fr(;)g(\024)p Fv(\))p Fk(g)3477 4759 y Fv(\(2)p Fr(:)p Fv(96\))50 4904 y(This)28 b(concludes)i(the)h(pro)s(of)e(of)i (Lemma)f(2.7.)p Fk(})50 5123 y Fa(Pro)s(of)36 b(of)f(Lemma)d(2.8:)88 b Fv(Again)30 b(using)f(renew)m(al)h(as)h(in)e(\(2.69\),)764 5391 y Fh(R)823 5354 y Fi(\016)886 5318 y Ff(\000)928 5391 y Fr(\034)978 5354 y Ft(0)968 5414 y Fj(x)1044 5391 y Fk(\024)c Fr(\034)1190 5354 y Ft(0)1180 5414 y Fj(J)1231 5318 y Ff(\001)1298 5391 y Fv(=)1405 5324 y Fh(R)1464 5291 y Fi(\016)1527 5250 y Ff(\000)1569 5324 y Fr(\034)1619 5291 y Ft(0)1609 5346 y Fj(x)1685 5324 y Fk(\024)g Fr(\034)1831 5291 y Ft(0)1821 5349 y Fj(J)6 b Fi([)p Ft(0)1957 5250 y Ff(\001)p 1405 5370 594 4 v 1453 5454 a Fh(R)1513 5427 y Fi(\016)1576 5454 y Fv(\()p Fr(\034)1661 5422 y Ft(0)1651 5480 y Fj(J)1727 5454 y Fr(<)25 b(\034)1873 5422 y Ft(0)1863 5478 y(0)1914 5454 y Fv(\))2034 5391 y(=)2266 5324 y Fh(R)2326 5291 y Fi(\016)2389 5250 y Ff(\000)2431 5324 y Fr(\034)2481 5291 y Ft(0)2471 5346 y Fj(x)2547 5324 y Fk(\024)g Fr(\034)2693 5291 y Ft(0)2683 5349 y Fj(J)6 b Fi([)p Ft(0)2818 5250 y Ff(\001)p 2141 5370 844 4 v 2141 5390 a(P)2237 5486 y Fj(y)r Fi(2)p Fj(J)2391 5458 y Fh(R)2451 5432 y Fi(\016)2514 5385 y Ff(\000)2556 5458 y Fr(\034)2606 5432 y Ft(0)2596 5481 y Fj(y)2672 5458 y Fk(\024)25 b Fr(\034)2818 5427 y Ft(0)2808 5485 y Fj(J)6 b Fi([)p Ft(0)2943 5385 y Ff(\001)3477 5391 y Fv(\(2)p Fr(:)p Fv(97\))p eop %%Page: 28 28 28 27 bop 50 295 a Fo(28)3270 b Fl(Se)l(ction)27 b Fo(2)50 513 y Fv(so)h(that)g(w)m(e)g(are)g(left)f(to)h(b)s(ound)e(a)i(term)f (of)h(the)g(form)f Fh(R)1989 480 y Fi(\016)2051 439 y Ff(\000)2093 513 y Fr(\034)2143 480 y Ft(0)2133 535 y Fj(y)2209 513 y Fk(\024)e Fr(\034)2355 480 y Ft(0)2345 538 y Fj(J)6 b Fi([)p Ft(0)2481 439 y Ff(\001)2523 513 y Fv(,)28 b Fr(y)g Fk(2)d Fr(J)9 b Fv(.)40 b(T)-8 b(o)28 b(do)f(so)h(observ)m(e)g(that)801 761 y Fh(R)861 723 y Fi(\016)924 687 y Ff(\000)966 761 y Fr(\034)1016 723 y Ft(0)1006 783 y Fj(y)1082 761 y Fk(\024)d Fr(\034)1228 723 y Ft(0)1218 783 y Fj(J)6 b Fi([)p Ft(0)1353 687 y Ff(\001)1420 761 y Fv(=)25 b Fh(R)1576 723 y Fi(\016)1639 687 y Ff(\000)1681 761 y Fr(\034)1731 723 y Ft(0)1721 783 y Fj(y)1797 761 y Fr(<)g(\034)1943 723 y Ft(0)1933 783 y(0)1983 687 y Ff(\001)2045 761 y Fk(\000)20 b Fh(R)2196 723 y Fi(\016)2259 660 y Ff(\020)2313 761 y Fr(\034)2363 723 y Ft(0)2353 786 y Fj(J)6 b Fi(n)p Fj(y)2505 761 y Fr(<)25 b(\034)2651 723 y Ft(0)2641 783 y Fj(y)2717 761 y Fr(<)g(\034)2863 723 y Ft(0)2853 783 y(0)2904 660 y Ff(\021)3477 761 y Fv(\(2)p Fr(:)p Fv(98\))50 1006 y(and)30 b(that)736 1152 y Fh(R)796 1114 y Fi(\016)859 1051 y Ff(\020)913 1152 y Fr(\034)963 1114 y Ft(0)953 1176 y Fj(J)6 b Fi(n)p Fj(y)1105 1152 y Fr(<)25 b(\034)1251 1114 y Ft(0)1241 1174 y Fj(y)1317 1152 y Fr(<)g(\034)1463 1114 y Ft(0)1453 1174 y(0)1504 1051 y Ff(\021)1584 1152 y Fv(=)1718 1065 y Ff(X)1680 1265 y Fj(z)s Fi(2)p Fj(J)6 b Fi(n)p Fj(y)1904 1152 y Fh(R)1963 1114 y Fi(\016)2026 1078 y Ff(\000)2068 1152 y Fr(\034)2118 1114 y Ft(0)2108 1174 y Fj(z)2184 1152 y Fk(\024)25 b Fr(\034)2330 1114 y Ft(0)2320 1174 y Fj(J)6 b Fi([)p Ft(0)2456 1078 y Ff(\001)2512 1152 y Fh(R)2572 1114 y Fi(\016)2635 1078 y Ff(\000)2677 1152 y Fr(\034)2727 1114 y Fj(z)2717 1174 y(y)2794 1152 y Fr(<)25 b(\034)2940 1114 y Fj(z)2930 1174 y Ft(0)2982 1078 y Ff(\001)3477 1152 y Fv(\(2)p Fr(:)p Fv(99\))50 1435 y(By)33 b(assumption,)g(the)h(probabilities)29 b Fh(R)1448 1402 y Fi(\016)1511 1362 y Ff(\000)1553 1435 y Fr(\034)1603 1402 y Fj(z)1593 1458 y(y)1670 1435 y Fr(<)c(\034)1816 1402 y Fj(z)1806 1458 y Ft(0)1858 1362 y Ff(\001)1933 1435 y Fv(in)32 b(the)h(r.h.s.)49 b(ab)s(o)m(v)m(e)34 b(ob)s(ey)f(the)h(b)s(ound)d(\(2.35\))36 b(of)50 1581 y(Lemma)30 b(2.7.)42 b(Th)m(us)810 1794 y Fh(R)870 1757 y Fi(\016)933 1693 y Ff(\020)987 1794 y Fr(\034)1037 1757 y Ft(0)1027 1819 y Fj(J)6 b Fi(n)p Fj(y)1179 1794 y Fr(<)25 b(\034)1325 1757 y Ft(0)1315 1817 y Fj(y)1391 1794 y Fr(<)g(\034)1537 1757 y Ft(0)1527 1817 y(0)1578 1693 y Ff(\021)1658 1794 y Fk(\024)f Fr(e)1795 1757 y Fi(\000)p Fj(N)7 b(h)p Ft(\()p Fj(\016)n(;\024)p Ft(\))2171 1708 y Ff(X)2132 1907 y Fj(z)s Fi(2)p Fj(J)f Fi(n)p Fj(y)2356 1794 y Fh(R)2415 1757 y Fi(\016)2478 1721 y Ff(\000)2520 1794 y Fr(\034)2570 1757 y Ft(0)2560 1817 y Fj(z)2636 1794 y Fk(\024)25 b Fr(\034)2782 1757 y Ft(0)2772 1817 y Fj(J)6 b Fi([)p Ft(0)2908 1721 y Ff(\001)1658 2054 y Fk(\024)24 b Fr(e)1795 2016 y Fi(\000)p Fj(N)7 b(h)p Ft(\()p Fj(\016)n(;\024)p Ft(\))2117 2054 y Fh(R)2176 2016 y Fi(\016)2239 1980 y Ff(\000)2281 2054 y Fr(\034)2331 2016 y Ft(0)2321 2076 y Fj(J)2397 2054 y Fr(<)25 b(\034)2543 2016 y Ft(0)2533 2076 y(0)2584 1980 y Ff(\001)3432 1910 y Fv(\(2)p Fr(:)p Fv(100\))50 2263 y(F)-8 b(rom)31 b(\(2.98\))h(and)e (\(2.100\))j(w)m(e)e(deduce)f(that)323 2508 y Fh(R)382 2470 y Fi(\016)445 2434 y Ff(\000)487 2508 y Fr(\034)537 2470 y Ft(0)527 2530 y Fj(y)603 2508 y Fr(<)25 b(\034)749 2470 y Ft(0)739 2530 y(0)790 2434 y Ff(\001)852 2508 y Fk(\000)19 b Fr(e)984 2470 y Fi(\000)p Fj(N)7 b(h)p Ft(\()p Fj(\016)n(;\024)p Ft(\))1306 2508 y Fh(R)1365 2470 y Fi(\016)1428 2434 y Ff(\000)1470 2508 y Fr(\034)1520 2470 y Ft(0)1510 2530 y Fj(J)1586 2508 y Fr(<)25 b(\034)1732 2470 y Ft(0)1722 2530 y(0)1773 2434 y Ff(\001)1840 2508 y Fk(\024)g Fh(R)1995 2470 y Fi(\016)2058 2434 y Ff(\000)2100 2508 y Fr(\034)2150 2470 y Ft(0)2140 2530 y Fj(y)2216 2508 y Fk(\024)g Fr(\034)2362 2470 y Ft(0)2352 2530 y Fj(J)6 b Fi([)p Ft(0)2488 2434 y Ff(\001)2555 2508 y Fk(\024)24 b Fh(R)2710 2470 y Fi(\016)2773 2434 y Ff(\000)2815 2508 y Fr(\034)2865 2470 y Ft(0)2855 2530 y Fj(y)2931 2508 y Fr(<)h(\034)3077 2470 y Ft(0)3067 2530 y(0)3118 2434 y Ff(\001)3432 2508 y Fv(\(2)p Fr(:)p Fv(101\))50 2753 y(and,)30 b(summing)e(o)m(v)m(er)k Fr(y)c Fk(2)d Fr(J)9 b Fv(,)162 2912 y Ff(X)161 3107 y Fj(y)r Fi(2)p Fj(J)310 2998 y Fh(R)370 2961 y Fi(\016)433 2925 y Ff(\000)475 2998 y Fr(\034)525 2961 y Ft(0)515 3021 y Fj(y)591 2998 y Fk(\024)25 b Fr(\034)737 2961 y Ft(0)727 3021 y(0)777 2925 y Ff(\001)839 2998 y Fk(\000)20 b(j)p Fr(J)9 b Fk(j)p Fr(e)1081 2961 y Fi(\000)p Fj(N)e(h)p Ft(\()p Fj(\016)n(;\024)p Ft(\))1403 2998 y Fh(R)1463 2961 y Fi(\016)1525 2925 y Ff(\000)1567 2998 y Fr(\034)1617 2961 y Ft(0)1607 3021 y Fj(J)1683 2998 y Fr(<)25 b(\034)1829 2961 y Ft(0)1819 3021 y(0)1870 2925 y Ff(\001)1937 2998 y Fk(\024)g Fh(R)2092 2961 y Fi(\016)2155 2925 y Ff(\000)2197 2998 y Fr(\034)2247 2961 y Ft(0)2237 3021 y Fj(J)2313 2998 y Fk(\024)g Fr(\034)2459 2961 y Ft(0)2449 3021 y(0)2500 2925 y Ff(\001)2567 2998 y Fk(\024)2664 2912 y Ff(X)2663 3107 y Fj(y)r Fi(2)p Fj(J)2812 2998 y Fh(R)2872 2961 y Fi(\016)2935 2925 y Ff(\000)2977 2998 y Fr(\034)3027 2961 y Ft(0)3017 3021 y Fj(y)3093 2998 y Fr(<)g(\034)3239 2961 y Ft(0)3229 3021 y(0)3279 2925 y Ff(\001)3432 2998 y Fv(\(2)p Fr(:)p Fv(102\))50 3306 y(Inserting)k(the)h(b)s(ounds)f(\(2.101\))k(and)c (\(2.102\))34 b(in)m(to)c(\(2.97\),)j(and)d(using)e(that)1542 3514 y Fh(R)1602 3481 y Fi(\016)1665 3441 y Ff(\000)1706 3514 y Fr(\034)1756 3481 y Ft(0)1746 3540 y Fj(J)1822 3514 y Fk(\024)d Fr(\034)1968 3481 y Ft(0)1958 3537 y(0)2009 3441 y Ff(\001)p 1417 3561 759 4 v 1417 3581 a(P)1513 3676 y Fj(y)r Fi(2)p Fj(J)1667 3649 y Fh(R)1727 3623 y Fi(\016)1790 3575 y Ff(\000)1831 3649 y Fr(\034)1881 3623 y Ft(0)1871 3672 y Fj(y)1947 3649 y Fr(<)g(\034)2093 3618 y Ft(0)2083 3673 y(0)2134 3575 y Ff(\001)2212 3582 y Fk(\024)g Fv(1)1079 b(\(2)p Fr(:)p Fv(103\))50 3858 y(w)m(e)31 b(arriv)m(e)f(at:)797 4004 y Fr(R)21 b Fk(\000)f Fr(e)1020 3966 y Fi(\000)p Fj(N)7 b(h)p Ft(\()p Fj(\016)n(;\024)p Ft(\))1367 4004 y Fk(\024)24 b Fh(R)1522 3966 y Fi(\016)1585 3930 y Ff(\000)1627 4004 y Fr(\034)1677 3966 y Ft(0)1667 4026 y Fj(x)1743 4004 y Fk(\024)h Fr(\034)1889 3966 y Ft(0)1879 4026 y Fj(J)1930 3930 y Ff(\001)1996 4004 y Fk(\024)g Fr(R)2177 3876 y Ff(\022)2547 3942 y Fv(1)p 2255 3983 630 4 v 2255 4068 a(1)c Fk(\000)f(j)p Fr(J)9 b Fk(j)p Fr(e)2563 4041 y Fi(\000)p Fj(N)e(h)p Ft(\()p Fj(\016)n(;\024)p Ft(\))2895 3876 y Ff(\023)3432 4004 y Fv(\(2)p Fr(:)p Fv(104\))50 4222 y(where)1394 4376 y Fr(R)26 b Fk(\021)1721 4309 y Fh(R)1781 4276 y Fi(\016)1843 4235 y Ff(\000)1885 4309 y Fr(\034)1935 4276 y Ft(0)1925 4331 y Fj(x)2001 4309 y Fk(\024)f Fr(\034)2147 4276 y Ft(0)2137 4331 y(0)2188 4235 y Ff(\001)p 1596 4355 759 4 v 1596 4375 a(P)1692 4470 y Fj(y)r Fi(2)p Fj(J)1846 4443 y Fh(R)1906 4417 y Fi(\016)1969 4369 y Ff(\000)2010 4443 y Fr(\034)2060 4417 y Ft(0)2050 4466 y Fj(y)2126 4443 y Fk(\024)g Fr(\034)2272 4412 y Ft(0)2262 4467 y(0)2313 4369 y Ff(\001)3432 4376 y Fv(\(2)p Fr(:)p Fv(105\))50 4614 y(T)-8 b(o)31 b(estimate)g(the)f(ab)s(o)m(v)m(e)i(ratio)e(w)m(e)h (use)f(\014rst)g(that,)h(b)m(y)f(rev)m(ersibilit)m(y)-8 b(,)1268 4871 y Fr(R)26 b Fv(=)1591 4809 y Fh(Q)1656 4776 y Fi(\016)1656 4834 y Fj(N)1731 4809 y Fv(\()p Fr(x)p Fv(\))p Fh(R)1913 4776 y Fi(\016)1976 4809 y Fv(\()p Fr(\034)2061 4776 y Fj(x)2051 4832 y Ft(0)2132 4809 y Fk(\024)f Fr(\034)2278 4776 y Fj(x)2268 4832 y(x)2324 4809 y Fv(\))p 1470 4850 1011 4 v 1470 4865 a Ff(P)1566 4960 y Fj(y)r Fi(2)p Fj(J)1720 4933 y Fh(Q)1785 4902 y Fi(\016)1785 4960 y Fj(N)1860 4933 y Fv(\()p Fr(y)s Fv(\))p Fh(R)2038 4907 y Fi(\016)2101 4933 y Fv(\()p Fr(\034)2186 4889 y Fj(y)2176 4957 y Ft(0)2255 4933 y Fk(\024)g Fr(\034)2401 4889 y Fj(y)2391 4944 y(y)2445 4933 y Fv(\))3432 4871 y(\(2)p Fr(:)p Fv(106\))50 5145 y(and)30 b(next)g(that,)h(b)m(y)g(Lemma)f(2.6,)1350 5413 y Fr(#)p 1404 5340 70 4 v(R)c Fk(\024)f Fh(R)1654 5376 y Fi(\016)1717 5339 y Ff(\000)1759 5413 y Fr(\034)1809 5376 y Ft(0)1799 5436 y Fj(x)1875 5413 y Fk(\024)g Fr(\034)2021 5376 y Ft(0)2011 5436 y Fj(J)2062 5339 y Ff(\001)2129 5413 y Fk(\024)p 2282 5279 V 2282 5352 a Fr(R)p 2235 5392 164 4 v 2235 5475 a(#)p Fk(j)p Fr(J)9 b Fk(j)3432 5413 y Fv(\(2)p Fr(:)p Fv(107\))p eop %%Page: 29 29 29 28 bop 50 295 a Fl(A)l(ging)27 b(in)h(the)g(REM.)f(Part)h(1.)2661 b Fo(29)50 513 y Fv(where)30 b Fr(#)g Fv(is)f(de\014ned)g(in)g (\(2.37\))k(and)p 1519 601 70 4 v 1519 674 a Fr(R)26 b Fk(\021)1844 613 y Fh(Q)1909 580 y Fi(\016)1909 638 y Fj(N)1984 613 y Fv(\()p Fr(x)p Fv(\))p 1721 653 509 4 v 1721 668 a Ff(P)1817 764 y Fj(y)r Fi(2)p Fj(J)1971 737 y Fh(Q)2036 705 y Fi(\016)2036 763 y Fj(N)2111 737 y Fv(\()p Fr(y)s Fv(\))3432 674 y(\(2)p Fr(:)p Fv(108\))50 925 y(No)m(w)31 b(since)f Fr(J)k Fk(\022)25 b Fr(\015)707 939 y Fj(I)749 925 y Fv(\()p Fr(I)7 b Fv(\),)31 b(and)f(since)g Fh(Q)1386 892 y Fi(\016)1386 950 y Fj(N)1461 925 y Fv(\()p Fr(y)s Fv(\))c(=)f(2)1746 892 y Fi(\000)p Fj(N)1903 925 y Fv(for)30 b(all)f Fr(y)f Fk(2)d Fr(\015)2374 939 y Fj(I)2416 925 y Fv(\()p Fr(I)7 b Fv(\),)p 1719 1116 70 4 v 1719 1189 a Fr(R)26 b Fv(=)1953 1127 y(1)p 1920 1168 110 4 v 1920 1251 a Fk(j)p Fr(J)9 b Fk(j)3432 1189 y Fv(\(2)p Fr(:)p Fv(109\))50 1460 y(Collecting)29 b(\(2.104\),)k (\(2.107\),)h(and)c(\(2.109\))j(yields)28 b(\(2.38\),)33 b(concluding)c(the)h(pro)s(of)g(of)g(Lemma)h(2.8.)42 b Fk(})50 1660 y Fs(2.3.)47 b(Pro)s(of)36 b(of)f(Prop)s(osition)h(2.1) 159 1933 y Fv(While)i(the)h(estimates)h(of)f(Section)g(2.2)h(will)d (furnish)f(all)i(the)h(basic)f(ingredien)m(ts)g(to)i(the)f(pro)s(of)g (of)50 2078 y(Prop)s(osition)29 b(2.1,)k(they)e(dep)s(end)f(up)s(on)g (the)h(c)m(hoice)h(of)g(the)f(mapping)f Fr(\015)2578 2092 y Fj(I)2651 2078 y Fv(through)h(sev)m(eral)g(quan)m(tities.)50 2223 y(T)-8 b(o)30 b(put)f(them)h(to)g(use)f(w)m(e)h(still)e(ha)m(v)m (e)j(to)f(iden)m(tify)e(whic)m(h)h(mappings)f Fr(\015)2522 2237 y Fj(I)2593 2223 y Fv(will)f(b)s(e)i(needed)g(and)g(establish)50 2369 y(the)i(prop)s(erties)e(of)h(all)g(related)g(ob)5 b(jects.)42 b(T)-8 b(aking)30 b(a)h(lo)s(ok)g(at)g(Prop)s(osition)d (2.1)k(in)d(the)i(ligh)m(t)f(of)h(Lemma)50 2514 y(2.5)25 b(tells)d(us)h(at)h(once)h(that)f(w)m(e)g(will)d(b)s(e)i(concerned)h (with)e(t)m(w)m(o)j(cases)g(only:)36 b(the)24 b(case)g(where)g(the)f (mapping)50 2660 y Fr(\015)97 2674 y Fj(I)166 2660 y Fv(is)j(induced)f(b)m(y)i(the)g(elemen)m(ts)g(of)g(the)h(top)f(\(as)g (required)f(for)g(the)i(pro)s(of)e(of)h(the)g(\014rst)g(four)f (assertions\))50 2805 y(or)31 b(the)g(top)g(augmen)m(ted)h(b)m(y)e(a)i (non-random)e(elemen)m(t)h(of)g Fk(S)2153 2819 y Fj(N)2253 2805 y Fv(\(whic)m(h)f(is)g(needed)h(for)f(the)h(pro)s(of)f(of)i(the)50 2951 y(last)e(one\).)41 b(These)31 b(t)m(w)m(o)g(cases)h(are)e (analysed)g(b)s(elo)m(w.)50 3205 y Fs(Lump)s(ed)k(c)m(hain)h(induced)h (b)m(y)f(the)f(T)-9 b(op)159 3423 y Fv(Let)334 3390 y Fj(t)366 3423 y Fr(\030)49 b Fv(=)d(\()p Fr(\030)651 3390 y Ft(1)692 3423 y Fr(;)15 b(\030)776 3390 y Ft(2)817 3423 y Fr(;)g(:)g(:)g(:)h(;)f(\030)1062 3390 y Fi(j)p Fj(T)10 b Ft(\()p Fj(E)s Ft(\))p Fi(j)1277 3423 y Fv(\))43 b(b)s(e)f(the)g(matrix)g(formed)g(of)h(the)f(elemen)m(ts)h(of)g(the)g (top)f(ordered)50 3569 y(according)30 b(to)h(the)g(magnitude)e(of)i Fr(X)1355 3583 y Fj(\033)1404 3569 y Fv(:)430 3823 y Fr(T)13 b Fv(\()p Fr(E)5 b Fv(\))26 b(=)f Fk(f)p Fr(\030)849 3786 y Ft(1)891 3823 y Fr(;)15 b(\030)975 3786 y Ft(2)1016 3823 y Fr(;)g(:)g(:)g(:)h(;)f(\030)1261 3786 y Fi(j)p Fj(T)10 b Ft(\()p Fj(E)s Ft(\))p Fi(j)1476 3823 y Fk(g)p Fr(;)107 b Fv(where)90 b Fr(X)2051 3839 y Fj(\030)2086 3821 y Fn(1)2152 3823 y Fk(\025)25 b Fr(X)2323 3839 y Fj(\030)2358 3821 y Fn(2)2424 3823 y Fk(\025)g(\001)15 b(\001)g(\001)26 b(\025)f Fr(X)2822 3844 y Fj(\030)2857 3826 y Fb(j)p Fm(T)8 b Fn(\()p Fm(E)r Fn(\))p Fb(j)3432 3823 y Fv(\(2)p Fr(:)p Fv(110\))50 4078 y(Th)m(us)28 b Fr(\030)33 b Fv(is)c(here)g(a)g(random)g(v)-5 b(ariable)28 b(on)h(the)h(probabilit)m(y)c(space)k(\(\012)p Fr(;)15 b Fk(F)9 b Fr(;)15 b(P)e Fv(\).)42 b(One)29 b(easily)f(v)m(eri\014es)h (that)50 4223 y(the)c(conditional)f(distribution)e(of)j Fr(\030)t Fv(,)i(giv)m(en)e(that)h(the)g(top)f(con)m(tains)h(exactly)g Fr(M)35 b Fv(p)s(oin)m(ts,)26 b(is)e(the)i(uniform)50 4369 y(distribution)h(o)m(v)m(er)k(the)g(set)1057 4346 y Ff(e)1039 4369 y Fk(S)1094 4383 y Fj(M)7 b Fi(\002)p Fj(N)1326 4369 y Fv(of)31 b Fr(M)10 b Fv(-tuples)29 b(of)i(m)m(utually) d(distinct)h(p)s(oin)m(ts)g(of)i Fk(S)3074 4383 y Fj(N)3143 4369 y Fv(,)g(i.e.:)752 4667 y Fr(P)d Fv(\()p Fr(\030)i Fv(=)24 b Fr(\020)32 b Fk(j)26 b(j)p Fr(T)13 b Fv(\()p Fr(E)5 b Fv(\))p Fk(j)26 b Fv(=)f Fr(M)10 b Fv(\))26 b(=)1796 4512 y Ff(\()1910 4562 y Ft(\(2)1974 4534 y Fm(N)2033 4562 y Fi(\000)p Fj(M)7 b Ft(\)!)p 1910 4585 306 4 v 1977 4638 a(\(2)2041 4620 y Fm(N)2100 4638 y Ft(\)!)2423 4606 y Fv(if)90 b Fr(\020)32 b Fk(2)2743 4583 y Ff(e)2725 4606 y Fk(S)2780 4620 y Fj(M)7 b Fi(\002)p Fj(N)1899 4753 y Fv(0)p Fr(;)454 b Fv(otherwise)3432 4667 y(\(2)p Fr(:)p Fv(111\))50 4962 y(where)636 5084 y Ff(e)618 5107 y Fk(S)673 5121 y Fj(M)7 b Fi(\002)p Fj(N)900 5107 y Fk(\021)996 5006 y Ff(n)1057 5107 y Fr(\020)31 b Fk(2)25 b(S)1269 5121 y Fj(M)7 b Fi(\002)p Fj(N)1502 5003 y Ff(\014)1502 5057 y(\014)1502 5112 y(\014)1562 5107 y Fr(\020)1609 5070 y Fj(\026)1683 5107 y Fk(6)p Fv(=)25 b Fr(\020)1826 5070 y Fj(\027)1961 5107 y Fv(for)30 b(all)90 b(1)26 b Fk(\024)f Fr(\026;)15 b(\027)31 b Fk(\024)25 b Fr(M)5 b(;)15 b(\026)25 b Fk(6)p Fv(=)g Fr(\027)3081 5006 y Ff(o)3432 5107 y Fv(\(2)p Fr(:)p Fv(112\))50 5316 y(Set)1448 5462 y Fr(\016)s Fv(\()p Fr(N)10 b Fv(\))27 b Fk(\021)e Fv(\()p Fr(d=)-5 b(N)10 b Fv(\))2007 5424 y Ft(1)p Fj(=)p Ft(2)2138 5462 y Fv(ln)k Fr(N)1130 b Fv(\(2)p Fr(:)p Fv(113\))p eop %%Page: 30 30 30 29 bop 50 295 a Fo(30)3270 b Fl(Se)l(ction)27 b Fo(2)50 513 y Fv(and)j(let)p 358 440 62 4 v 30 w Fk(S)420 527 y Fj(M)7 b Fi(\002)p Fj(N)652 513 y Fv(b)s(e)30 b(de\014ned)f(through)p 181 659 V 181 732 a Fk(S)242 746 y Fj(M)7 b Fi(\002)p Fj(N)470 732 y Fk(\021)566 631 y Ff(n)626 732 y Fr(\020)32 b Fk(2)25 b(S)839 746 y 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1589 y Fk(\021)1448 1516 y Ff(\010)1501 1589 y Fr(!)d Fk(2)d Fv(\012)g Fk(j)h Fr(\030)t Fv(\()p Fr(!)s Fv(\))f Fk(2)p 2099 1516 62 4 v 25 w(S)2161 1606 y Fi(j)p Fj(T)10 b Ft(\()p Fj(E)s Ft(\))p Fi(j\002)p Fj(N)2496 1516 y Ff(\011)3432 1589 y Fv(\(2)p Fr(:)p Fv(116\))159 1856 y(It)30 b(is)g(easy)h(to)g(see,)g (using)e(the)i(pro)s(of)e(of)i(Lemma)f(4.2)i(of)e([G],)i(that:)50 2074 y Fc(Lemma)38 b(2.11:)1690 2219 y Fr(P)28 b Fv(\()p Fk(E)8 b Fv(\))26 b(=)f(1)1363 b(\(2)p Fr(:)p Fv(117\))159 2631 y(W)-8 b(e)34 b(will)d(need)i(a)g(certain)g(n)m(um)m(b)s(er)f(of)h (geometric)h(prop)s(erties)e(of)h(the)g(set)h Fr(T)13 b Fv(\()p Fr(E)5 b Fv(\),)35 b(whic)m(h)d(w)m(e)h(collect)50 2776 y(b)s(elo)m(w.)50 2994 y Fc(Lemma)40 b(2.12:)98 b Fq(F)-7 b(or)35 b(al)5 b(l)34 b Fv(0)28 b Fk(\024)f Fr(")h(<)g Fv(1)p Fr(=)p Fv(2)p Fq(,)35 b(al)5 b(l)34 b Fr(!)d Fk(2)c(E)1986 3008 y Fj(N)2055 2994 y Fq(,)34 b(and)h(lar)-5 b(ge)35 b(enough)f Fr(N)44 b Fq(the)34 b(fol)5 b(lowing)35 b(holds:)50 3140 y(for)e(al)5 b(l)33 b Fr(\021)c Fk(6)p Fv(=)i(\026)-51 b Fr(\021)s Fq(,)33 b Fr(\021)28 b Fk(2)d Fr(T)13 b Fq(,)39 b Fv(\026)-52 b Fr(\021)29 b Fk(2)c Fr(T)13 b Fq(,)1330 3285 y Fk(B)1390 3260 y Fi(p)p 1449 3260 99 4 v 1449 3315 a Fj("N)1553 3285 y Fv(\()p Fr(\021)s Fv(\))21 b Fk(\\)f(B)1833 3260 y Fi(p)p 1892 3260 V 1892 3315 a Fj("N)1995 3285 y Fv(\()7 b(\026)-52 b Fr(\021)t Fv(\))26 b(=)f Fk(;)p Fr(;)1126 b Fv(\(2)p Fr(:)p Fv(118\))50 3479 y Fq(and)1542 3593 y Fv(1)p 1523 3633 83 4 v 1523 3717 a Fr(N)1665 3541 y Fj(N)1632 3568 y Ff(X)1639 3761 y Fj(i)p Ft(=1)1778 3654 y Fr(\021)1823 3668 y Fj(i)1860 3654 y Fv(\026)-51 b Fr(\021)1899 3668 y Fj(i)1954 3654 y Fk(\024)25 b Fr(\016)s Fv(\()p Fr(N)10 b Fv(\))1186 b(\(2)p Fr(:)p Fv(119\))50 4066 y Fa(Pro)s(of:)51 b Fv(With)32 b(the)g(notation)g(of)g(\(2.110\))j (let)d Fr(\030)1735 4033 y Fj(\026)1812 4066 y Fk(6)p Fv(=)27 b Fr(\030)1954 4033 y Fj(\027)2032 4066 y Fv(b)s(e)k(an)m(y)h (t)m(w)m(o)i(distinct)c(elemen)m(ts)i(of)h Fr(T)13 b Fv(.)45 b(F)-8 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Fj(N)605 513 y Fv(b)m(y)31 b(assumption,)e(it)h(follo)m(ws)f(from)h (\(2.114\))j(that)1047 769 y Ft(1)p 1033 784 65 4 v 1033 836 a Fj(N)1157 691 y(N)1124 718 y Ff(X)1130 912 y Fj(i)p Ft(=1)1270 804 y Fr(\030)1314 767 y Fj(\027)1310 827 y(i)1359 804 y Fr(\030)1403 761 y Fj(\026)1399 830 y(i)1477 804 y Fv(=)1585 769 y Ft(1)p 1584 784 38 4 v 1584 836 a Fj(d)1695 691 y(d)1648 718 y Ff(X)1648 914 y Fj(k)r Ft(=1)1795 804 y Fr(e)1837 767 y Fj(\027)1837 827 y(k)1883 804 y Fr(e)1925 761 y Fj(\026)1925 832 y(k)1994 804 y Fv(+)2097 769 y Ft(1)p 2096 784 V 2096 836 a Fj(d)2207 691 y(d)2160 718 y Ff(X)2160 914 y Fj(k)r Ft(=1)2307 804 y Fr(\025)2360 818 y Fj(k)2405 804 y Fv(\()p Fr(N)10 b Fv(\))p Fr(e)2600 767 y Fj(\027)2600 827 y(k)2646 804 y Fr(e)2688 761 y Fj(\026)2688 832 y(k)1477 1126 y Fv(=)1585 1090 y Ft(1)p 1584 1105 V 1584 1157 a Fj(d)1695 1012 y(d)1648 1040 y Ff(X)1648 1235 y Fj(k)r Ft(=1)1795 1126 y Fr(\025)1848 1140 y Fj(k)1893 1126 y Fv(\()p Fr(N)g Fv(\))p Fr(e)2088 1089 y Fj(\027)2088 1149 y(k)2134 1126 y Fr(e)2176 1082 y Fj(\026)2176 1154 y(k)1477 1358 y Fk(\024)25 b Fr(\016)s Fv(\()p Fr(N)10 b Fv(\))3432 1036 y(\(2)p Fr(:)p Fv(122\))50 1560 y(where)31 b(the)h(second)g(equalit)m (y)f(follo)m(ws)g(from)h(Lemma)g(2.1)h(of)f([G].)g(Th)m(us)f(\(2.119\)) k(is)c(pro)m(v)m(en.)45 b(Inserting)50 1706 y(\(2.122\))33 b(in)c(\(2.120\))k(yields,)1147 1944 y Fk(k)p Fr(\033)24 b Fk(\000)c Fr(\030)1403 1907 y Fj(\027)1448 1944 y Fk(k)1493 1958 y Ft(2)1560 1944 y Fk(\025)1656 1862 y Ff(p)p 1747 1862 554 4 v 82 x Fv(2)p Fr(N)10 b Fv(\(1)21 b Fk(\000)f Fr(\016)s Fv(\()p Fr(N)10 b Fv(\)\))22 b Fk(\000)2411 1863 y(p)p 2487 1863 126 4 v 81 x Fr("N)830 b Fv(\(2)p Fr(:)p Fv(123\))50 2182 y(whic)m(h)29 b(w)m(ould)g(en)m(tail)h (\(2.118\))j(if)c(w)m(e)i(had)1240 2339 y Ff(p)p 1331 2339 554 4 v 82 x Fv(2)p Fr(N)10 b Fv(\(1)22 b Fk(\000)e Fr(\016)s Fv(\()p Fr(N)10 b Fv(\)\))22 b Fk(\000)1996 2339 y(p)p 2071 2339 126 4 v 2071 2421 a Fr("N)36 b(>)2318 2339 y Fk(p)p 2394 2339 V 82 x Fr("N)923 b 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3594 y Fj(d)p 2286 3609 V 2286 3661 a(N)2361 3556 y Ff(\001)2403 3573 y Ft(1)p Fj(=)p Ft(2)2532 3630 y Fv(ln)14 b Fr(N)2706 3529 y Ff(\021)1278 3860 y Fv(max)1235 3922 y Fj(k)r Fi(2K)p Ft(\()p Fj(T)c Ft(\))1505 3860 y Fk(j)p Fv(\003)1593 3874 y Fj(k)1638 3860 y Fv(\()p Fr(T)j Fv(\))p Fk(j)26 b(\024)1932 3824 y Fj(N)p 1932 3839 V 1945 3891 a(d)2022 3759 y Ff(\020)2077 3860 y Fv(1)20 b(+)2233 3786 y Ff(\000)2299 3824 y Fj(d)p 2286 3839 V 2286 3891 a(N)2361 3786 y Ff(\001)2403 3802 y Ft(1)p Fj(=)p Ft(2)2532 3860 y Fv(ln)14 b Fr(N)2706 3759 y Ff(\021)672 b Fv(\(2)p Fr(:)p Fv(126\))50 4081 y(Of)40 b(course)g Fr(\024)i Fv(=)g Fr(d)e Fv(and)g(the)h(pro)5 b(jection)40 b Fr(\031)j Fv(de\014ned)c(in)g(the)h(line)f(preceeding)h (\(2.33\))i(simply)c(is)i(the)50 4227 y(iden)m(tit)m(y)-8 b(.)40 b(Kno)m(wing)30 b(this)f(w)m(e)i(ha)m(v)m(e:)50 4445 y Fc(Lemma)38 b(2.13:)91 b Fq(Assume)33 b(that)h Fr(!)28 b Fk(2)d(E)1527 4459 y Fj(N)1595 4445 y Fq(.)42 b(Then,)33 b(for)g(al)5 b(l)33 b Fr(N)43 b Fq(lar)-5 b(ge)33 b(enough,)47 4663 y(i\))47 b(F)-7 b(or)34 b(al)5 b(l)33 b Fr(\033)28 b Fk(2)d Fr(W)711 4677 y Fj(")750 4663 y Fv(\()p Fr(T)13 b Fv(\))p Fq(,)1187 4934 y Fv(inf)1098 4997 y Fj(x)p Fi(2)p Fj(\015)1226 5006 y Fm(T)1277 4997 y Ft(\()p Fj(T)d Ft(\))1402 4934 y Fk(k)p Fr(\031)s(x)21 b Fk(\000)e Fr(\031)s(\015)1767 4948 y Fj(T)1825 4934 y Fv(\()p Fr(\033)s Fv(\))p Fk(k)1995 4948 y Ft(2)2062 4934 y Fk(\025)2169 4873 y Fr(")2211 4795 y Fk(p)p 2287 4795 48 4 v 78 x Fr(d)p 2169 4913 166 4 v 2229 4996 a Fv(2)2345 4934 y(\(1)i Fk(\000)f Fr(\016)s Fv(\()p Fr(N)10 b Fv(\)\))664 b(\(2)p Fr(:)p Fv(127\))20 5265 y Fq(ii\))46 b(F)-7 b(or)34 b(al)5 b(l)33 b Fr(\033)28 b Fk(2)d Fr(T)13 b Fq(,)1197 5410 y Fv(inf)994 5473 y Fj(x)p Fi(2)p Fj(\015)1122 5482 y Fm(T)1173 5473 y Ft(\()p Fj(T)d Ft(\))p Fi(n)p Fj(\015)1357 5482 y Fm(T)1408 5473 y Ft(\()p Fj(\033)r Ft(\))1525 5410 y Fk(k)p Fr(\031)s(x)20 b Fk(\000)g Fr(\031)s(\015)1890 5424 y Fj(T)1948 5410 y Fv(\()p Fr(\033)s Fv(\))p Fk(k)2118 5424 y Ft(2)2185 5410 y Fk(\025)25 b Fv(\(1)c Fk(\000)f Fv(2)p Fr(\016)s Fv(\()p Fr(N)10 b Fv(\)\))2749 5328 y Fk(p)p 2827 5328 48 4 v 2827 5410 a Fr(d)558 b Fv(\(2)p Fr(:)p Fv(128\))p eop %%Page: 32 32 32 31 bop 50 295 a Fo(32)3270 b Fl(Se)l(ction)27 b Fo(2)50 731 y Fa(Pro)s(of:)65 b Fv(As)38 b(a)h(consequence)g(of)f(\(2.126\))j (and)d(assertion)g(iv\))f(of)i(Lemma)f(2.2)h(w)m(e)g(ha)m(v)m(e,)j(for) c(all)f Fr(\033)k Fk(2)50 876 y Fr(W)136 890 y Fj(")175 876 y Fv(\()p Fr(T)13 b Fv(\),)1030 995 y(inf)941 1058 y Fj(x)p Fi(2)p Fj(\015)1069 1067 y Fm(T)1119 1058 y Ft(\()p Fj(T)d Ft(\))1244 995 y Fk(k)p Fr(\031)s(x)21 b Fk(\000)f Fr(\031)s(\015)1610 1009 y Fj(T)1667 995 y Fv(\()p Fr(\033)s Fv(\))p Fk(k)1837 1009 y Ft(2)1905 995 y Fv(=)103 b(inf)1990 1058 y Fj(x)p Fi(2)p Fj(\015)2118 1067 y Fm(T)2169 1058 y Ft(\()p Fj(T)10 b Ft(\))2294 995 y Fk(k)p Fr(x)20 b Fk(\000)g Fr(\015)2549 1009 y Fj(T)2607 995 y Fv(\()p Fr(\033)s Fv(\))p Fk(k)2777 1009 y Ft(2)1905 1243 y Fk(\025)2277 1182 y Fr("N)p 1986 1222 708 4 v 1986 1318 a Fv(2)2031 1240 y Fk(p)p 2107 1240 48 4 v 78 x Fr(d)c Fv(max)2339 1332 y Fj(k)2399 1318 y Fk(j)p Fv(\003)2487 1332 y Fj(k)2532 1318 y Fv(\()p Fr(T)d Fv(\))p Fk(j)1905 1523 y(\025)1986 1462 y Fr(")2028 1384 y Fk(p)p 2104 1384 V 78 x Fr(d)p 1986 1502 166 4 v 2046 1586 a Fv(2)2163 1523 y(\(1)21 b Fk(\000)f Fr(\016)s Fv(\()p Fr(N)10 b Fv(\)\))2586 1486 y Fi(\000)p Ft(1)3432 1274 y Fv(\(2)p Fr(:)p Fv(129\))50 1736 y(whic)m(h)29 b(yields)f(\(2.127\).)44 b(Similarly)27 b(note)k(that)g(if)e Fr(\033)f Fk(2)d Fr(T)43 b Fv(then,)31 b(b)m(y)f(Lemma)g(2.12,)1261 2001 y(inf)1205 2063 y Fj(\021)r Fi(2)p Fj(T)10 b Fi(n)p Fj(\033)1443 2001 y Fk(k)p Fr(\021)24 b Fk(\000)c Fr(\033)s Fk(k)1748 2015 y Ft(2)1815 2001 y Fk(\025)1911 1918 y Ff(p)p 2002 1918 554 4 v 83 x Fv(2)p Fr(N)10 b Fv(\(1)21 b Fk(\000)f Fr(\016)s Fv(\()p Fr(N)10 b Fv(\)\))879 b(\(2)p Fr(:)p Fv(130\))50 2285 y(Just)30 b(as)i(in)e(\(2.129\))k(this)c(prop)s 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b Fk(T)59 b Fv(=)37 b Fr(\015)2806 3285 y Fj(T)2863 3271 y Fv(\()p Fr(T)13 b Fv(\),)40 b Fr(y)f Fv(=)e Fr(\015)3303 3285 y Fj(T)3360 3271 y Fv(\()p Fr(\033)s Fv(\),)j Fr(x)d Fv(=)50 3417 y Fr(\015)97 3431 y Fj(T)154 3417 y Fv(\()p Fr(\021)s Fv(\))32 b(and)j(\026)-51 b Fr(x)26 b Fv(=)e Fr(\015)700 3431 y Fj(T)758 3417 y Fv(\()7 b(\026)-52 b Fr(\021)t Fv(\).)41 b(It)30 b(will)e(moreo)m(v)m(er)k(b)s(e)d (assumed)h(that)h Fr(!)d Fk(2)d(E)2509 3431 y Fj(N)2578 3417 y Fv(.)50 3635 y Fa(Pro)s(of)36 b(of)f(Prop)s(osition)h(2.1,)f (i\):)87 b Fv(Using)29 b(in)h(turn)f(assertion)h(i\))g(of)g(Lemma)h (2.4)g(and)f(Lemma)h(2.5,)851 3899 y Fh(P)921 3798 y Ff(\020)976 3899 y Fr(\034)1026 3862 y Fj(\033)1016 3922 y(\021)1100 3899 y Fr(<)25 b(\034)1246 3862 y Fj(\033)1236 3924 y(T)10 b Fi(n)p Fj(\021)1370 3798 y Ff(\021)1450 3899 y Fv(=)24 b Fh(P)1600 3862 y Fi(\016)1658 3798 y Ff(\020)1712 3899 y Fr(\034)1762 3862 y Fj(\033)1752 3922 y(\021)1837 3899 y Fr(<)g(\034)1982 3862 y Fj(\033)1972 3924 y(T)10 b 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y(x)2315 4505 y Fr(<)25 b(\034)2461 4461 y Fj(y)2451 4529 y Ft(0)2505 4505 y Fk(g)2550 4404 y Ff(\021)3432 4405 y Fv(\(2)p Fr(:)p Fv(133\))50 4751 y Fh(R)109 4718 y Fi(\016)172 4650 y Ff(\020)227 4751 y Fr(\034)277 4718 y Fj(y)267 4773 y(x)346 4751 y Fr(<)g(\034)492 4707 y Fj(y)482 4782 y Fi(T)16 b(n)p Fj(x)622 4650 y Ff(\021)707 4751 y Fv(ma)m(y)31 b(b)s(e)f(decomp)s(osed)g(as)1340 5042 y Fh(R)1400 5005 y Fi(\016)1463 4941 y Ff(\020)1517 5042 y Fr(\034)1567 5005 y Fj(y)1557 5065 y(x)1636 5042 y Fr(<)25 b(\034)1782 4999 y Fj(y)1772 5073 y Fi(T)17 b(n)p Fj(x)1913 4941 y Ff(\021)1993 5042 y Fv(=)24 b Fr(R)2157 5056 y Ft(1)2218 5042 y Fv(+)c Fr(R)2378 5056 y Ft(2)3432 5042 y Fv(\(2)p Fr(:)p Fv(134\))50 5307 y(Ob)m(viously)1430 5462 y(0)26 b Fk(\024)f Fr(R)1666 5476 y Ft(2)1732 5462 y Fk(\024)g Fh(R)1887 5424 y Fi(\016)1950 5462 y Fv(\()q Fr(\034)2036 5424 y Fj(y)2026 5484 y(x)2105 5462 y Fr(<)g(\034)2251 5418 y Fj(y)2241 5486 y Ft(0)2294 5462 y Fv(\))1103 b(\(2)p Fr(:)p Fv(135\))p eop %%Page: 33 33 33 32 bop 50 295 a Fl(A)l(ging)27 b(in)h(the)g(REM.)f(Part)h(1.)2661 b Fo(33)50 513 y Fv(while)1091 626 y Fr(R)1160 640 y Ft(1)1226 626 y Fv(=)p Fh(R)1356 589 y Fi(\016)1419 525 y Ff(\020)1474 626 y Fr(\034)1524 583 y Fj(y)1514 651 y Ft(0)1593 626 y Fr(<)25 b(\034)1739 589 y Fj(y)1729 649 y(x)1808 626 y Fr(<)g(\034)1954 583 y Fj(y)1944 657 y Fi(T)16 b(n)p Fj(x)2084 525 y Ff(\021)1226 826 y Fv(=)p Fh(R)1356 789 y Fi(\016)1419 826 y Fv(\()q Fr(\034)1505 783 y Fj(y)1495 851 y Ft(0)1574 826 y Fr(<)25 b(\034)1720 783 y Fj(y)1710 853 y Fi(T)1772 826 y Fv(\))15 b Fh(R)1882 789 y Fi(\016)1945 725 y Ff(\020)1999 826 y Fr(\034)2049 789 y Ft(0)2039 849 y Fj(x)2115 826 y Fr(<)25 b(\034)2261 789 y Ft(0)2251 851 y Fi(T)17 b(n)p Fj(x)2392 725 y Ff(\021)1226 1026 y Fv(=)e([1)21 b Fk(\000)f Fh(R)1553 989 y Fi(\016)1616 1026 y Fv(\()q Fr(\034)1702 983 y Fj(y)1692 1053 y Fi(T)1779 1026 y Fr(<)25 b(\034)1925 983 y Fj(y)1915 1051 y Ft(0)1969 1026 y Fv(\)])16 b Fh(R)2104 989 y Fi(\016)2167 926 y Ff(\020)2222 1026 y Fr(\034)2272 989 y Ft(0)2262 1049 y Fj(x)2338 1026 y Fr(<)25 b(\034)2484 989 y Ft(0)2474 1051 y Fi(T)16 b(n)p Fj(x)2614 926 y Ff(\021)3432 826 y Fv(\(2)p Fr(:)p Fv(136\))50 1213 y(whic)m(h,)29 b(together)j(with)d (the)i(b)s(ound)1211 1448 y Fh(R)1270 1411 y Fi(\016)1333 1448 y Fv(\()q Fr(\034)1419 1404 y Fj(y)1409 1475 y Fi(T)1496 1448 y Fr(<)25 b(\034)1642 1404 y Fj(y)1632 1472 y Ft(0)1686 1448 y Fv(\))h Fk(\024)1863 1362 y Ff(X)1843 1557 y Fj(x)1884 1539 y Fb(0)1908 1557 y Fi(2T)2030 1448 y Fh(R)2090 1411 y Fi(\016)2153 1448 y Fv(\()p Fr(\034)2238 1404 y Fj(y)2228 1475 y(x)2269 1457 y Fb(0)2324 1448 y Fr(<)f(\034)2470 1404 y Fj(y)2460 1472 y Ft(0)2513 1448 y Fv(\))884 b(\(2)p Fr(:)p Fv(137\))50 1733 y(yields)397 1981 y Fh(R)457 1944 y Fi(\016)520 1880 y Ff(\020)574 1981 y Fr(\034)624 1944 y Ft(0)614 2004 y Fj(x)690 1981 y Fr(<)25 b(\034)836 1944 y Ft(0)826 2006 y Fi(T)17 b(n)p Fj(x)967 1880 y Ff(\021)1036 1853 y(\024)1084 1981 y Fv(1)k Fk(\000)f Fr(M)43 b Fv(sup)1354 2057 y Fj(x)1395 2039 y Fb(0)1420 2057 y Fi(2T)1542 1981 y Fh(R)1602 1944 y Fi(\016)1665 1981 y Fv(\()p Fr(\034)1750 1937 y Fj(y)1740 2008 y(x)1781 1990 y Fb(0)1835 1981 y Fr(<)25 b(\034)1981 1937 y Fj(y)1971 2005 y Ft(0)2025 1981 y Fv(\))2060 1853 y Ff(\025)2134 1981 y Fk(\024)g Fr(R)2299 1995 y Ft(1)2365 1981 y Fk(\024)g Fh(R)2520 1944 y Fi(\016)2583 1880 y Ff(\020)2638 1981 y Fr(\034)2688 1944 y Ft(0)2678 2004 y Fj(x)2754 1981 y Fr(<)g(\034)2900 1944 y Ft(0)2890 2006 y Fi(T)16 b(n)p Fj(x)3030 1880 y Ff(\021)3432 1981 y Fv(\(2)p Fr(:)p Fv(138\))50 2275 y(W)-8 b(e)37 b(are)f(th)m(us)f(left)h(to)g(b)s(ound)e (the)i(quan)m(tities)f(sup)1865 2297 y Fj(x)1906 2279 y Fb(0)1931 2297 y Fi(2T)2058 2275 y Fh(R)2117 2242 y Fi(\016)2180 2275 y Fv(\()p Fr(\034)2265 2231 y Fj(y)2255 2301 y(x)2296 2283 y Fb(0)2351 2275 y Fr(<)25 b(\034)2497 2231 y Fj(y)2487 2299 y Ft(0)2541 2275 y Fv(\))36 b(and)f Fh(R)2853 2242 y Fi(\016)2916 2174 y Ff(\020)2971 2275 y Fr(\034)3021 2242 y Ft(0)3011 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Fv(\))i(and)41 b(\026)-51 b Fr(x)33 b Fv(=)h Fr(\015)3193 5072 y Fj(T)3250 5058 y Fv(\()7 b(\026)-52 b Fr(\021)t Fv(\))35 b(w)m(e)h(ma)m(y)50 5204 y(write,)30 b(using)f(in)g(turn)g(assertion)h(ii\))f(of)i(Lemma)f (2.4)i(and)d(Lemma)i(2.5,)335 5439 y Fh(P)405 5338 y Ff(\020)460 5439 y Fr(\034)516 5402 y Ft(\026)-42 b Fj(\021)500 5462 y(\021)580 5439 y Fr(<)25 b(\034)731 5395 y Ft(\026)-41 b Fj(\021)716 5470 y(T)10 b Fi(n)p Fj(\021)850 5338 y Ff(\021)929 5439 y Fv(=)25 b Fr(e)1067 5401 y Fi(\000)p Fj(\014)1168 5347 y Fi(p)p 1228 5347 V 54 x Fj(N)7 b(E)1349 5372 y Fn(+)1351 5420 y(\026)-36 b Fm(\021)1406 5439 y Fh(P)1461 5402 y Fi(\016)1518 5338 y Ff(\020)1572 5439 y Fr(\034)1628 5402 y Ft(\026)-42 b Fj(\021)1612 5462 y(\021)1692 5439 y Fr(<)25 b(\034)1844 5395 y Ft(\026)-42 b Fj(\021)1828 5470 y(T)10 b Fi(n)p Fj(\021)1962 5338 y Ff(\021)2041 5439 y Fv(=)25 b Fr(e)2179 5401 y Fi(\000)p Fj(\014)2280 5347 y Fi(p)p 2341 5347 V 2341 5401 a Fj(N)6 b(E)2461 5372 y Fn(+)2463 5420 y(\026)-36 b Fm(\021)2518 5439 y Fh(R)2577 5402 y Fi(\016)2640 5338 y Ff(\020)2695 5439 y Fr(\034)2750 5402 y Ft(\026)-41 b Fj(x)2735 5462 y(x)2816 5439 y Fr(<)25 b(\034)2967 5402 y Ft(\026)-41 b Fj(x)2952 5464 y Fi(T)16 b(n)p Fj(x)3092 5338 y Ff(\021)3432 5439 y Fv(\(2)p Fr(:)p Fv(142\))p eop %%Page: 34 34 34 33 bop 50 295 a Fo(34)3270 b Fl(Se)l(ction)27 b Fo(2)50 526 y Fv(W)-8 b(e)28 b(then)f(decomp)s(ose)h Fh(R)921 493 y Fi(\016)984 425 y Ff(\020)1039 526 y Fr(\034)1094 493 y Ft(\026)-41 b Fj(x)1079 549 y(x)1160 526 y Fr(<)25 b(\034)1311 493 y Ft(\026)-41 b Fj(x)1296 556 y Fi(T)16 b(n)p Fj(x)1436 425 y Ff(\021)1518 526 y Fv(as)27 b(in)f(\(2.134\),)31 b(and)c(b)s(ound)e Fr(R)2580 540 y Ft(2)2648 526 y Fv(as)j(in)e (\(2.135\).)42 b(As)27 b(for)g Fr(R)3538 540 y Ft(1)3606 526 y Fv(w)m(e)50 672 y(write,)j(just)f(as)i(in)e(\(2.136\),)1066 895 y Fr(R)1135 909 y Ft(1)1201 895 y Fv(=)1297 822 y Ff(\002)1335 895 y Fv(1)21 b Fk(\000)e Fh(R)1551 858 y Fi(\016)1614 822 y Ff(\000)1656 895 y Fr(\034)1711 858 y Ft(\026)-41 b Fj(x)1696 918 y Fi(T)1783 895 y Fr(<)25 b(\034)1934 858 y Ft(\026)-41 b Fj(x)1919 918 y Ft(0)1975 822 y Ff(\001\003)2070 895 y Fh(R)2129 858 y Fi(\016)2192 794 y Ff(\020)2247 895 y Fr(\034)2297 858 y Ft(0)2287 918 y Fj(x)2363 895 y Fr(<)25 b(\034)2509 858 y Ft(0)2499 920 y Fi(T)16 b(n)p Fj(x)2639 794 y Ff(\021)3432 895 y Fv(\(2)p Fr(:)p Fv(143\))50 1119 y(but)29 b(this)h(time)g(use)g (\(2.137\))j(to)e(deduce)f(that)455 1324 y(1)20 b Fk(\000)g Fh(R)671 1287 y Fi(\016)734 1251 y Ff(\000)775 1324 y Fr(\034)830 1287 y Ft(\026)-41 b Fj(x)815 1347 y Fi(T)903 1324 y Fr(<)25 b(\034)1054 1287 y Ft(\026)-41 b Fj(x)1039 1347 y Ft(0)1095 1251 y Ff(\001)1162 1324 y Fk(\025)24 b Fv(1)d Fk(\000)1435 1238 y Ff(X)1414 1433 y Fj(x)1455 1415 y Fb(0)1480 1433 y Fi(2T)1602 1324 y Fh(R)1662 1287 y Fi(\016)1725 1251 y Ff(\000)1766 1324 y Fr(\034)1821 1287 y Ft(\026)-41 b Fj(x)1806 1347 y(x)1847 1328 y Fb(0)1902 1324 y Fr(<)25 b(\034)2053 1287 y Ft(\026)-41 b Fj(x)2038 1347 y Ft(0)2093 1251 y 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y Fh(R)2554 1793 y Fi(\016)2617 1756 y Ff(\000)2659 1830 y Fr(\034)2714 1793 y Ft(\026)f Fj(x)2699 1853 y(x)2740 1834 y Fb(0)2794 1830 y Fr(<)25 b(\034)2945 1793 y Ft(\026)-41 b Fj(x)2930 1853 y Ft(0)2986 1756 y Ff(\001)3432 1605 y Fv(\(2)p Fr(:)p Fv(144\))50 2086 y(Therefore)134 2332 y Fh(R)194 2295 y Fi(\016)257 2231 y Ff(\020)311 2332 y Fr(\034)361 2295 y Ft(0)351 2355 y Fj(x)427 2332 y Fr(<)25 b(\034)573 2295 y Ft(0)563 2357 y Fi(T)17 b(n)p Fj(x)704 2231 y Ff(\021)773 2177 y(")826 2332 y Fh(R)886 2295 y Fi(\016)949 2259 y Ff(\000)991 2332 y Fr(\034)1046 2295 y Ft(\026)-41 b Fj(x)1031 2355 y Ft(0)1112 2332 y Fr(<)25 b(\034)1263 2295 y Ft(\026)-41 b Fj(x)1252 2355 y Ft(\026)h Fj(x)1303 2259 y Ff(\001)1365 2332 y Fk(\000)20 b Fv(\()p Fr(M)31 b Fk(\000)20 b Fv(1\))73 b(sup)1796 2413 y Fj(x)1837 2395 y Fb(0)1863 2413 y Fi(2T)17 b(n)t Ft(\026)-40 b Fj(x)2063 2332 y Fh(R)2123 2295 y Fi(\016)2186 2259 y Ff(\000)2227 2332 y Fr(\034)2282 2295 y Ft(\026)f Fj(x)2267 2355 y(x)2308 2336 y Fb(0)2363 2332 y Fr(<)25 b(\034)2514 2295 y Ft(\026)-41 b Fj(x)2499 2355 y Ft(0)2555 2259 y Ff(\001)2596 2177 y(#)2674 2332 y Fk(\024)25 b Fr(R)2839 2346 y Ft(1)2905 2332 y Fk(\024)g Fh(R)3061 2295 y Fi(\016)3124 2231 y Ff(\020)3178 2332 y Fr(\034)3228 2295 y Ft(0)3218 2355 y Fj(x)3294 2332 y Fr(<)g(\034)3440 2295 y Ft(0)3430 2357 y Fi(T)17 b(n)p Fj(x)3571 2231 y Ff(\021)3432 2523 y Fv(\(2)p Fr(:)p Fv(145\))50 2669 y(Ha)m(ving)30 b(already)f(estimated)i(the)f (probabilit)m(y)d Fh(R)1775 2636 y Fi(\016)1838 2568 y Ff(\020)1892 2669 y Fr(\034)1942 2636 y Ft(0)1932 2691 y Fj(x)2008 2669 y Fr(<)e(\034)2154 2636 y Ft(0)2144 2698 y Fi(T)17 b(n)p Fj(x)2285 2568 y Ff(\021)2369 2669 y Fv(in)29 b(\(2.140\),)k(w)m(e)d(are)g(left)g(to)h(treat)g(the)50 2814 y(terms)25 b Fh(R)358 2781 y Fi(\016)421 2814 y Fv(\()p Fr(\034)511 2781 y Ft(\026)-41 b Fj(x)496 2837 y Ft(0)577 2814 y Fr(<)25 b(\034)728 2781 y Ft(\026)-41 b Fj(x)717 2836 y Ft(\026)h Fj(x)769 2814 y Fv(\))26 b(and)e(sup)1138 2836 y Fj(x)1179 2818 y Fb(0)1204 2836 y Fi(2T)17 b(n)t Ft(\026)-40 b Fj(x)1409 2814 y Fh(R)1469 2781 y Fi(\016)1532 2814 y Fv(\()p Fr(\034)1622 2781 y Ft(\026)f Fj(x)1607 2839 y(x)1648 2821 y Fb(0)1702 2814 y Fr(<)25 b(\034)1853 2781 y Ft(\026)-41 b Fj(x)1838 2837 y Ft(0)1894 2814 y Fv(\))q(.)39 b(The)25 b(probabilities)c Fh(R)2751 2781 y Fi(\016)2814 2814 y Fv(\()p Fr(\034)2904 2781 y Ft(\026)-41 b Fj(x)2889 2839 y(x)2930 2821 y Fb(0)2985 2814 y Fr(<)25 b(\034)3136 2781 y Ft(\026)-41 b Fj(x)3121 2837 y Ft(0)3177 2814 y Fv(\))25 b(en)m(tering)h(the)50 2959 y(latter)35 b(term)h(are)f(easily)f(dealt)h(with)f(b)m(y)h(means)g (of)h(Lemma)f(2.7:)51 b(note)36 b(that)g(for)f(all)f Fr(x)3159 2926 y Fi(0)3218 2959 y Fk(2)e(T)46 b(n)24 b Fr(x)35 b Fv(and)56 3105 y(\026)-51 b Fr(x)25 b Fk(2)g(T)d Fv(,)k(it)d(follo)m(ws)g(from)g(\(2.128\))j(that)e Fr(\016)k Fv(in)22 b(Lemma)i(2.7)h(ma)m(y)f(b)s(e)f(c)m(hosen)h(as)g Fr(\016)29 b Fv(=)2897 3027 y Fk(p)p 2973 3027 48 4 v 78 x Fr(d)p Fv(,)c(so)f(that)h(inserting)50 3250 y(the)30 b(b)s(ound)f(\(2.126\))k(in)c(\(2.35\))j(yields)914 3474 y Fh(R)974 3436 y Fi(\016)1037 3400 y Ff(\000)1078 3474 y Fr(\034)1133 3436 y Ft(\026)-41 b Fj(x)1118 3496 y(x)1159 3478 y Fb(0)1214 3474 y Fr(<)25 b(\034)1365 3436 y Ft(\026)-41 b Fj(x)1350 3496 y Ft(0)1405 3400 y Ff(\001)1472 3474 y Fk(\024)25 b Fr(e)1610 3436 y Fi(\000)p Fj(N)7 b(h)1775 3409 y Fb(0)1799 3436 y Ft(\()p Fj(d)p Ft(\))1898 3474 y Fr(;)197 b Fv(for)31 b(all)90 b Fr(x)2499 3436 y Fi(0)2549 3474 y Fk(2)25 b(T)43 b(n)26 b Fv(\026)-50 b Fr(x)586 b Fv(\(2)p Fr(:)p Fv(146\))50 3697 y(for)28 b(some)h(constan)m(t)g Fr(h)829 3664 y Fi(0)855 3697 y Fv(\()p Fr(d)p Fv(\))e Fr(>)e Fv(0)j(and)g(large)h(enough)f Fr(N)10 b Fv(.)40 b(T)-8 b(o)29 b(b)s(ound)d Fh(R)2493 3664 y Fi(\016)2555 3697 y Fv(\()q Fr(\034)2646 3664 y Ft(\026)-41 b Fj(x)2631 3720 y Ft(0)2712 3697 y Fr(<)25 b(\034)2863 3664 y Ft(\026)-41 b Fj(x)2852 3720 y Ft(\026)h Fj(x)2904 3697 y Fv(\))28 b(w)m(e)h(simply)d(use)i(that)50 3843 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(this)e(third)g(assertion)h(is)f(v)m(ery)i(similar)d(to)50 5216 y(that)j(of)f(assertion)g(i\).)41 b(Using)29 b(in)g(turn)h (assertion)g(i\))f(of)i(Lemma)g(2.4)g(and)f(Lemma)g(2.5,)650 5439 y Fh(P)720 5338 y Ff(\020)775 5439 y Fr(\034)831 5402 y Ft(\026)-42 b Fj(\021)815 5462 y(\021)895 5439 y Fr(<)24 b(\034)1046 5395 y Ft(\026)-42 b Fj(\021)1030 5470 y(T)10 b Fi(nf)p Fj(\021)r(;)c Ft(\026)-42 b Fj(\021)s Fi(g)1300 5338 y Ff(\021)1380 5439 y Fv(=)25 b Fh(P)1531 5402 y Fi(\016)1588 5338 y Ff(\020)1643 5439 y Fr(\034)1698 5402 y Ft(\026)-41 b Fj(\021)1683 5462 y(\021)1762 5439 y Fr(<)25 b(\034)1914 5395 y Ft(\026)-42 b Fj(\021)1898 5470 y(T)10 b Fi(nf)p Fj(\021)r(;)c Ft(\026)-42 b Fj(\021)s Fi(g)2168 5338 y Ff(\021)2247 5439 y Fv(=)25 b Fh(R)2403 5402 y Fi(\016)2466 5338 y Ff(\020)2520 5439 y Fr(\034)2575 5402 y Ft(\026)-41 b Fj(x)2560 5462 y(x)2641 5439 y Fr(<)25 b(\034)2792 5402 y Ft(\026)-41 b Fj(x)2777 5464 y Fi(T)17 b(nf)p Fj(x;)t Ft(\026)-40 b Fj(x)q Fi(g)3055 5338 y Ff(\021)3432 5439 y Fv(\(2)p Fr(:)p Fv(149\))p eop %%Page: 35 35 35 34 bop 50 295 a Fl(A)l(ging)27 b(in)h(the)g(REM.)f(Part)h(1.)2661 b Fo(35)50 513 y Fv(De\014ning)1083 644 y Fr(R)1152 658 y Ft(1)1218 644 y Fk(\021)25 b Fh(R)1374 606 y Fi(\016)1437 543 y Ff(\020)1491 644 y Fk(f)p Fr(\034)1591 606 y Ft(\026)-41 b Fj(x)1576 666 y(x)1658 644 y Fr(<)25 b(\034)1809 606 y Ft(\026)-41 b Fj(x)1794 669 y Fi(T)16 b(nf)p Fj(x;)t Ft(\026)-40 b Fj(x)q Fi(g)2072 644 y Fk(g)20 b(\\)g(f)p Fr(\034)2318 606 y Ft(\026)-41 b Fj(x)2303 666 y Ft(0)2385 644 y Fr(<)25 b(\034)2536 606 y Ft(\026)-41 b Fj(x)2521 666 y(x)2577 644 y Fk(g)2622 543 y Ff(\021)1083 844 y Fr(R)1152 858 y Ft(2)1218 844 y Fk(\021)25 b Fh(R)1374 807 y Fi(\016)1437 743 y Ff(\020)1491 844 y Fk(f)p Fr(\034)1591 807 y Ft(\026)-41 b Fj(x)1576 867 y(x)1658 844 y Fr(<)25 b(\034)1809 807 y Ft(\026)-41 b Fj(x)1794 869 y Fi(T)16 b(nf)p Fj(x;)t Ft(\026)-40 b Fj(x)q Fi(g)2072 844 y Fk(g)20 b(\\)g(f)p Fr(\034)2318 807 y Ft(\026)-41 b Fj(x)2303 867 y(x)2385 844 y Fr(<)25 b(\034)2536 807 y Ft(\026)-41 b Fj(x)2521 867 y Ft(0)2577 844 y Fk(g)2622 743 y Ff(\021)3432 744 y Fv(\(2)p Fr(:)p Fv(150\))50 1031 y(w)m(e)31 b(ha)m(v)m(e)1271 1177 y Fh(R)1330 1139 y Fi(\016)1393 1076 y Ff(\020)1448 1177 y Fr(\034)1503 1139 y Ft(\026)-41 b Fj(x)1488 1199 y(x)1569 1177 y Fr(<)25 b(\034)1720 1139 y Ft(\026)-41 b Fj(x)1705 1201 y Fi(T)16 b(nf)p Fj(x;)t Ft(\026)-40 b Fj(x)q Fi(g)1983 1076 y Ff(\021)2062 1177 y Fv(=)25 b Fr(R)2227 1191 y Ft(1)2288 1177 y Fv(+)20 b Fr(R)2448 1191 y Ft(2)3432 1177 y Fv(\(2)p Fr(:)p Fv(151\))50 1378 y(Next,)31 b(just)f(as)h(in)e(\(2.135\))k(w)m(e)e(write)1422 1615 y(0)25 b Fk(\024)g Fr(R)1657 1629 y Ft(2)1723 1615 y Fk(\024)g Fh(R)1879 1577 y Fi(\016)1942 1541 y Ff(\000)1984 1615 y Fr(\034)2039 1577 y Ft(\026)-41 b Fj(x)2024 1637 y(x)2105 1615 y Fr(<)25 b(\034)2256 1577 y Ft(\026)-41 b Fj(x)2241 1637 y Ft(0)2296 1541 y Ff(\001)3432 1615 y Fv(\(2)p Fr(:)p Fv(152\))50 1852 y(while)20 b(pro)s(ceeding)i(as)g (in)f(\(2.136\))k(and)d(\(2.137\))j(to)f(treat)f(the)g(term)f Fr(R)2444 1866 y Ft(1)2507 1852 y Fv(yields,)h(in)e(analogy)i(with)e (\(2.138\),)234 2134 y Fh(R)293 2097 y Fi(\016)356 2033 y Ff(\020)410 2134 y Fr(\034)460 2097 y Ft(0)450 2157 y Fj(x)527 2134 y Fr(<)j(\034)672 2097 y Ft(0)662 2159 y Fi(T)17 b(nf)p Fj(x;)t Ft(\026)-40 b Fj(x)q Fi(g)941 2033 y Ff(\021)1010 1979 y(")1063 2134 y Fv(1)21 b Fk(\000)f Fv(\()p Fr(M)30 b Fk(\000)20 b Fv(1\))73 b(sup)1559 2215 y Fj(x)1600 2197 y Fb(0)1626 2215 y Fi(2T)17 b(n)t Ft(\026)-40 b Fj(x)1826 2134 y Fh(R)1886 2097 y Fi(\016)1949 2061 y Ff(\000)1991 2134 y Fr(\034)2046 2097 y Ft(\026)f Fj(x)2031 2157 y(x)2072 2139 y Fb(0)2126 2134 y Fr(<)25 b(\034)2277 2097 y Ft(\026)-41 b Fj(x)2262 2157 y Ft(0)2318 2061 y Ff(\001)2360 1979 y(#)2438 2134 y Fk(\024)25 b Fr(R)2603 2148 y Ft(1)2669 2134 y Fk(\024)g Fh(R)2824 2097 y Fi(\016)2887 2033 y Ff(\020)2942 2134 y Fr(\034)2992 2097 y Ft(0)2982 2157 y Fj(x)3058 2134 y Fr(<)g(\034)3204 2097 y Ft(0)3194 2159 y Fi(T)16 b(nf)p Fj(x;)t Ft(\026)-40 b Fj(x)q Fi(g)3472 2033 y Ff(\021)3432 2325 y Fv(\(2)p Fr(:)p Fv(153\))50 2471 y(Since)22 b(the)h(probabilities)d Fh(R)1002 2438 y Fi(\016)1065 2471 y Fv(\()q Fr(\034)1156 2438 y Ft(\026)-41 b Fj(x)1141 2496 y(x)1182 2477 y Fb(0)1236 2471 y Fr(<)25 b(\034)1387 2438 y Ft(\026)-41 b Fj(x)1372 2493 y Ft(0)1428 2471 y Fv(\),)25 b Fr(x)1565 2438 y Fi(0)1616 2471 y Fk(2)g(T)j(n)11 b Fv(\026)-50 b Fr(x)p Fv(,)24 b(app)s(earing)e(in)g (\(2.152\))k(and)c(\(2.153\))k(ha)m(v)m(e)e(already)50 2616 y(b)s(een)30 b(b)s(ounded)f(in)h(\(2.146\),)k(w)m(e)d(are)h(left)f (to)g(estimate)h Fh(R)2061 2583 y Fi(\016)2124 2515 y Ff(\020)2178 2616 y Fr(\034)2228 2583 y Ft(0)2218 2639 y Fj(x)2294 2616 y Fr(<)25 b(\034)2440 2583 y Ft(0)2430 2645 y Fi(T)17 b(nf)p Fj(x;)t Ft(\026)-40 b Fj(x)p Fi(g)2708 2515 y Ff(\021)2762 2616 y Fv(.)43 b(T)-8 b(o)32 b(do)f(this)e(w)m(e)j (pro)s(ceed)50 2762 y(exactly)i(as)g(in)e(the)i(pro)s(of)f(of)g (\(2.140\))k(\(the)d(only)e(di\013erence)h(b)s(eing)g(that)h(the)f(set) i Fr(J)42 b Fv(in)32 b(Lemma)i(2.8)h(is)50 2907 y(here)30 b(giv)m(en)g(b)m(y)h Fr(J)j Fv(=)25 b Fk(T)43 b(n)21 b(f)p Fr(x;)g Fv(\026)-51 b Fr(x)p Fk(g)31 b Fv(so)g(that)g Fk(j)p Fr(J)9 b Fk(j)26 b Fv(=)f Fr(M)30 b Fk(\000)20 b Fv(1\))31 b(and)f(obtain)436 3030 y Ff(\014)436 3085 y(\014)436 3139 y(\014)436 3194 y(\014)467 3162 y Fh(R)526 3125 y Fi(\016)589 3061 y Ff(\020)644 3162 y Fr(\034)694 3125 y Ft(0)684 3185 y Fj(x)760 3162 y Fr(<)25 b(\034)906 3125 y Ft(0)896 3187 y Fi(T)16 b(nf)p Fj(x;)t Ft(\026)-40 b Fj(x)q Fi(g)1174 3061 y Ff(\021)1248 3162 y Fk(\000)1454 3101 y Fv(1)p 1350 3141 255 4 v 1350 3225 a Fr(M)30 b Fk(\000)20 b Fv(1)1615 3030 y Ff(\014)1615 3085 y(\014)1615 3139 y(\014)1615 3194 y(\014)1671 3162 y Fk(\024)1958 3101 y Fr(d)p 1778 3141 409 4 v 1778 3225 a(N)10 b Fv(\()p Fr(M)30 b Fk(\000)20 b Fv(1\))2212 3061 y Ff(\020)2267 3162 y Fv(1)g(+)g Fr(c)2462 3176 y Ft(0)2518 3089 y Ff(\000)2584 3126 y Fj(d)p 2571 3141 65 4 v 2571 3194 a(N)2646 3089 y Ff(\001)2688 3105 y Ft(1)p Fj(=)p Ft(2)2817 3162 y Fv(ln)14 b Fr(N)2991 3061 y Ff(\021)3432 3162 y Fv(\(2)p Fr(:)p Fv(154\))50 3417 y(for)30 b(some)h(constan)m(t)g Fr(c)822 3431 y Ft(0)889 3417 y Fr(>)25 b Fv(0.)41 b(Collecting)29 b(our)h(b)s(ounds)e(yields)h(the)h(claim)g(of)g(assertion)g(iii\).)39 b Fk(})50 3636 y Fa(Pro)s(of)46 b(of)f(Prop)s(osition)h(2.1,)i(iv\):) 124 b Fv(This)37 b(assertion)i(is)f(nothing)g(but)h(a)g(direct)g (consequence)h(of)50 3781 y(assertion)30 b(ii\))f(since)1134 3926 y Fh(P)1204 3826 y Ff(\020)1259 3926 y Fr(\034)1309 3883 y Fj(\021)1299 3957 y(T)10 b Fi(n)p Fj(\021)1458 3926 y Fr(<)25 b(\034)1604 3889 y Fj(\021)1594 3949 y(\021)1648 3826 y Ff(\021)1728 3926 y Fv(=)1867 3840 y Ff(X)1829 4039 y Ft(\026)-41 b Fj(\021)r Fi(2)p Fj(T)10 b Fi(n)p Fj(\021)2057 3926 y Fh(P)2127 3826 y Ff(\020)2182 3926 y Fr(\034)2232 3883 y Fj(\021)2228 3946 y Ft(\026)-42 b Fj(\021)2302 3926 y Fr(<)25 b(\034)2448 3883 y Fj(\021)2438 3957 y(T)10 b Fi(n)c Ft(\026)-42 b Fj(\021)2572 3826 y Ff(\021)3432 3926 y Fv(\(2)p Fr(:)p Fv(155\))50 4197 y(Th)m(us)32 b(\(2.6\))j(is)e(pro)m(v)m(en.)50 b(F)-8 b(or)34 b(later)f(use)g(\(see)i(the)e(pro)s(of)g(of)h(assertion)f (v\)\))h(let)f(us)g(ho)m(w)m(ev)m(er)h(giv)m(e)g(a)g(full)50 4343 y(deriv)-5 b(ation)29 b(of)h(the)g(lo)m(w)m(er)h(b)s(ound)d(in)g (\(2.6\):)43 b(again,)30 b(with)f(the)i(same)f(notation)g(as)h(in)e (the)h(pro)s(ofs)f(of)i(the)50 4488 y(\014rst)e(t)m(w)m(o)j (assertions,)e(using)e(in)h(turn)g(assertion)h(ii\))f(of)h(Lemma)g(2.4) h(and)f(Lemma)g(2.5,)h(it)f(follo)m(ws)f(from)50 4634 y(\(2.155\))k(that)335 4871 y Fh(P)405 4770 y Ff(\020)460 4871 y Fr(\034)510 4827 y Fj(\021)500 4902 y(T)10 b Fi(n)p Fj(\021)659 4871 y Fr(<)25 b(\034)805 4833 y Fj(\021)795 4893 y(\021)850 4770 y Ff(\021)929 4871 y Fv(=)g Fr(e)1067 4833 y Fi(\000)p Fj(\014)1168 4779 y Fi(p)p 1228 4779 V 54 x Fj(N)7 b(E)1349 4806 y Fn(+)1346 4851 y Fm(\021)1406 4871 y Fh(P)1461 4833 y Fi(\016)1518 4770 y Ff(\020)1572 4871 y Fr(\034)1622 4827 y Fj(\021)1612 4902 y(T)j Fi(n)p Fj(\021)1772 4871 y Fr(<)25 b(\034)1918 4833 y Fj(\021)1908 4893 y(\021)1962 4770 y Ff(\021)2041 4871 y Fv(=)g Fr(e)2179 4833 y Fi(\000)p Fj(\014)2280 4779 y Fi(p)p 2341 4779 V 2341 4833 a Fj(N)6 b(E)2461 4806 y Fn(+)2458 4851 y Fm(\021)2518 4871 y Fh(R)2577 4833 y Fi(\016)2640 4770 y Ff(\020)2695 4871 y Fr(\034)2745 4833 y Fj(x)2735 4895 y Fi(T)16 b(n)p Fj(x)2901 4871 y Fr(<)25 b(\034)3047 4833 y Fj(x)3037 4893 y(x)3092 4770 y Ff(\021)3432 4871 y Fv(\(2)p Fr(:)p Fv(156\))50 5108 y(De\014ning)1110 5239 y Fr(R)1179 5253 y Ft(1)1245 5239 y Fk(\021)g Fh(R)1400 5201 y Fi(\016)1463 5138 y Ff(\020)1517 5239 y Fk(f)p Fr(\034)1612 5201 y Fj(x)1602 5264 y Fi(T)18 b(n)p Fj(x)1769 5239 y Fr(<)25 b(\034)1915 5201 y Fj(x)1905 5261 y(x)1961 5239 y Fk(g)20 b(\\)g(f)p Fr(\034)2202 5201 y Fj(x)2192 5261 y Ft(0)2274 5239 y Fr(<)25 b(\034)2420 5201 y Fj(x)2410 5264 y Fi(T)16 b(n)p Fj(x)2550 5239 y Fk(g)2595 5138 y Ff(\021)1110 5439 y Fr(R)1179 5453 y Ft(2)1245 5439 y Fk(\021)25 b Fh(R)1400 5402 y Fi(\016)1463 5338 y Ff(\020)1517 5439 y Fk(f)p Fr(\034)1612 5402 y Fj(x)1602 5464 y Fi(T)18 b(n)p Fj(x)1769 5439 y Fr(<)25 b(\034)1915 5402 y Fj(x)1905 5462 y(x)1961 5439 y Fk(g)20 b(\\)g(f)p Fr(\034)2202 5402 y Fj(x)2192 5464 y Fi(T)d(n)p Fj(x)2359 5439 y Fr(<)25 b(\034)2505 5402 y Fj(x)2495 5462 y Ft(0)2550 5439 y Fk(g)2595 5338 y Ff(\021)3432 5339 y Fv(\(2)p Fr(:)p Fv(157\))p eop %%Page: 36 36 36 35 bop 50 295 a Fo(36)3270 b Fl(Se)l(ction)27 b Fo(2)50 513 y Fv(w)m(e)k(ha)m(v)m(e)1141 658 y(1)25 b Fk(\025)g Fh(R)1367 621 y Fi(\016)1430 557 y Ff(\020)1484 658 y Fr(\034)1534 621 y Fj(x)1524 683 y Fi(T)17 b(n)p Fj(x)1690 658 y Fr(<)25 b(\034)1836 621 y Fj(x)1826 681 y(x)1882 557 y Ff(\021)1961 658 y Fv(=)g Fr(R)2126 672 y Ft(1)2187 658 y Fv(+)20 b Fr(R)2347 672 y Ft(2)2413 658 y Fk(\025)25 b Fr(R)2578 672 y Ft(1)3432 658 y Fv(\(2)p Fr(:)p Fv(158\))50 867 y(and)30 b(since)969 981 y Fr(R)1038 995 y Ft(1)1104 981 y Fv(=)p Fh(R)1234 943 y Fi(\016)1297 880 y Ff(\020)1351 981 y Fr(\034)1401 943 y Fj(x)1391 1003 y Ft(0)1472 981 y Fr(<)25 b(\034)1618 943 y Fj(x)1608 1006 y Fi(T)17 b(n)p Fj(x)1774 981 y Fr(<)25 b(\034)1920 943 y Fj(x)1910 1003 y(x)1966 880 y Ff(\021)1104 1181 y Fv(=)p Fh(R)1234 1143 y Fi(\016)1297 1181 y Fv(\()p Fr(\034)1382 1143 y Fj(x)1372 1203 y Ft(0)1454 1181 y Fr(<)g(\034)1600 1143 y Fj(x)1590 1203 y Fi(T)1652 1181 y Fv(\))15 b Fh(R)1762 1143 y Fi(\016)1825 1080 y Ff(\020)1879 1181 y Fr(\034)1929 1143 y Ft(0)1919 1206 y Fi(T)i(n)p Fj(x)2085 1181 y Fr(<)25 b(\034)2231 1143 y Ft(0)2221 1203 y Fj(x)2272 1080 y Ff(\021)1104 1381 y Fv(=)15 b([1)21 b Fk(\000)f Fh(R)1431 1343 y Fi(\016)1494 1381 y Fv(\()p Fr(\034)1579 1343 y Fj(x)1569 1403 y Fi(T)1657 1381 y Fr(<)25 b(\034)1803 1343 y Fj(x)1793 1403 y Ft(0)1849 1381 y Fv(\)])1924 1280 y Ff(h)1967 1381 y Fv(1)c Fk(\000)f Fh(R)2184 1343 y Fi(\016)2247 1280 y Ff(\020)2301 1381 y Fr(\034)2351 1343 y Ft(0)2341 1403 y Fj(x)2417 1381 y Fr(<)25 b(\034)2563 1343 y Ft(0)2553 1406 y Fi(T)17 b(n)p Fj(x)2694 1280 y Ff(\021i)3432 1181 y Fv(\(2)p Fr(:)p Fv(159\))50 1581 y(w)m(e)31 b(obtain,)f(pro)s(ceeding)f(as)i(in)e(\(2.144\))k(to)e(b)s (ound)d(1)21 b Fk(\000)f Fh(R)2082 1548 y Fi(\016)2145 1581 y Fv(\()q Fr(\034)2231 1548 y Fj(x)2221 1606 y Fi(T)2308 1581 y Fr(<)25 b(\034)2454 1548 y Fj(x)2444 1604 y Ft(0)2500 1581 y Fv(\),)259 1883 y Fr(R)328 1897 y Ft(1)394 1883 y Fk(\025)490 1728 y Ff(")543 1883 y Fh(R)603 1846 y Fi(\016)666 1883 y Fv(\()p Fr(\034)751 1846 y Fj(x)741 1906 y Ft(0)822 1883 y Fr(<)g(\034)968 1846 y Fj(x)958 1906 y(x)1014 1883 y Fv(\))c Fk(\000)f Fv(\()p Fr(M)30 b Fk(\000)20 b Fv(1\))73 b(sup)1500 1964 y Fj(x)1541 1946 y Fb(0)1567 1964 y Fi(2T)17 b(n)p Fj(x)1768 1883 y Fh(R)1827 1846 y Fi(\016)1890 1883 y Fv(\()p Fr(\034)1975 1846 y Fj(x)1965 1906 y(x)2006 1888 y Fb(0)2061 1883 y Fr(<)25 b(\034)2207 1846 y Fj(x)2197 1906 y Ft(0)2253 1883 y Fv(\))2288 1728 y Ff(#)2356 1782 y(h)2399 1883 y Fv(1)c Fk(\000)f Fh(R)2615 1846 y Fi(\016)2678 1782 y Ff(\020)2733 1883 y Fr(\034)2783 1846 y Ft(0)2773 1906 y Fj(x)2849 1883 y Fr(<)25 b(\034)2995 1846 y Ft(0)2985 1908 y Fi(T)16 b(n)p Fj(x)3125 1782 y Ff(\021)q(i)3432 1883 y Fv(\(2)p Fr(:)p Fv(160\))50 2183 y(No)m(w)42 b(all)e(the)i (probabilities)c(en)m(tering)k(the)f(ab)s(o)m(v)m(e)i(expression)d(ha)m (v)m(e)j(already)e(b)s(een)g(estimated)h(\(see)50 2329 y(resp)s(ectiv)m(ely)e(\(2.147\),)46 b(\(2.146\))e(and)c(\(2.140\))j (for)e(the)g(estimates)g(on)g Fh(R)2649 2296 y Fi(\016)2712 2329 y Fv(\()p Fr(\034)2797 2296 y Fj(x)2787 2351 y Ft(0)2869 2329 y Fr(<)24 b(\034)3014 2296 y Fj(x)3004 2351 y(x)3060 2329 y Fv(\))q(,)43 b Fh(R)3224 2296 y Fi(\016)3287 2329 y Fv(\()p Fr(\034)3372 2296 y Fj(x)3362 2354 y(x)3403 2336 y Fb(0)3457 2329 y Fr(<)25 b(\034)3603 2296 y Fj(x)3593 2351 y Ft(0)3649 2329 y Fv(\))q(,)50 2474 y Fr(x)102 2441 y Fi(0)153 2474 y Fk(2)f(T)43 b(n)21 b Fr(x)p Fv(,)30 b(and)g Fh(R)741 2441 y Fi(\016)804 2373 y Ff(\020)858 2474 y Fr(\034)908 2441 y Ft(0)898 2497 y Fj(x)974 2474 y Fr(<)25 b(\034)1120 2441 y Ft(0)1110 2503 y Fi(T)17 b(n)p Fj(x)1251 2373 y Ff(\021)1305 2474 y Fv(\).)41 b(Plugging)29 b(these)i(estimates)g(in)e(\(2.160\))k(w)m(e)e(obtain)728 2757 y Fh(R)788 2719 y Fi(\016)851 2656 y Ff(\020)905 2757 y Fr(\034)955 2719 y Fj(x)945 2781 y Fi(T)17 b(n)p Fj(x)1111 2757 y Fr(<)25 b(\034)1257 2719 y Fj(x)1247 2779 y(x)1303 2656 y Ff(\021)1383 2757 y Fk(\025)1478 2683 y Ff(\000)1520 2757 y Fv(1)c Fk(\000)1708 2721 y Ft(1)p 1688 2736 77 4 v 1688 2788 a Fj(M)1775 2683 y Ff(\001)1831 2656 y(\020)1886 2757 y Fv(1)f Fk(\000)2067 2721 y Fj(d)p 2053 2736 65 4 v 2053 2788 a(N)2144 2656 y Ff(\020)2198 2757 y Fv(1)h(+)f Fr(c)2394 2771 y Ft(4)2450 2683 y Ff(\000)2516 2721 y Fj(d)p 2502 2736 V 2502 2788 a(N)2578 2683 y Ff(\001)2619 2699 y Ft(1)p Fj(=)p Ft(2)2749 2757 y Fv(ln)14 b Fr(N)2923 2656 y Ff(\021\021)3432 2757 y Fv(\(2)p Fr(:)p Fv(161\))50 3011 y(for)30 b(some)h(constan)m(t)g Fr(c)822 3025 y Ft(4)889 3011 y Fr(>)25 b Fv(0.)41 b(Inserting)29 b(\(2.161\))k(in)c(\(2.156\))k(pro)m(v)m(es)e(the)g(lo)m(w)m(er)f(b)s (ound)f(of)h(\(2.6\).)43 b Fk(})159 3229 y Fv(As)30 b(is)f(b)m(y)h(no)m (w)g(routine,)f(the)i(pro)s(of)e(of)h(assertion)g(v\))g(of)g(Prop)s (osition)e(2.1)k(b)s(egins)c(as)i(in)f(\(2.156\):)43 b(w)m(e)50 3375 y(\014rst)29 b(in)m(v)m(ok)m(e)j(Lemma)e(2.4)i(to)f (write)1147 3629 y Fh(P)15 b Fv(\()r Fr(\034)1304 3592 y Fj(\033)1294 3652 y(T)1378 3629 y Fr(<)25 b(\034)1524 3592 y Fj(\033)1514 3652 y(\033)1573 3629 y Fv(\))g(=)g Fr(e)1771 3592 y Fi(\000)p Fj(\014)1872 3538 y Fi(p)p 1933 3538 V 1933 3592 a Fj(N)6 b(E)2053 3564 y Fn(+)2050 3610 y Fm(\033)2110 3629 y Fh(P)2165 3592 y Fi(\016)2222 3629 y Fv(\()q Fr(\034)2308 3592 y Fj(\033)2298 3652 y(T)2382 3629 y Fr(<)25 b(\034)2528 3592 y Fj(\033)2518 3652 y(\033)2577 3629 y Fv(\))820 b(\(2)p Fr(:)p Fv(162\))50 3884 y(and)30 b(next)g(use)g(Lemma)h(2.5)g(to)g(express)f(the)h(last)f 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y(b)m(y)29 b(a)g(non-random)g(p)s(oin)m(t)f Fr(\033)s Fv(.)40 b(T)-8 b(o)30 b(pro)s(ceed)f(an)m(y)g(further)f(w)m(e)i(th)m(us)e(need)h(to)h (in)m(v)m(estigate)h(its)d(prop)s(erties.)p eop %%Page: 37 37 37 36 bop 50 295 a Fl(A)l(ging)27 b(in)h(the)g(REM.)f(Part)h(1.)2661 b Fo(37)50 513 y Fs(Lump)s(ed)34 b(c)m(hain)h(induced)h(b)m(y)f(the)f (T)-9 b(op)36 b(and)e(a)h(non-random)g(p)s(oin)m(t)159 731 y Fv(In)f(order)g(to)h(study)f(the)h(mapping)e Fr(\015)1475 745 y Fj(T)10 b Fi([)p Fj(\033)1660 731 y Fv(w)m(e)35 b(m)m(ust)g(go)g(bac)m(k)g(to)h(its)e(de\014nition)e(\(see)k(Section)e (2.1\).)50 876 y(Most)e(of)f(the)h(results)d(w)m(e)j(will)d(establish)g (b)s(elo)m(w)h(rely)h(on)g(the)g(simple)e(observ)-5 b(ation)31 b(that)h(the)f(partition)50 1022 y Fk(P)113 1036 y Fj(T)10 b Fi([)p Fj(\033)264 1022 y Fv(\(\003\))41 b(induced)d(b)m(y)i Fr(T)f Fk([)26 b Fr(\033)43 b Fv(ma)m(y)e(b)s(e)e(constructed)i(b)m(y)f (\014rst)f(constructing)h(the)g(partition)e Fk(P)3518 1036 y Fj(T)3576 1022 y 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5087 y Fv(\))3275 5104 y Fj(k)3316 5085 y Fb(0)3340 5104 y Fi(2K)p Ft(\()p Fj(T)10 b Fi([)p Fj(\033)r Ft(\))3685 5087 y Fv(.)50 5233 y(F)-8 b(or)31 b(eac)m(h)g Fr(k)e Fk(2)c(f)p Fv(1)p Fr(;)15 b(:)g(:)g(:)i(;)e(d)p Fk(g)p Fv(,)32 b(let)e Fr(s)1194 5247 y Fi(\003)1261 5233 y Fk(2)25 b(f\000)p Fv(1)p Fr(;)15 b Fv(1)p Fk(g)32 b Fv(b)s(e)e(de\014ned)f(through)1464 5462 y Fk(j)p Fv(\003)1552 5421 y Fj(s)1586 5430 y Fb(\003)1552 5489 y Fj(k)1629 5462 y Fv(\()p Fr(T)13 b Fv(\))p Fk(j)26 b(\025)f(j)p Fv(\003)2000 5421 y Fi(\000)p Fj(s)2091 5430 y Fb(\003)2000 5489 y Fj(k)2134 5462 y Fv(\()p Fr(T)13 b Fv(\))p Fk(j)1137 b Fv(\(2)p Fr(:)p Fv(176\))p eop %%Page: 39 39 39 38 bop 50 295 a Fl(A)l(ging)27 b(in)h(the)g(REM.)f(Part)h(1.)2661 b Fo(39)50 513 y Fv(and)30 b(set)688 667 y Fk(D)e Fv(=)d Fk(f)p Fr(k)977 629 y Fi(0)1028 667 y Fk(2)g(f)p Fv(1)p Fr(;)15 b(:)g(:)g(:)j(;)d(d)1454 629 y Fi(0)1480 667 y Fk(g)25 b(j)h Fv(\003)1664 681 y Fj(k)1705 662 y Fb(0)1733 667 y Fv(\()p Fr(T)34 b Fk([)20 b 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b(F)-7 b(or)34 b(al)5 b(l)33 b Fr(\021)28 b Fk(2)d Fr(T)13 b Fq(,)1555 1620 y Fr(\031)1610 1583 y Fi(\003)1652 1620 y Fr(\015)1699 1634 y Fj(T)d Fi([)p Fj(\033)1850 1620 y Fv(\()p Fr(\021)s Fv(\))26 b(=)f Fr(\015)2137 1634 y Fj(T)2194 1620 y Fv(\()p Fr(\021)s Fv(\))1120 b(\(2)p Fr(:)p Fv(178\))159 1838 y Fq(F)-7 b(or)34 b Fv(0)25 b Fk(\024)g Fr(")h(<)670 1802 y Ft(1)p 670 1817 37 4 v 670 1869 a(2)717 1838 y Fq(,)32 b(de\014ne)860 2101 y Fk(A)933 2115 y Fj(")971 2101 y Fv(\()p Fr(\033)s Fv(\))26 b(=)1218 2000 y Ff(n)1279 2101 y Fr(\021)i Fk(2)d Fr(T)1529 1996 y Ff(\014)1529 2051 y(\014)1529 2105 y(\014)1585 2101 y Fk(k)p Fr(\031)1685 2063 y Fi(\003)1727 2101 y Fr(\015)1774 2115 y Fj(T)10 b Fi([)p Fj(\033)1925 2101 y Fv(\()p Fr(\021)s Fv(\))21 b Fk(\000)f Fr(\031)2210 2063 y Fi(\003)2252 2101 y Fr(\015)2299 2115 y Fj(T)10 b Fi([)p Fj(\033)2450 2101 y Fv(\()p Fr(\033)s Fv(\))p Fk(k)2620 2115 y Ft(2)2687 2101 y Fk(\024)25 b Fr(")2825 2019 y Fk(p)p 2901 2019 48 4 v 82 x Fr(d)2949 2000 y 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Fv(\))26 b(=)f(\003)2094 4287 y Fj(s)2094 4347 y(k)2139 4325 y Fv(\()p Fr(T)13 b Fv(\))1157 b(\(2)p Fr(:)p Fv(182\))50 4588 y(Fix)30 b Fr(k)263 4555 y Fi(0)314 4588 y Fk(2)24 b(f)p Fv(1)p Fr(;)15 b(:)g(:)g(:)j(;)d(d)739 4555 y Fi(0)765 4588 y Fk(g)p Fv(.)41 b(It)31 b(follo)m(ws)e(from)h(de\014nition)e (\(2.16\))33 b(and)d(\(2.182\))j(that)1313 4858 y Fr(\015)1365 4820 y Fj(k)1406 4793 y Fb(0)1360 4880 y Fj(T)10 b Fi([)p Fj(\033)1510 4858 y Fv(\()p Fr(\021)s Fv(\))27 b(=)1886 4796 y(1)p 1762 4837 295 4 v 1762 4920 a Fk(j)p Fv(\003)1850 4889 y Fj(s)1850 4947 y(k)1895 4920 y Fv(\()p Fr(T)13 b Fv(\))p Fk(j)2154 4771 y Ff(X)2083 4970 y Fj(i)p Fi(2)p Ft(\003)2208 4949 y Fm(s)2208 4995 y(k)2247 4970 y Ft(\()p Fj(T)d Ft(\))2371 4858 y Fr(\021)2416 4872 y Fj(i)3432 4858 y Fv(\(2)p Fr(:)p Fv(183\))50 5199 y(No)m(w)31 b(b)m(y)f (\(2.166\),)j(w)m(e)e(ha)m(v)m(e,)1104 5462 y(\003)1167 5424 y Fj(s)1167 5484 y(k)1212 5462 y Fv(\()p Fr(T)13 b Fv(\))26 b Fk(\022)e Fv(\003)1532 5476 y Fj(k)1577 5462 y Fv(\()p Fr(T)13 b Fv(\))92 b(for)30 b(eac)m(h)92 b Fr(s)25 b Fk(2)g(f\000)p Fv(1)p Fr(;)15 b Fv(1)p Fk(g)778 b Fv(\(2)p Fr(:)p Fv(184\))p eop %%Page: 40 40 40 39 bop 50 295 a Fo(40)3270 b Fl(Se)l(ction)27 b Fo(2)50 513 y Fv(But)j(assertion)g(iii\))f(of)h(Lemma)h(2.2)g(states)h(that,) 1248 781 y Fr(\021)1293 795 y Fj(i)1348 781 y Fv(=)25 b Fr(\015)1496 744 y Fj(k)1491 804 y(T)1549 781 y Fv(\()p Fr(\021)s Fv(\))p Fr(;)107 b Fv(for)30 b(all)90 b Fr(i)26 b Fk(2)f Fv(\003)2331 795 y Fj(k)2375 781 y Fv(\()p Fr(T)13 b Fv(\))921 b(\(2)p Fr(:)p Fv(185\))50 1050 y(Hence,)31 b(com)m(bining)e(\(2.185\))k(and)d(\(2.183\))j(w)m(e)e(get)1078 1318 y Fr(\015)1130 1281 y Fj(k)1171 1253 y Fb(0)1125 1341 y Fj(T)10 b Fi([)p Fj(\033)1276 1318 y Fv(\()p Fr(\021)s Fv(\))26 b(=)f Fr(\015)1568 1281 y Fj(k)1563 1341 y(T)1621 1318 y Fv(\()p Fr(\021)s Fv(\))92 b(for)30 b(eac)m(h)92 b Fr(s)25 b Fk(2)g(f\000)p Fv(1)p Fr(;)15 b Fv(1)p Fk(g)752 b Fv(\(2)p Fr(:)p Fv(186\))50 1587 y(Since)32 b(\(2.186\))k(holds)c (for)h(eac)m(h)i Fr(s)30 b Fk(2)g(f\000)p Fv(1)p Fr(;)15 b Fv(1)p Fk(g)35 b Fv(it)e(holds)f(true)h(for)g Fr(s)d Fv(=)g Fr(s)2571 1601 y Fi(\003)2613 1587 y Fv(.)49 b(W)-8 b(e)35 b(therefore)f(ha)m(v)m(e)g(pro)m(v)m(en)50 1732 y(that)f(for)f(eac)m(h)h Fr(k)646 1699 y Fi(0)700 1732 y Fk(2)28 b(f)p Fv(1)p Fr(;)15 b(:)g(:)g(:)i(;)e(d)1128 1699 y Fi(0)1155 1732 y Fk(g)32 b Fv(and)g(eac)m(h)h Fr(k)f Fk(2)c(f)p Fv(1)p Fr(;)15 b(:)g(:)g(:)i(;)e(d)p Fk(g)34 b Fv(related)e(through)f(\003)2916 1746 y Fj(k)2957 1728 y Fb(0)2986 1732 y Fv(\()p Fr(T)j Fk([)21 b Fr(\033)s Fv(\))29 b(=)f(\003)3471 1692 y Fj(s)3505 1701 y Fb(\003)3471 1760 y Fj(k)3548 1732 y Fv(\()p Fr(T)13 b Fv(\),)50 1878 y Fr(\015)102 1845 y Fj(k)143 1817 y Fb(0)97 1903 y Fj(T)d Fi([)p Fj(\033)248 1878 y Fv(\()p Fr(\021)s Fv(\))31 b(=)e Fr(\015)549 1845 y Fj(k)544 1903 y(T)601 1878 y Fv(\()p Fr(\021)s Fv(\).)50 b(But)34 b(this,)f(in)f(view)g(of)h (\(2.177\),)k(implies)30 b(that)k Fr(\031)2514 1845 y Fi(\003)2556 1878 y Fr(\015)2603 1892 y Fj(T)10 b Fi([)p Fj(\033)2754 1878 y Fv(\()p Fr(\021)s Fv(\))31 b(=)e Fr(\015)3050 1892 y Fj(T)3107 1878 y Fv(\()p Fr(\021)s Fv(\),)36 b(concluding)50 2023 y(the)30 b(pro)s(of)g(of)h(assertion)f (i\).)159 2241 y(W)-8 b(e)31 b(no)m(w)g(turn)e(to)i(the)f(pro)s(of)g (of)g(assertion)g(ii\).)39 b(Note)32 b(that)f(b)m(y)f(\(2.178\),)j Fk(A)2795 2255 y Fj(")2834 2241 y Fv(\()p Fr(\033)s Fv(\))e(ma)m(y)g(b) s(e)e(written)g(as)900 2510 y Fk(A)973 2524 y Fj(")1012 2510 y Fv(\()p Fr(\033)s Fv(\))d(=)1259 2409 y Ff(n)1319 2510 y Fr(\021)j Fk(2)c Fr(T)1570 2405 y Ff(\014)1570 2460 y(\014)1570 2514 y(\014)1625 2510 y Fk(k)p Fr(\015)1717 2524 y Fj(T)1775 2510 y Fv(\()p Fr(\021)s Fv(\))d Fk(\000)d Fr(\031)2060 2472 y Fi(\003)2102 2510 y Fr(\015)2149 2524 y Fj(T)10 b Fi([)p Fj(\033)2300 2510 y Fv(\()p Fr(\033)s Fv(\))p Fk(k)2470 2524 y Ft(2)2537 2510 y Fk(\024)25 b Fr(")2675 2428 y Fk(p)p 2752 2428 48 4 v 2752 2510 a Fr(d)2799 2409 y Ff(o)3432 2510 y Fv(\(2)p Fr(:)p Fv(187\))50 2786 y(Assume)33 b(that)i Fk(A)664 2800 y Fj(")702 2786 y Fv(\()p Fr(\033)s Fv(\))d Fk(6)p Fv(=)f Fk(;)p Fv(.)52 b(Then)33 b(there)h(exists)g Fr(\021)g Fk(2)d Fr(T)46 b Fv(suc)m(h)34 b(that)h Fk(k)p Fr(\015)2587 2800 y Fj(T)2644 2786 y Fv(\()p Fr(\021)s Fv(\))24 b Fk(\000)e Fr(\031)2934 2753 y Fi(\003)2976 2786 y Fr(\015)3023 2800 y Fj(T)10 b Fi([)p Fj(\033)3174 2786 y Fv(\()p Fr(\033)s Fv(\))p Fk(k)3344 2800 y Ft(2)3417 2786 y Fk(\024)31 b Fr(")3561 2708 y Fk(p)p 3637 2708 V 78 x Fr(d)q Fv(.)50 2931 y(Th)m(us,)713 3061 y(inf)p 659 3086 40 4 v 659 3124 a Fj(\021)s Fi(2)p Fj(T)10 b Fi(n)p Fj(\021)893 3061 y Fk(k)p Fr(\015)985 3075 y Fj(T)1042 3061 y Fv(\()p 1077 3011 49 4 v Fr(\021)t Fv(\))21 b Fk(\000)f Fr(\031)1328 3024 y Fi(\003)1370 3061 y Fr(\015)1417 3075 y Fj(T)10 b Fi([)p Fj(\033)1567 3061 y Fv(\()p Fr(\033)s Fv(\))p Fk(k)1737 3075 y Ft(2)1805 3061 y Fk(\025)79 b Fv(inf)p 1901 3086 40 4 v 1901 3124 a Fj(\021)r Fi(2)p Fj(T)10 b Fi(n)p Fj(\021)2134 3061 y Fk(k)p Fr(\015)2226 3075 y Fj(T)2284 3061 y Fv(\()p 2319 3011 49 4 v Fr(\021)t Fv(\))20 b Fk(\000)g Fr(\015)2561 3075 y Fj(T)2619 3061 y Fv(\()p Fr(\021)s Fv(\))p Fk(k)2782 3075 y Ft(2)2844 3061 y Fk(\000)g Fr(")2977 2979 y Fk(p)p 3053 2979 48 4 v 82 x Fr(d)1805 3275 y Fk(\025)25 b Fv(\(1)c Fk(\000)f Fv(2)p Fr(\016)s Fv(\()p Fr(N)10 b Fv(\)\))2369 3193 y Fk(p)p 2447 3193 V 2447 3275 a Fr(d)20 b Fk(\000)g Fr(")2647 3193 y Fk(p)p 2723 3193 V 82 x Fr(d)3432 3162 y Fv(\(2)p Fr(:)p Fv(188\))50 3470 y(where)25 b(the)i(last)f(inequalit) m(y)e(follo)m(ws)h(from)h(Lemma)g(2.13.)41 b(Since)25 b(for)h(all)f(0)h Fk(\024)f Fr(")g(<)2928 3434 y Ft(1)p 2928 3449 37 4 v 2928 3502 a(2)2975 3470 y Fv(,)i(1)12 b Fk(\000)g Fv(2)p Fr(\016)s Fv(\()p Fr(N)e Fv(\))i Fk(\000)g Fr(")26 b(>)f(")50 3616 y Fv(pro)m(vided)33 b(that)i Fr(N)44 b Fv(is)33 b(su\016cien)m(tly)g(large,)j(w)m(e)f(conclude)f (that)h Fk(jA)2368 3630 y Fj(")2406 3616 y Fv(\()p Fr(\033)s Fv(\))p Fk(j)f Fv(=)d(1.)53 b(The)34 b(claim)g(of)g(assertion)50 3761 y(ii\))29 b(is)g(th)m(us)h(pro)m(v)m(en)h(and)f(it)g(remains)f(to) i(pro)m(v)m(e)g(iii\).)159 3979 y(T)-8 b(o)41 b(do)g(so)g(note)g(that)h (pro)s(ceeding)e(just)g(as)h(in)e(the)i(pro)s(of)f(of)h(\(2.171\),)47 b(w)m(e)41 b(easily)f(see)h(that)h Fk(D)j(\022)50 4125 y(K)q Fv(\()p Fr(T)34 b Fk([)20 b Fr(\033)s Fv(\).)41 b(Hence,)31 b(for)f(all)g Fr(y)s(;)15 b(y)1176 4092 y Fi(0)1226 4125 y Fk(2)25 b Fh(R)1372 4092 y Fj(d)1410 4064 y Fb(0)1445 4125 y Fv(,)1278 4393 y Fk(k)p Fr(\031)1378 4356 y Fi(0)1404 4393 y Fr(y)1452 4356 y Fi(0)1497 4393 y Fk(\000)20 b Fr(\031)1643 4356 y Fi(0)1669 4393 y Fr(y)s Fk(k)1762 4407 y Ft(2)1828 4393 y Fk(\025)25 b(k)p Fr(\031)2024 4356 y Fi(\003)2066 4393 y Fr(y)2114 4356 y Fi(0)2160 4393 y Fk(\000)20 b Fr(\031)2306 4356 y Fi(\003)2348 4393 y Fr(y)s Fk(k)2441 4407 y Ft(2)3432 4393 y Fv(\(2)p Fr(:)p Fv(189\))50 4662 y(No)m(w)26 b(\(2.180\))j(is)24 b(an)i(immediate)f(consequence)h(of)g(\(2.189\))i(and)d(the)h (de\014nition)e(of)h Fk(A)3009 4676 y Fj(")3048 4662 y Fv(\()p Fr(\033)s Fv(\))h(while)e(\(2.181\))50 4807 y(results)40 b(from)i(the)g(com)m(bination)f(of)h(\(2.189\))i(and)d (\(2.128\))k(of)d(Lemma)g(2.13.)76 b(Assertion)41 b(iii\))f(b)s(eing)50 4953 y(pro)m(v)m(en,)31 b(the)f(pro)s(of)g(of)h(the)f(lemma)g(is)f (done.)41 b Fk(})159 5171 y Fv(W)-8 b(e)45 b(are)f(no)m(w)g(ready)g(to) g(pro)m(v)m(e)h(the)f(last)g(t)m(w)m(o)h(assertions)e(of)h(Prop)s (osition)e(2.1.)82 b(The)43 b(follo)m(wing)50 5316 y(notation)34 b(will)c(b)s(e)j(used)g(throughout:)47 b Fr(I)37 b Fk(\021)30 b Fr(T)35 b Fk([)22 b Fr(\033)s Fv(,)35 b Fk(I)i(\021)30 b Fr(\015)2145 5330 y Fj(I)2187 5316 y Fv(\()p Fr(I)7 b Fv(\),)35 b Fr(y)e Fk(\021)d Fr(\015)2590 5330 y Fj(I)2632 5316 y Fv(\()p Fr(\033)s Fv(\),)36 b(and)j(\026)-51 b Fr(y)33 b Fk(\021)d Fr(\015)3224 5330 y Fj(I)3266 5316 y Fv(\()5 b(\026)-50 b Fr(\033)s Fv(\).)51 b(It)33 b(will)50 5462 y(moreo)m(v)m(er)f(b)s(e)d(assumed)h(that)h Fr(!)d Fk(2)d(E)1346 5476 y Fj(N)1415 5462 y Fv(.)p eop %%Page: 41 41 41 40 bop 50 295 a Fl(A)l(ging)27 b(in)h(the)g(REM.)f(Part)h(1.)2661 b Fo(41)50 513 y Fa(Pro)s(of)33 b(of)f(Prop)s(osition)i(2.1,)f(v\))f (and)h(vi\):)84 b Fv(With)27 b(the)i(notation)f(in)m(tro)s(duced)f(ab)s (o)m(v)m(e,)j(\(2.163\))g(and)50 658 y(\(2.164\))j(read,)d(resp)s (ectiv)m(ely)-8 b(,)1262 804 y Fh(P)1317 766 y Fi(\016)1374 804 y Fv(\()q Fr(\034)1460 766 y Fj(\033)1450 826 y(T)1534 804 y Fr(<)25 b(\034)1680 766 y Fj(\033)1670 826 y(\033)1729 804 y Fv(\))h(=)f Fh(R)1945 766 y Fi(\016)2008 703 y Ff(\020)2062 804 y Fr(\034)2112 760 y Fj(y)2102 835 y Fi(I)t(n)p Fj(y)2254 804 y Fr(<)g(\034)2400 766 y Fj(y)2390 826 y(y)2443 703 y Ff(\021)3432 804 y Fv(\(2)p Fr(:)p Fv(190\))50 1001 y(and)1277 1146 y Fh(P)1347 1073 y Ff(\000)1389 1146 y Fr(\034)1443 1109 y Ft(\026)-40 b Fj(\033)1429 1169 y(\033)1513 1146 y Fk(\024)25 b Fr(\034)1663 1109 y Ft(\026)-40 b Fj(\033)1649 1169 y(T)1708 1073 y Ff(\001)1775 1146 y Fv(=)25 b Fh(R)1931 1109 y Fi(\016)1994 1045 y Ff(\020)2048 1146 y Fr(\034)2103 1109 y Ft(\026)-41 b Fj(y)2088 1169 y(y)2167 1146 y Fr(<)25 b(\034)2318 1103 y Ft(\026)-41 b Fj(y)2303 1177 y Fi(I)t(n)p Fj(y)2429 1045 y Ff(\021)3432 1146 y Fv(\(2)p Fr(:)p Fv(191\))50 1344 y(W)-8 b(e)34 b(ma)m(y)f(no)m(w)f(distinguish)d(t)m(w)m(o)34 b(cases)f(since,)g(according)g(to)g(assertion)f(ii\))f(of)i(Lemma)g (2.16,)i(either)d Fr(\033)50 1489 y Fv(is)d(suc)m(h)h(that,)i(case)f (1\),)g Fk(A)970 1503 y Fj(")1009 1489 y Fv(\()p Fr(\033)s Fv(\))26 b(=)f Fk(;)p Fv(,)31 b(or)f(else)g(case)i(2\),)f Fk(A)2042 1503 y Fj(")2081 1489 y Fv(\()p Fr(\033)s Fv(\))26 b(=)f Fk(f)p Fr(\021)s Fk(g)32 b Fv(for)e(some)g Fr(\021)f Fk(2)c Fr(T)13 b Fv(.)159 1707 y(In)36 b(case)j(1\),)h(a)d(simple)e (adaptation)j(of)f(the)h(pro)s(of)e(of)h(the)h(lo)m(w)m(er)f(b)s(ound)e (\(2.161\))40 b(of)e(assertion)f(iii\))50 1853 y(yields)741 1998 y Fh(R)801 1961 y Fi(\016)864 1897 y Ff(\020)918 1998 y Fr(\034)968 1955 y Fj(y)958 2029 y Fi(I)t(n)p Fj(y)1109 1998 y Fr(<)25 b(\034)1255 1961 y Fj(y)1245 2021 y(y)1299 1897 y Ff(\021)1379 1998 y Fk(\025)1475 1897 y Ff(\020)1529 1998 y Fv(1)c Fk(\000)1762 1962 y Ft(1)p 1696 1977 169 4 v 1696 2030 a Fj(M)7 b Ft(+1)1875 1897 y Ff(\021)1945 1925 y(\000)1987 1998 y Fv(1)20 b Fk(\000)2167 1962 y Fj(d)p 2154 1977 65 4 v 2154 2030 a(N)2245 1998 y Fv(\(1)h(+)e Fr(c)2475 2012 y Ft(5)2517 1998 y Fv(\(1)i(+)e Fr(\016)s Fv(\()p Fr(N)10 b Fv(\)\)\))2977 1925 y Ff(\001)3432 1998 y Fv(\(2)p Fr(:)p Fv(192\))50 2196 y(for)33 b(some)h(constan)m(t)h Fr(c)832 2210 y Ft(5)903 2196 y Fr(>)30 b Fv(0.)51 b(Similarly)-8 b(,)31 b(retracing)j(the)f(pro)s(of)g(of)h(the)f(upp)s(er)f(b)s(ound)f(of)j (assertion)f(i\),)50 2341 y(w)m(e)e(readily)d(obtain)i(that)877 2464 y Ff(\014)877 2519 y(\014)877 2574 y(\014)908 2569 y Fh(R)967 2531 y Fi(\016)1030 2468 y Ff(\020)1085 2569 y Fr(\034)1140 2531 y Ft(\026)-41 b Fj(y)1125 2591 y(y)1204 2569 y Fr(<)25 b(\034)1355 2525 y Ft(\026)-41 b Fj(y)1340 2600 y Fi(I)t(n)p Fj(y)1466 2468 y Ff(\021)1540 2569 y Fk(\000)1708 2533 y Ft(1)p 1642 2548 169 4 v 1642 2600 a Fj(M)7 b Ft(+1)1821 2464 y Ff(\014)1821 2519 y(\014)1821 2574 y(\014)1877 2569 y Fk(\024)2035 2533 y Fj(d)p 1983 2548 141 4 v 1983 2600 a(N)g(M)2150 2569 y Fv(\(1)21 b(+)f Fr(c)2381 2583 y Ft(6)2422 2569 y Fv(\(1)h(+)f Fr(\016)s Fv(\()p Fr(N)10 b Fv(\)\)\))552 b(\(2)p Fr(:)p Fv(193\))50 2797 y(for)30 b(some)h(constan)m(t)g Fr(c)822 2811 y Ft(6)889 2797 y Fr(>)25 b Fv(0.)159 3015 y(Case)39 b(2\))h(will)c(also)j(b)s(e)f(brough)m(t)g(bac)m(k)i(to)g(w)m(ell)d (kno)m(wn)i(situations)e(once)j(observ)m(ed)f(that,)j(setting)50 3161 y Fr(x)25 b Fk(\021)g Fr(\015)270 3175 y Fj(I)312 3161 y Fv(\()p Fr(\021)s Fv(\),)1138 3306 y Fh(R)1198 3269 y Fi(\016)1261 3205 y Ff(\020)1315 3306 y Fr(\034)1365 3262 y Fj(y)1355 3337 y Fi(I)t(n)p Fj(y)1507 3306 y Fr(<)g(\034)1653 3269 y Fj(y)1643 3329 y(y)1696 3205 y Ff(\021)1776 3306 y Fk(\025)g Fh(R)1931 3269 y Fi(\016)1994 3205 y Ff(\020)2049 3306 y Fr(\034)2099 3262 y Fj(y)2089 3337 y Fi(I)t(nf)p Fj(y)r(;x)p Fi(g)2377 3306 y Fr(<)g(\034)2523 3269 y Fj(y)2513 3329 y(y)2567 3205 y Ff(\021)3432 3306 y Fv(\(2)p Fr(:)p Fv(194\))50 3503 y(while)1138 3649 y Fh(R)1198 3611 y Fi(\016)1261 3548 y Ff(\020)1315 3649 y Fr(\034)1371 3611 y Ft(\026)-42 b Fj(y)1355 3671 y(y)1434 3649 y Fr(<)25 b(\034)1586 3605 y Ft(\026)-42 b Fj(y)1570 3680 y Fi(I)t(n)p Fj(y)1696 3548 y Ff(\021)1776 3649 y Fk(\024)25 b Fh(R)1931 3611 y Fi(\016)1994 3548 y Ff(\020)2049 3649 y Fr(\034)2104 3611 y Ft(\026)-41 b Fj(y)2089 3671 y(y)2168 3649 y Fr(<)25 b(\034)2319 3605 y Ft(\026)-41 b Fj(y)2304 3680 y Fi(I)t(nf)p Fj(y)r(;x)p Fi(g)2567 3548 y Ff(\021)3432 3649 y Fv(\(2)p Fr(:)p Fv(195\))50 3846 y(Then,)29 b(pro)s(o)s(ceeding)h(as)g(in)f(the) i(pro)s(of)f(of)g(\(2.192\))j(w)m(e)e(obtain)f(that)731 4074 y Fh(R)791 4037 y Fi(\016)854 3973 y Ff(\020)908 4074 y Fr(\034)958 4030 y Fj(y)948 4105 y Fi(I)t(nf)p Fj(y)r(;x)p Fi(g)1237 4074 y Fr(<)25 b(\034)1383 4037 y Fj(y)1373 4097 y(y)1426 3973 y Ff(\021)1506 4074 y Fk(\025)1602 4001 y Ff(\000)1644 4074 y Fv(1)20 b 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Fv(1\))1923 1870 y Ff(\022)1990 1998 y Fv(1)h(+)e Fk(V)2202 2015 y Fj(\014)s(;N)s(;T)10 b Ft(\()p Fj(E)s Ft(\))2521 1998 y Fr(e)2563 1961 y Fj(E)s(=)p Ft(2)2770 1937 y Fr(\013)21 b Fk(\000)f Fv(1)p 2709 1977 337 4 v 2709 1995 a Fk(p)p 2785 1995 261 4 v 72 x Fv(2)p Fr(\013)h Fk(\000)f Fv(1)3057 1870 y Ff(\023)3523 1998 y Fv(\(3)p Fr(:)p Fv(3\))50 2261 y(where)30 b Fr(M)35 b Fv(=)25 b Fk(j)p Fr(T)13 b Fv(\()p Fr(E)5 b Fv(\))p Fk(j)p Fv(.)50 2479 y Fs(Remark:)42 b Fv(A)23 b(remark)g(is)f(in)f(order)i(concerning)g(the)g(random)f(v)-5 b(ariables)22 b(de\014ned)g(in)f(\(3.1\))k(to)f(\(3.3\).)40 b(The)50 2624 y(b)s(eha)m(vior)33 b(of)h Fr(Z)594 2638 y Fj(\014)s(;N)729 2624 y Fv(\()p Fr(T)830 2591 y Fj(c)867 2624 y Fv(\()p Fr(E)5 b Fv(\)\))36 b(will)31 b(b)s(e)j(studied)e(in)h (Lemma)h(3.3.)53 b(It)34 b(will)e(in)h(particular)f(b)s(e)h (established)50 2770 y(that)42 b Fr(Z)320 2784 y Fj(\014)s(;N)456 2770 y Fv(\()p Fr(T)557 2737 y Fj(c)594 2770 y Fv(\()p Fr(E)5 b Fv(\)\))46 b(=)e Fr(M)10 b 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Fj(\014)s(;N)1836 5357 y Fv(\()p Fr(\033)s Fv(\))c(+)2177 5271 y Ff(X)2073 5470 y Fj(\033)2117 5452 y Fb(0)2142 5470 y Fi(2)p Ft(\()p Fj(I)5 b Fi([)p Fj(\033)r Ft(\))2377 5452 y Fm(c)2427 5357 y Fr(\026)2482 5371 y Fj(\014)s(;N)2617 5357 y Fv(\()p Fr(\033)2707 5320 y Fi(0)2733 5357 y Fv(\))p Fh(P)p Fv(\()p Fr(\034)2908 5320 y Fj(\033)2952 5292 y Fb(0)2898 5380 y Fj(\033)3008 5357 y Fr(<)25 b(\034)3154 5320 y Fj(\033)3198 5292 y Fb(0)3144 5380 y Fj(I)3228 5357 y Fv(\))3263 5175 y Ff(3)3263 5338 y(5)3523 5357 y Fv(\(3)p Fr(:)p Fv(5\))p eop %%Page: 43 43 43 42 bop 50 295 a Fl(A)l(ging)27 b(in)h(the)g(REM.)f(Part)h(1.)2661 b Fo(43)50 513 y Fv(Therefore,)30 b(b)m(y)g(de\014nition)e(of)j Fr(\026)1176 527 y Fj(\014)s(;N)1311 513 y Fv(,)f(\(3.4\))j(reads)125 737 y Fh(E)11 b Fv(\()p Fr(\034)265 700 y Fj(\033)255 760 y(I)345 737 y Fk(j)26 b Fr(\034)446 700 y Fj(\033)436 760 y(I)520 737 y Fk(\024)f Fr(\034)666 700 y Fj(\033)656 760 y(J)715 737 y Fv(\))g(=)773 930 y(1)p 378 971 836 4 v 378 1068 a 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Fj(\033)3498 870 y Fb(0)3444 955 y Fj(J)3528 930 y Fv(\))p 3068 971 495 4 v 3093 1054 a Fh(P)p Fv(\()p Fr(\034)3233 1023 y Fj(\033)3223 1081 y(I)3308 1054 y Fk(\024)g Fr(\034)3454 1023 y Fj(\033)3444 1081 y(J)3503 1054 y Fv(\))3574 809 y Ff(3)3574 973 y(5)3523 1210 y Fv(\(3)p Fr(:)p Fv(6\))50 1355 y(and)30 b(similarly)-8 b(,)442 1681 y Fh(E)10 b Fv(\()q Fr(\034)582 1643 y Fj(\033)572 1703 y(I)637 1681 y Fv(\))25 b(=)1160 1619 y(1)p 804 1660 758 4 v 804 1757 a Fr(e)846 1731 y Fj(\014)890 1677 y Fi(p)p 951 1677 65 4 v 951 1731 a Fj(N)6 b(E)1068 1740 y Fm(\033)1116 1757 y Fh(P)p Fv(\()p Fr(\034)1256 1726 y Fj(\033)1246 1784 y(I)1331 1757 y Fr(<)25 b(\034)1477 1731 y Fj(\033)1467 1780 y(\033)1526 1757 y Fv(\))1588 1498 y Ff(2)1588 1661 y(4)1648 1681 y Fr(e)1690 1643 y Fj(\014)1734 1589 y Fi(p)p 1795 1589 V 1795 1643 a Fj(N)6 b(E)1912 1652 y Fm(\033)1981 1681 y Fv(+)2176 1594 y Ff(X)2071 1793 y Fj(\033)2115 1775 y Fb(0)2140 1793 y Fi(2)p Ft(\()p Fj(I)f Fi([)p Fj(\033)r Ft(\))2375 1775 y Fm(c)2426 1681 y Fr(e)2468 1643 y Fj(\014)2512 1589 y Fi(p)p 2573 1589 V 2573 1643 a Fj(N)h(E)2690 1660 y Fm(\033)2729 1647 y Fb(0)2763 1681 y Fh(P)p Fv(\()p Fr(\034)2903 1643 y Fj(\033)2947 1616 y Fb(0)2893 1703 y Fj(\033)3002 1681 y Fr(<)25 b(\034)3148 1643 y Fj(\033)3192 1616 y Fb(0)3138 1703 y Fj(I)3222 1681 y Fv(\))3257 1498 y Ff(3)3257 1661 y(5)3523 1681 y Fv(\(3)p Fr(:)p Fv(7\))50 2014 y(Applying)37 b(\(3.6\))k(and)e(\(3.7\))i(to)f(the)f(quan)m (tities)g Fh(E)11 b Fv(\()p Fr(\034)1945 1970 y Fj(\021)1941 2033 y Ft(\026)-42 b Fj(\021)2035 2014 y Fk(j)41 b Fr(\034)2151 1970 y Fj(\021)2147 2033 y Ft(\026)-42 b Fj(\021)2235 2014 y Fk(\024)40 b Fr(\034)2396 1970 y Fj(\021)2386 2045 y(T)10 b Ft(\()p Fj(E)s Ft(\))p Fi(n)p Fj(\021)2634 2014 y Fv(\),)42 b Fh(E)11 b Fv(\()p Fr(\034)2876 1970 y Fj(\021)2866 2045 y(T)f Fi(n)p Fj(\021)3006 2014 y Fv(\))40 b(and)e Fh(E)11 b Fv(\()p Fr(\034)3406 1981 y Fj(\033)3396 2039 y(T)3461 2014 y Fv(\),)42 b(and)50 2159 y(inserting)30 b(the)j(probabilit)m(y)d(estimates)j(of)f(Prop)s (osition)f(2.1)i(in)e(the)i(resulting)d(expressions,)i(the)g(pro)s(of) 50 2305 y(of)21 b(Prop)s(osition)e(1.4)j(essen)m(tially)e(reduces)g(to) i(studying)d(the)j(b)s(eha)m(vior)e(of)h(the)g(random)f(v)-5 b(ariable)20 b Fr(Z)7 b Fv(\()p Fr(T)3542 2272 y Fj(c)3578 2305 y Fv(\))26 b(=)50 2382 y Ff(P)146 2477 y Fj(\033)190 2459 y Fb(0)215 2477 y Fi(2)p Fj(T)317 2459 y Fm(c)370 2450 y Fr(e)412 2417 y Fj(\014)456 2363 y Fi(p)p 517 2363 V 517 2417 a Fj(N)6 b(E)634 2434 y Fm(\033)673 2421 y Fb(0)707 2450 y Fv(.)41 b(W)-8 b(e)31 b(start)g(b)m(y)f(pro)m(ving)g (the)g(\014rst)g(assertion)g(of)h(the)f(prop)s(osition.)50 2668 y Fa(Pro)s(of)h(of)f(assertion)g(i\))f(of)h(Theorem)f(1.4:)83 b Fv(W)-8 b(e)27 b(will)c(assume)j(throughout)g(that)g(the)g (assumptions)50 2814 y(of)k(Prop)s(osition)f(2.1)i(are)g(satis\014ed.) 40 b(It)30 b(follo)m(ws)g(from)g(\(3.7\))i(that,)f(for)f(all)f Fr(\021)g Fk(2)c Fr(T)13 b Fv(,)459 3117 y Fh(E)d Fv(\()q Fr(\034)599 3073 y Fj(\021)589 3148 y(T)g Fi(n)p Fj(\021)729 3117 y Fv(\))25 b(=)1286 3055 y(1)p 896 3096 825 4 v 896 3195 a Fr(e)938 3167 y Fj(\014)982 3113 y Fi(p)p 1043 3113 65 4 v 1043 3167 a Fj(N)6 b(E)1160 3176 y Fm(\021)1205 3195 y Fh(P)p Fv(\()p Fr(\034)1345 3151 y Fj(\021)1335 3226 y(T)k Fi(n)p Fj(\021)1495 3195 y Fr(<)25 b(\034)1641 3151 y Fj(\021)1631 3205 y(\021)1686 3195 y Fv(\))1747 2961 y Ff(")1800 3117 y Fr(e)1842 3079 y Fj(\014)1886 3025 y Fi(p)p 1946 3025 V 54 x Fj(N)7 b(E)2064 3088 y Fm(\021)2129 3117 y Fv(+)2244 3030 y Ff(X)2220 3225 y Fj(\033)r Fi(2)p Fj(T)2366 3207 y Fm(c)2415 3117 y Fr(e)2457 3079 y Fj(\014)2501 3025 y Fi(p)p 2562 3025 V 2562 3079 a Fj(N)f(E)2679 3088 y Fm(\033)2727 3117 y Fh(P)p Fv(\()p Fr(\034)2867 3079 y Fj(\033)2857 3139 y(\021)2942 3117 y Fr(<)25 b(\034)3088 3079 y Fj(\033)3078 3141 y(T)10 b Fi(n)p Fj(\021)3212 3117 y Fv(\))3247 2961 y Ff(#)3523 3117 y Fv(\(3)p Fr(:)p Fv(8\))50 3431 y(The)29 b(factor)i(in)e(fron)m (t)h(of)g(the)g(square)g(brac)m(k)m(ets)i(w)m(as)e(estimated)g(in)f (Prop)s(osition)f(2.1,)j(iv\).)40 b(Plugging)29 b(in)50 3576 y(this)g(estimate)i(yields)467 3874 y Fh(E)11 b Fv(\()p Fr(\034)607 3831 y Fj(\021)597 3905 y(T)f Fi(n)p Fj(\021)737 3874 y Fv(\))25 b(=)1009 3813 y(1)p 904 3853 255 4 v 904 3941 a(1)c Fk(\000)1092 3905 y Ft(1)p 1072 3920 77 4 v 1072 3973 a Fj(M)1185 3719 y Ff(")1238 3874 y Fr(e)1280 3837 y Fj(\014)1324 3783 y Fi(p)p 1385 3783 65 4 v 1385 3837 a Fj(N)6 b(E)1502 3846 y Fm(\021)1567 3874 y Fv(+)1682 3788 y Ff(X)1658 3983 y Fj(\033)r Fi(2)p Fj(T)1804 3965 y Fm(c)1853 3874 y Fr(e)1895 3837 y Fj(\014)1939 3783 y Fi(p)p 2000 3783 V 2000 3837 a Fj(N)g(E)2117 3846 y Fm(\033)2165 3874 y Fh(P)p Fv(\()p Fr(\034)2305 3837 y Fj(\033)2295 3897 y(\021)2381 3874 y Fr(<)24 b(\034)2526 3837 y Fj(\033)2516 3899 y(T)10 b Fi(n)p Fj(\021)2650 3874 y Fv(\))2685 3719 y Ff(#)2754 3874 y Fv(\(1)21 b(+)f Fr(O)s Fv(\(1)p Fr(=)-5 b(N)10 b Fv(\)\))232 b(\(3)p Fr(:)p Fv(9\))50 4181 y(and)30 b(w)m(e)g(are)h(left)f(to)h(study)f(the) g(term)1261 4439 y Fr(I)i Fk(\021)1454 4352 y Ff(X)1429 4547 y Fj(\033)r Fi(2)p Fj(T)1575 4529 y Fm(c)1624 4439 y Fr(e)1666 4401 y Fj(\014)1710 4347 y Fi(p)p 1771 4347 V 1771 4401 a Fj(N)6 b(E)1888 4410 y Fm(\033)1937 4439 y Fh(P)1992 4401 y Fi(\016)2034 4439 y Fv(\()p Fr(\034)2119 4401 y Fj(\033)2109 4461 y(\021)2193 4439 y Fr(<)25 b(\034)2339 4401 y Fj(\033)2329 4463 y(T)10 b Fi(n)p Fj(\021)2463 4439 y Fv(\))979 b(\(3)p Fr(:)p Fv(10\))50 4751 y(T)-8 b(o)31 b(do)g(so,)g(w)m(e)h(pro)s(ceed)f(as)g(follo)m(ws:)41 b(for)30 b Fr(")d(>)f Fv(0)31 b(a)g(constan)m(t,)i(let)e Fk(B)2409 4725 y Fi(p)p 2468 4725 99 4 v 2468 4780 a Fj("N)2571 4751 y Fv(\()p Fr(\021)s Fv(\))h(and)f Fr(W)2985 4765 y Fj(")3023 4751 y Fv(\()p Fr(T)13 b Fv(\))32 b(b)s(e)e(de\014ned) f(as)50 4896 y(in)g(\(2.1\))j(and)e(\(2.2\))i(and)e(set)1312 5044 y Fr(V)1365 5058 y Fj(")1404 5044 y Fv(\()p Fr(T)13 b Fv(\))26 b Fk(\021)1682 4958 y Ff([)1662 5153 y Fj(\021)r Fi(2)p Fj(T)1818 4971 y Ff(\000)1860 5044 y Fr(T)1926 5007 y Fj(c)1983 5044 y Fk(\\)20 b(B)2124 5019 y Fi(p)p 2183 5019 V 2183 5074 a Fj("N)2287 5044 y Fv(\()p Fr(\021)s Fv(\))2405 4971 y Ff(\001)3477 5044 y Fv(\(3)p Fr(:)p Fv(11\))50 5313 y(Observing)28 b(that)1473 5462 y Fr(T)1539 5424 y Fj(c)1601 5462 y Fv(=)d Fr(V)1750 5476 y Fj(")1788 5462 y Fv(\()p Fr(T)13 b Fv(\))21 b Fk([)f Fr(W)2112 5476 y Fj(")2151 5462 y Fv(\()p Fr(T)13 b Fv(\))1190 b(\(3)p Fr(:)p Fv(12\))p eop %%Page: 44 44 44 43 bop 50 295 a Fo(44)3270 b Fl(Se)l(ction)27 b Fo(3)50 513 y Fr(I)37 b Fv(ma)m(y)31 b(b)s(e)f(decomp)s(osed)g(as)1659 658 y Fr(I)j Fv(=)25 b Fr(I)1868 672 y Ft(1)1929 658 y Fv(+)20 b Fr(I)2060 672 y Ft(2)3477 658 y Fv(\(3)p Fr(:)p Fv(13\))50 855 y(with)1181 963 y Fr(I)1221 977 y Ft(1)1287 963 y Fk(\021)1458 876 y Ff(X)1383 1075 y Fj(\033)r Fi(2)p Fj(V)1519 1084 y Fm(")1554 1075 y Ft(\()p Fj(T)10 b Ft(\))1679 963 y Fr(e)1721 925 y Fj(\014)1765 871 y Fi(p)p 1826 871 65 4 v 1826 925 a Fj(N)c(E)1943 934 y Fm(\033)1991 963 y Fh(P)2046 925 y Fi(\016)2088 963 y Fv(\()p Fr(\034)2173 925 y Fj(\033)2163 985 y(\021)2248 963 y Fr(<)25 b(\034)2394 925 y Fj(\033)2384 987 y(T)10 b Fi(n)p Fj(\021)2518 963 y Fv(\))1181 1235 y Fr(I)1221 1249 y Ft(2)1287 1235 y Fk(\021)1470 1148 y Ff(X)1383 1348 y Fj(\033)r Fi(2)p Fj(W)1544 1357 y Fm(")1580 1348 y Ft(\()p Fj(T)g Ft(\))1704 1235 y Fr(e)1746 1197 y Fj(\014)1790 1143 y Fi(p)p 1851 1143 V 1851 1197 a Fj(N)c(E)1968 1206 y Fm(\033)2017 1235 y Fh(P)2072 1197 y Fi(\016)2114 1235 y Fv(\()p Fr(\034)2199 1197 y Fj(\033)2189 1257 y(\021)2273 1235 y Fr(<)25 b(\034)2419 1197 y Fj(\033)2409 1260 y(T)10 b Fi(n)p Fj(\021)2543 1235 y Fv(\))3477 1141 y(\(3)p Fr(:)p Fv(14\))50 1497 y(No)m(w)31 b(ob)m(viously)-8 b(,)1391 1642 y(0)26 b Fk(\024)f Fr(I)1598 1656 y Ft(1)1664 1642 y Fk(\024)1835 1556 y Ff(X)1760 1755 y Fj(\033)r Fi(2)p Fj(V)1896 1764 y Fm(")1931 1755 y Ft(\()p Fj(T)10 b Ft(\))2056 1642 y Fr(e)2098 1605 y Fj(\014)2142 1550 y Fi(p)p 2203 1550 V 2203 1605 a Fj(N)c(E)2320 1614 y Fm(\033)3477 1642 y Fv(\(3)p Fr(:)p Fv(15\))50 1909 y(while,)28 b(b)m(y)j(Prop)s(osition)d(2.1,)k(i\),)e(for)g(all)f Fr(!)f Fk(2)d(E)39 b Fv(and)30 b(large)g(enough)g Fr(N)10 b Fv(,)31 b Fr(I)2630 1923 y Ft(2)2701 1909 y Fv(ob)s(eys)f(the)g(b)s (ound)975 2023 y Ff(\014)975 2078 y(\014)975 2133 y(\014)975 2187 y(\014)975 2242 y(\014)975 2296 y(\014)1005 2210 y Fr(I)1045 2224 y Ft(2)1106 2210 y Fk(\000)1234 2148 y Fv(1)p 1208 2189 99 4 v 1208 2272 a Fr(M)1420 2123 y Ff(X)1332 2323 y Fj(\033)r Fi(2)p Fj(W)1493 2332 y Fm(")1529 2323 y Ft(\()p Fj(T)10 b Ft(\))1654 2210 y Fr(e)1696 2172 y Fj(\014)1740 2118 y Fi(p)p 1801 2118 65 4 v 1801 2172 a Fj(N)c(E)1918 2181 y Fm(\033)1966 2023 y Ff(\014)1966 2078 y(\014)1966 2133 y(\014)1966 2187 y(\014)1966 2242 y(\014)1966 2296 y(\014)2022 2210 y Fk(\024)2195 2148 y Fr(d)p 2128 2189 182 4 v 2128 2272 a(N)k(M)2320 2210 y Fv(\(1)21 b(+)f Fr(c\016)s Fv(\()p Fr(N)10 b Fv(\)\))695 b(\(3)p Fr(:)p Fv(16\))50 2505 y(Therefore,)30 b(setting)1331 2630 y Fr(Z)7 b Fv(\()p Fr(V)1488 2644 y Fj(")1527 2630 y Fv(\()p Fr(T)13 b Fv(\)\))26 b Fk(\021)1895 2544 y Ff(X)1820 2743 y Fj(\033)r Fi(2)p Fj(V)1956 2752 y Fm(")1992 2743 y Ft(\()p Fj(T)10 b Ft(\))2116 2630 y Fr(e)2158 2593 y Fj(\014)2202 2538 y Fi(p)p 2263 2538 65 4 v 2263 2593 a Fj(N)c(E)2380 2602 y Fm(\033)1457 2903 y Fr(Z)h Fv(\()p Fr(T)1627 2865 y Fj(c)1664 2903 y Fv(\))25 b Fk(\021)1845 2816 y Ff(X)1820 3011 y Fj(\033)r Fi(2)p Fj(T)1966 2993 y Fm(c)2015 2903 y Fr(e)2057 2865 y Fj(\014)2101 2811 y Fi(p)p 2162 2811 V 2162 2865 a Fj(N)6 b(E)2279 2874 y Fm(\033)3477 2801 y Fv(\(3)p Fr(:)p Fv(17\))50 3154 y(and)30 b(com)m(bining)e(\(3.15\))33 b(and)d(\(3.16\))i(together)g(with)d(\(3.13\),)k(w)m(e)e(arriv)m(e)f (at)704 3268 y Ff(\014)704 3323 y(\014)704 3378 y(\014)704 3432 y(\014)734 3400 y Fr(I)e Fk(\000)930 3339 y Fv(1)p 903 3379 99 4 v 903 3463 a Fr(M)1012 3400 y(Z)7 b Fv(\()p Fr(T)1182 3363 y Fj(c)1219 3400 y Fv(\))1269 3272 y Ff(\024)1318 3400 y Fv(1)20 b(+)g(\()p Fr(M)31 b Fk(\000)20 b Fv(1\))1810 3339 y Fr(Z)7 b Fv(\()p Fr(V)1967 3353 y Fj(")2006 3339 y Fv(\()p Fr(T)13 b Fv(\)\))p 1810 3379 368 4 v 1873 3463 a Fr(Z)7 b Fv(\()p Fr(T)2043 3436 y Fj(c)2080 3463 y Fv(\))2189 3272 y Ff(\025)2237 3268 y(\014)2237 3323 y(\014)2237 3378 y(\014)2237 3432 y(\014)2293 3400 y Fk(\024)2466 3339 y Fr(d)p 2399 3379 182 4 v 2399 3463 a(N)j(M)2591 3400 y Fv(\(1)21 b(+)f Fr(c\016)s Fv(\()p Fr(N)10 b Fv(\)\))424 b(\(3)p Fr(:)p Fv(18\))50 3646 y(and)31 b(it)h(remains)f(to)i(study)e(the)h(b)s(eha)m(vior)f(of)h(the) h(random)e(v)-5 b(ariables)31 b Fr(Z)7 b Fv(\()p Fr(V)2693 3660 y Fj(")2732 3646 y Fv(\()p Fr(T)13 b Fv(\)\))32 b(and)g Fr(Z)7 b Fv(\()p Fr(T)3284 3613 y Fj(c)3321 3646 y Fv(\).)46 b(As)32 b(this)50 3792 y(dep)s(ends)c(on)j(the)f (cardinalit)m(y)f(of)h Fr(V)1295 3806 y Fj(")1334 3792 y Fv(\()p Fr(T)13 b Fv(\),)31 b(w)m(e)g(will)d(\014rst)i(establish)e (that:)50 4010 y Fc(Lemma)38 b(3.1:)91 b Fq(Assume)32 b(that)i Fv(0)26 b Fr(<)f(")h(<)f Fv(1)p Fr(=)p Fv(2)33 b Fq(and)h(set)1048 4238 y Fr(J)9 b Fv(\()p Fr(x)p Fv(\))26 b(=)f(\(1)c Fk(\000)f Fr(x)p Fv(\))15 b(ln)1797 4202 y Ft(1)p 1748 4217 135 4 v 1748 4269 a(1)p Fi(\000)p Fj(x)1913 4238 y Fv(+)20 b Fr(x)15 b Fv(ln)2175 4202 y Ft(1)p 2173 4217 42 4 v 2173 4269 a Fj(x)2225 4238 y Fr(;)76 b Fv(0)26 b Fr(<)f(x)g(<)g Fv(1)766 b(\(3)p Fr(:)p Fv(19\))50 4466 y Fq(Then,)33 b(for)g(al)5 b(l)33 b Fr(!)28 b Fk(2)d(E)41 b Fq(and)33 b(lar)-5 b(ge)34 b(enough)f Fr(N)10 b Fq(,)32 b(ther)-5 b(e)34 b(exist)e(c)-5 b(onstants,)35 b Fv(0)25 b Fr(<)g(c)2769 4433 y Fi(\000)2856 4466 y Fk(\024)g Fr(c)2991 4433 y Ft(+)3077 4466 y Fr(<)g Fk(1)p Fq(,)33 b(such)f(that)768 4694 y Fr(c)807 4656 y Fi(\000)869 4694 y Fr(M)10 b(N)1050 4656 y Fi(\000)p Ft(1)p Fj(=)p Ft(2)1221 4694 y Fr(e)1263 4656 y Fj(N)d(J)f Ft(\()p Fj("=)p Ft(4\))1564 4694 y Fk(\000)20 b Fr(M)35 b Fk(\024)25 b(j)p Fr(V)1952 4708 y Fj(")1991 4694 y Fv(\()p Fr(T)13 b Fv(\))p Fk(j)26 b(\024)f Fr(c)2313 4656 y Ft(+)2374 4694 y Fr(M)10 b(N)2555 4656 y Ft(1)p Fj(=)p Ft(2)2669 4694 y Fr(e)2711 4656 y Fj(N)d(J)f Ft(\()p Fj("=)p Ft(4\))3477 4694 y Fv(\(3)p Fr(:)p Fv(20\))50 5140 y Fa(Pro)s(of:)47 b Fv(Under)30 b(the)g(assumptions)f(of)i(Lemma)f (2.12,)345 5368 y Fk(j)p Fr(V)423 5382 y Fj(")463 5368 y Fv(\()p Fr(T)13 b Fv(\))p Fk(j)25 b Fv(=)750 5282 y Ff(X)745 5477 y Fj(\021)r Fi(2)p Fj(T)902 5368 y Fk(j)p Fr(T)993 5331 y Fj(c)1050 5368 y Fk(\\)20 b(B)1191 5343 y Fi(p)p 1251 5343 99 4 v 54 x Fj("N)1354 5368 y Fv(\()p Fr(\021)s Fv(\))p Fk(j)26 b Fv(=)1624 5282 y Ff(X)1619 5477 y Fj(\021)r Fi(2)p Fj(T)1776 5368 y Fk(jB)1861 5343 y Fi(p)p 1921 5343 V 54 x Fj("N)2024 5368 y Fv(\()p Fr(\021)s Fv(\))21 b Fk(n)g Fr(\021)s Fk(j)26 b Fv(=)2429 5282 y Ff(X)2424 5477 y Fj(\021)r Fi(2)p Fj(T)2580 5368 y Fk(jB)2665 5343 y Fi(p)p 2725 5343 V 54 x Fj("N)2828 5368 y Fv(\()p Fr(\021)s Fv(\))p Fk(j)c(\000)e Fr(M)305 b Fv(\(3)p Fr(:)p Fv(21\))p eop %%Page: 45 45 45 44 bop 50 295 a Fl(A)l(ging)27 b(in)h(the)g(REM.)f(Part)h(1.)2661 b Fo(45)50 513 y Fv(No)m(w,)31 b(for)f(all)g Fr(\021)e Fk(2)d Fr(T)13 b Fv(,)1146 543 y Ff(\022)1276 610 y Fr(N)1213 734 y("N)r(=)p Fv(4)1422 543 y Ff(\023)1514 671 y Fk(\024)25 b(jB)1695 646 y Fi(p)p 1755 646 99 4 v 54 x Fj("N)1858 671 y Fv(\()p Fr(\021)s Fv(\))p Fk(j)i(\024)2135 610 y Fr("N)p 2135 650 126 4 v 2174 734 a Fv(4)2271 543 y Ff(\022)2401 610 y Fr(N)2338 734 y("N)r(=)p Fv(4)2546 543 y Ff(\023)3477 671 y Fv(\(3)p Fr(:)p Fv(22\))50 906 y(where)41 b(w)m(e)h(used)e(that)895 832 y Ff(\000)937 866 y Fj(N)949 937 y(k)1001 832 y Ff(\001)1084 906 y Fv(is)h(an)g(increasing)f(function)g(of)h Fr(k)k Fv(for)c(0)j Fk(\024)f Fr(k)k Fk(\024)c Fr("N)r(=)p Fv(4.)76 b(By)42 b(Stirling's)50 1052 y(form)m(ula,)30 b(for)g(large)g(enough)g Fr(N)10 b Fv(,)31 b(there)f(exist)h(constan)m(ts,)g(0)26 b Fr(<)f(a)2312 1019 y Fi(\000)2399 1052 y Fk(\024)g Fr(a)2543 1019 y Ft(+)2628 1052 y Fr(<)g Fk(1)31 b Fv(suc)m(h)f(that) 925 1242 y Fr(a)973 1209 y Fi(\000)p 705 1282 549 4 v 705 1300 a Ff(p)p 796 1300 459 4 v 78 x Fr(\031)s(")p Fv(\(1)22 b Fk(\000)e Fr("=)p Fv(4\))1265 1303 y Fr(e)1307 1265 y Fj(N)7 b(J)f Ft(\()p Fj("=)p Ft(4\))1613 1303 y Fk(\024)1709 1175 y Ff(\022)1838 1242 y Fr(N)1775 1365 y("N)r(=)p Fv(4)1984 1175 y Ff(\023)2076 1303 y Fk(\024)2403 1242 y Fr(a)2451 1209 y Ft(+)p 2183 1282 549 4 v 2183 1300 a Ff(p)p 2274 1300 459 4 v 78 x Fr(\031)s(")p Fv(\(1)22 b Fk(\000)d Fr("=)p Fv(4\))2743 1303 y Fr(e)2785 1265 y Fj(N)7 b(J)f Ft(\()p Fj("=)p Ft(4\))3477 1303 y Fv(\(3)p Fr(:)p Fv(23\))50 1574 y(Inserting)29 b(\(3.23\))k(in)c(\(3.22\))j(and) e(using)f(that,)i(b)m(y)f(assumption,)f(0)d Fr(<)f(")h(<)f Fv(1)p Fr(=)p Fv(2)31 b(w)m(e)g(obtain)914 1810 y Fr(c)953 1773 y Fi(\000)1015 1810 y Fr(N)1098 1773 y Fi(\000)p Ft(1)p Fj(=)p Ft(2)1269 1810 y Fr(e)1311 1773 y Fj(N)7 b(J)f Ft(\()p Fj("=)p Ft(4\))1617 1810 y Fk(\024)25 b(jB)1798 1785 y Fi(p)p 1857 1785 99 4 v 1857 1839 a Fj("N)1961 1810 y Fv(\()p Fr(\021)s Fv(\))p Fk(j)h(\024)f Fr(c)2265 1773 y Ft(+)2326 1810 y Fr(N)2409 1773 y 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5094 235 4 v 82 x Fr(N)g Fv(ln)14 b(2)2096 5071 y Ff(\014)2096 5126 y(\014)2096 5180 y(\014)2142 5176 y Fr(T)f Fv(\()p Fr(!)s Fv(\))25 b(=)g Fr(T)2525 5075 y Ff(\021)1087 5376 y Fk(\024)1205 5289 y Ff(Y)1173 5484 y Fj(\033)r Fi(2)p Fj(T)1319 5466 y Fm(c)1368 5376 y Fr(P)1454 5275 y Ff(\020)1509 5376 y Fr(E)1576 5390 y Fj(\033)1650 5376 y Fr(<)1746 5294 y Fk(p)p 1822 5294 V 82 x Fr(N)g Fv(ln)14 b(2)2071 5271 y Ff(\014)2071 5326 y(\014)2071 5380 y(\014)2117 5376 y Fr(E)2184 5390 y Fj(\033)2258 5376 y Fr(<)25 b(u)2406 5390 y Fj(N)2475 5376 y Fv(\()p Fr(E)5 b Fv(\))2617 5275 y Ff(\021)3477 5207 y Fv(\(3)p Fr(:)p Fv(44\))p eop %%Page: 48 48 48 47 bop 50 295 a Fo(48)3270 b Fl(Se)l(ction)27 b Fo(3)50 513 y Fv(Since)287 435 y Fk(p)p 363 435 235 4 v 78 x Fr(N)e Fv(ln)14 b(2)26 b Fr(<)f(u)771 527 y Fj(N)840 513 y Fv(\()p Fr(E)5 b Fv(\))31 b(for)f(\014xed)g Fr(E)35 b Fv(and)30 b(large)h(enough)f Fr(N)10 b Fv(,)30 b(w)m(e)h(ha)m(v)m(e,) 688 827 y Fr(P)774 726 y Ff(\020)828 827 y Fr(E)895 841 y 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54 x Fj(N)19 b Ft(ln)13 b(2)587 1438 y(0)893 1355 y Fr(e)935 1317 y Fi(\000)1003 1293 y Fm(x)1039 1274 y Fn(2)p 1003 1303 72 4 v 1023 1339 a(2)1090 1355 y Fr(dx)25 b Fv(=)g(1)c Fk(\000)1543 1293 y Fv(2)p 1478 1334 177 4 v 1478 1352 a Fk(p)p 1554 1352 101 4 v 75 x Fv(2)p Fr(\031)1680 1231 y Ff(Z)1771 1253 y Fi(1)1731 1383 y(p)p 1790 1383 187 4 v 1790 1438 a Fj(N)e Ft(ln)13 b(2)1996 1355 y Fr(e)2038 1317 y Fi(\000)2106 1293 y Fm(x)2142 1274 y Fn(2)p 2106 1303 72 4 v 2127 1339 a(2)2193 1355 y Fr(dx)26 b Fk(\025)f Fv(1)20 b Fk(\000)2661 1293 y Fv(1)p 2581 1334 205 4 v 2581 1429 a(4)2626 1352 y Fk(p)p 2703 1352 83 4 v 2703 1429 a Fr(N)2796 1355 y(e)2838 1317 y Fi(\000)p Fj(N)7 b Ft(\(ln)13 b(2\))p Fj(=)p Ft(2)3477 1355 y Fv(\(3)p Fr(:)p Fv(46\))50 1602 y(that)277 1887 y Fr(P)363 1786 y Ff(\020)418 1887 y Fk(8)469 1901 y Fj(\033)r Fi(2)p Fj(T)615 1883 y Fm(c)653 1887 y Fr(E)720 1901 y Fj(\033)794 1887 y Fr(<)890 1805 y Fk(p)p 966 1805 235 4 v 82 x Fr(N)25 b Fv(ln)14 b(2)1215 1783 y Ff(\014)1215 1837 y(\014)1215 1892 y(\014)1261 1887 y Fr(T)f Fv(\()p Fr(!)s Fv(\))25 b(=)g Fr(T)1644 1786 y Ff(\021)1724 1887 y Fk(\024)1820 1732 y Ff( )1903 1802 y Fv(1)20 b Fk(\000)2132 1767 y Ft(1)p 2070 1782 161 4 v 2070 1846 a(4)2106 1791 y Fi(p)p 2166 1791 65 4 v 55 x Fj(N)2241 1802 y Fr(e)2283 1769 y Fi(\000)p Fj(N)7 b Ft(\(ln)13 b(2\))p Fj(=)p Ft(2)p 1903 1866 747 4 v 2032 1950 a Fv(1)20 b Fk(\000)g Fv(2)2233 1923 y Fi(\000)p Fj(N)2359 1950 y Fr(e)2401 1923 y Fi(\000)p Fj(E)2660 1732 y Ff(!)2732 1744 y Fi(j)p Fj(T)2807 1717 y Fm(c)2840 1744 y Fi(j)1724 2205 y Fk(\024)1820 2077 y Ff(\022)1887 2205 y Fv(1)g Fk(\000)2133 2144 y Fv(1)p 2054 2184 205 4 v 2054 2279 a(8)2099 2202 y Fk(p)p 2175 2202 83 4 v 77 x Fr(N)2269 2205 y(e)2311 2168 y Fi(\000)p Fj(N)7 b Ft(\(ln)13 b(2\))p Fj(=)p Ft(2)2677 2077 y Ff(\023)2744 2096 y Ft(2)2780 2068 y Fm(N)2838 2096 y Ft(\(1)p Fi(\000)p Ft(2)2995 2068 y Fb(\000)p Fm(N)3104 2096 y Fj(M)7 b Ft(\))1724 2433 y Fk(\024)25 b Fv(exp)1974 2332 y Ff(\020)2028 2433 y Fk(\000)p Fr(e)2141 2395 y Fj(N)7 b Ft(\(ln)13 b(2\))p Fj(=)p Ft(2)2450 2433 y Fv(\(1)21 b Fk(\000)f Fv(2)2687 2395 y Fi(\000)p Fj(N)2813 2433 y Fr(M)10 b Fv(\))p Fr(=)p Fv(8)3036 2351 y Fk(p)p 3113 2351 V 3113 2433 a Fr(N)3196 2332 y Ff(\021)1724 2633 y Fk(\024)25 b Fv(exp)1974 2532 y Ff(\020)2028 2633 y Fk(\000)p Fr(e)2141 2595 y Fj(N)7 b Ft(\(ln)13 b(2\))p Fj(=)p Ft(4)2450 2532 y Ff(\021)3477 2217 y Fv(\(3)p Fr(:)p Fv(47\))50 2855 y(This)28 b(pro)m(v)m(es)j(\(3.26\))i(and)d(concludes)f(the)i(pro)s(of) f(of)g(Lemma)h(3.3.)p Fk(})159 3073 y Fv(W)-8 b(e)34 b(are)g(no)m(w)f(ready)g(to)h(complete)g(the)f(pro)s(of)f(of)i (assertion)e(i\))h(of)g(Theorem)g(1.4.)50 b(F)-8 b(or)34 b Fr(\014)h Fk(\025)3437 2995 y(p)p 3513 2995 197 4 v 78 x Fv(2)15 b(ln)f(2)50 3218 y(and)30 b(\014xed)f(0)d Fr(<)f(")h(<)f Fv(1)p Fr(=)p Fv(2)31 b(\(to)h(b)s(e)d(c)m(hosen)i (appropriately)e(later\),)i(let)2485 3195 y Ff(e)2472 3218 y Fk(E)2520 3232 y Fj(N)2614 3218 y Fk(\021)2723 3195 y Ff(e)2710 3218 y Fk(E)2758 3232 y Fj(N)2827 3218 y Fv(\()p Fr(\014)5 b(;)15 b(")p Fv(\))32 b(b)s(e)e(de\014ned)f(as)322 3427 y Ff(e)309 3450 y Fk(E)357 3464 y Fj(N)451 3450 y Fk(\021)c(E)j(\\)704 3349 y Ff(n)765 3450 y Fr(!)g Fk(2)d Fv(\012)g Fk(j)g Fr(Z)7 b Fv(\()p Fr(V)1234 3464 y Fj(")1273 3450 y Fv(\()p Fr(T)13 b Fv(\)\))26 b Fk(\024)f(j)p Fr(V)1644 3464 y Fj(")1683 3450 y Fv(\()p Fr(T)13 b Fv(\))p Fk(j)p Fr(e)1886 3412 y Ft(2)p Fj(\014)1966 3350 y Fk(p)p 2044 3350 381 4 v 2044 3412 a Fj(N)19 b Ft(ln)13 b Fi(j)p Fj(V)2259 3421 y Fm(")2293 3412 y Ft(\()p Fj(T)d Ft(\))p Fi(j)2429 3450 y Fr(;)15 b(Z)7 b Fv(\()p Fr(T)2639 3412 y Fj(c)2676 3450 y Fv(\))26 b Fk(\025)f Fr(e)2875 3412 y Fj(\014)s(N)2984 3357 y Fi(p)p 3044 3357 110 4 v 55 x Ft(ln)13 b(2)3158 3349 y Ff(o)3477 3450 y Fv(\(3)p Fr(:)p Fv(48\))50 3681 y(where)30 b Fk(E)38 b Fv(is)30 b(tak)m(en)h(from)f(Prop)s(osition)e(2.1.)42 b(No)m(w)31 b(set)1573 3889 y Ff(e)1560 3912 y Fk(E)i(\021)1737 3826 y Ff([)1741 4021 y Fj(N)1799 4030 y Fn(0)1910 3826 y Ff(\\)1853 4021 y Fj(N)7 b(>N)2033 4030 y Fn(0)2096 3889 y Ff(e)2083 3912 y Fk(E)2131 3926 y Fj(N)3477 3912 y Fv(\(3)p Fr(:)p Fv(49\))50 4202 y(Ob)m(viously)-8 b(,)29 b(b)m(y)h(\(3.25\),)j(\(3.26\))g(and)c(Lemma)i(2.11,)1671 4433 y Fr(P)1757 4333 y Ff(\020)1824 4410 y(e)1811 4433 y Fk(E)1868 4333 y Ff(\021)1947 4433 y Fv(=)25 b(1)1389 b(\(3)p Fr(:)p Fv(50\))50 4676 y(Assume)28 b(from)g(no)m(w)h(on)f(that) h Fr(!)f Fk(2)1289 4653 y Ff(e)1276 4676 y Fk(E)1324 4690 y Fj(N)1422 4676 y Fv(\(and)g(that)h Fr(N)39 b Fv(is)27 b(large)i(enough\).)40 b(By)29 b(Lemma)g(3.1)h(and)e(Lemma)50 4822 y(3.2)1494 4967 y Fr(Z)7 b Fv(\()p Fr(V)1651 4981 y Fj(")1690 4967 y Fv(\()p Fr(T)13 b Fv(\)\))26 b Fk(\024)f Fr(e)2025 4929 y Fj(c\014)s(N)2167 4883 y Fi(p)p 2227 4883 35 4 v 46 x Fj(")3477 4967 y Fv(\(3)p Fr(:)p Fv(51\))50 5166 y(for)30 b(some)h(n)m(umerical)e(constan)m(t)i(0)26 b Fr(<)f(c)h(<)f Fk(1)p Fv(.)40 b(Th)m(us,)30 b(b)m(y)g(\(3.26\))j(of)d (Lemma)h(3.3,)1373 5413 y(0)26 b Fr(<)1551 5352 y(Z)7 b Fv(\()p Fr(V)1708 5366 y Fj(")1746 5352 y Fv(\()p Fr(T)13 b Fv(\)\))p 1551 5392 368 4 v 1613 5475 a Fr(Z)7 b Fv(\()p Fr(T)1783 5449 y Fj(c)1820 5475 y Fv(\))1954 5413 y Fr(<)25 b(e)2092 5376 y Fi(\000)p Fj(c)p Ft(\()p Fj(")p Ft(\))p Fj(\014)s(N)3477 5413 y Fv(\(3)p Fr(:)p Fv(52\))p eop %%Page: 49 49 49 48 bop 50 295 a Fl(A)l(ging)27 b(in)h(the)g(REM.)f(Part)h(1.)2661 b Fo(49)50 513 y Fv(where)38 b Fr(c)p Fv(\()p Fr(")p Fv(\))j(=)e(ln)14 b(2)27 b Fk(\000)e Fr(c)921 447 y Fk(p)p 997 447 43 4 v 66 x Fr(")q Fv(.)66 b(Let)40 b Fr(")1345 527 y Ft(0)1425 513 y Fv(b)s(e)e(de\014ned)g(through)g Fr(c)p Fv(\()p Fr(")2347 527 y Ft(0)2389 513 y Fv(\))i(=)f(0.)67 b(Then,)40 b(c)m(ho)s(osing)f(0)h Fr(<)f(")h(<)50 658 y Fv(\(1)p Fr(=)p Fv(2)22 b Fk(^)e Fr(")365 672 y Ft(0)406 658 y Fv(\),)608 853 y(1)p 582 894 99 4 v 582 977 a Fr(M)691 915 y(Z)7 b Fv(\()p Fr(T)861 877 y Fj(c)898 915 y Fv(\))948 787 y Ff(\024)996 915 y Fv(1)21 b(+)f(\()p Fr(M)31 b Fk(\000)19 b Fv(1\))1488 853 y 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2768 y Fv(\))1498 2509 y Ff(2)1498 2672 y(4)1559 2692 y Fr(e)1601 2654 y Fj(\014)1645 2600 y Fi(p)p 1705 2600 V 54 x Fj(N)7 b(E)1823 2663 y Fm(\033)1891 2692 y Fv(+)2059 2605 y Ff(X)1982 2804 y Fj(\033)2026 2786 y Fb(0)2051 2804 y Fi(2)p Fj(T)2153 2786 y Fm(c)2187 2804 y Fi(n)p Fj(\033)2283 2692 y Fr(e)2325 2654 y Fj(\014)2369 2600 y Fi(p)p 2430 2600 V 2430 2654 a Fj(N)f(E)2547 2671 y Fm(\033)2586 2658 y Fb(0)2620 2692 y Fh(P)p Fv(\()p Fr(\034)2760 2654 y Fj(\033)2804 2627 y Fb(0)2750 2714 y Fj(\033)2860 2692 y Fr(<)25 b(\034)3006 2654 y Fj(\033)3050 2627 y Fb(0)2996 2714 y Fj(T)3079 2692 y Fv(\))3114 2509 y Ff(3)3114 2672 y(5)3477 2692 y Fv(\(3)p Fr(:)p Fv(56\))50 2998 y(Assuming)j(again)h (that)h(the)g(assumptions)d(of)j(Prop)s(osition)d(2.1)k(are)f (satis\014ed,)f(it)g(follo)m(ws)f(from)h(Prop)s(o-)50 3143 y(sition)g(2.1,)j(v\),)e(that)639 3287 y Ff(h)682 3388 y Fr(e)724 3351 y Fj(\014)768 3296 y Fi(p)p 829 3296 V 829 3351 a Fj(N)6 b(E)946 3360 y Fm(\033)1014 3388 y Fv(+)20 b Fr(I)1152 3351 y Fi(0)1178 3287 y Ff(i)1246 3388 y Fk(\024)25 b Fh(E)11 b Fv(\()p Fr(\034)1482 3351 y Fj(\033)1472 3411 y(T)1537 3388 y Fv(\))26 b Fk(\024)1809 3327 y Fv(1)p 1704 3367 255 4 v 1704 3455 a(1)21 b Fk(\000)1892 3419 y Ft(1)p 1872 3434 77 4 v 1872 3486 a Fj(M)1985 3287 y Ff(h)2028 3388 y Fr(e)2070 3351 y Fj(\014)2114 3296 y Fi(p)p 2174 3296 65 4 v 55 x Fj(N)7 b(E)2292 3360 y Fm(\033)2360 3388 y Fv(+)20 b Fr(I)2498 3351 y Fi(0)2524 3287 y Ff(i)2582 3388 y Fv(\(1)h(+)f Fr(O)s Fv(\(1)p Fr(=)-5 b(N)10 b Fv(\)\))358 b(\(3)p Fr(:)p Fv(57\))50 3652 y(where)1216 3797 y Fr(I)1263 3760 y Fi(0)1314 3797 y Fk(\021)1488 3711 y Ff(X)1410 3910 y Fj(\033)1454 3892 y Fb(0)1479 3910 y Fi(2)p Fj(T)1581 3892 y Fm(c)1615 3910 y Fi(n)p Fj(\033)1712 3797 y Fr(e)1754 3760 y Fj(\014)1798 3705 y Fi(p)p 1858 3705 V 55 x Fj(N)7 b(E)1976 3776 y Fm(\033)2015 3763 y Fb(0)2049 3797 y Fh(P)p Fv(\()p Fr(\034)2189 3760 y Fj(\033)2233 3732 y Fb(0)2179 3820 y Fj(\033)2288 3797 y Fr(<)25 b(\034)2434 3760 y Fj(\033)2478 3732 y Fb(0)2424 3820 y Fj(T)2508 3797 y Fv(\))934 b(\(3)p Fr(:)p Fv(58\))50 4069 y(Then,)29 b(decomp)s(osing)h Fr(I)898 4036 y Fi(0)954 4069 y Fv(as)1647 4214 y Fr(I)1694 4177 y Fi(0)1744 4214 y Fv(=)25 b Fr(I)1887 4177 y Fi(0)1880 4237 y Ft(1)1941 4214 y Fv(+)20 b Fr(I)2079 4177 y Fi(0)2072 4237 y Ft(2)3477 4214 y Fv(\(3)p Fr(:)p Fv(59\))50 4416 y(with)29 b(\(for)h Fr(")c(>)f Fv(0)31 b(a)f(constan)m(t)i(and)e(with)f Fr(V)1550 4430 y Fj(")1589 4416 y Fv(\()p Fr(T)13 b Fv(\))31 b(and)e Fr(W)2018 4430 y Fj(")2057 4416 y Fv(\()p Fr(T)13 b Fv(\))31 b(de\014ned)e(as)i(in)e(\(3.11\))j(and)e(\(3.12\)\))1143 4639 y Fr(I)1190 4602 y Fi(0)1183 4662 y Ft(1)1249 4639 y Fk(\021)1479 4553 y Ff(X)1345 4752 y Fj(\033)1389 4734 y Fb(0)1414 4752 y Fi(2)p Fj(V)1506 4761 y Fm(")1541 4752 y Ft(\()p Fj(T)10 b Fi([)p Fj(\033)r Ft(\))1760 4639 y Fr(e)1802 4602 y Fj(\014)1846 4548 y Fi(p)p 1906 4548 V 54 x Fj(N)c(E)2023 4619 y Fm(\033)2062 4606 y Fb(0)2096 4639 y Fh(P)p Fv(\()p Fr(\034)2236 4602 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Fr(e)2145 5323 y Fj(\014)2189 5268 y Fi(p)p 2249 5268 V 55 x Fj(N)d(E)2367 5339 y Fm(\033)2406 5326 y Fb(0)3477 5360 y Fv(\(3)p Fr(:)p Fv(61\))p eop %%Page: 50 50 50 49 bop 50 295 a Fo(50)3270 b Fl(Se)l(ction)27 b Fo(3)50 513 y Fv(while)h(b)m(y)i(Lemma)h(2.1,)h(vi\),)e Fr(I)1121 480 y Fi(0)1114 535 y Ft(2)1185 513 y Fv(ob)s(eys)g(the)g(b)s(ounds)856 787 y Fr(I)903 749 y Fi(0)896 809 y Ft(2)962 787 y Fk(\025)1173 726 y Fv(1)p 1068 766 255 4 v 1068 849 a Fr(M)h Fv(+)20 b(1)1496 701 y Ff(X)1349 900 y Fj(\033)1393 882 y Fb(0)1418 900 y Fi(2)p Fj(W)1535 909 y Fm(")1571 900 y Ft(\()p Fj(T)10 b Fi([)p Fj(\033)r Ft(\))1789 787 y Fr(e)1831 749 y Fj(\014)1875 695 y Fi(p)p 1935 695 65 4 v 54 x Fj(N)d(E)2053 766 y Fm(\033)2092 753 y Fb(0)2141 659 y Ff(\022)2208 787 y Fv(1)20 b(+)2393 726 y Fr(d)p 2375 766 83 4 v 2375 849 a(N)2469 787 y Fv(\(1)h Fk(\000)f Fr(c\016)s Fv(\()p Fr(N)10 b Fv(\)\))2931 659 y Ff(\023)760 1093 y Fk(\024)25 b Fr(I)903 1056 y Fi(0)896 1116 y Ft(2)962 1093 y Fk(\024)1095 1032 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Fv(1)h Fk(\000)f Fv(\()p Fr(M)31 b Fk(\000)20 b Fv(1\))1606 1632 y Fr(Z)7 b Fv(\()p Fr(V)1763 1646 y Fj(")1802 1632 y Fv(\()p Fr(T)33 b Fk([)20 b Fr(\033)s Fv(\)\))p 1606 1672 524 4 v 1634 1756 a Fr(Z)7 b Fv(\(\()p Fr(T)33 b Fk([)20 b Fr(\033)s Fv(\))2030 1729 y Fj(c)2067 1756 y Fv(\))2141 1565 y Ff(\025)15 b(\022)2271 1693 y Fv(1)21 b(+)2505 1632 y Fr(d)p 2439 1672 182 4 v 2439 1756 a(N)10 b(M)2631 1693 y Fv(\(1)21 b(+)e Fr(c\016)s Fv(\()p Fr(N)10 b Fv(\)\))3092 1565 y Ff(\023)366 1948 y Fr(I)413 1910 y Fi(0)464 1948 y Fk(\024)570 1886 y Fr(Z)d Fv(\(\()p Fr(T)34 b Fk([)19 b Fr(\033)s Fv(\))966 1853 y Fj(c)1004 1886 y Fv(\))p 570 1927 469 4 v 756 2010 a Fr(M)1065 1820 y Ff(\024)1113 1948 y Fv(1)i(+)f(\()p Fr(M)31 b Fk(\000)20 b Fv(1\))1606 1886 y Fr(Z)7 b Fv(\()p Fr(V)1763 1900 y Fj(")1802 1886 y Fv(\()p Fr(T)33 b Fk([)20 b Fr(\033)s Fv(\)\))p 1606 1927 524 4 v 1634 2010 a Fr(Z)7 b Fv(\(\()p Fr(T)33 b Fk([)20 b Fr(\033)s Fv(\))2030 1984 y Fj(c)2067 2010 y Fv(\))2141 1820 y Ff(\025)15 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b(ii\))f(follo)m(ws)h(step)g(b)m(y)g(step)h (that)g(of)f(the)h(\014rst)e(assertion.)50 2877 y Fa(Pro)s(of)40 b(of)g(assertion)f(iii\))g(of)h(Theorem)e(1.4:)105 b Fv(As)34 b(is)f(the)i(pro)s(of)e(of)i(the)f(\014rst)g(t)m(w)m(o)h (assertions)f(w)m(e)50 3022 y(will)d(assume)j(that)h(the)f(assumptions) f(of)h(Prop)s(osition)e(2.1)j(are)g(satis\014ed.)51 b(By)34 b(\(3.6\))i(w)m(e)e(ha)m(v)m(e,)j(for)d(all)50 3168 y Fr(\021)s(;)22 b Fv(\026)-52 b Fr(\021)29 b Fk(2)c Fr(T)13 b Fv(\()p Fr(E)5 b Fv(\),)31 b Fr(\021)e Fk(6)p Fv(=)j(\026)-52 b Fr(\021)t Fv(,)164 3387 y Fh(E)10 b Fv(\()p Fr(\034)304 3344 y Fj(\021)299 3407 y Ft(\026)-42 b Fj(\021)379 3387 y Fk(j)26 b Fr(\034)480 3344 y Fj(\021)476 3407 y Ft(\026)-42 b Fj(\021)549 3387 y Fk(\024)25 b Fr(\034)695 3344 y Fj(\021)685 3418 y(T)10 b Fi(n)p Fj(\021)819 3387 y Fv(\))26 b(=)806 3568 y(1)p 417 3609 825 4 v 417 3708 a Fr(e)459 3680 y Fj(\014)503 3626 y Fi(p)p 563 3626 65 4 v 54 x Fj(N)7 b(E)681 3689 y Fm(\021)726 3708 y Fh(P)p Fv(\()p Fr(\034)866 3664 y Fj(\021)856 3739 y(T)j Fi(n)p Fj(\021)1016 3708 y Fr(<)25 b(\034)1162 3664 y Fj(\021)1152 3719 y(\021)1206 3708 y Fv(\))1268 3474 y Ff(")1321 3630 y Fr(e)1363 3592 y Fj(\014)1407 3538 y Fi(p)p 1467 3538 V 54 x Fj(N)7 b(E)1585 3601 y Fm(\021)1650 3630 y Fv(+)1765 3543 y Ff(X)1741 3738 y Fj(\033)r Fi(2)p Fj(T)1887 3720 y Fm(c)1936 3630 y Fr(e)1978 3592 y Fj(\014)2022 3538 y Fi(p)p 2082 3538 V 54 x Fj(N)g(E)2200 3601 y Fm(\033)2248 3630 y Fh(P)p Fv(\()p Fr(\034)2388 3592 y Fj(\033)2378 3652 y(\021)2463 3630 y Fr(<)25 b(\034)2609 3592 y Fj(\033)2599 3654 y(T)10 b Fi(n)p Fj(\021)2733 3630 y Fv(\))2779 3549 y Fh(P)p Fv(\()p Fr(\034)2919 3516 y Fj(\033)2915 3572 y Ft(\026)-42 b Fj(\021)2995 3549 y Fk(\024)24 b Fr(\034)3140 3516 y Fj(\033)3130 3578 y(T)10 b Fi(n)p Fj(\021)3264 3549 y Fv(\))p 2779 3609 521 4 v 2781 3692 a Fh(P)p Fv(\()p Fr(\034)2921 3648 y Fj(\021)2917 3712 y Ft(\026)-42 b Fj(\021)2992 3692 y Fk(\024)25 b Fr(\034)3138 3648 y Fj(\021)3128 3723 y(T)10 b Fi(n)p Fj(\021)3262 3692 y Fv(\))3311 3474 y Ff(#)3477 3555 y Fv(\(3)p Fr(:)p Fv(64\))50 3943 y(Recalling)35 b(the)h(de\014nition)e(of)j Fr(I)43 b Fv(from)36 b(\(3.10\))j(and)c(comparing)h(equations)g(\(3.64\))j(and) c(\(3.8\),)40 b(w)m(e)d(see)50 4088 y(that)e(their)f(righ)m(t)h(hand)f (sides)g(are)h(iden)m(tical)f(up)g(to)h(the)h(extra)f(factor)h Fh(P)p Fv(\()p Fr(\034)2737 4055 y Fj(\033)2733 4111 y Ft(\026)-42 b Fj(\021)2820 4088 y Fk(\024)33 b Fr(\034)2974 4055 y Fj(\033)2964 4117 y(T)10 b Fi(n)p Fj(\021)3098 4088 y Fv(\))p Fr(=)p Fh(P)p Fv(\()p Fr(\034)3318 4044 y Fj(\021)3314 4108 y Ft(\026)-42 b Fj(\021)3397 4088 y Fk(\024)32 b Fr(\034)3550 4044 y Fj(\021)3540 4119 y(T)10 b Fi(n)p Fj(\021)3674 4088 y Fv(\))50 4234 y(that)31 b(m)m(ultiplies)c(eac)m(h)32 b(of)e(the)h(terms)f(of)g(the)h(sum)e(in)g Fr(I)7 b Fv(.)41 b(Mimic)m(king)29 b(the)i(pro)s(of)e(of)i(assertion)f (i\),)g(set)1647 4490 y Fr(I)1694 4452 y Fi(0)1744 4490 y Fk(\021)25 b Fr(I)1887 4452 y Fi(0)1880 4512 y Ft(1)1941 4490 y 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5442 y Ft(\026)-42 b Fj(\021)2531 5423 y Fk(\024)25 b Fr(\034)2677 5379 y Fj(\021)2667 5454 y(T)10 b Fi(n)p Fj(\021)2801 5423 y Fv(\))3477 5216 y(\(3)p Fr(:)p Fv(66\))p eop %%Page: 51 51 51 50 bop 50 295 a Fl(A)l(ging)27 b(in)h(the)g(REM.)f(Part)h(1.)2661 b Fo(51)50 513 y Fv(It)30 b(follo)m(ws)g(from)g(Prop)s(osition)e(2.1,)k (iii\))c(that,)j(for)g Fr(\033)d Fk(2)d Fr(V)2045 527 y Fj(")2084 513 y Fv(\()p Fr(T)13 b Fv(\),)1033 787 y(0)25 b Fk(\024)1210 706 y Fh(P)p Fv(\()p Fr(\034)1350 673 y Fj(\033)1346 729 y Ft(\026)-42 b Fj(\021)1425 706 y Fk(\024)25 b Fr(\034)1571 673 y Fj(\033)1561 736 y(T)10 b Fi(n)p Fj(\021)1695 706 y Fv(\))p 1210 766 521 4 v 1212 849 a Fh(P)p Fv(\()p Fr(\034)1352 806 y Fj(\021)1348 869 y Ft(\026)-42 b Fj(\021)1423 849 y Fk(\024)25 b Fr(\034)1569 806 y Fj(\021)1559 880 y(T)10 b Fi(n)p Fj(\021)1693 849 y Fv(\))1767 787 y Fk(\024)25 b Fv(\()p Fr(M)31 b Fk(\000)19 b Fv(1\)\(1)j(+)e Fr(O)s Fv(\(1)p Fr(=)-5 b(N)10 b Fv(\)\))752 b(\(3)p Fr(:)p Fv(67\))50 1067 y(where)38 b(w)m(e)h(trivially)d(used)h (that)i(0)h Fk(\024)e Fh(P)p Fv(\()p Fr(\034)1569 1034 y Fj(\033)1565 1090 y Ft(\026)-42 b Fj(\021)1658 1067 y Fk(\024)38 b Fr(\034)1817 1034 y Fj(\033)1807 1096 y(T)10 b Fi(n)p Fj(\021)1941 1067 y Fv(\))40 b Fk(\024)e Fv(1)h(in)e(the)i(n)m(umerator.)65 b(In)38 b(the)h(case)g(where)50 1213 y Fr(\033)e Fk(2)c Fr(W)319 1227 y Fj(")358 1213 y Fv(\()p Fr(T)13 b Fv(\),)38 b(observing)c(that)i Fh(P)p Fv(\()p Fr(\034)1311 1180 y Fj(\033)1307 1235 y Ft(\026)-42 b Fj(\021)1395 1213 y Fk(\024)34 b Fr(\034)1550 1180 y Fj(\033)1540 1242 y(T)10 b Fi(n)p Fj(\021)1674 1213 y Fv(\))34 b(=)g Fh(P)p Fv(\()p Fr(\034)1988 1180 y Fj(\033)1984 1235 y Ft(\026)-42 b Fj(\021)2071 1213 y Fr(<)34 b(\034)2226 1180 y Fj(\033)2216 1242 y Ft(\()p Fj(T)10 b Fi(n)p Fj(\021)r Ft(\))p Fi(n)c Ft(\026)-42 b Fj(\021)2484 1213 y Fv(\),)37 b(and)e(making)g(use)g(of)h(assertion)50 1358 y(i\))e(of)i(Prop)s (osition)d(2.1)j(with)d Fr(T)48 b Fv(replaced)35 b(b)m(y)g Fr(T)h Fk(n)23 b 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Fr(M)g Fv([)p Fr(Z)d Fv(\()p Fr(V)1810 3341 y Fj(")1849 3327 y Fv(\()p Fr(T)13 b Fv(\)\))p Fr(=)-5 b(Z)7 b Fv(\()p Fr(V)2217 3341 y Fj(")2257 3327 y Fv(\()p Fr(T)13 b Fv(\)\)])p 1098 3368 1789 4 v 1098 3459 a(1)20 b Fk(\000)g Fv([)p Fr(Z)7 b Fv(\()p Fr(V)1436 3473 y Fj(")1475 3459 y Fv(\()p Fr(T)13 b Fv(\)\))p Fr(=)-5 b(Z)7 b Fv(\()p Fr(V)1843 3473 y Fj(")1883 3459 y Fv(\()p Fr(T)13 b Fv(\)\)])2094 3385 y Ff(\000)2137 3459 y Fv(1)21 b(+)2318 3423 y Fj(d)p 2305 3438 65 4 v 2305 3490 a(N)2380 3459 y Fv(\(1)g(+)f Fr(c\016)s Fv(\()p Fr(N)10 b Fv(\)\))2842 3385 y Ff(\001)3477 3264 y Fv(\(3)p Fr(:)p Fv(70\))50 3682 y(With)305 3659 y Ff(e)292 3682 y Fk(E)340 3696 y Fj(N)447 3682 y Fv(de\014ned)36 b(as)i(in)f(\(3.48\),)42 b(c)m(ho)s(osing)37 b Fr(")i Fv(as)f(in)e(the)i(line)e(follo)m(wing)g(\(3.52\),)42 b(and)c(inserting)e(the)50 3827 y(b)s(ound)28 b(\(3.52\))33 b(in)c(\(3.69\))j(w)m(e)f(get,)h(for)e(large)h(enough)f Fr(N)10 b Fv(,)1355 4015 y Fr(I)1395 4029 y Ft(1)p 1355 4055 81 4 v 1355 4139 a Fr(I)1395 4153 y Ft(2)1472 4076 y Fk(\024)25 b Fv(2)p Fr(e)1655 4039 y Fi(\000)p Fj(c)p Ft(\()p Fj(")p Ft(\))p Fj(\014)s(N)1950 4076 y Fr(;)106 b Fv(on)2220 4053 y Ff(e)2207 4076 y Fk(E)2255 4090 y Fj(N)3477 4076 y Fv(\(3)p Fr(:)p Fv(71\))50 4323 y(F)-8 b(rom)31 b(this)e(and)h(\(3.55\),)j(\(3.69\))f(yields)1474 4562 y Fk(j)p Fr(I)1546 4525 y Fi(0)1592 4562 y Fk(\000)20 b Fr(I)7 b Fk(j)26 b(\024)f(W)7 b Fr(O)s Fv(\(1)p Fr(=)-5 b(N)10 b Fv(\))1193 b(\(3)p Fr(:)p Fv(72\))50 4802 y(Finally)-8 b(,)29 b(com)m(bining)g(\(3.64\))j(and)e(\(3.8\),)j(and)c(using)g(the)i (previous)e(b)s(ound,)163 4943 y Ff(\014)163 4997 y(\014)163 5052 y(\014)193 5047 y Fh(E)11 b Fv(\()p Fr(\034)333 5003 y Fj(\021)329 5067 y Ft(\026)-42 b Fj(\021)409 5047 y Fk(j)25 b Fr(\034)509 5003 y Fj(\021)505 5067 y Ft(\026)-42 b Fj(\021)579 5047 y Fk(\024)25 b Fr(\034)725 5003 y Fj(\021)715 5078 y(T)10 b Fi(n)p Fj(\021)849 5047 y Fv(\))20 b Fk(\000)g Fh(E)11 b Fv(\()p Fr(\034)1135 5003 y Fj(\021)1125 5078 y(T)f Fi(n)p 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Fo(52)3270 b Fl(Se)l(ction)27 b Fo(3)50 513 y Fa(Pro)s(of)35 b(of)g(Theorem)e(1.5:)87 b Fv(The)29 b(pro)s(of)g(of)h(this)f(Theorem)h(is)f(v)m(ery)h(similar)d(to)k(that)f (of)g(assertion)g(\(i\))50 658 y(of)c(Theorem)g(1.4.)40 b(The)26 b(only)f(di\013erence)h(is)f(that)i(this)e(time,)i(the)f (partial)f(partition)g(function)g Fr(Z)7 b Fv(\()p Fr(V)3497 672 y Fj(\017)3531 658 y Fv(\()p Fr(E)e Fv(\)\))50 804 y(is)28 b(negligible)g(compared)h(to)i Fr(Z)1130 818 y Fj(\014)s(;N)1265 804 y Fv(.)40 b(Finally)-8 b(,)28 b(for)i Fr(\014)g(<)1972 726 y Fk(p)p 2048 726 197 4 v 78 x Fv(2)15 b(ln)f(2)q(,)30 b Fr(Z)2362 818 y Fj(\014)s(;N)2522 804 y Fv(=)25 b Fr(e)2660 771 y Fj(N)7 b(\014)2769 743 y Fn(2)2805 771 y Fj(=)p Ft(2)2883 804 y Fv(\(1)19 b(+)g Fr(O)s Fv(\()p Fr(N)3262 771 y Fi(\000)p Ft(1)p Fj(=)p Ft(4)3433 804 y Fv(\)\))30 b(with)50 949 y(probabilit)m(y)k(tending)h (to)j(one)f(faster)g(than)f(an)m(y)h(p)s(olynomial,)f(as)h(follo)m(ws)e (from)h(easy)i(estimates)f(\(see)50 1095 y([BKL])31 b(or)f([Bo]\),)i (and)e(this)f(pro)m(v)m(es)i(the)g(theorem.)p Fk(})50 1313 y Fs(Ac)m(kno)m(wledgemen)m(ts:)43 b Fv(W)-8 b(e)27 b(thank)e(the)g(W)-8 b(eierstrass)26 b(Institute)e(and)h(the)g (Mathematics)h(departmen)m(t)50 1458 y(of)k(the)h(EPFL)f(for)g (\014nancial)f(supp)s(ort)g(and)g(m)m(utual)h(hospitalit)m(y)-8 b(.)p eop %%Page: 53 53 53 52 bop 50 295 a Fl(A)l(ging)27 b(in)h(the)g(REM.)f(Part)h(1.)2661 b Fo(53)50 513 y Fu(A.)40 b(App)s(endix)159 731 y Fv(W)-8 b(e)31 b(state)h(and)e(pro)m(v)m(e)h(a)g(simple)d(lemma)i(that)h(is)f (used)f(in)g(Section)h(2.)50 949 y Fc(Lemma)41 b(6.1:)103 b Fq(L)-5 b(et)36 b Fv(\001)976 963 y Fj(k)1050 949 y Fk(\032)29 b Fv(\000)1207 963 y Fj(N)1276 949 y Fq(,)35 b Fv(1)30 b Fk(\024)g Fr(k)i Fk(\024)e Fr(K)7 b Fq(,)35 b(b)-5 b(e)35 b(a)g(c)-5 b(ol)5 b(le)-5 b(ction)37 b(of)e(sub)-5 b(gr)g(aphs)37 b(of)e Fv(\000)3127 963 y Fj(N)3231 949 y Fq(and)h(let)3548 926 y Ff(e)3540 949 y Fh(R)3600 916 y Fi(\016)3600 974 y Ft(\001)3660 984 y Fm(k)50 1095 y Fq(denote)d(the)g(law)h(of)f(the)g(Markov)g(chain)g(with)g(tr)-5 b(ansition)35 b(r)-5 b(ates)570 1367 y Ff(e)-53 b Fr(r)612 1329 y Fi(\016)609 1389 y Ft(\001)669 1399 y Fm(k)713 1367 y Fv(\()p Fr(x)800 1329 y Fi(0)826 1367 y Fr(;)15 b(x)918 1329 y Fi(00)965 1367 y Fv(\))25 b(=)1121 1239 y Ff(\032)1220 1299 y Fr(r)1264 1266 y Fi(\016)1261 1324 y Fj(N)1330 1299 y Fv(\()p Fr(x)1417 1266 y Fi(0)1443 1299 y Fr(;)15 b(x)1535 1266 y Fi(00)1582 1299 y Fv(\))p Fr(;)186 b Fq(if)93 b Fr(x)2029 1266 y Fi(0)2079 1299 y Fk(6)p Fv(=)25 b Fr(x)2227 1266 y Fi(00)2289 1299 y Fr(;)15 b Fq(and)32 b Fv(\()p Fr(x)2591 1266 y Fi(0)2617 1299 y Fr(;)15 b(x)2709 1266 y Fi(00)2755 1299 y Fv(\))26 b Fk(2)f Fr(E)5 b Fv(\(\001)3085 1313 y Fj(k)3130 1299 y Fv(\))1220 1435 y(0)p Fr(;)538 b Fq(otherwise)3523 1367 y Fv(\(6)p Fr(:)p Fv(1\))50 1639 y Fq(Assume)32 b(that)901 1784 y Fr(E)5 b Fv(\(\001)1084 1798 y Fj(k)1129 1784 y Fv(\))21 b Fk(\\)e Fr(E)5 b Fv(\(\001)1448 1798 y Fj(k)1489 1780 y Fb(0)1518 1784 y Fv(\))26 b(=)f Fk(;)p Fr(;)76 b Fk(8)p Fr(k)s(;)15 b(k)2012 1747 y Fi(0)2063 1784 y Fk(2)25 b(f)p Fv(1)p Fr(;)15 b(:)g(:)g(:)j(;)d(K)7 b Fk(g)p Fr(;)15 b(k)29 b Fk(6)p Fv(=)c Fr(k)2833 1747 y Fi(0)3523 1784 y Fv(\(6)p Fr(:)p Fv(2\))50 1994 y Fq(and)33 b(that)1548 2169 y Fr(y)s(;)15 b(x)26 b Fk(2)1833 2055 y Fj(K)1815 2082 y Ff(\\)1800 2278 y Fj(k)r Ft(=1)1947 2169 y Fr(V)20 b Fv(\(\001)2131 2183 y Fj(k)2176 2169 y Fv(\))1312 b(\(6)p Fr(:)p Fv(3\))50 2423 y Fq(Then)1200 2598 y Fh(R)1259 2561 y Fi(\016)1322 2525 y Ff(\000)1364 2598 y Fr(\034)1414 2561 y Fj(y)1404 2621 y(x)1483 2598 y Fr(<)25 b(\034)1629 2561 y Fj(y)1619 2621 y(y)1673 2525 y Ff(\001)1740 2598 y Fk(\025)1869 2485 y Fj(K)1836 2512 y Ff(X)1836 2708 y Fj(k)r Ft(=1)1991 2575 y Ff(e)1983 2598 y Fh(R)2043 2561 y Fi(\016)2043 2621 y Ft(\001)2103 2631 y Fm(k)2168 2525 y Ff(\000)2209 2598 y Fr(\034)2259 2561 y Fj(y)2249 2621 y(x)2328 2598 y Fr(<)g(\034)2474 2561 y Fj(y)2464 2621 y(y)2518 2525 y Ff(\001)3523 2598 y Fv(\(6)p Fr(:)p Fv(4\))50 3026 y Fa(Pro)s(of:)55 b Fv(This)32 b(lemma)h(is)f(a)i(straigh)m(tforw)m(ard)g(generalisation)e (of)i(Lemma)g(2.1)h(of)e([BEGK1].)52 b(Let)34 b Fk(H)3665 2993 y Fj(y)3664 3048 y(x)50 3171 y Fv(denote)d(the)f(space)h(of)g (functions)1026 3426 y Fk(H)1104 3388 y Fj(y)1103 3448 y(x)1174 3426 y Fk(\021)25 b(f)p Fr(h)h Fv(:)g(\000)1501 3440 y Fj(N)1595 3426 y Fk(!)f Fv([0)p Fr(;)15 b Fv(1])27 b(:)e Fr(h)p Fv(\()p Fr(y)s Fv(\))h(=)f(0)p Fr(;)15 b(h)p Fv(\()p Fr(x)p Fv(\))28 b(=)d(1)p Fk(g)790 b Fv(\(6)p Fr(:)p Fv(5\))50 3680 y(and)30 b(de\014ne)f(the)i(Diric)m(hlet)e(forms) 855 3894 y(\010)921 3908 y Fj(N)989 3894 y Fv(\()p Fr(h)p Fv(\))e Fk(\021)1321 3808 y Ff(X)1224 4003 y Fj(x)1265 3984 y Fb(0)1289 4003 y Fj(;x)1352 3984 y Fb(00)1397 4003 y Fi(2)p Ft(\000)1491 4012 y Fm(N)1564 3894 y Fh(Q)1629 3908 y Fj(N)1704 3894 y Fv(\()p Fr(x)1791 3856 y Fi(0)1817 3894 y Fv(\))p Fr(r)1896 3856 y Fi(\016)1893 3916 y Fj(N)1962 3894 y Fv(\()p Fr(x)2049 3856 y Fi(0)2075 3894 y Fr(;)15 b(x)2167 3856 y Fi(00)2214 3894 y Fv(\)[)p Fr(h)p Fv(\()p Fr(x)2413 3856 y Fi(0)2440 3894 y Fv(\))20 b Fk(\000)g Fr(h)p Fv(\()p Fr(x)2725 3856 y Fi(0)q(0)2772 3894 y Fv(\)])2832 3856 y Ft(2)820 4156 y Fv(\010)886 4170 y Ft(\001)946 4180 y Fm(k)989 4156 y Fv(\()p Fr(h)p Fv(\))27 b Fk(\021)1319 4070 y Ff(X)1224 4265 y Fj(x)1265 4246 y Fb(0)1289 4265 y Fj(;x)1352 4246 y Fb(00)1397 4265 y Fi(2)p Ft(\001)1506 4275 y Fm(k)1571 4132 y Ff(e)1560 4156 y Fh(Q)1625 4118 y Fi(\016)1625 4178 y Ft(\001)1685 4188 y Fm(k)1735 4156 y Fv(\()p Fr(x)1822 4118 y Fi(0)1848 4156 y Fv(\))p Fr(r)1927 4118 y Fi(\016)1924 4178 y Ft(\001)1984 4188 y Fm(k)2028 4156 y Fv(\()p Fr(x)2115 4118 y Fi(0)2141 4156 y Fr(;)15 b(x)2233 4118 y Fi(0)q(0)2280 4156 y Fv(\)[)p Fr(h)p Fv(\()p Fr(x)2479 4118 y Fi(0)2506 4156 y Fv(\))20 b Fk(\000)g Fr(h)p Fv(\()p Fr(x)2791 4118 y Fi(0)q(0)2838 4156 y Fv(\)])2898 4118 y Ft(2)3523 4065 y Fv(\(6)p Fr(:)p Fv(6\))50 4493 y(where)323 4470 y Ff(e)313 4493 y Fh(Q)377 4460 y Fi(\016)377 4518 y Ft(\001)437 4528 y Fm(k)487 4493 y Fv(\()p Fr(y)s Fv(\))26 b(=)f Fh(Q)792 4507 y Fj(N)867 4493 y Fv(\()p Fr(y)s Fv(\))p Fr(=)p Fh(Q)1096 4507 y Fj(N)1170 4493 y Fv(\(\001)1281 4507 y Fj(k)1326 4493 y Fv(\).)41 b(Then)30 b(\010)1731 4507 y Fj(N)1799 4493 y Fv(\()p Fr(h)p Fv(\))d Fk(\025)2044 4425 y Ff(P)2140 4448 y Fj(K)2140 4521 y(k)r Ft(=1)2291 4493 y Fh(Q)2356 4507 y Fj(N)2431 4493 y Fv(\(\001)2542 4507 y Fj(k)2587 4493 y Fv(\)\010)2688 4507 y Ft(\001)2748 4517 y Fm(k)2792 4493 y Fv(\()p Fr(h)p Fv(\),)32 b(implying)27 b(that)579 4812 y(inf)537 4878 y Fj(h)p Fi(2H)691 4847 y Fm(y)691 4890 y(x)746 4812 y Fv(\010)812 4826 y Fj(N)881 4812 y Fv(\()p Fr(h)p Fv(\))f Fk(\025)67 b Fv(inf)1125 4878 y Fj(h)p Fi(2H)1279 4847 y Fm(y)1279 4890 y(x)1368 4699 y Fj(K)1335 4726 y Ff(X)1335 4922 y Fj(k)r Ft(=1)1482 4812 y Fh(Q)1547 4826 y Fj(N)1622 4812 y Fv(\(\001)1733 4826 y Fj(k)1778 4812 y Fv(\)\010)1879 4826 y Ft(\001)1939 4836 y Fm(k)1983 4812 y Fv(\()p Fr(h)p Fv(\))26 b Fk(\025)2260 4699 y Fj(K)2228 4726 y Ff(X)2227 4922 y Fj(k)r Ft(=1)2374 4812 y Fh(Q)2439 4826 y Fj(N)2514 4812 y Fv(\(\001)2625 4826 y Fj(k)2670 4812 y Fv(\))57 b(inf)2720 4878 y Fj(h)p Fi(2H)2874 4847 y Fm(y)2874 4890 y(x)2930 4812 y Fv(\010)2996 4826 y Ft(\001)3056 4836 y Fm(k)3100 4812 y Fv(\()p Fr(h)p Fv(\))301 b(\(6)p Fr(:)p Fv(7\))50 5117 y(from)30 b(whic)m(h)f(the)h (lemma)g(follo)m(ws)g(b)m(y)g(an)g(application)f(of)i(Theorem)f(2.2)h (of)g([BEGK1].)42 b Fk(})p eop %%Page: 54 54 54 53 bop 50 295 a Fo(54)3225 b Fl(R)l(efer)l(enc)l(es)50 513 y Fu(References)-1 731 y Fv([B])46 b(J.P)-8 b(.)36 b(Bouc)m(haud,)g(\\W)-8 b(eak)38 b(ergo)s(dicit)m(y)d(breaking)g(and)g (aging)h(in)e(disordered)g(systems",)i Fq(J.)h(Phys.)h(I)159 876 y(\(F)-7 b(r)i(anc)g(e\))p Fv(,)32 b Fs(2)p Fv(,)f(1705)h (\(1992\).)-47 1095 y([Bo])47 b(A.)41 b(Bo)m(vier,)h(\\Statistical)f (mec)m(hanics)g(of)g(disordered)e(systems",)i(MaPh)m(ySto)h(Lecture)g (Notes)g(10,)159 1240 y(Aarh)m(us,)30 b(2001.)-137 1458 y([BBG])47 b(G.)22 b(Ben)g(Arous,)f(A.)h(Bo)m(vier,)h(and)d(V.)i(Ga)m (yrard,)h(\\Glaub)s(er)d(dynamics)g(of)i(the)g(random)f(energy)h(mo)s (del.)159 1604 y(I)s(I.)30 b(Aging)g(b)s(elo)m(w)f(the)i(critical)e (temp)s(erature",)i(\(2001\).)-221 1822 y([BCKM])46 b(J.P)-8 b(.)30 b(Bouc)m(haud,)g(L.)f(Cugliandolo,)e(J.)j(Kurc)m(han,)e(M.)i(M)m (\023)-43 b(ezard,)32 b(\\Out-of-equilibrium)25 b(dynamics)j(in)159 1967 y(spi-glasses)23 b(and)h(other)h(glassy)f(systems",)h(in)f Fq(Spin-glasses)k(and)g(r)-5 b(andom)29 b(\014elds)d Fv(\(A.P)-8 b(.)25 b(Y)-8 b(oung,)26 b(Ed.\).)159 2113 y(W)-8 b(orld)30 b(Scien)m(ti\014c,)g(Singap)s(ore)e(\(1998\).)-71 2331 y([BD])47 b(J.P)-8 b(.)31 b(Bouc)m(haud)f(and)g(D.)h(Dean,\\Aging) h(on)e(P)m(arisi's)f(tree")j Fq(J.)g(Phys.)h(I)f(\(F)-7 b(r)i(anc)g(e\))p Fv(,)33 b Fs(5)p Fv(,)e(265)g(\(1995\).)-142 2549 y([BDG])47 b(G.)42 b(Ben)f(Arous,)g(A.)g(Dem)m(b)s(o)h(and)e(A.)i (Guionnet,)f(\\Aging)g(of)g(spherical)f(spin)f(glasses",)j Fq(Pr)-5 b(ob)g(ab.)159 2695 y(The)g(or.)34 b(R)-5 b(el.)33 b(Fields)e Fs(120)p Fv(,)g(1-67)h(\(2001\).)-251 2913 y([BEGK1])47 b(A.)35 b(Bo)m(vier,)h(M.)f(Ec)m(kho\013,)h(V.)g(Ga)m (yrard)f(and)f(M.)i(Klein,)d(\\Metastabilit)m(y)j(in)d(sto)s(c)m (hastic)j(dynamics)159 3058 y(of)30 b(disordered)f(mean)h(\014eld)f(mo) s(dels",)h Fq(Pr)-5 b(ob.)33 b(The)-5 b(or.)34 b(R)-5 b(el.)33 b(Fields)e Fs(119)p Fv(,)h(99-161)h(\(2001\).)-251 3276 y([BEGK2])47 b(A.)30 b(Bo)m(vier,)h(M.)g(Ec)m(kho\013,)g(V.)f(Ga)m (yrard)g(and)g(M.)g(Klein,)f(\\Metastabilit)m(y)h(and)g(lo)m(w)g(lying) e(sp)s(ectra)i(in)159 3422 y(rev)m(ersible)f(Mark)m(o)m(v)j(c)m (hains",)e(to)i(app)s(ear)d(in)g Fq(Commun.)34 b(Math.)f(Phys.)e Fv(\(2001\).)-129 3640 y([BKL])46 b(A.)39 b(Bo)m(vier,)h(I.)f(Kurk)m(o) m(v)-5 b(a,)40 b(M.)f(L\177)-45 b(ow)m(e,)40 b(\\Fluctuations)f(of)g (the)g(free)g(energy)g(in)f(the)h(REM)g(and)g(the)159 3785 y Fr(p)p Fv(-spin)28 b(SK)i(mo)s(dels",)g(to)h(app)s(ear)f(in)f Fq(A)n(nn.)i(Pr)-5 b(ob)g(ab.)32 b Fv(\(2001\).)-85 4004 y([BM])47 b(J.P)-8 b(.)22 b(Bouc)m(haud,)g(C.)g(Mon)m(th)m(us,)h(\\Mo)s (dels)e(of)h(traps)g(and)f(glass)h(phenomenology",)g Fq(J.)i(Phys.)h(A-Math.)159 4149 y(Gen.)30 b Fs(29)p Fv(,)h(\(14\))h(3847-3869)j(\(1996\).)-152 4367 y([BMR])47 b(J.P)-8 b(.)34 b(Bouc)m(haud,)g(P)-8 b(.)34 b(Maass,)h(B.)f(Rinn,)e (\\Multiple)g(scaling)h(regimes)g(in)f(simple)g(aging)i(mo)s(dels",)f (q)159 4513 y Fq(Phys.)g(R)-5 b(ev.)32 b(L)-5 b(etts.)31 b Fs(84)p Fv(,)h(\(23\))f(5403-5406)k(\(2000\).)-68 4731 y([BR])46 b(C.J.)32 b(Burk)m(e)h(and)f(M.)i(Rosen)m(blatt,)f(\\A)g (Mark)m(o)m(vian)h(function)d(of)i(a)g(Mark)m(o)m(v)i(Chain",)c Fq(A)n(nn.)j(Math.)159 4876 y(Statist.)d Fs(29)p Fv(,)g(1112-1122)k (\(1958\).)-72 5095 y([CD])46 b(L.)34 b(Cugliandolo)e(and)h(D.)i(Dean,) g(\\F)-8 b(ull)34 b(dynamical)e(solution)h(for)h(a)g(spherical)f (spin-glass)f(mo)s(del",)159 5240 y Fq(J.)g(Phys.)h(A)f(-)g(Math.)h (Gen.)e Fs(28)p Fv(,)g(\(15\))h(4213-4234)i(\(1995\).)-52 5458 y([D1])47 b(B.)33 b(Derrida,)f(Random)g(energy)h(mo)s(del:)e (\\Limit)g(of)i(a)g(family)e(of)h(disordered)f(mo)s(dels",)h Fq(Phys.)j(R)-5 b(ev.)p eop %%Page: 55 55 55 54 bop 50 295 a Fl(A)l(ging)27 b(in)h(the)g(REM.)f(Part)h(1.)2661 b Fo(55)159 513 y Fq(L)-5 b(etts.)31 b Fs(45)p Fv(,)g(79-82)h (\(1980\).)-52 731 y([D2])47 b(B.)35 b(Derrida,)g(Random)f(energy)h(mo) s(del:)f(\\An)h(exactly)g(solv)-5 b(able)34 b(mo)s(del)g(of)h (disordered)e(disordered)159 876 y(systems",)e Fq(Phys.)i(R)-5 b(ev.)33 b(B)d Fs(24)p Fv(,)h(2613-2626)j(\(1981\).)-24 1095 y([Ei])45 b(Th.)30 b(Eisele,)f(\\On)h(a)h(third-order)d(phase)i (transition")g Fq(Comm.)j(Math.)g(Phys.)e Fs(90)p Fv(,)g(125-159)i (\(1983\).)-162 1313 y([FIKP])46 b(L.R.G.)39 b(F)-8 b(on)m(tes,)39 b(M.)g(Isopi,)e(Y.)h(Koha)m(y)m(ak)-5 b(a)m(w)m(a,)41 b(and)c(P)-8 b(.)39 b(Picco,)f(\\The)g(sp)s(ectral)f(gap)i(of)f(the)g (REM)159 1458 y(under)29 b(Metrop)s(olis)g(dynamics",)h Fq(A)n(nn.)i(Appl.)h(Pr)-5 b(ob)g(ab.)31 b Fs(8)p Fv(,)g(917-943)i (\(2001\).)-8 1676 y([G])46 b(V.)c(Ga)m(yrard,)g(\\Thermo)s(dynamic)d (limit)g(of)j(the)f Fr(q)s Fv(-state)i(P)m(otts-Hop\014eld)f(mo)s(del)e (with)g(in\014nitely)159 1822 y(man)m(y)30 b(patterns",)h Fq(J.)i(Stat.)g(Phys.)e Fs(68)p Fv(,)g(977-1011)i(\(1992\).)-154 2040 y([GMP])47 b(A.)30 b(Galv)m(ez,)h(S.)e(Martinez,)h(and)e(P)-8 b(.)30 b(Picco,)g(\\Fluctuations)f(in)f(Derrida's)h(random)g(energy)g (and)g(gen-)159 2185 y(eralized)g(random)h(energy)h(mo)s(dels",)f Fq(J.)i(Stat.)h(Phys.)e Fv(54,)g(515-529)i(\(1989\).)-73 2404 y([KT])45 b(S.)26 b(Karlin)e(and)i(H.M.)i(T)-8 b(a)m(ylor,)27 b Fq(A)h(\014rst)i(c)-5 b(ourse)30 b(in)f(sto)-5 b(chastic)31 b(pr)-5 b(o)g(c)g(esses)p Fv(,)28 b(Academic)f(Press,)g(Boston,)159 2549 y(1975.)-19 2767 y([Li])45 b(T.M.)31 b(Liggett,)g Fq(Inter)-5 b(acting)34 b(p)-5 b(article)34 b(systems)p Fv(,)e(Springer,)c(Berlin,)h(1985.)-118 2985 y([LLR])46 b(M.R.)34 b(Leadb)s(etter,)f(G.)h(Lindgren,)d(H.)j(Ro)s(otz)m(\023)-43 b(en,)35 b Fq(Extr)-5 b(emes)36 b(and)g(R)-5 b(elate)g(d)37 b(Pr)-5 b(op)g(erties)37 b(of)e(R)-5 b(andom)159 3131 y(Se)g(quenc)g(es)32 b(and)i(Pr)-5 b(o)g(c)g(esses)p Fv(,)32 b(Springer,)c(Berlin-Heidelb)s(erg-New)g(Y)-8 b(ork,)32 b(1983.)-20 3349 y([M])46 b(P)-8 b(.)32 b(Mathieu,)g(\\Con)m (v)m(ergence)i(to)e(equilibrium)27 b(for)32 b(spin)d(glasses",)k(to)f (app)s(ear)f(in)g(Comm)m(un.)g(Math.)159 3495 y(Ph)m(ys.)f(\(2001\).) -82 3713 y([MP])46 b(P)-8 b(.)35 b(Mathieu)g(and)f(P)-8 b(.)35 b(Pico)s(o,)g(\\Con)m(v)m(ergence)h(to)g(equilibrium)30 b(for)k(\014nite)g(Mark)m(o)m(v)i(pro)s(cesses,)f(with)159 3858 y(application)29 b(to)i(the)f(Random)g(Energy)g(Mo)s(del",)h (preprin)m(t)d(CPT-2000/P)-8 b(.3982)35 b(\(2000\).)-70 4076 y([OP])46 b(E.Olivieri)40 b(and)k(P)-8 b(.)44 b(Picco,)g(\\On)g (the)g(existence)g(of)g(thermo)s(dynamics)e(for)h(the)h(random)f (energy)159 4222 y(mo)s(del",)30 b Fq(Comm.)j(Math.)g(Phys.)p Fs(96)p Fv(,125-144)i(\(1984\).)-54 4440 y([Ru])45 b(D.)h(Ruelle,)f (\\A)h(mathematical)g(reform)m(ulation)e(of)i(Derrida's)f(REM)g(and)g (GREM",)i Fq(Commun.)159 4585 y(Math.)33 b(Phys.)e Fs(108)p Fv(,)g(225-239)i(\(1987\).)-33 4804 y([So])46 b(P)-8 b(.M.)23 b(Soardi,)e Fq(Potential)26 b(the)-5 b(ory)26 b(on)g(in\014nite)f(networks)p Fv(,)e(LNM)f(1590,)i(Springer,)c (Berlin-Heidelb)s(erg-)159 4949 y(New)30 b(Y)-8 b(ork,)32 b(1994.)-38 5167 y([Sp])45 b(F.)34 b(Spitzer,)f Fq(Principles)j(of)g(r) -5 b(andom)38 b(walks)p Fv(,)d(Second)f(edition.)e(Graduate)j(T)-8 b(exts)34 b(in)e(Mathematics,)159 5313 y(V)-8 b(ol.)31 b(34.)g(Springer-V)-8 b(erlag,)30 b(New)g(Y)-8 b(ork-Heidelb)s(erg,)30 b(1976.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF