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y(\(1.1\))h(and)h(\(1.2\))f(are)h(coupled.)59 b(In)38 b(conclusion)f(the)h(PFE)g(describ)s(e)g(a)f(c)m(hange)i(of)e (phase)i(including)0 1196 y(laten)m(t)29 b(heat)h(e\013ects)i(and,)e(b) s(ecause)h(of)f(that,)g(unlik)m(e)f(for)h(the)g(A)m(C)g(equation,)h (there)f(is)g(a)f(conserv)-5 b(ation)0 1312 y(la)m(w:)41 b(the)29 b(in)m(tegral)d(of)i Fq(m)13 b Fu(+)g Fq(h)30 b Fu(is)d(in)h(fact)g(in)m(v)-5 b(arian)m(t)27 b(under)i(PFE.)g(The)g (noise)f(term)g(in)f(\(1.1\))h(tak)m(es)i(in)m(to)0 1428 y(accoun)m(t)f(some)f(external)g(\015uctuating)g(force)g(and)g(the)g (resulting)f(equation)h(is)g(kno)m(wn)h(in)f(the)g(ph)m(ysical)0 1544 y(literature)j(as)i(mo)s(del)e(C)i(of)f(Hohen)m(b)s(erg)h(and)g (Halp)s(erin,)e([14].)100 1660 y(Since)56 b(\026)-67 b Fq(m)38 b Fu(solv)m(es)h(\(1.14\),)g(the)g(state)f Fq(m)g Fu(=)55 b(\026)-67 b Fq(m)p Fu(,)40 b Fq(h)d Fu(=)g(0)h(is)f(a)h (stationary)g(solution)e(of)i(PFE,)g(whic)m(h)0 1777 y(is)e(therefore)g(in)m(terpreted,)i(lik)m(e)d(in)h(A)m(C,)h(as)f(the)h 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Fq(\025)27 b Fu(=)h(0)k(and)f Fq(\015)i(>)28 b Fu(0\),)j(but)i(the)0 614 y(noise,)e(in)e(a)i(p)s (olynomial)26 b(scale,)31 b(cannot)g(driv)m(e)g(to)s(o)e(far)h(a)m(w)m (a)m(y)i(from)d Fo(M)h Fu(b)s(ecause)i(of)e(the)h(exp)s(onen)m(tial)0 731 y(attraction)h(of)i(the)g(deterministic)e(part)i(of)f(the)h (equation:)46 b(in)33 b([10])h(it)e(is)i(sho)m(wn)h(that)f(for)f(an)m (y)h Fq(\034)42 b(>)29 b Fu(0)0 847 y(and)k Fq(\020)h(>)28 b Fu(0,)1016 1110 y(lim)1010 1169 y Fl(\015)t Fn(!)p Fm(0)1173 1110 y Fj(P)1249 939 y Fp( )1377 1110 y Fu(sup)1329 1197 y Fl(t)p Fn(\024)p Fl(\015)1449 1178 y Fh(\000)p Fi(1)1533 1197 y Fl(\034)1588 1110 y Fu(dist\()p Fq(m)p Fu(\()p Fq(t)p Fu(\))p Fq(;)17 b Fo(M)p Fu(\))27 b Fo(\025)h Fq(\015)2370 1068 y Fm(1)p Fl(=)p Fm(2)p Fn(\000)p Fl(\020)2571 939 y Fp(!)2677 1110 y Fu(=)g(0)809 b(\(2.1\))0 1394 y(where)32 b(\\dist")f(denotes)h(distance)f(in)g(sup)h(norm.)42 b(The)32 b(analysis)e(extends)j(to)e(our)g(case)h(with)f Fq(\025)c Fu(=)3701 1332 y Fo(p)p 3784 1332 57 4 v 62 x Fq(\015)0 1510 y Fu(and)33 b(\(1.1\))o({\(1.2\))f(coupled,)h(as)g (stated)g(in)f(\(1.11\))f(and)i(pro)m(v)m(ed)h(in)e(Section)g(8.)100 1626 y(Going)d(bac)m(k)k(to)e(A)m(C)h(without)f(noise,)h(observ)m(e)h (that)f(stabilit)m(y)d(of)i Fo(M)g Fu(do)s(es)h(not)g(mean)f(stabilit)m (y)e(of)0 1743 y(the)34 b(single)f(instan)m(ton:)46 b(let)33 b Fq(m)h Fu(b)s(e)g(a)g(small)d(deviation)i(from)51 b(\026)-68 b Fq(m)2391 1758 y Fl(\030)2429 1743 y Fu(,)35 b(then)f(for)f(what)i (said)e(ab)s(o)m(v)m(e)h(it)f(will)0 1859 y(relax)f(under)i(A)m(C)g(to) e(some)51 b(\026)-67 b Fq(m)1141 1874 y Fl(\030)1175 1855 y Fh(0)1201 1859 y Fu(,)33 b(but)g Fq(\030)1488 1823 y Fn(0)1543 1859 y Fu(is)g(close)g(but)g(not)f(necessarily)i (equal)e(to)h Fq(\030)5 b Fu(.)43 b(In)33 b(the)h(space)f(of)0 1975 y(all)d(pro\014les)i Fq(m)p Fu(,)50 b(\026)-67 b Fq(m)700 1990 y Fl(\030)770 1975 y Fu(is)31 b(marginally)e(stable)i (along)g(the)h(direction)f Fo(M)g Fu(while)g(all)f(the)j(other)f (directions)0 2091 y(are)k(stable.)54 b(It)37 b(is)e(then)i(natural)e (to)h(asso)s(ciate)g(to)g(eac)m(h)h Fq(m)f Fu(\(as)g(ab)s(o)m(v)m(e\))h (the)g(v)-5 b(alue)35 b Fq(\030)3237 2055 y Fn(0)3296 2091 y Fu(of)h(the)h(cen)m(ter)0 2208 y(of)30 b(the)g(limit)c(instan)m (ton.)43 b(In)30 b(practice,)h(ho)m(w)m(ev)m(er,)i(it)c(is)g(b)s(etter) i(to)e(w)m(ork)i(with)f(a)f(more)h(\\geometrical")0 2324 y(de\014nition.)52 b(F)-8 b(ollo)m(wing)33 b([10)o(,)j(9],)h(w)m(e)f (de\014ne)h(a)f(cen)m(ter)h Fq(\030)j Fu(of)35 b Fq(m)h Fu(as)g(a)g(real)e(n)m(um)m(b)s(er)i(whic)m(h)h(minimizes)0 2440 y(the)c Fq(L)234 2455 y Fm(2)274 2440 y Fu({norm)e(of)h Fq(m)23 b Fo(\000)41 b Fu(\026)-68 b Fq(m)980 2455 y Fl(x)1057 2440 y Fu(\(as)33 b(a)f(function)g(of)g Fq(x)p Fu(\).)44 b(Then)34 b Fq(\030)j Fu(is)32 b(suc)m(h)i(that)1529 2658 y Fo(h)p Fq(m)22 b Fo(\000)41 b Fu(\026)-67 b Fq(m)1860 2673 y Fl(\030)1898 2658 y Fq(;)35 b Fu(\026)-67 b Fq(m)2027 2617 y Fn(0)2027 2682 y Fl(\030)2065 2658 y Fo(i)27 b Fu(=)h(0)p Fq(:)1328 b 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y(the)e(negativ)m(e)f(axis)g(strictly)g(a)m(w)m(a)m(y)i(from)d(0.) 43 b(Then,)34 b(if)d Fq(m)h Fu(has)h(cen)m(ter)h Fq(\030)5 b Fu(,)32 b(the)h(deviation)e Fq(v)g Fu(=)d Fq(m)22 b Fo(\000)40 b Fu(\026)-67 b Fq(m)3802 3639 y Fl(\030)0 3741 y Fu(b)m(y)32 b(\(2.2\))e(has)g(no)g(comp)s(onen)m(t)g(along)47 b(\026)-67 b Fq(m)1508 3705 y Fn(0)1508 3767 y Fl(\030)1546 3741 y Fu(,)30 b(hence)i(the)e(linearized)e(ev)m(olution)i(starting)f (from)f Fq(v)34 b Fu(deca)m(ys)0 3857 y(exp)s(onen)m(tially)47 b(fast)g(and)h(corresp)s(ondingly)f Fq(m)h Fu(con)m(v)m(erges)i(to)66 b(\026)-67 b Fq(m)2549 3872 y Fl(\030)2587 3857 y Fu(,)51 b(so)d(that)g(the)g(cen)m(ter)h(of)e Fq(m)h Fu(is)0 3973 y(also)34 b(the)h(cen)m(ter)g(of)f(the)h(instan)m(ton)g(to)f(whic)m(h)h (the)g(linearized)e(A)m(C)i(ev)m(olution)f(con)m(v)m(erges.)52 b(The)35 b(pair)0 4089 y Fo(f)p Fq(\030)5 b(;)17 b(m)k Fo(\000)41 b Fu(\026)-67 b Fq(m)433 4104 y Fl(\030)471 4089 y Fo(g)32 b Fu(is)h(kno)m(wn)g(as)g(the)g(F)-8 b(ermi)31 b(co)s(ordinates)h(of)g Fq(m)p Fu(.)100 4206 y(This)27 b(notion)f(of)h(a)g(cen)m(ter)h(of)f(a)g(function)f(pla)m(ys)i(an)f (imp)s(ortan)m(t)e(role)h(also)g(in)h(our)g(pro)s(ofs,)h(so)f(w)m(e)h (will)0 4322 y(sp)s(end)34 b(a)e(few)h(more)f(w)m(ords,)h(recalling)d (Prop)s(osition)h(3.2)h(in)g([9],)h(whic)m(h)g(giv)m(es)g(a)f (su\016cien)m(t)h(condition)0 4438 y(for)f Fq(m)c Fu(=)g Fq(m)p Fu(\()p Fq(x)p Fu(\))33 b(to)f(ha)m(v)m(e)i(a)e(cen)m(ter.)0 4791 y Fv(Prop)s(osition)51 b(2.1.)e Ft(Ther)-5 b(e)46 b(is)h(a)g(c)-5 b(onstant)47 b Fq(\016)54 b(>)d Fu(0)c Ft(such)g(that)g(if)g(ther)-5 b(e)47 b(exists)g Fq(x)3212 4806 y Fm(0)3303 4791 y Fo(2)k Fj(R)58 b Ft(so)47 b(that)0 4907 y Fo(k)p Fq(m)24 b Fo(\000)43 b Fu(\026)-67 b Fq(m)346 4922 y Fl(x)386 4931 y Fi(0)425 4907 y Fo(k)475 4922 y Fn(1)582 4907 y Fo(\024)33 b Fq(\016)41 b Ft(then)d(the)f(fol)5 b(lowing)36 b(holds)h(for)g(some)g(c)-5 b(onstant)37 b Fq(C)j Fu(=)32 b Fq(C)7 b Fu(\()p Fq(\016)t Fu(\))38 b Ft(indep)-5 b(endent)36 b(of)h Fq(x)3800 4922 y Fm(0)0 5023 y Ft(and)d Fq(m)p Ft(.)170 5190 y Fu(\(i\))41 b Ft(The)34 b(function)g Fq(m)i Ft(has)e(a)h(unique)f(c)-5 b(enter)35 b Fq(x)g Ft(and)1379 5408 y Fo(j)p Fq(x)22 b Fo(\000)h Fq(x)1639 5423 y Fm(0)1679 5408 y Fo(j)k(\024)h Fq(C)7 b Fo(k)p Fq(m)22 b Fo(\000)41 b Fu(\026)-67 b Fq(m)2258 5423 y Fl(x)2298 5432 y Fi(0)2337 5408 y Fo(k)2387 5423 y Fn(1)3639 5408 y Fu(\(2.4\))1901 5532 y Fk(5)p eop %%Page: 6 7 6 6 bop 143 266 a Fu(\(ii\))40 b Ft(the)34 b(c)-5 b(enter)35 b Fq(x)g Ft(has)g(the)g(exp)-5 b(ansion)321 507 y Fq(x)28 b Fu(=)f Fq(x)562 522 y Fm(0)624 507 y Fo(\000)734 440 y Fu(3)p 734 484 49 4 v 734 576 a(4)793 507 y Fo(h)18 b Fu(\026)-67 b Fq(m)917 466 y Fn(0)917 532 y Fl(x)957 541 y Fi(0)995 507 y Fq(;)17 b(m)22 b Fo(\000)41 b Fu(\026)-67 b Fq(m)1331 522 y Fl(x)1371 531 y Fi(0)1410 507 y Fo(i)22 b(\000)1605 440 y Fu(9)p 1580 484 98 4 v 1580 576 a(16)1688 507 y Fo(h)c Fu(\026)-67 b Fq(m)1812 466 y Fn(0)1812 532 y Fl(x)1852 541 y Fi(0)1890 507 y Fq(;)17 b(m)23 b Fo(\000)40 b Fu(\026)-67 b Fq(m)2226 522 y Fl(x)2266 531 y Fi(0)2305 507 y Fo(ih)18 b Fu(\026)-67 b Fq(m)2468 466 y Fn(00)2468 532 y Fl(x)2508 541 y Fi(0)2546 507 y Fq(;)17 b(m)22 b Fo(\000)41 b Fu(\026)-67 b Fq(m)2882 522 y Fl(x)2922 531 y Fi(0)2961 507 y Fo(i)22 b Fu(+)g Fq(R)q Fu(\()p Fq(m)g Fo(\000)41 b Fu(\026)-67 b Fq(m)3525 522 y Fl(x)3565 531 y Fi(0)3604 507 y Fu(\))p Fq(;)321 697 y Fo(j)o Fq(R)q Fu(\()p Fq(m)23 b Fo(\000)41 b Fu(\026)-67 b Fq(m)754 712 y Fl(x)794 721 y Fi(0)832 697 y Fu(\))p Fo(j)28 b(\024)g Fq(C)7 b Fo(k)p Fq(m)22 b Fo(\000)41 b Fu(\026)-67 b Fq(m)1450 712 y Fl(x)1490 721 y Fi(0)1528 697 y Fo(k)1578 656 y Fm(3)1578 722 y Fn(1)3639 697 y Fu(\(2.5\))100 1126 y(Th)m(us,)28 b(b)m(y)f(\(2.1\))d(and)h(\(1.11\))f (and)i(with)e(the)h(help)g(of)g(Prop)s(osition)e(2.1,)j(w)m(e)g(can)f (talk)f(unam)m(bigously)-8 b(,)0 1243 y(with)32 b(probabilit)m(y)e (going)h(to)h(1,)g(of)f(the)i(cen)m(ter)h Fq(\030)5 b Fu(\()p Fq(t)p Fu(\))31 b(of)h Fq(m)p Fu(\()p Fq(t)p 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f(as)h(in)f(Theorem)g(1.1)h(and)f(it)g(leads)g(to)g(the)h(same)g(limit) c(la)m(w)j(but)0 2409 y(without)30 b(drift,)h(i.e.)f(a)h(Bro)m(wnian)f (motion.)41 b(T)-8 b(o)31 b(explain)f(heuristically)f(the)j(result,)f (let)f(us)i(regard)f(the)0 2525 y(forcing)e(term)551 2463 y Fo(p)p 634 2463 57 4 v 62 x Fq(\015)5 b(dW)847 2489 y Fm(\()p Fl(a)p Fm(\))943 2525 y Fu(\()p Fq(t)p Fu(\))30 b(as)h(a)f(\\source)h(of)f(small)e(kic)m(ks")j(whic)m(h)g(w)m (e)h(decomp)s(ose)e(in)g(a)g(comp)s(onen)m(t)0 2641 y(along)j Fo(M)h Fu(and)g(another)h(one)g(orthogonal)d(to)i Fo(M)p Fu(.)49 b(The)35 b(latter)e(\014gh)m(ts)j(against)d(the)i(A)m(C)g (drift)e(whic)m(h)0 2757 y(pushes)44 b(bac)m(k)f(to)m(w)m(ard)f Fo(M)p Fu(,)i(and)e(b)s(ecause)h(of)f(the)g(small)e(factor)2507 2695 y Fo(p)p 2590 2695 V 62 x Fq(\015)5 b Fu(,)44 b(to)e(a)g(\014rst)g (order,)i(w)m(e)f(forget)0 2874 y(ab)s(out)31 b(orthogonal)f(comp)s (onen)m(ts.)43 b(On)32 b(the)g(con)m(trary)g(the)g(kic)m(ks)h(along)d Fo(M)h 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Fm(0)1919 495 y Fq(p)1968 510 y Fl(t)p Fn(\000)p Fl(s)2085 495 y Fq(dm)2221 453 y Fm(\()p Fl(\025)p Fm(\))2321 495 y Fu(\()p Fq(s)p Fu(\))1148 b(\(2.11\))0 713 y(Using)32 b(again)f(\(2.7\))h(and)h(taking)f(the)h(scalar)f(pro)s(duct)h(with)50 b(\026)-67 b Fq(m)2376 677 y Fn(0)2376 743 y Fl(\030)s Fm(\()p Fl(t)p Fm(\))2527 713 y Fu(w)m(e)33 b(get)g(from)e(\(2.11\)) 1027 963 y Fo(h)18 b Fu(\026)-67 b Fq(m)1151 922 y Fn(0)1151 988 y Fl(\030)s Fm(\()p Fl(t)p Fm(\))1269 963 y Fq(;)17 b(h)p Fu(\()p Fq(t)p Fu(\))p Fo(i)28 b(\031)g(\000)1746 827 y Fp(Z)1845 854 y Fl(t)1801 1053 y Fm(0)1892 963 y Fo(h)18 b Fu(\026)-67 b Fq(m)2016 922 y Fn(0)2016 988 y Fl(\030)s Fm(\()p Fl(t)p Fm(\))2134 963 y Fq(;)17 b(p)2227 978 y Fl(t)p Fn(\000)p Fl(s)2362 963 y Fu(\026)-67 b Fq(m)2429 922 y Fn(0)2429 988 y Fl(\030)s Fm(\()p Fl(s)p Fm(\))2554 963 y Fo(i)p Fq(d\030)5 b Fu(\()p Fq(s)p Fu(\))777 b(\(2.12\))0 1182 y(The)27 b(appro)m(ximate)e(system)j(\(2.10\))o ({\(2.12\))d(\(with)h Fo(\031)h 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b(a)d(naiv)m(e)g(estimate)f(giv)m(es)i(a) f(non{negligible)d(con)m(tribution.)63 b(The)40 b(fact)f(that)0 5316 y(they)d(are)g(indeed)g(negligible)c(m)m(ust)k(then)g(come)f(from) g(cancellations,)f(and)i(the)g(true)g(problem)e(is)h(to)0 5433 y(\014nd)e(and)g(tak)m(e)g(them)f(in)m(to)g(prop)s(er)h(accoun)m (t.)1901 5532 y Fk(7)p eop %%Page: 8 9 8 8 bop 100 266 a Fu(This)28 b(is)g(more)g(clearly)g(seen)h(in)f(the)h (sto)s(c)m(hastic)g(A)m(C)g(equation.)42 b(F)-8 b(ollo)m(wing)26 b([10)o(],)k(w)m(e)f(split)e(the)i(time)0 382 y(axis)24 b(in)m(to)g(in)m(terv)-5 b(als)23 b(of)h(length)g Fq(T)14 b Fu(,)26 b(taking)e Fq(T)38 b Fu(long,)25 b(y)m(et)g(v)m(ery)h(small)c (as)j(compared)f(to)g(the)h(macroscopic)0 498 y(time)e Fq(\015)270 462 y Fn(\000)p Fm(1)364 498 y Fq(\034)11 b Fu(,)27 b(for)d(instance)h Fq(T)41 b Fu(=)27 b Fq(\015)1241 462 y Fn(\000)p Fm(1)p Fl(=)p Fm(10)1442 498 y Fu(.)40 b(The)26 b(crucial)d(p)s(oin)m(t)g(is)h(an)h(iterativ)m(e)e(pro)s (cedure)j(for)e(whic)m(h)h(the)0 614 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(ha)m(v)m(e)h(already)e(de\014ned)i(for)e(all)f Fq(k)h Fo(\024)f Fq(n)24 b Fo(\000)h Fu(1)35 b(b)s(oth)g Fq(x)3604 3372 y Fl(k)3683 3357 y Fu(and)0 3474 y Fq(v)51 3437 y Fm(\()p Fl(k)r Fm(\))148 3474 y Fu(\()p Fq(t)p Fu(\))d(and)f(that)h (\(3.1\))f(holds)g(for)g(suc)m(h)i Fq(k)s Fu('s.)44 b(If)31 b(there)h(is)f Fq(k)g Fo(\024)d Fq(n)20 b Fo(\000)h Fu(1)31 b(suc)m(h)i(that)e Fo(k)p Fq(v)3147 3437 y Fm(\()p Fl(k)r Fm(\))3244 3474 y Fu(\()p Fq(T)3339 3489 y Fl(k)r Fm(+1)3472 3474 y Fu(\))p Fo(k)3560 3489 y Fn(1)3662 3474 y Fq(>)d(\016)t Fu(,)0 3593 y(w)m(e)43 b(set)f Fq(x)369 3608 y Fl(n)460 3593 y Fu(=)h(0)f(and)g Fq(v)920 3557 y Fm(\()p Fl(n)p Fm(\))1021 3593 y Fu(\()p Fq(t)p Fu(\))i(=)f(0,)h Fq(t)f Fo(2)h Fu([)p Fq(T)1687 3608 y Fl(n)1734 3593 y Fq(;)17 b(T)1835 3608 y Fl(n)p Fm(+1)1972 3593 y Fu(].)72 b(Otherwise)42 b(w)m(e)h(de\014ne)g Fq(x)3067 3608 y Fl(n)3156 3593 y Fu(as)f(the)g(cen)m(ter)h(of)18 3712 y(\026)-67 b Fq(m)85 3727 y Fl(x)125 3736 y Fg(n)p Fh(\000)p Fi(1)277 3712 y Fu(+)27 b Fq(v)431 3676 y Fm(\()p Fl(n)p Fn(\000)p Fm(1\))623 3712 y Fu(\()p Fq(T)718 3727 y Fl(n)765 3712 y Fu(\),)42 b(whic)m(h)f(b)m(y)g(Prop)s(osition)d(2.1)i(is)f(w)m(ell)h (de\014ned,)j(as)d Fo(k)p Fq(v)2910 3676 y Fm(\()p Fl(n)p Fn(\000)p Fm(1\))3102 3712 y Fu(\()p Fq(T)3197 3727 y Fl(n)3244 3712 y Fu(\))p Fo(k)3332 3727 y Fn(1)3447 3712 y Fo(\024)h Fq(\016)t Fu(.)66 b(W)-8 b(e)0 3837 y(then)36 b(set,)h(according)e(to)g(\(3.1\))o(,)h Fq(v)1286 3801 y Fm(\()p Fl(n)p Fm(\))1388 3837 y Fu(\()p Fq(t)p Fu(\))d(=)f Fq(m)p Fu(\()p Fq(t)p Fu(\))24 b Fo(\000)43 b Fu(\026)-67 b Fq(m)2047 3852 y Fl(x)2087 3860 y Fg(n)2134 3837 y Fu(,)36 b Fq(t)d Fo(2)f Fu([)p Fq(T)2447 3852 y Fl(n)2495 3837 y Fq(;)17 b(T)2596 3852 y Fl(n)p Fm(+1)2733 3837 y Fu(],)36 b(and)f(ha)m(v)m(e)i(\(alw)m(a)m(ys)f(under)0 3956 y(the)d(assumption)f(that)g Fo(k)p Fq(v)996 3920 y Fm(\()p Fl(k)r Fm(\))1093 3956 y Fu(\()p Fq(T)1188 3971 y Fl(k)r Fm(+1)1321 3956 y Fu(\))p Fo(k)1409 3971 y Fn(1)1511 3956 y Fo(\024)d Fq(\016)36 b Fu(for)c(all)f Fq(k)f Fo(\024)f Fq(n)22 b Fo(\000)h Fu(1\))425 4190 y Fq(m)p Fu(\()p Fq(T)605 4205 y Fl(n)653 4190 y Fu(\))k(=)46 b(\026)-67 b Fq(m)907 4205 y Fl(x)947 4214 y Fg(n)p Fh(\000)p Fi(1)1094 4190 y Fu(+)22 b Fq(v)1243 4149 y Fm(\()p Fl(n)p Fn(\000)p Fm(1\))1435 4190 y Fu(\()p Fq(T)1530 4205 y Fl(n)1577 4190 y Fu(\))27 b(=)46 b(\026)-67 b Fq(m)1831 4205 y Fl(x)1871 4213 y Fg(n)1940 4190 y Fu(+)22 b Fq(v)2089 4149 y Fm(\()p Fl(n)p Fm(\))2190 4190 y Fu(\()p Fq(T)2285 4205 y Fl(n)2333 4190 y Fu(\))p Fq(;)211 b Fo(h)18 b Fu(\026)-67 b Fq(m)2733 4149 y Fn(0)2733 4215 y Fl(x)2773 4223 y Fg(n)2820 4190 y Fq(;)17 b(v)2915 4149 y Fm(\()p Fl(n)p Fm(\))3016 4190 y Fu(\()p Fq(T)3111 4205 y Fl(n)3158 4190 y Fu(\))p Fo(i)27 b Fu(=)h(0)224 b(\(3.2\))100 4416 y(Let)1227 4600 y Fq(h)1283 4559 y Fm(\()p Fl(n)p Fm(\))1385 4600 y Fu(\()p Fq(t)p Fu(\))27 b(=)h Fq(h)p Fu(\()p Fq(t)p Fu(\))p Fq(;)212 b(t)28 b Fo(2)g Fu([)p Fq(T)2274 4615 y Fl(n)2321 4600 y Fq(;)17 b(T)2422 4615 y Fl(n)p Fm(+1)2559 4600 y Fu(])p Fq(;)1026 b Fu(\(3.3\))0 4806 y(in)m(tro)s(duce)33 b(the)g(stopping)f(time)f(w.r.t.)i(the)g(discrete)g(\014ltration)e Fo(F)2485 4821 y Fl(T)2526 4830 y Fg(n)p Fi(+1)1189 5045 y Fq(N)1267 5060 y Fl(\016)1333 5045 y Fu(:=)d(inf)1598 4965 y Fp(\010)1656 5045 y Fq(k)48 b Fu(:)c Fo(k)p Fq(v)1927 5004 y Fm(\()p Fl(k)r Fm(\))2024 5045 y Fu(\()p Fq(T)2119 5060 y Fl(k)r Fm(+1)2252 5045 y Fu(\))p Fo(k)2340 5060 y Fn(1)2442 5045 y Fq(>)28 b(\016)2593 4965 y Fp(\011)3639 5045 y Fu(\(3.4\))0 5299 y(\()p Fq(\016)41 b Fu(as)36 b(in)g(Prop)s(osition)f(2.1\))h(and)h(de\014ne)h Fq(g)1622 5248 y Fm(\()p Fl(n)p Fm(\))1618 5321 y Fl(t)1758 5299 y Fu(:=)d(exp)q Fo(f)p Fq(tL)2196 5314 y Fl(x)2236 5322 y Fg(n)2282 5299 y Fo(g)p Fu(,)j Fq(t)d Fo(2)g Fj(R)2634 5314 y Fm(+)2699 5299 y Fu(,)j(with)e Fq(L)3056 5314 y Fl(x)3096 5322 y Fg(n)3179 5299 y Fu(as)h(in)f(\(2.3\).)56 b(W)-8 b(e)0 5433 y(shortand)33 b Fq(g)454 5448 y Fl(t)511 5433 y Fu(=)28 b Fq(g)666 5382 y Fm(\(0\))662 5455 y Fl(t)759 5433 y Fu(.)1882 5532 y Fk(11)p eop %%Page: 12 13 12 12 bop 100 266 a Fu(After)40 b(expanding)g Fq(V)917 230 y Fn(0)940 266 y Fu(\()p Fq(m)p Fu(\()p Fq(t)p Fu(\)\))h(in)e (\(1.8\))h(and)g(\(1.10\))g(ab)s(out)58 b(\026)-67 b Fq(m)2474 281 y Fl(x)2514 289 y Fg(n)2560 266 y Fu(,)43 b(for)c Fq(t)i Fo(2)h Fu([)p Fq(T)3054 281 y Fl(n)3101 266 y Fq(;)17 b(T)3202 281 y Fl(n)p Fm(+1)3339 266 y Fu(])40 b(as)h(long)e(as)0 385 y Fq(n)28 b Fo(\024)g Fq(N)269 400 y Fl(\016)307 385 y Fu(,)33 b Fq(v)418 349 y Fm(\()p Fl(n)p Fm(\))519 385 y Fu(\()p Fq(t)p Fu(\))g(and)g Fq(h)909 349 y Fm(\()p Fl(n)p Fm(\))1011 385 y Fu(\()p Fq(t)p Fu(\))f(are)h(giv)m(en)f(as)h(the)g(solution)e(of)451 573 y Fq(dv)553 532 y Fm(\()p Fl(n)p Fm(\))654 573 y Fu(\()p Fq(t)p Fu(\))d(=)896 492 y Fp(\002)938 573 y Fq(L)1004 588 y Fl(x)1044 596 y Fg(n)1090 573 y Fq(v)1141 532 y Fm(\()p Fl(n)p Fm(\))1243 573 y Fu(\()p Fq(t)p Fu(\))22 b(+)g Fq(\025h)1587 532 y Fm(\()p Fl(n)p Fm(\))1689 573 y Fu(\()p Fq(t)p Fu(\))g Fo(\000)h Fu(3)18 b(\026)-67 b Fq(m)2056 588 y Fl(x)2096 596 y Fg(n)2142 573 y Fq(v)2193 532 y Fm(\()p Fl(n)p Fm(\))2295 573 y Fu(\()p Fq(t)p Fu(\))2406 532 y Fm(2)2468 573 y Fo(\000)22 b Fq(v)2618 532 y Fm(\()p Fl(n)p Fm(\))2720 573 y Fu(\()p Fq(t)p Fu(\))2831 532 y Fm(3)2870 492 y Fp(\003)2929 573 y Fq(dt)g Fu(+)g Fq(\025dW)3349 532 y Fm(\()p Fl(a)p Fm(\))3444 573 y Fu(\()p Fq(t)p Fu(\))84 b(\(3.5\))451 795 y Fq(h)507 754 y Fm(\()p Fl(n)p Fm(\))609 795 y Fu(\()p Fq(t)p Fu(\))27 b(=)h Fq(p)900 810 y Fl(t)p Fn(\000)p Fl(T)1021 818 y Fg(n)1068 795 y Fq(h)1124 754 y Fm(\()p Fl(n)p Fn(\000)p Fm(1\))1316 795 y Fu(\()p Fq(T)1411 810 y Fl(n)1458 795 y Fu(\))22 b Fo(\000)h Fq(v)1669 754 y Fm(\()p Fl(n)p Fm(\))1770 795 y Fu(\()p Fq(t)p Fu(\))g(+)f Fq(p)2051 810 y Fl(t)p Fn(\000)p Fl(T)2172 818 y Fg(n)2219 795 y Fq(v)2270 754 y Fm(\()p Fl(n)p Fm(\))2371 795 y Fu(\()p Fq(T)2466 810 y Fl(n)2513 795 y Fu(\))h(+)2672 660 y Fp(Z)2771 686 y Fl(t)2727 885 y(T)2768 893 y Fg(n)2815 795 y Fq(ds)2938 728 y(@)5 b(p)3043 743 y Fl(t)p Fn(\000)p Fl(s)p 2938 772 224 4 v 2998 864 a Fq(@)g(s)3171 795 y(v)3222 754 y Fm(\()p Fl(n)p Fm(\))3323 795 y Fu(\()p Fq(s)p Fu(\))194 b(\(3.6\))0 1051 y(with)42 b(initial)d(condition)j Fq(v)1014 1015 y Fm(\()p Fl(n)p Fm(\))1115 1051 y Fu(\()p Fq(T)1210 1066 y Fl(n)1257 1051 y Fu(\))h(for)f(the)i(\014rst)f (equation.)74 b(Recall)41 b(that)h Fq(v)2930 1015 y Fm(\()p Fl(n)p Fm(\))3032 1051 y Fu(\()p Fq(T)3127 1066 y Fl(n)3174 1051 y Fu(\))h(is)f(determined)0 1171 y(from)31 b Fq(v)281 1134 y Fm(\()p Fl(n)p Fn(\000)p Fm(1\))473 1171 y Fu(\()p Fq(T)568 1186 y Fl(n)615 1171 y Fu(\))i(via)e(\(3.2\),)i(and)f(that)h (from)e(\(3.3\))h(w)m(e)i(ha)m(v)m(e)g Fq(h)2395 1134 y Fm(\()p Fl(n)p Fn(\000)p Fm(1\))2587 1171 y Fu(\()p Fq(T)2682 1186 y Fl(n)2729 1171 y Fu(\))27 b(=)h Fq(h)2954 1134 y Fm(\()p Fl(n)p Fm(\))3056 1171 y Fu(\()p Fq(T)3151 1186 y Fl(n)3198 1171 y Fu(\).)100 1305 y(W)-8 b(e)36 b(will)d(write)j(in)f(the)h(next)h(section)f Fq(v)1602 1268 y Fm(\()p Fl(n)p Fm(\))1704 1305 y Fu(\()p Fq(t)p Fu(\))g(as)g(a)f(sum)h(of)g(10)f(terms,)i Fq(v)2870 1268 y Fm(\()p Fl(n)p Fm(\))2971 1305 y Fu(\()p Fq(t)p Fu(\))d(=)f(\000)3286 1254 y Fm(\()p Fl(n)p Fm(\))3286 1329 y(1)3388 1305 y Fu(\()p Fq(t)p Fu(\))24 b(+)h Fo(\001)17 b(\001)g(\001)22 b Fu(+)0 1439 y(\000)61 1388 y Fm(\()p Fl(n)p Fm(\))61 1463 y(10)163 1439 y Fu(\()p Fq(t)p Fu(\),)34 b(but)g(for)f(the)g (momen)m(t)g(it)g(is)g(more)f(con)m(v)m(enien)m(t)k(to)d(k)m(eep)i (some)e(of)h(the)f(terms)h(together.)47 b(W)-8 b(e)0 1555 y(are)33 b(going)e(to)h(pro)m(v)m(e)i(that)e(for)g Fq(n)c(<)g(N)1433 1570 y Fl(\016)1471 1555 y Fu(,)103 1747 y Fq(v)154 1706 y Fm(\()p Fl(n)p Fm(\))255 1747 y Fu(\()p Fq(t)p Fu(\))g(=)g(\000)559 1696 y Fm(\()p Fl(n)p Fm(\))559 1771 y(1)660 1747 y Fu(\()p Fq(t)p Fu(\))23 b(+)f(\000)953 1696 y Fm(\()p Fl(n)p Fm(\))953 1771 y(2)1055 1747 y Fu(\()p Fq(t)p Fu(\))g(+)g(\003)1354 1696 y Fm(\()p Fl(n)p Fm(\))1354 1771 y(3)1455 1747 y Fu(\()p Fq(t)p Fu(\))h(+)f(\000)1748 1696 y Fm(\()p Fl(n)p Fm(\))1748 1771 y(4)1849 1747 y Fu(\()p Fq(t)p Fu(\))h(+)f(\000)2142 1696 y Fm(\()p Fl(n)p Fm(\))2142 1771 y(5)2243 1747 y Fu(\()p Fq(t)p Fu(\))h(+)f(\000)2536 1696 y Fm(\()p Fl(n)p Fm(\))2536 1771 y(6)2638 1747 y Fu(\()p Fq(t)p Fu(\))g(+)g(\000)2930 1696 y Fm(\()p Fl(n)p Fm(\))2930 1771 y(9)3032 1747 y Fu(\()p Fq(t)p Fu(\))g(+)g(\000)3324 1696 y Fm(\()p Fl(n)p Fm(\))3324 1771 y(10)3426 1747 y Fu(\()p Fq(t)p Fu(\))102 b(\(3.7\))0 1927 y(where)773 2107 y(\000)834 2056 y Fm(\()p Fl(n)p Fm(\))834 2131 y(1)936 2107 y Fu(\()p Fq(t)p Fu(\))83 b(:=)g Fq(g)1367 2056 y Fm(\()p Fl(n)p Fm(\))1363 2134 y Fl(t)p Fn(\000)p Fl(T)1484 2142 y Fg(n)1531 2107 y Fq(v)1582 2066 y Fm(\()p Fl(n)p Fm(\))1683 2107 y Fu(\()p Fq(T)1778 2122 y Fl(n)1825 2107 y Fu(\))1776 b(\(3.8\))773 2330 y(\000)834 2279 y Fm(\()p Fl(n)p Fm(\))834 2354 y(2)936 2330 y Fu(\()p Fq(t)p Fu(\))83 b(:=)g Fq(\025z)1422 2289 y Fm(\()p Fl(n)p Fm(\))1524 2330 y Fu(\()p Fq(t)p Fu(\))28 b(:=)g Fq(\025)1868 2194 y Fp(Z)1967 2220 y Fl(t)1923 2420 y(T)1964 2428 y Fg(n)2010 2330 y Fq(g)2061 2279 y Fm(\()p Fl(n)p Fm(\))2057 2352 y Fl(t)p Fn(\000)p Fl(s)2191 2330 y Fq(dW)2348 2289 y Fm(\()p Fl(a)p Fm(\))2444 2330 y Fu(\()p Fq(s)p Fu(\))1073 b(\(3.9\))766 2628 y(\003)834 2578 y Fm(\()p Fl(n)p Fm(\))834 2653 y(3)936 2628 y Fu(\()p Fq(t)p Fu(\))83 b(:=)g Fo(\000)p Fq(\025)1517 2504 y Fl(n)1467 2534 y Fp(X)1474 2746 y Fl(k)r Fm(=1)1627 2493 y Fp(Z)1727 2519 y Fl(t)1683 2718 y(T)1724 2726 y Fg(n)1770 2628 y Fq(ds)17 b(g)1935 2578 y Fm(\()p Fl(n)p Fm(\))1931 2651 y Fl(t)p Fn(\000)p Fl(s)2048 2628 y Fq(p)2097 2643 y Fl(s)p Fn(\000)p Fl(T)2226 2655 y Fg(k)2267 2628 y Fu([)i(\026)-68 b Fq(m)2379 2643 y Fl(x)2419 2655 y Fg(k)2484 2628 y Fo(\000)41 b Fu(\026)-67 b Fq(m)2669 2643 y Fl(x)2709 2655 y Fg(k)q Fh(\000)p Fi(1)2829 2628 y Fu(])735 b(\(3.10\))773 2943 y(\000)834 2893 y Fm(\()p Fl(n)p Fm(\))834 2968 y(4)936 2943 y Fu(\()p Fq(t)p Fu(\))83 b(:=)g Fq(\025)1390 2808 y Fp(Z)1489 2834 y Fl(t)1445 3033 y(T)1486 3041 y Fg(n)1532 2943 y Fq(ds)17 b(g)1697 2893 y Fm(\()p Fl(n)p Fm(\))1693 2966 y Fl(t)p Fn(\000)p Fl(s)1826 2808 y Fp(Z)1926 2834 y Fl(s)1881 3033 y(T)1922 3041 y Fg(n)1986 2943 y Fq(ds)2083 2902 y Fn(0)2132 2876 y Fq(@)5 b(p)2237 2891 y Fl(s)p Fn(\000)p Fl(s)2358 2872 y Fh(0)p 2132 2921 253 4 v 2196 3012 a Fq(@)g(s)2298 2983 y Fn(0)2395 2943 y Fq(v)2446 2902 y Fm(\()p Fl(n)p Fm(\))2547 2943 y Fu(\()p Fq(s)2631 2902 y Fn(0)2655 2943 y Fu(\))898 b(\(3.11\))773 3229 y(\000)834 3178 y Fm(\()p Fl(n)p Fm(\))834 3254 y(5)936 3229 y Fu(\()p Fq(t)p Fu(\))83 b(:=)g Fo(\000)p Fq(\025)1467 3094 y Fp(Z)1566 3120 y Fl(t)1522 3319 y(T)1563 3327 y Fg(n)1610 3229 y Fq(ds)17 b(g)1775 3178 y Fm(\()p Fl(n)p Fm(\))1771 3252 y Fl(t)p Fn(\000)p Fl(s)1887 3229 y Fq(v)1938 3188 y Fm(\()p Fl(n)p Fm(\))2040 3229 y Fu(\()p Fq(s)p Fu(\))1429 b(\(3.12\))773 3528 y(\000)834 3477 y Fm(\()p Fl(n)p Fm(\))834 3552 y(6)936 3528 y Fu(\()p Fq(t)p Fu(\))83 b(:=)g Fo(\000)p Fq(\025)1517 3403 y Fl(n)1467 3433 y Fp(X)1474 3645 y Fl(k)r Fm(=1)1627 3392 y Fp(Z)1727 3419 y Fl(t)1683 3618 y(T)1724 3626 y Fg(n)1770 3528 y Fq(ds)17 b(g)1935 3477 y Fm(\()p Fl(n)p Fm(\))1931 3550 y Fl(t)p Fn(\000)p Fl(s)2064 3392 y Fp(Z)2164 3419 y Fl(T)2205 3431 y Fg(k)2119 3618 y Fl(T)2160 3630 y Fg(k)q Fh(\000)p Fi(1)2281 3528 y Fq(ds)2378 3487 y Fn(0)2428 3461 y Fq(@)5 b(p)2533 3476 y Fl(s)p Fn(\000)p Fl(s)2654 3457 y Fh(0)p 2428 3505 V 2503 3596 a Fq(@)g(s)2690 3528 y(v)2741 3487 y Fm(\()p Fl(k)r Fn(\000)p Fm(1\))2929 3528 y Fu(\()p Fq(s)3013 3487 y Fn(0)3036 3528 y Fu(\))517 b(\(3.13\))773 3843 y(\000)834 3792 y Fm(\()p Fl(n)p Fm(\))834 3867 y(9)936 3843 y Fu(\()p Fq(t)p Fu(\))83 b(:=)g Fo(\000)p Fu(3)1459 3707 y Fp(Z)1558 3734 y Fl(t)1514 3933 y(T)1555 3941 y Fg(n)1602 3843 y Fq(ds)17 b(g)1767 3792 y Fm(\()p Fl(n)p Fm(\))1763 3865 y Fl(t)p Fn(\000)p Fl(s)1896 3762 y Fp(\002)1955 3843 y Fu(\026)-67 b Fq(m)2022 3858 y Fl(x)2062 3866 y Fg(n)2109 3843 y Fq(v)2160 3802 y Fm(\()p Fl(n)p Fm(\))2261 3843 y Fu(\()p Fq(s)p Fu(\))2383 3802 y Fm(2)2423 3762 y Fp(\003)3591 3843 y Fu(\(3.14\))773 4129 y(\000)834 4078 y Fm(\()p Fl(n)p Fm(\))834 4153 y(10)936 4129 y Fu(\()p Fq(t)p Fu(\))83 b(:=)g Fo(\000)1410 3993 y Fp(Z)1510 4019 y Fl(t)1465 4219 y(T)1506 4227 y Fg(n)1553 4129 y Fq(ds)17 b(g)1718 4078 y Fm(\()p Fl(n)p Fm(\))1714 4151 y Fl(t)p Fn(\000)p Fl(s)1847 4048 y Fp(\002)1888 4129 y Fq(v)1939 4088 y Fm(\()p Fl(n)p Fm(\))2041 4129 y Fu(\()p Fq(s)p Fu(\))2163 4088 y Fm(3)2202 4048 y Fp(\003)3591 4129 y Fu(\(3.15\))0 4454 y Ft(Pr)-5 b(o)g(of)35 b(of)55 b Fu(\(3.7\))p Ft(.)43 b Fu(By)33 b(applying)e(\()p Fq(@)1311 4469 y Fl(t)1364 4454 y Fo(\000)22 b Fq(L)1529 4469 y Fl(x)1569 4477 y Fg(n)1616 4454 y Fu(\))1654 4406 y Fn(\000)p Fm(1)1781 4454 y Fu(to)32 b(\(3.5\))g(w)m(e)i(get)539 4701 y Fq(v)590 4660 y Fm(\()p Fl(n)p Fm(\))692 4701 y Fu(\()p Fq(t)p Fu(\))28 b(=)f(\000)995 4650 y Fm(\()p Fl(n)p Fm(\))995 4726 y(1)1097 4701 y Fu(\()p Fq(t)p Fu(\))22 b(+)g(\000)1389 4650 y Fm(\()p Fl(n)p Fm(\))1389 4726 y(2)1491 4701 y Fu(\()p Fq(t)p Fu(\))g(+)g(\000)1783 4650 y Fm(\()p Fl(n)p Fm(\))1783 4726 y(9)1885 4701 y Fu(\()p Fq(t)p Fu(\))g(+)g(\000)2177 4650 y Fm(\()p Fl(n)p Fm(\))2177 4726 y(10)2279 4701 y Fu(\()p Fq(t)p Fu(\))g(+)h Fq(\025)2585 4566 y Fp(Z)2684 4592 y Fl(t)2639 4791 y(T)2680 4799 y Fg(n)2727 4701 y Fq(ds)17 b(g)2892 4650 y Fm(\()p Fl(n)p Fm(\))2888 4724 y Fl(t)p Fn(\000)p Fl(s)3021 4701 y Fq(h)3077 4660 y Fm(\()p Fl(n)p Fm(\))3179 4701 y Fu(\()p Fq(s)p Fu(\))0 4957 y(W)-8 b(e)33 b(use)h(for)e Fq(h)542 4921 y Fm(\()p Fl(n)p Fm(\))644 4957 y Fu(\()p Fq(s)p Fu(\))g(the)h(expression)h(giv)m(en)e(b)m(y)j(\(3.6\),)d(and)h(get)545 5155 y Fq(v)596 5114 y Fm(\()p Fl(n)p Fm(\))698 5155 y Fu(\()p Fq(t)p Fu(\))83 b(=)g(\000)1112 5104 y Fm(\()p Fl(n)p Fm(\))1112 5180 y(1)1214 5155 y Fu(\()p Fq(t)p Fu(\))22 b(+)g(\000)1506 5104 y Fm(\()p Fl(n)p Fm(\))1506 5180 y(2)1608 5155 y Fu(\()p Fq(t)p Fu(\))g(+)g(\000)1900 5104 y Fm(\()p Fl(n)p Fm(\))1900 5180 y(4)2002 5155 y Fu(\()p Fq(t)p Fu(\))g(+)g(\000)2294 5104 y Fm(\()p Fl(n)p Fm(\))2294 5180 y(5)2396 5155 y Fu(\()p Fq(t)p Fu(\))g(+)g(\000)2688 5104 y Fm(\()p Fl(n)p Fm(\))2688 5180 y(9)2790 5155 y Fu(\()p Fq(t)p Fu(\))g(+)g(\000)3082 5104 y Fm(\()p Fl(n)p Fm(\))3082 5180 y(10)3184 5155 y Fu(\()p Fq(t)p Fu(\))1083 5368 y(+)p Fq(\025)1233 5232 y Fp(Z)1332 5258 y Fl(t)1288 5458 y(T)1329 5466 y Fg(n)1376 5368 y Fq(ds)17 b(g)1541 5317 y Fm(\()p Fl(n)p Fm(\))1537 5390 y Fl(t)p Fn(\000)p Fl(s)1653 5368 y Fq(p)1702 5383 y Fl(s)p Fn(\000)p Fl(T)1831 5391 y Fg(n)1894 5287 y Fp(\002)1935 5368 y Fq(h)1991 5326 y Fm(\()p Fl(n)p Fn(\000)p Fm(1\))2183 5368 y Fu(\()p Fq(T)2278 5383 y Fl(n)2325 5368 y Fu(\))22 b(+)g Fq(v)2534 5326 y Fm(\()p Fl(n)p Fm(\))2636 5368 y Fu(\()p Fq(T)2731 5383 y Fl(n)2778 5368 y Fu(\))2816 5287 y Fp(\003)1882 5532 y Fk(12)p eop %%Page: 13 14 13 13 bop 0 266 a Fu(T)-8 b(o)33 b(complete)e(the)i(pro)s(of)f(of)39 b(\(3.7\))32 b(w)m(e)i(th)m(us)g(need)f(to)f(sho)m(w)668 508 y Fq(\025)742 372 y Fp(Z)841 398 y Fl(t)797 598 y(T)838 606 y Fg(n)884 508 y Fq(ds)17 b(g)1049 457 y Fm(\()p Fl(n)p Fm(\))1045 530 y Fl(t)p Fn(\000)p Fl(s)1161 508 y Fq(p)1210 523 y Fl(s)p Fn(\000)p Fl(T)1339 531 y Fg(n)1402 427 y Fp(\002)1444 508 y Fq(h)1500 466 y Fm(\()p Fl(n)p Fn(\000)p Fm(1\))1692 508 y Fu(\()p Fq(T)1787 523 y Fl(n)1834 508 y Fu(\))22 b(+)g Fq(v)2043 466 y Fm(\()p Fl(n)p Fm(\))2145 508 y Fu(\()p Fq(T)2240 523 y Fl(n)2287 508 y Fu(\))2325 427 y Fp(\003)2394 508 y Fu(=)27 b(\003)2565 457 y Fm(\()p Fl(n)p Fm(\))2565 532 y(3)2667 508 y Fu(\()p Fq(t)p Fu(\))22 b(+)g(\000)2959 457 y Fm(\()p Fl(n)p Fm(\))2959 532 y(6)3061 508 y Fu(\()p Fq(t)p Fu(\))419 b(\(3.16\))0 747 y(F)-8 b(rom)31 b(\(3.2\),)h(w)m(e)i(ha)m(v)m(e)558 930 y Fq(h)614 889 y Fm(\()p Fl(n)p Fn(\000)p Fm(1\))806 930 y Fu(\()p Fq(T)901 945 y Fl(n)948 930 y Fu(\))22 b(+)g Fq(v)1157 889 y Fm(\()p Fl(n)p Fm(\))1259 930 y Fu(\()p Fq(T)1354 945 y Fl(n)1401 930 y Fu(\))27 b(=)h Fq(h)1626 889 y Fm(\()p Fl(n)p Fn(\000)p Fm(1\))1818 930 y Fu(\()p Fq(T)1913 945 y Fl(n)1960 930 y Fu(\))22 b(+)g Fq(v)2169 889 y Fm(\()p Fl(n)p Fn(\000)p Fm(1\))2361 930 y Fu(\()p Fq(T)2456 945 y Fl(n)2503 930 y Fu(\))g Fo(\000)h Fu(\()18 b(\026)-67 b Fq(m)2786 945 y Fl(x)2826 953 y Fg(n)2895 930 y Fo(\000)40 b Fu(\026)-67 b Fq(m)3079 945 y Fl(x)3119 954 y Fg(n)p Fh(\000)p Fi(1)3244 930 y Fu(\))309 b(\(3.17\))0 1105 y(W)-8 b(e)27 b(use)h(this)e(iden)m(tit)m(y)h(in)f(the)h(l.h.s.)g(of)33 b(\(3.16\).)41 b(The)28 b(di\013erence)g(\()18 b(\026)-67 b Fq(m)2541 1120 y Fl(x)2581 1128 y Fg(n)2638 1105 y Fo(\000)29 b Fu(\026)-68 b Fq(m)2810 1120 y Fl(x)2850 1129 y Fg(n)p Fh(\000)p Fi(1)2976 1105 y Fu(\))26 b(repro)s(duces)i (the)g(last)0 1245 y(term)k(of)f(the)i(sum)f(in)f(\(3.10\))g(\(i.e.)h (in)f(the)h(de\014nition)f(of)h(\003)2198 1194 y Fm(\()p Fl(n)p Fm(\))2198 1269 y(3)2300 1245 y Fu(\()p Fq(t)p Fu(\)\).)43 b(F)-8 b(or)31 b(the)i(term)e(with)h Fq(h)3371 1209 y Fm(\()p Fl(n)p Fn(\000)p Fm(1\))3563 1245 y Fu(\()p Fq(T)3658 1260 y Fl(n)3705 1245 y Fu(\))21 b(+)0 1364 y Fq(v)51 1328 y Fm(\()p Fl(n)p Fn(\000)p Fm(1\))243 1364 y Fu(\()p Fq(T)338 1379 y Fl(n)385 1364 y Fu(\),)32 b(w)m(e)i(use)g(\(3.6\))e(to)g(write)399 1547 y Fq(h)455 1506 y Fm(\()p Fl(n)p Fn(\000)p Fm(1\))647 1547 y Fu(\()p Fq(T)742 1562 y Fl(n)789 1547 y Fu(\))22 b(+)g Fq(v)998 1506 y Fm(\()p Fl(n)p Fn(\000)p Fm(1\))1190 1547 y Fu(\()p Fq(T)1285 1562 y Fl(n)1332 1547 y Fu(\))654 1764 y(=)50 b Fq(p)829 1779 y Fl(T)870 1787 y Fg(n)913 1779 y Fn(\000)p Fl(T)1009 1788 y Fg(n)p Fh(\000)p Fi(1)1150 1683 y Fp(\010)1208 1764 y Fq(h)1264 1723 y Fm(\()p Fl(n)p Fn(\000)p Fm(1\))1456 1764 y Fu(\()p Fq(T)1551 1779 y Fl(n)p Fn(\000)p Fm(1)1689 1764 y Fu(\))22 b(+)g Fq(v)1898 1723 y Fm(\()p Fl(n)p Fn(\000)p Fm(1\))2090 1764 y Fu(\()p Fq(T)2185 1779 y Fl(n)p Fn(\000)p Fm(1)2322 1764 y Fu(\))2360 1683 y Fp(\011)2440 1764 y Fu(+)2538 1628 y Fp(Z)2638 1654 y Fl(T)2679 1662 y Fg(n)2593 1854 y Fl(T)2634 1863 y Fg(n)p Fh(\000)p Fi(1)2759 1764 y Fq(ds)2856 1723 y Fn(0)2906 1696 y Fq(@)5 b(p)3011 1711 y Fl(T)3052 1719 y Fg(n)3095 1711 y Fn(\000)p Fl(s)3183 1692 y Fh(0)p 2906 1741 304 4 v 2995 1832 a Fq(@)g(s)3097 1803 y Fn(0)3219 1764 y Fq(v)3270 1723 y Fm(\()p Fl(n)p Fn(\000)p Fm(1\))3462 1764 y Fu(\()p Fq(s)3546 1723 y Fn(0)3569 1764 y Fu(\))0 2011 y(The)31 b(con)m(tribution)e(of)g(the)i(last)e(in)m(tegral)f(to)i(the)g(l.h.s.)g (of)36 b(\(3.16\))30 b(giv)m(es)g(the)g(last)g(term)f(of)g(the)i(sum)f (in)0 2145 y(\(3.13\))o(,)37 b(i.e.)f(the)g(de\014nition)g(of)f(\000) 1258 2094 y Fm(\()p Fl(n)p Fm(\))1258 2170 y(6)1360 2145 y Fu(\()p Fq(t)p Fu(\).)54 b(By)37 b(iteration)d(w)m(e)j(then)g(get)f (\(3.16\))f(and)h(\(3.7\))g(is)f(therefore)0 2262 y(pro)m(v)m(ed.)3453 b Fd(\003)1454 2850 y Fu(4.)49 b Fr(A)38 b(priori)f(bounds)100 3025 y Fu(W)-8 b(e)45 b(will)d(use)j(the)g(represen)m(tation)h(\(3.7\)) e(to)g(pro)m(v)m(e)i(in)e(Prop)s(osition)f(4.1)h(b)s(elo)m(w)g(some)h Fe(a)f(priori)0 3144 y Fu(b)s(ounds)29 b(on)g Fq(v)518 3108 y Fm(\()p Fl(n)p Fm(\))619 3144 y Fu(\()p Fq(t)p 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(the)h(displacemen)m(t)g Fq(x)2818 3857 y Fl(k)2880 3842 y Fo(\000)19 b Fq(x)3031 3857 y Fl(k)r Fn(\000)p Fm(1)3195 3842 y Fu(of)30 b(the)h(cen)m(ter)i(in)0 3958 y(the)g(time)e(in)m(terv) -5 b(al)31 b([)p Fq(T)826 3973 y Fl(k)r Fn(\000)p Fm(1)960 3958 y Fq(;)17 b(T)1061 3973 y Fl(k)1103 3958 y Fu(].)100 4075 y(W)-8 b(e)33 b(set)548 4249 y Fq(V)605 4264 y Fl(n)680 4249 y Fu(:=)142 b(sup)810 4335 y Fl(t)p Fn(2)p Fm([)p Fl(T)943 4343 y Fg(n)986 4335 y Fl(;T)1047 4344 y Fg(n)p Fi(+1)1166 4335 y Fm(])1203 4165 y Fp(\015)1203 4224 y(\015)1258 4249 y Fq(v)1309 4208 y Fm(\()p Fl(n)p Fm(\))1410 4249 y Fu(\()p Fq(t)p Fu(\))1521 4165 y Fp(\015)1521 4224 y(\015)1577 4288 y Fn(1)1668 4249 y Fq(;)212 b(V)1964 4264 y Fl(n;)p Fn(\003)2093 4249 y Fu(=)28 b(sup)2202 4331 y Fl(k)r Fn(\024)p Fl(n)2360 4249 y Fq(V)2417 4264 y Fl(k)2460 4249 y Fq(;)127 b(V)2671 4264 y Fn(\003)2711 4249 y Fu(\()p Fq(\034)11 b Fu(\))28 b(:=)f Fq(V)3055 4265 y Fl(n)3098 4277 y Fg(\025)3139 4265 y Fm(\()p Fl(\034)8 b Fm(\))p Fl(;)p Fn(\003)3639 4249 y Fu(\(4.2\))0 4498 y(and)1357 4632 y Fq(\016)1404 4591 y Fn(\003)1444 4632 y Fu(\()p Fq(\034)j Fu(\))28 b(:=)90 b(sup)1732 4718 y Fl(k)r Fn(\024)p Fl(n)1869 4730 y Fg(\025)1909 4718 y Fm(\()p Fl(\034)8 b Fm(\))2019 4632 y Fo(j)p Fq(x)2102 4647 y Fl(k)r Fm(+1)2257 4632 y Fo(\000)23 b Fq(x)2412 4647 y Fl(k)2455 4632 y Fo(j)1156 b Fu(\(4.3\))0 4860 y(Let)33 b(also)1355 4994 y Fq(Z)1422 5009 y Fl(n)1496 4994 y Fu(:=)146 b(sup)1627 5080 y Fl(s)p Fn(2)p Fm([)p Fl(T)1768 5088 y Fg(n)1810 5080 y Fl(;T)1871 5089 y Fg(n)p Fi(+1)1990 5080 y Fm(])2026 4909 y Fp(\015)2026 4969 y(\015)2082 4994 y Fq(z)2131 4953 y Fm(\()p Fl(n)p Fm(\))2233 4994 y Fu(\()p Fq(s)p Fu(\))2355 4909 y Fp(\015)2355 4969 y(\015)2410 5033 y Fn(1)3639 4994 y Fu(\(4.4\))0 5222 y(and)913 5363 y Fq(Z)980 5378 y Fl(n;)p Fn(\003)1110 5363 y Fu(=)27 b(sup)1218 5445 y Fl(k)r Fn(\024)p Fl(n)1377 5363 y Fq(Z)1444 5378 y Fl(k)1486 5363 y Fq(;)212 b Fo(B)1793 5312 y Fm(\(1\))1790 5390 y Fl(\025;\034)1922 5363 y Fu(:=)2052 5252 y Fp(n)2119 5363 y Fq(Z)2186 5378 y Fl(n)2229 5390 y Fg(\025)2269 5378 y Fm(\()p Fl(\034)8 b Fm(\))p Fl(;)p Fn(\003)2450 5363 y Fo(\024)28 b Fq(\025)2612 5321 y Fn(\000)p Fl(\020)2707 5273 y Fo(p)p 2790 5273 71 4 v 90 x Fq(T)2860 5252 y Fp(o)3639 5363 y Fu(\(4.5\))1882 5532 y Fk(13)p eop %%Page: 14 15 14 14 bop 0 266 a Fu(W)-8 b(e)33 b(will)e(pro)m(v)m(e)j(in)e(App)s (endix)i(B)f(the)g(follo)m(wing)d(Gaussian)j(estimate:)43 b(for)32 b(eac)m(h)i Fq(\034)6 b(;)17 b(\020)8 b(;)17 b(q)31 b(>)d Fu(0)33 b(there)g(is)0 382 y(a)f(constan)m(t)i Fq(C)g Fu(=)28 b Fq(C)7 b Fu(\()p Fq(\034)f(;)17 b(\020)8 b(;)17 b(q)t Fu(\))31 b(suc)m(h)j(that)e(for)g(an)m(y)h Fq(\025)28 b(>)f Fu(0)1499 575 y Fj(P)1575 464 y Fp(\020)1636 575 y Fo(B)1704 524 y Fm(\(1\))1701 603 y Fl(\025;\034)1806 464 y Fp(\021)1893 575 y Fo(\025)h Fu(1)22 b Fo(\000)h Fq(C)7 b(\025)2303 534 y Fl(q)3639 575 y Fu(\(4.6\))100 768 y(The)33 b(next)h(prop)s(osition)d(is)h(the)i(k)m(ey)g(ingredien)m (t)e(for)h(the)g(b)s(ound)g(\(1.11\))o(,)g(see)h(the)f(b)s(eginning)f (of)g Fo(x)p Fu(8)0 884 y(for)g(the)h(conclusion)f(of)g(the)h(pro)s (of.)0 1237 y Fv(Prop)s(osition)44 b(4.1.)i Ft(F)-7 b(or)41 b(e)-5 b(ach)41 b Fq(\034)6 b(;)17 b(\020)48 b(>)40 b Fu(0)i Ft(ther)-5 b(e)42 b(is)g(a)f(c)-5 b(onstant)42 b Fq(C)48 b Fu(=)40 b Fq(C)7 b Fu(\()p Fq(\034)f(;)17 b(\020)8 b Fu(\))41 b Ft(such)g(that,)j(for)e(any)0 1353 y Fq(\025)28 b(>)f Fu(0)p Ft(,)665 1486 y Fq(V)722 1501 y Fn(\003)761 1486 y Fu(\()p Fq(\034)11 b Fu(\))28 b Fo(\024)h Fq(C)7 b(\025)1158 1445 y Fm(1)p Fn(\000)p Fl(\020)1287 1397 y Fo(p)p 1370 1397 71 4 v 89 x Fq(T)14 b(;)116 b(\016)1631 1445 y Fn(\003)1671 1486 y Fu(\()p Fq(\034)11 b Fu(\))28 b Fo(\024)g Fq(C)7 b(\025)2067 1445 y Fm(1)p Fn(\000)p Fl(\020)2197 1397 y Fo(p)p 2280 1397 V 89 x Fq(T)213 b Fu(on)32 b(the)h(set)h Fo(B)3074 1435 y Fm(\(1\))3071 1514 y Fl(\025;\034)3639 1486 y Fu(\(4.7\))0 1629 y Ft(In)g(p)-5 b(articular,)35 b(for)g Fq(\025)27 b(>)h Fu(0)34 b Ft(smal)5 b(l)34 b(enough,)1268 1803 y Fq(N)1346 1818 y Fl(\016)1411 1803 y Fq(>)28 b(n)1573 1818 y Fl(\025)1619 1803 y Fu(\()p Fq(\034)11 b Fu(\))199 b(on)33 b(the)g(set)g Fo(B)2471 1752 y Fm(\(1\))2468 1831 y Fl(\025;\034)3639 1803 y Fu(\(4.8\))0 2208 y Ft(Pr)-5 b(o)g(of.)42 b Fu(Let)28 b(\()p Fq(m)592 2223 y Fl(\025)637 2208 y Fq(;)17 b(t)716 2223 y Fl(\025)762 2208 y Fu(\),)29 b Fq(t)891 2223 y Fl(\025)964 2208 y Fo(2)f Fu([)p Fq(T)1142 2223 y Fl(m)1204 2235 y Fg(\025)1250 2208 y Fq(;)17 b(T)1351 2223 y Fl(m)1413 2235 y Fg(\025)1454 2223 y Fm(+1)1548 2208 y Fu(],)29 b Fq(m)1716 2223 y Fl(\025)1790 2208 y Fo(\024)f Fq(n)1953 2223 y Fl(\025)1998 2208 y Fu(\()p Fq(\034)11 b Fu(\),)30 b(b)s(e)e(the)h(stopping)f(times)f(\()p Fq(m)3247 2223 y Fl(\025)3321 2208 y Fu(w.r.t.)i Fo(F)3649 2223 y Fl(T)3690 2232 y Fg(n)p Fi(+1)3813 2208 y Fu(,)0 2325 y Fq(t)35 2340 y Fl(\025)114 2325 y Fu(w.r.t.)34 b Fo(F)447 2340 y Fl(t)476 2325 y Fu(\))g(so)f(that)h(\()p Fq(m)1004 2340 y Fl(\025)1049 2325 y Fq(;)17 b(t)1128 2340 y Fl(\025)1174 2325 y Fu(\))29 b(=)g(\()p Fq(n)1442 2340 y Fl(\025)1487 2325 y Fu(\()p Fq(\034)11 b Fu(\))p Fq(;)17 b(T)1717 2340 y Fl(n)1760 2352 y Fg(\025)1801 2340 y Fm(\()p Fl(\034)8 b Fm(\)+1)1989 2325 y Fu(\))34 b(if)e Fq(V)2208 2340 y Fn(\003)2247 2325 y Fu(\()p Fq(\034)11 b Fu(\))30 b Fq(<)f(\025T)14 b Fu(,)34 b(otherwise)g(\()p Fq(m)3258 2340 y Fl(\025)3303 2325 y Fq(;)17 b(t)3382 2340 y Fl(\025)3428 2325 y Fu(\))29 b(=)g(\()p Fq(n;)17 b(t)p Fu(\),)0 2451 y(where)34 b Fq(n)f Fu(and)f Fq(t)h Fu(are)g(the)g(\014rst)g(ones)g(for)f(whic)m(h)h Fo(k)p Fq(v)1908 2415 y Fm(\()p Fl(n)p Fm(\))2009 2451 y Fu(\()p Fq(t)p Fu(\))p Fo(k)2170 2466 y Fn(1)2273 2451 y Fo(\025)28 b Fq(\025T)14 b Fu(.)100 2567 y(In)38 b(the)g(follo)m(wing)d(w)m(e)k (ma)m(y)e(assume)i Fq(\025)e Fu(is)h(small)d(enough,)40 b(otherwise)e(\(4.7\))f(holds)h(trivially)-8 b(.)56 b(W)-8 b(e)0 2683 y(claim)30 b(there)j(is)g(a)f(constan)m(t)h Fq(C)40 b Fu(so)32 b(that)1479 2845 y Fo(k)p Fq(v)1580 2804 y Fm(\()p Fl(m)1669 2816 y Fg(\025)1711 2804 y Fm(\))1742 2845 y Fu(\()p Fq(t)1815 2860 y Fl(\025)1861 2845 y Fu(\))p Fo(k)1949 2860 y Fn(1)2051 2845 y Fo(\024)c Fq(C)7 b(\025T)1292 b Fu(\(4.9\))0 3009 y(In)29 b(fact,)g(if)e Fq(t)454 3024 y Fl(\025)527 3009 y Fo(2)h Fu(\()p Fq(T)716 3024 y Fl(m)778 3036 y Fg(\025)823 3009 y Fq(;)17 b(T)924 3024 y Fl(m)986 3036 y Fg(\025)1028 3024 y Fm(+1)1122 3009 y Fu(],)29 b(\(4.9\))f(follo)m(ws)f(from)g(the)h(con)m(tin)m(uit)m(y)h(of)e Fo(k)p Fq(v)2803 2973 y Fm(\()p Fl(m)2892 2985 y Fg(\025)2934 2973 y Fm(\))2965 3009 y Fu(\()p Fq(t)p Fu(\))p Fo(k)3126 3024 y Fn(1)3201 3009 y Fu(;)j(otherwise,)f(i.e.)0 3126 y(if)i Fq(t)124 3141 y Fl(\025)198 3126 y Fu(=)c Fq(T)358 3141 y Fl(m)420 3153 y Fg(\025)466 3126 y Fu(,)117 3297 y Fo(k)p Fq(v)218 3256 y Fm(\()p Fl(m)307 3268 y Fg(\025)348 3256 y Fm(\))379 3297 y Fu(\()p Fq(T)474 3312 y Fl(m)536 3324 y Fg(\025)582 3297 y Fu(\))p Fo(k)670 3312 y Fn(1)772 3297 y Fo(\024)h(k)p Fq(v)978 3256 y Fm(\()p Fl(m)1067 3268 y Fg(\025)1109 3256 y Fn(\000)p Fm(1\))1230 3297 y Fu(\()p Fq(T)1325 3312 y Fl(m)1387 3324 y Fg(\025)1433 3297 y Fu(\))p Fo(k)1521 3312 y Fn(1)1617 3297 y Fu(+)22 b Fo(k)c Fu(\026)-67 b Fq(m)1850 3312 y Fl(m)1912 3324 y Fg(\025)1980 3297 y Fo(\000)41 b Fu(\026)-67 b Fq(m)2165 3312 y Fl(m)2227 3324 y Fg(\025)2268 3312 y Fn(\000)p Fm(1)2363 3297 y Fo(k)27 b(\024)h Fq(\025T)36 b Fu(+)22 b Fo(j)p Fq(x)2876 3312 y Fl(m)2938 3324 y Fg(\025)3006 3297 y Fo(\000)g Fq(x)3160 3312 y Fl(m)3222 3324 y Fg(\025)3264 3312 y Fn(\000)p Fm(1)3358 3297 y Fo(j)28 b(\024)g Fq(C)7 b(\025T)0 3463 y Fu(from)31 b(\(3.2\))h(and)h(\(2.4\),)f(since)h Fo(k)18 b Fu(\026)-67 b Fq(m)1287 3427 y Fn(0)1311 3463 y Fo(k)1361 3478 y Fn(1)1463 3463 y Fu(=)27 b(1)33 b(and)f Fo(k)p Fq(v)1938 3427 y Fm(\()p Fl(m)2027 3439 y Fg(\025)2069 3427 y Fn(\000)p Fm(1\))2190 3463 y Fu(\()p Fq(T)2285 3478 y Fl(m)2347 3490 y Fg(\025)2393 3463 y Fu(\))p Fo(k)2481 3478 y Fn(1)2583 3463 y Fo(\024)c Fq(\025T)14 b Fu(.)100 3579 y(W)-8 b(e)48 b(will)e(study)j(the)f(ev)m(olution)f(till)e(time)i Fq(t)1828 3594 y Fl(\025)1873 3579 y Fu(,)52 b(so)c(that)g(our)g(next)h (considerations)e(are)h(tacitly)0 3696 y(restricted)33 b(to)f Fq(n)c Fo(\024)g Fq(m)829 3711 y Fl(\025)908 3696 y Fu(and)k Fq(t)c Fo(\024)g Fq(t)1300 3711 y Fl(\025)1346 3696 y Fu(.)100 3812 y(By)43 b(\(4.9\))o(,)h Fq(N)612 3827 y Fl(\016)692 3812 y Fq(>)f(m)896 3827 y Fl(\025)983 3812 y Fu(\(for)d Fq(\025)h Fu(small)e(enough\))j(so)g(that)f(w)m(e)h (can)g(use)g(\(3.7\))o(,)i(regarding)c(it)h(as)g(an)0 3931 y(in)m(tegral)33 b(equation)i(in)g Fq(v)926 3895 y Fm(\()p Fl(n)p Fm(\))1027 3931 y Fu(\()p Fq(t)p Fu(\).)51 b(W)-8 b(e)36 b(are)f(going)e(to)i(b)s(ound)g(one)h(after)f(the)g (other)g(all)e(eigh)m(t)i(terms)g(on)0 4047 y(the)e(r.h.s.)g(of)39 b(\(3.7\).)100 4164 y(W)-8 b(e)27 b(will)e(use)k(in)d(the)i(sequel)g (the)g(follo)m(wing)d(b)s(ounds)j(\(see)g(e.g.)g([9])f(and)h (references)h(therein\).)42 b(There)0 4280 y(is)32 b Fq(C)j(<)27 b Fo(1)32 b Fu(so)h(that,)g(for)f(an)m(y)h(measurable)f (function)g Fq(')p Fu(,)1539 4441 y Fo(k)p Fq(g)1636 4456 y Fl(t)1665 4441 y Fq(')p Fo(k)1779 4456 y Fn(1)1881 4441 y Fo(\024)c Fq(C)7 b Fo(k)p Fq(')p Fo(k)2227 4456 y Fn(1)3591 4441 y Fu(\(4.10\))0 4603 y(Moreo)m(v)m(er,)35 b(b)m(y)f(the)f(P)m(erron{F)-8 b(rob)s(enius)33 b(Theorem,)h(there)g (are)f Fq(\013)c(>)f Fu(0)33 b(and)g Fq(C)j(<)28 b Fo(1)33 b Fu(so)g(that)g(for)f(an)m(y)0 4719 y Fq(')g Fu(orthogonal)f(to)51 b(\026)-68 b Fq(m)788 4683 y Fn(0)812 4719 y Fu(,)33 b Fo(h)18 b Fu(\026)-67 b Fq(m)996 4683 y Fn(0)1019 4719 y Fq(;)17 b(')p Fo(i)27 b Fu(=)h(0,)1451 4881 y Fo(k)p Fq(g)1548 4896 y Fl(t)1577 4881 y Fq(')p Fo(k)1691 4896 y Fn(1)1793 4881 y Fo(\024)h Fq(C)7 b(e)2021 4840 y Fn(\000)p Fl(\013t)2151 4881 y Fo(k)p Fq(')p Fo(k)2315 4896 y Fn(1)3591 4881 y Fu(\(4.11\))100 5042 y(Then,)33 b(b)m(y)h(the)f(last)f(iden)m (tit)m(y)g(in)g(\(3.2\))o(,)h(for)f Fq(t)c Fo(2)g Fu([)p Fq(T)1994 5057 y Fl(n)2041 5042 y Fq(;)17 b(T)2142 5057 y Fl(n)p Fm(+1)2279 5042 y Fu(])23 5121 y Fp(\015)23 5181 y(\015)23 5240 y(\015)79 5235 y Fu(\000)140 5185 y Fm(\()p Fl(n)p Fm(\))140 5260 y(1)242 5235 y Fu(\()p Fq(t)p Fu(\))353 5121 y Fp(\015)353 5181 y(\015)353 5240 y(\015)408 5304 y Fn(1)566 5235 y Fo(\024)83 b Fq(C)7 b(e)848 5194 y Fn(\000)p Fl(\013)p Fm(\()p Fl(t)p Fn(\000)p Fl(T)1096 5202 y Fg(n)1140 5194 y Fm(\))1172 5235 y Fo(k)p Fq(v)1273 5194 y Fm(\()p Fl(n)p Fm(\))1374 5235 y Fu(\()p Fq(T)1469 5250 y Fl(n)1516 5235 y Fu(\))p Fo(k)1604 5250 y Fn(1)1706 5235 y Fo(\024)28 b Fq(C)7 b(e)1933 5194 y Fn(\000)p Fl(\013)p Fm(\()p Fl(t)p Fn(\000)p Fl(T)2181 5202 y Fg(n)2225 5194 y Fm(\))2273 5155 y Fp(\000)2319 5235 y Fo(k)p Fq(v)2420 5194 y Fm(\()p Fl(n)p Fn(\000)p Fm(1\))2612 5235 y Fu(\()p Fq(T)2707 5250 y Fl(n)2754 5235 y Fu(\))p Fo(k)2842 5250 y Fn(1)2938 5235 y Fu(+)22 b Fo(k)c Fu(\026)-67 b Fq(m)3171 5250 y Fl(x)3211 5258 y Fg(n)3280 5235 y Fo(\000)41 b Fu(\026)-67 b Fq(m)3465 5250 y Fl(x)3505 5259 y Fg(n)p Fh(\000)p Fi(1)3630 5235 y Fo(k)3680 5250 y Fn(1)3754 5155 y Fp(\001)566 5433 y Fo(\024)83 b Fq(C)7 b(e)848 5392 y Fn(\000)p Fl(\013)p Fm(\()p Fl(t)p Fn(\000)p Fl(T)1096 5400 y Fg(n)1140 5392 y Fm(\))1172 5433 y Fo(k)p Fq(v)1273 5392 y Fm(\()p Fl(n)p Fn(\000)p Fm(1\))1464 5433 y Fu(\()p Fq(T)1559 5448 y Fl(n)1606 5433 y Fu(\))p Fo(k)1694 5448 y Fn(1)3591 5433 y Fu(\(4.12\))1882 5532 y Fk(14)p eop %%Page: 15 16 15 15 bop 0 266 a Fu(The)32 b(second)g(inequalit)m(y)e(follo)m(ws)g (from)g(\(3.2\))g(and)h(the)h(last)e(one)h(follo)m(ws)f(from)g(\(2.4\)) o(,)i(pro)s(ceeding)f(as)0 382 y(in)h(the)h(pro)s(of)f(of)39 b(\(4.9\))o(.)100 498 y(By)33 b(de\014nition)e(\(4.4\))h(w)m(e)i(get) 1410 687 y(sup)1296 772 y Fl(t)p Fn(2)p Fm([)p Fl(T)1429 780 y Fg(n)1471 772 y Fl(;T)1532 781 y Fg(n)p Fi(+1)1652 772 y Fm(])1704 572 y Fp(\015)1704 632 y(\015)1704 692 y(\015)1760 687 y Fu(\000)1821 636 y Fm(\()p Fl(n)p Fm(\))1821 711 y(2)1923 687 y Fu(\()p Fq(t)p Fu(\))2034 572 y Fp(\015)2034 632 y(\015)2034 692 y(\015)2089 756 y Fn(1)2191 687 y Fo(\024)29 b Fq(C)7 b(\025Z)2498 702 y Fl(n)3591 687 y Fu(\(4.13\))0 949 y(W)-8 b(e)33 b(next)g(b)s(ound)g(\003)756 898 y Fm(\()p Fl(n)p Fm(\))756 973 y(3)858 949 y Fu(.)43 b(W)-8 b(e)33 b(ha)m(v)m(e)1406 1154 y Fo(k)p Fq(p)1505 1169 y Fl(t)1553 1154 y Fu(\026)-67 b Fq(m)1620 1113 y Fn(0)1620 1178 y Fl(z)1660 1154 y Fo(k)1710 1169 y Fn(1)1812 1154 y Fo(\024)2016 1086 y Fu(2)p 1927 1131 226 4 v 1927 1151 a Fo(p)p 2010 1151 143 4 v 82 x Fu(2)p Fq(\031)t(t)2191 1154 y Fo(\024)2340 1086 y Fu(1)p 2306 1131 119 4 v 2306 1151 a Fo(p)p 2389 1151 36 4 v 81 x Fq(t)3591 1154 y Fu(\(4.14\))0 1386 y(b)s(ecause)34 b Fq(p)410 1401 y Fl(t)440 1386 y Fu(\()p Fq(x;)17 b(y)t Fu(\))26 b Fo(\024)j Fu(1)p Fq(=)897 1303 y Fo(p)p 979 1303 143 4 v 979 1386 a Fu(2)p Fq(\031)t(t)k Fu(and)f Fo(k)18 b Fu(\026)-67 b Fq(m)1479 1350 y Fn(0)1479 1410 y Fl(z)1519 1386 y Fo(k)1569 1401 y Fm(1)1636 1386 y Fu(=)28 b(2)k(for)g(an)m(y)h Fq(z)g Fo(2)28 b Fj(R)5 b Fu(.)49 b(Then)711 1549 y Fp(\015)711 1609 y(\015)767 1634 y Fq(p)816 1649 y Fl(t)845 1634 y Fu(\()18 b(\026)-67 b Fq(m)968 1649 y Fl(x)1008 1661 y Fg(k)1073 1634 y Fo(\000)40 b Fu(\026)-67 b Fq(m)1257 1649 y Fl(x)1297 1661 y Fg(k)q Fh(\000)p Fi(1)1418 1634 y Fu(\))1456 1549 y Fp(\015)1456 1609 y(\015)1511 1673 y Fn(1)1614 1634 y Fo(\024)1719 1460 y Fp(\014)1719 1520 y(\014)1719 1579 y(\014)1719 1639 y(\014)1719 1699 y(\014)1752 1498 y(Z)1852 1525 y Fl(x)1892 1537 y Fg(k)1807 1724 y Fl(x)1847 1736 y Fg(k)q Fh(\000)p Fi(1)1984 1634 y Fq(dz)21 b Fo(k)p Fq(p)2200 1649 y Fl(t)2248 1634 y Fu(\026)-67 b Fq(m)2315 1593 y Fn(0)2315 1659 y Fl(z)2355 1634 y Fo(k)2405 1649 y Fn(1)2479 1460 y Fp(\014)2479 1520 y(\014)2479 1579 y(\014)2479 1639 y(\014)2479 1699 y(\014)2540 1634 y Fo(\024)2655 1567 y(j)p Fq(x)2738 1582 y Fl(k)2803 1567 y Fo(\000)23 b Fq(x)2958 1582 y Fl(k)r Fn(\000)p Fm(1)3091 1567 y Fo(j)p 2655 1611 464 4 v 2828 1631 a(p)p 2911 1631 36 4 v 81 x Fq(t)3591 1634 y Fu(\(4.15\))0 1882 y(so)33 b(that,)f(b)m(y)j(\(4.10\))o(,)e(w)m(e)h(get)1095 2005 y Fp(\015)1095 2065 y(\015)1095 2124 y(\015)1151 2119 y Fu(\003)1219 2068 y Fm(\()p Fl(n)p Fm(\))1219 2144 y(3)1320 2119 y Fu(\()p Fq(t)p Fu(\))1431 2005 y Fp(\015)1431 2065 y(\015)1431 2124 y(\015)1487 2188 y Fn(1)1589 2119 y Fo(\024)28 b Fq(C)7 b(\025)1895 1995 y Fl(n)1845 2025 y Fp(X)1852 2237 y Fl(k)r Fm(=1)2005 1984 y Fp(Z)2105 2010 y Fl(t)2060 2209 y(T)2101 2217 y Fg(n)2148 2119 y Fq(ds)2271 2052 y Fo(j)p Fq(x)2354 2067 y Fl(k)2419 2052 y Fo(\000)23 b Fq(x)2574 2067 y Fl(k)r Fn(\000)p Fm(1)2707 2052 y Fo(j)p 2271 2096 464 4 v 2328 2116 a(p)p 2411 2116 268 4 v 77 x Fq(s)f Fo(\000)h Fq(T)2636 2208 y Fl(k)0 2371 y Fu(By)34 b(\(2.4\),)f(w)m(e)g(then)g(get)1135 2452 y Fp(\015)1135 2512 y(\015)1135 2572 y(\015)1191 2567 y Fu(\003)1259 2516 y Fm(\()p Fl(n)p Fm(\))1259 2591 y(3)1360 2567 y Fu(\()p Fq(t)p Fu(\))1471 2452 y Fp(\015)1471 2512 y(\015)1471 2572 y(\015)1527 2636 y Fn(1)1629 2567 y Fo(\024)28 b Fq(C)7 b(\025)1868 2478 y Fo(p)p 1951 2478 71 4 v 89 x Fq(T)2089 2442 y Fl(n)2038 2472 y Fp(X)2046 2684 y Fl(k)r Fm(=1)2357 2500 y Fq(V)2414 2515 y Fl(k)r Fn(\000)p Fm(1)p 2209 2544 487 4 v 2209 2564 a Fo(p)p 2292 2564 403 4 v 81 x Fq(n)22 b Fo(\000)h Fq(k)i Fu(+)d(1)3591 2567 y(\(4.16\))100 2824 y(W)-8 b(e)33 b(will)d(next)j(b)s(ound)g(\000)1033 2773 y Fm(\()p Fl(n)p Fm(\))1033 2848 y(4)1135 2824 y Fu(\()p Fq(t)p Fu(\).)43 b(W)-8 b(e)33 b(write)338 3048 y Fq(g)389 2997 y Fm(\()p Fl(n)p Fm(\))385 3070 y Fl(t)p Fn(\000)p Fl(s)530 3048 y Fu(=)27 b Fq(p)682 3063 y Fl(t)p Fn(\000)p Fl(s)822 3048 y Fo(\000)921 2912 y Fp(Z)1021 2939 y Fl(t)977 3138 y(s)1067 3048 y Fq(ds)1164 3007 y Fn(00)1277 2980 y Fq(d)p 1233 3025 140 4 v 1233 3116 a(ds)1330 3087 y Fn(00)1382 2937 y Fp(\020)1441 3048 y Fq(g)1492 2997 y Fm(\()p Fl(n)p Fm(\))1488 3076 y Fl(t)p Fn(\000)p Fl(s)1601 3057 y Fh(00)1646 3048 y Fq(p)1695 3063 y Fl(s)1728 3044 y Fh(00)1768 3063 y Fn(\000)p Fl(s)1860 2937 y Fp(\021)1947 3048 y Fu(=)g Fq(p)2099 3063 y Fl(t)p Fn(\000)p Fl(s)2239 3048 y Fu(+)2337 2912 y Fp(Z)2436 2939 y Fl(t)2392 3138 y(s)2482 3048 y Fq(ds)2579 3007 y Fn(00)2638 3048 y Fq(g)2689 2997 y Fm(\()p Fl(n)p Fm(\))2685 3076 y Fl(t)p Fn(\000)p Fl(s)2798 3057 y Fh(00)2843 3048 y Fu([1)22 b Fo(\000)g Fu(3)c(\026)-67 b Fq(m)3174 3007 y Fm(2)3174 3073 y Fl(x)3214 3081 y Fg(n)3261 3048 y Fu(])p Fq(p)3337 3063 y Fl(s)3370 3044 y Fh(00)3410 3063 y Fn(\000)p Fl(s)0 3262 y Fu(ha)m(ving)32 b(used)i(\(2.3\))e(in)g(the)h(second)h(equalit)m(y)-8 b(.)43 b(Then)511 3486 y(\000)572 3435 y Fm(\()p Fl(n)p Fm(\))572 3510 y(4)674 3486 y Fu(\()p Fq(t)p Fu(\))83 b(=)g Fq(\025)1101 3350 y Fp(Z)1200 3376 y Fl(t)1155 3576 y(T)1196 3584 y Fg(n)1243 3486 y Fq(ds)1357 3350 y Fp(Z)1456 3376 y Fl(s)1412 3576 y(T)1453 3584 y Fg(n)1516 3486 y Fq(ds)1613 3444 y Fn(0)1663 3418 y Fq(@)5 b(p)1768 3433 y Fl(t)p Fn(\000)p Fl(s)1881 3414 y Fh(0)p 1663 3463 246 4 v 1722 3554 a Fq(@)g(s)1824 3525 y Fn(0)1918 3486 y Fq(v)1969 3444 y Fm(\()p Fl(n)p Fm(\))2071 3486 y Fu(\()p Fq(s)2155 3444 y Fn(0)2178 3486 y Fu(\))1027 3771 y(+)22 b Fq(\025)1199 3636 y Fp(Z)1298 3662 y Fl(t)1253 3861 y(T)1294 3869 y Fg(n)1341 3771 y Fq(ds)1455 3636 y Fp(Z)1554 3662 y Fl(t)1510 3861 y(s)1600 3771 y Fq(ds)1697 3730 y Fn(00)1756 3771 y Fq(g)1807 3720 y Fm(\()p Fl(n)p Fm(\))1803 3800 y Fl(t)p Fn(\000)p Fl(s)1916 3781 y Fh(00)1960 3771 y Fu([1)g Fo(\000)h Fu(3)18 b(\026)-67 b Fq(m)2292 3730 y Fm(2)2292 3796 y Fl(x)2332 3804 y Fg(n)2379 3771 y Fu(])2423 3636 y Fp(Z)2522 3662 y Fl(s)2478 3861 y(T)2519 3869 y Fg(n)2582 3771 y Fq(ds)2679 3730 y Fn(0)2728 3704 y Fq(@)5 b(p)2833 3719 y Fl(s)2866 3700 y Fh(00)2908 3719 y Fn(\000)p Fl(s)2996 3700 y Fh(0)p 2728 3748 294 4 v 2812 3840 a Fq(@)g(s)2914 3811 y Fn(0)3032 3771 y Fq(v)3083 3730 y Fm(\()p Fl(n)p Fm(\))3184 3771 y Fu(\()p Fq(s)3268 3730 y Fn(0)3291 3771 y Fu(\))0 3993 y(By)33 b(using)h(\(4.10\))e(and)g(\(see)i(App)s(endix)f(A\))1472 4067 y Fp(\015)1472 4127 y(\015)1472 4187 y(\015)1472 4247 y(\015)1555 4144 y Fq(@)p 1537 4189 93 4 v 1537 4280 a(@)5 b(t)1656 4212 y(p)1705 4227 y Fl(t)1734 4212 y Fq(')1798 4067 y Fp(\015)1798 4127 y(\015)1798 4187 y(\015)1798 4247 y(\015)1854 4311 y Fn(1)1956 4212 y Fo(\024)2071 4144 y Fu(1)p 2071 4189 49 4 v 2078 4280 a Fq(t)2130 4212 y Fo(k)p Fq(')p Fo(k)2294 4227 y Fn(1)3591 4212 y Fu(\(4.17\))0 4446 y(since)33 b Fo(k)p Fq(v)340 4409 y Fm(\()p Fl(n)p Fm(\))441 4446 y Fu(\()p Fq(s)525 4409 y Fn(0)549 4446 y Fu(\))p Fo(k)637 4461 y Fn(1)739 4446 y Fo(\024)28 b Fq(V)901 4461 y Fl(n)948 4446 y Fu(,)k(w)m(e)i(get) 1450 4519 y Fp(\015)1450 4579 y(\015)1450 4639 y(\015)1505 4634 y Fu(\000)1566 4583 y Fm(\()p Fl(n)p Fm(\))1566 4658 y(4)1668 4634 y Fu(\()p Fq(t)p Fu(\))1779 4519 y Fp(\015)1779 4579 y(\015)1779 4639 y(\015)1835 4703 y Fn(1)1937 4634 y Fo(\024)28 b Fq(C)7 b(\025T)2247 4593 y Fm(2)2286 4634 y Fq(V)2343 4649 y Fl(n)3591 4634 y Fu(\(4.18\))0 4821 y(this)32 b(is)g(not)h(optimal)c(but)k(go)s(o)s(d)f (enough)h(for)f(our)g(purp)s(oses.)45 b(Analogously)359 4925 y Fp(\015)359 4985 y(\015)359 5045 y(\015)414 5040 y Fu(\000)475 4989 y Fm(\()p Fl(n)p Fm(\))475 5064 y(5)577 5040 y Fu(\()p Fq(t)p Fu(\))688 4925 y Fp(\015)688 4985 y(\015)688 5045 y(\015)743 5109 y Fn(1)901 5040 y Fo(\024)84 b Fq(C)7 b(\025)1213 4904 y Fp(Z)1312 4930 y Fl(t)1267 5130 y(T)1308 5138 y Fg(n)1355 5040 y Fq(ds)17 b Fo(k)p Fq(v)1570 4999 y Fm(\()p Fl(n)p Fm(\))1671 5040 y Fu(\()p Fq(s)p Fu(\))p Fo(k)1843 5055 y Fn(1)1945 5040 y Fo(\024)28 b Fq(C)7 b(\025T)14 b(V)2312 5055 y Fl(n)3591 5040 y Fu(\(4.19\))359 5224 y Fp(\015)359 5284 y(\015)359 5343 y(\015)414 5338 y Fu(\000)475 5287 y Fm(\()p Fl(n)p Fm(\))475 5363 y(6)577 5338 y Fu(\()p Fq(t)p Fu(\))688 5224 y Fp(\015)688 5284 y(\015)688 5343 y(\015)743 5407 y Fn(1)901 5338 y Fo(\024)84 b Fq(C)7 b(\025)1263 5214 y Fl(n)1212 5244 y Fp(X)1220 5456 y Fl(k)r Fm(=1)1373 5203 y Fp(Z)1472 5229 y Fl(t)1428 5428 y(T)1469 5436 y Fg(n)1516 5338 y Fq(ds)1630 5203 y Fp(Z)1729 5229 y Fl(T)1770 5241 y Fg(k)1684 5428 y Fl(T)1725 5440 y Fg(k)q Fh(\000)p Fi(1)1846 5338 y Fq(ds)1943 5297 y Fn(0)1976 5271 y Fo(k)p Fq(v)2077 5235 y Fm(\()p Fl(k)r Fn(\000)p Fm(1\))2264 5271 y Fu(\()p Fq(s)2348 5235 y Fn(0)2371 5271 y Fu(\))p Fo(k)2459 5286 y Fn(1)p 1976 5315 558 4 v 2136 5407 a Fq(s)22 b Fo(\000)h Fq(s)2350 5378 y Fn(0)2571 5338 y Fo(\024)29 b Fq(C)7 b(\025T)2948 5214 y Fl(n)2898 5244 y Fp(X)2905 5456 y Fl(k)r Fm(=1)3175 5271 y Fq(V)3232 5286 y Fl(k)r Fn(\000)p Fm(1)p 3068 5315 403 4 v 3068 5407 a Fq(n)23 b Fo(\000)f Fq(k)j Fu(+)d(1)3591 5338 y(\(4.20\))1882 5532 y Fk(15)p eop %%Page: 16 17 16 16 bop 0 266 a Fu(Finally)-8 b(,)30 b(using)i(again)f(\(4.10\),)h(w) m(e)i(ha)m(v)m(e)1436 364 y Fp(\015)1436 424 y(\015)1436 483 y(\015)1491 478 y Fu(\000)1552 427 y Fm(\()p Fl(n)p Fm(\))1552 503 y(9)1654 478 y Fu(\()p Fq(t)p Fu(\))1765 364 y Fp(\015)1765 424 y(\015)1765 483 y(\015)1820 547 y Fn(1)1978 478 y Fo(\024)84 b Fq(C)7 b(T)14 b(V)2365 437 y Fm(2)2344 503 y Fl(n)3591 478 y Fu(\(4.21\))1436 585 y Fp(\015)1436 644 y(\015)1436 704 y(\015)1491 699 y Fu(\000)1552 648 y Fm(\()p Fl(n)p Fm(\))1552 724 y(10)1654 699 y Fu(\()p Fq(t)p Fu(\))1765 585 y Fp(\015)1765 644 y(\015)1765 704 y(\015)1820 768 y Fn(1)1978 699 y Fo(\024)84 b Fq(C)7 b(T)14 b(V)2365 658 y Fm(3)2344 724 y Fl(n)3591 699 y Fu(\(4.22\))0 916 y(hence,)34 b(recalling)c(the)j(de\014nition)f (of)g(\()p Fq(m)1524 931 y Fl(\025)1570 916 y Fq(;)17 b(t)1649 931 y Fl(\025)1694 916 y Fu(\))32 b(and)h(\(4.9\))o(,)1160 1014 y Fp(\015)1160 1074 y(\015)1160 1134 y(\015)1216 1129 y Fu(\000)1277 1078 y Fm(\()p Fl(n)p Fm(\))1277 1153 y(9)1378 1129 y Fu(\()p Fq(t)p Fu(\))1489 1014 y Fp(\015)1489 1074 y(\015)1489 1134 y(\015)1545 1198 y Fn(1)1642 1129 y Fu(+)1740 1014 y Fp(\015)1740 1074 y(\015)1740 1134 y(\015)1795 1129 y Fu(\000)1856 1078 y Fm(\()p Fl(n)p Fm(\))1856 1153 y(10)1958 1129 y Fu(\()p Fq(t)p Fu(\))2069 1014 y Fp(\015)2069 1074 y(\015)2069 1134 y(\015)2124 1198 y Fn(1)2227 1129 y Fo(\024)28 b Fq(C)7 b(\025T)2537 1087 y Fm(2)2576 1129 y Fq(V)2633 1144 y Fl(n)3591 1129 y Fu(\(4.23\))0 1345 y(After)33 b(collecting)d(all)h(these)j(b)s (ounds,)f(w)m(e)h(ha)m(v)m(e,)g(for)e Fq(t)27 b Fo(2)i Fu([)p Fq(T)2226 1360 y Fl(n)2273 1345 y Fq(;)17 b(T)2374 1360 y Fl(n)p Fm(+1)2511 1345 y Fu(],)33 b Fq(n)27 b Fo(\024)i Fq(m)2874 1360 y Fl(\025)2919 1345 y Fu(,)k Fq(t)28 b Fo(\024)g Fq(t)3182 1360 y Fl(\025)3227 1345 y Fu(,)621 1473 y Fp(\015)621 1533 y(\015)676 1558 y Fq(v)727 1517 y Fm(\()p Fl(n)p Fm(\))829 1558 y Fu(\()p Fq(t)p Fu(\))940 1473 y Fp(\015)940 1533 y(\015)995 1597 y Fn(1)1153 1558 y Fo(\024)83 b Fq(C)1390 1447 y Fp(n)1457 1558 y Fq(e)1502 1517 y Fn(\000)p Fl(\013)p Fm(\()p Fl(t)p Fn(\000)p Fl(T)1750 1525 y Fg(n)1794 1517 y Fm(\))1842 1473 y Fp(\015)1842 1533 y(\015)1897 1558 y Fq(v)1948 1517 y Fm(\()p Fl(n)p Fn(\000)p Fm(1\))2140 1558 y Fu(\()p Fq(T)2235 1573 y Fl(n)2282 1558 y Fu(\))2320 1473 y Fp(\015)2320 1533 y(\015)2375 1597 y Fn(1)2472 1558 y Fu(+)22 b Fq(\025Z)2694 1573 y Fl(n)2763 1558 y Fu(+)g Fq(\025T)2989 1517 y Fm(2)3028 1558 y Fq(V)3085 1573 y Fl(n)1313 1823 y Fu(+)g Fq(\025)1468 1734 y Fo(p)p 1551 1734 71 4 v 89 x Fq(T)1689 1699 y Fl(n)1639 1729 y Fp(X)1646 1941 y Fl(k)r Fm(=1)1957 1756 y Fq(V)2014 1771 y Fl(k)r Fn(\000)p Fm(1)p 1809 1800 487 4 v 1809 1820 a Fo(p)p 1892 1820 403 4 v 81 x Fq(n)g Fo(\000)h Fq(k)i Fu(+)d(1)2327 1823 y(+)g Fq(\025T)2620 1699 y Fl(n)2569 1729 y Fp(X)2577 1941 y Fl(k)r Fm(=1)2847 1756 y Fq(V)2904 1771 y Fl(k)r Fn(\000)p Fm(1)p 2740 1800 V 2740 1892 a Fq(n)g Fo(\000)h Fq(k)i Fu(+)d(1)3153 1713 y Fp(o)3591 1823 y Fu(\(4.24\))0 2093 y(By)33 b(iterating)e(once)i (the)g(ab)s(o)m(v)m(e)g(inequalit)m(y)f(w)m(e)h(get)721 2330 y Fq(V)778 2345 y Fl(n)908 2330 y Fo(\024)83 b Fq(C)1145 2190 y Fp(\032)1220 2330 y Fq(e)1265 2289 y Fn(\000)p Fl(\013T)1420 2330 y Fq(V)1477 2345 y Fl(n)p Fn(\000)p Fm(2)1636 2330 y Fu(+)22 b Fq(\025)p Fu(\()p Fq(Z)1896 2345 y Fl(n)1965 2330 y Fu(+)g Fq(Z)2130 2345 y Fl(n)p Fn(\000)p Fm(1)2267 2330 y Fu(\))g(+)g Fq(\025T)2553 2289 y Fm(2)2609 2330 y Fu(\()p Fq(V)2704 2345 y Fl(n)2773 2330 y Fu(+)g Fq(V)2928 2345 y Fl(n)p Fn(\000)p Fm(1)3065 2330 y Fu(\))1068 2625 y(+)g Fq(\025)1223 2536 y Fo(p)p 1306 2536 71 4 v 89 x Fq(T)1444 2501 y Fl(n)1393 2531 y Fp(X)1401 2743 y Fl(k)r Fm(=1)1712 2558 y Fq(V)1769 2573 y Fl(k)r Fn(\000)p Fm(1)p 1564 2602 487 4 v 1564 2622 a Fo(p)p 1647 2622 403 4 v 81 x Fq(n)g Fo(\000)h Fq(k)i Fu(+)d(1)2082 2625 y(+)g Fq(\025T)2375 2501 y Fl(n)2324 2531 y Fp(X)2332 2743 y Fl(k)r Fm(=1)2601 2558 y Fq(V)2658 2573 y Fl(k)r Fn(\000)p Fm(1)p 2495 2602 V 2495 2694 a Fq(n)g Fo(\000)h Fq(k)i Fu(+)d(1)2908 2485 y Fp(\033)0 2901 y Fu(Recalling)31 b(\(4.5\),)i(w)m(e)g(ha)m(v)m(e)793 3113 y Fq(V)850 3128 y Fl(n)980 3113 y Fo(\024)84 b Fq(C)1218 3003 y Fp(n)1284 3113 y Fq(e)1329 3072 y Fn(\000)p Fl(\013T)1485 3113 y Fq(V)1542 3128 y Fl(n;)p Fn(\003)1665 3113 y Fu(+)22 b Fq(\025Z)1887 3128 y Fl(n;)p Fn(\003)2011 3113 y Fu(+)g Fq(\025T)2237 3072 y Fm(2)2276 3113 y Fq(V)2333 3128 y Fl(n;)p Fn(\003)1141 3378 y Fu(+)g Fq(\025)1296 3289 y Fo(p)p 1379 3289 71 4 v 89 x Fq(T)1517 3254 y Fl(n)1466 3284 y Fp(X)1474 3496 y Fl(k)r Fm(=1)1757 3311 y Fq(V)1814 3326 y Fl(k)r Fn(\000)p Fm(1)p Fl(;)p Fn(\003)p 1636 3356 487 4 v 1636 3375 a Fo(p)p 1719 3375 403 4 v 81 x Fq(n)h Fo(\000)f Fq(k)k Fu(+)c(1)2155 3378 y(+)g Fq(\025T)2447 3254 y Fl(n)2397 3284 y Fp(X)2404 3496 y Fl(k)r Fm(=1)2646 3311 y Fq(V)2703 3326 y Fl(k)r Fn(\000)p Fm(1)p Fl(;)p Fn(\003)p 2567 3356 V 2567 3447 a Fq(n)h Fo(\000)f Fq(k)j Fu(+)e(1)2980 3268 y Fp(o)0 3648 y Fu(and)33 b(since)g(the)g(r.h.s.)g (is)f(an)h(increasing)e(function)i(of)f Fq(n)p Fu(,)766 3855 y Fq(V)823 3870 y Fl(n;)p Fn(\003)1008 3855 y Fo(\024)83 b Fq(C)1245 3745 y Fp(n)1312 3855 y Fq(e)1357 3814 y Fn(\000)p Fl(\013T)1512 3855 y Fq(V)1569 3870 y Fl(n;)p Fn(\003)1693 3855 y Fu(+)22 b Fq(\025Z)1915 3870 y Fl(n;)p Fn(\003)2039 3855 y Fu(+)g Fq(\025T)2265 3814 y Fm(2)2304 3855 y Fq(V)2361 3870 y Fl(n;)p Fn(\003)1168 4121 y Fu(+)g Fq(\025)1323 4031 y Fo(p)p 1406 4031 71 4 v 90 x Fq(T)1544 3996 y Fl(n)1493 4026 y Fp(X)1501 4238 y Fl(k)r Fm(=1)1785 4053 y Fq(V)1842 4068 y Fl(k)r Fn(\000)p Fm(1)p Fl(;)p Fn(\003)p 1664 4098 487 4 v 1664 4118 a Fo(p)p 1747 4118 403 4 v 80 x Fq(n)g Fo(\000)h Fq(k)i Fu(+)d(1)2182 4121 y(+)g Fq(\025T)2475 3996 y Fl(n)2424 4026 y Fp(X)2432 4238 y Fl(k)r Fm(=1)2674 4053 y Fq(V)2731 4068 y Fl(k)r Fn(\000)p Fm(1)p Fl(;)p Fn(\003)p 2595 4098 V 2595 4189 a Fq(n)g Fo(\000)h Fq(k)i Fu(+)d(1)3008 4010 y Fp(o)3591 4121 y Fu(\(4.25\))0 4396 y(Inequalit)m(y)34 b(\(4.25\))e(yields,)g (for)g Fq(n)c Fo(\024)g Fq(m)1470 4411 y Fl(\025)1544 4396 y Fo(\024)g Fq(n)1707 4411 y Fl(\025)1752 4396 y Fu(\()p Fq(\034)11 b Fu(\),)354 4591 y Fp(\002)396 4671 y Fu(1)22 b Fo(\000)g Fq(C)660 4591 y Fp(\000)706 4671 y Fq(\025T)834 4630 y Fm(2)895 4671 y Fu(+)g Fq(e)1038 4630 y Fn(\000)p Fl(\013T)1215 4671 y Fu(+)g Fq(\025T)31 b Fu(log)16 b Fq(n)1658 4686 y Fl(\025)1704 4671 y Fu(\()p Fq(\034)11 b Fu(\))1833 4591 y Fp(\001\003)1937 4671 y Fq(V)1994 4686 y Fl(n;)p Fn(\003)2123 4671 y Fo(\024)28 b Fq(C)7 b(\025Z)2429 4686 y Fl(n;)p Fn(\003)2553 4671 y Fu(+)2657 4547 y Fl(n)p Fn(\000)p Fm(1)2651 4577 y Fp(X)2659 4789 y Fl(k)r Fm(=0)2836 4604 y Fq(C)g(\025)2970 4520 y Fo(p)p 3053 4520 71 4 v 84 x Fq(T)p 2822 4648 318 4 v 2822 4668 a Fo(p)p 2905 4668 235 4 v 81 x Fq(n)22 b Fo(\000)h Fq(k)3165 4671 y(V)3222 4686 y Fl(k)r(;)p Fn(\003)0 4947 y Fu(F)-8 b(or)27 b Fq(\025)h Fu(small)e(enough)i(the)h (square)g(brac)m(k)m(et)h(term)d(is)h(larger)f(than)h(1)p Fq(=)p Fu(2,)g(so)h(that)f(\(pro)m(vided)g(w)m(e)h(double)0 5063 y(the)k(v)-5 b(alue)32 b(of)g(the)h(constan)m(t)h Fq(C)7 b Fu(\))1229 5338 y Fq(V)1286 5353 y Fl(n;)p Fn(\003)1415 5338 y Fo(\024)28 b Fq(C)7 b(\025Z)1721 5353 y Fl(n;)p Fn(\003)1845 5338 y Fu(+)1948 5214 y Fl(n)p Fn(\000)p Fm(1)1943 5244 y Fp(X)1950 5456 y Fl(k)r Fm(=0)2128 5271 y Fq(C)g(\025)2262 5187 y Fo(p)p 2345 5187 71 4 v 84 x Fq(T)p 2113 5315 318 4 v 2113 5335 a Fo(p)p 2196 5335 235 4 v 81 x Fq(n)23 b Fo(\000)f Fq(k)2457 5338 y(V)2514 5353 y Fl(k)r(;)p Fn(\003)3591 5338 y Fu(\(4.26\))1882 5532 y Fk(16)p eop %%Page: 17 18 17 17 bop 0 266 a Fu(By)33 b(iteration)e(of)39 b(\(4.26\))32 b(w)m(e)h(get)684 517 y Fq(V)741 532 y Fl(n;)p Fn(\003)871 517 y Fo(\024)28 b Fq(C)7 b(\025Z)1177 532 y Fl(n;)p Fn(\003)1300 517 y Fu(+)1404 393 y Fl(n)p Fn(\000)p Fm(1)1398 423 y Fp(X)1406 635 y Fl(k)r Fm(=0)1569 450 y Fq(C)1646 414 y Fm(2)1685 450 y Fq(\025)1742 414 y Fm(2)1782 366 y Fo(p)p 1865 366 71 4 v 84 x Fq(T)p 1569 495 367 4 v 1593 514 a Fo(p)p 1677 514 235 4 v 1677 595 a Fq(n)22 b Fo(\000)h Fq(k)1962 517 y(Z)2029 532 y Fl(k)r(;)p Fn(\003)2148 517 y Fu(+)f Fq(C)2323 476 y Fm(2)2363 517 y Fq(\025)2420 476 y Fm(2)2459 517 y Fq(T)2552 393 y Fl(n)p Fn(\000)p Fm(2)2546 423 y Fp(X)2554 635 y Fl(k)r Fm(=0)2707 517 y Fq(\013)q Fu(\()p Fq(n;)17 b(k)s Fu(\))p Fq(V)3059 532 y Fl(k)r(;)p Fn(\003)3591 517 y Fu(\(4.27\))0 764 y(where)1250 950 y Fq(\013)q Fu(\()p Fq(n;)g(k)s Fu(\))27 b(=)1721 825 y Fl(n)p Fn(\000)p Fm(1)1716 855 y Fp(X)1675 1067 y Fl(h)p Fm(=)p Fl(k)r Fm(+1)2061 883 y Fu(1)p 1926 927 320 4 v 1926 947 a Fo(p)p 2009 947 237 4 v 81 x Fq(n)c Fo(\000)f Fq(h)2398 883 y Fu(1)p 2265 927 316 4 v 2265 947 a Fo(p)p 2348 947 232 4 v 81 x Fq(h)h Fo(\000)f Fq(k)3591 950 y Fu(\(4.28\))0 1183 y(is)32 b(a)g(b)s(ounded)i(function)e(of)g Fq(n)h Fu(and)f Fq(k)s Fu(.)100 1299 y(W)-8 b(e)33 b(b)s(ound)f(the)h (second)h(term)e(on)h(the)g(r.h.s.)g(of)39 b(\(4.27\))32 b(b)m(y)979 1551 y Fq(Z)1046 1566 y Fl(n;)p Fn(\003)1197 1426 y Fl(n)p Fn(\000)p Fm(1)1192 1456 y Fp(X)1200 1668 y Fl(k)r Fm(=0)1362 1483 y Fq(C)1439 1447 y Fm(2)1479 1483 y Fq(\025)1536 1447 y Fm(2)1575 1399 y Fo(p)p 1658 1399 71 4 v 84 x Fq(T)p 1362 1528 367 4 v 1387 1548 a Fo(p)p 1470 1548 235 4 v 81 x Fq(n)23 b Fo(\000)f Fq(k)1767 1551 y Fo(\024)28 b Fq(C)7 b(Z)2016 1566 y Fl(n;)p Fn(\003)2117 1551 y Fq(\025)2174 1510 y Fm(2)2214 1462 y Fo(p)p 2297 1462 129 4 v 89 x Fq(T)14 b(n)27 b Fo(\024)h Fq(C)7 b(\025Z)2759 1566 y Fl(n;)p Fn(\003)0 1807 y Fu(since)33 b Fq(n)28 b Fo(\024)g Fq(m)515 1822 y Fl(\025)588 1807 y Fo(\024)h Fu(\()p Fq(\025)789 1771 y Fm(2)828 1807 y Fq(T)14 b Fu(\))937 1771 y Fn(\000)p Fm(1)1031 1807 y Fq(\034)d Fu(.)44 b(W)-8 b(e)33 b(then)g(get)f(from)g(\(4.27\))1275 2059 y Fq(V)1332 2074 y Fl(n;)p Fn(\003)1462 2059 y Fo(\024)c Fq(C)7 b(\025Z)1768 2074 y Fl(n;)p Fn(\003)1891 2059 y Fu(+)22 b Fq(C)7 b(\025)2123 2017 y Fm(2)2163 2059 y Fq(T)2255 1934 y Fl(n)p Fn(\000)p Fm(2)2250 1964 y Fp(X)2258 2176 y Fl(k)r Fm(=0)2410 2059 y Fq(V)2467 2074 y Fl(k)r(;)p Fn(\003)3591 2059 y Fu(\(4.29\))0 2310 y(W)-8 b(e)33 b(p)s(ostp)s(one)g(the)g(pro)s(of)f(that)g(there)h(is)f Fq(C)j Fu(=)27 b Fq(C)7 b Fu(\()p Fq(\034)k Fu(\))33 b(suc)m(h)h(that)1200 2468 y Fq(V)1257 2483 y Fl(n;)p Fn(\003)1387 2468 y Fo(\024)28 b Fq(C)7 b(\025Z)1693 2483 y Fl(n;)p Fn(\003)1795 2468 y Fq(;)211 b Fu(for)32 b(all)f Fq(n)d Fo(\024)g Fq(m)2594 2483 y Fl(\025)3591 2468 y Fu(\(4.30\))0 2625 y(By)34 b(\(4.5\))e(and)h(\(4.30\))o(,)g(c)m (ho)s(osing)f Fq(\025)h Fu(small)d(enough,)j(w)m(e)h(ha)m(v)m(e,)g(for) e Fq(n)c Fo(\024)g Fq(m)2811 2640 y Fl(\025)2856 2625 y Fu(,)1155 2801 y Fq(V)1212 2816 y Fl(n;)p Fn(\003)1342 2801 y Fo(\024)g Fq(C)7 b(\025)1581 2760 y Fm(1)p Fn(\000)p Fl(\020)1711 2711 y Fo(p)p 1794 2711 71 4 v 90 x Fq(T)209 b Fu(on)32 b(the)h(set)g Fo(B)2583 2750 y Fm(\(1\))2580 2828 y Fl(\025;\034)0 2992 y Fu(whic)m(h)h(implies)d Fq(m)697 3007 y Fl(\025)771 2992 y Fu(=)e Fq(n)934 3007 y Fl(\025)979 2992 y Fu(\()p Fq(\034)11 b Fu(\))34 b(in)e(the)i(set)g Fo(B)1646 2941 y Fm(\(1\))1643 3019 y Fl(\025;\034)1747 2992 y Fu(.)46 b(Th)m(us)34 b(the)g(\014rst)g(b)s(ound)f(in)g(\(4.7\))g (follo)m(ws)e(from)h(\(4.5\))0 3115 y(and)38 b(\(4.30\))o(.)59 b(Since)38 b Fq(N)868 3130 y Fl(\016)942 3115 y Fq(>)e(m)1139 3130 y Fl(\025)1185 3115 y Fu(,)j(\(4.8\))e(also)g(follo)m(ws.)58 b(Since)37 b Fo(k)18 b Fu(\026)-67 b Fq(m)2458 3079 y Fn(0)2482 3115 y Fo(k)2532 3130 y Fn(1)2643 3115 y Fu(=)36 b(1,)j(the)f(second)h(estimate)e(in)0 3232 y(\(4.7\))32 b(follo)m(ws)f(directly)h(from)g(the)h(\014rst)g(one)g(and)f(Prop)s (osition)f(2.1.)100 3351 y(W)-8 b(e)37 b(are)h(left)e(with)i(the)f(pro) s(of)g(of)44 b(\(4.30\))o(.)58 b(In)38 b Fj(R)1925 3315 y Fl(n)1968 3327 y Fg(\025)2015 3315 y Fm(\()p Fl(\034)8 b Fm(\)+1)2203 3351 y Fu(,)38 b(for)f(an)m(y)h Fq(n)f Fo(\024)f Fq(n)2877 3366 y Fl(\025)2923 3351 y Fu(\()p Fq(\034)11 b Fu(\),)39 b(w)m(e)f(in)m(tro)s(duce)g(the)0 3467 y(seminorm)31 b Fo(j)p Fq(y)t Fo(j)553 3482 y Fl(n)626 3467 y Fu(:=)d(sup)904 3491 y Fm(0)p Fn(\024)p Fl(j)t Fn(\024)p Fl(n)1145 3467 y Fo(j)p Fq(y)1221 3482 y Fl(j)1256 3467 y Fo(j)33 b Fu(and)f(the)h(corresp)s(onding)g(seminorm)e(on)h (matrices)1140 3716 y Fo(k)p Fq(A)p Fo(k)1313 3731 y Fl(n)1388 3716 y Fu(:=)59 b(sup)1518 3802 y Fn(j)p Fl(y)r Fn(j)1595 3810 y Fg(n)1637 3802 y Fm(=1)1744 3716 y Fo(j)p Fq(Ay)t Fo(j)1925 3731 y Fl(n)1998 3716 y Fu(=)h(sup)2102 3797 y Fm(0)p Fn(\024)p Fl(i)p Fn(\024)p Fl(n)2381 3592 y(n)2330 3621 y Fp(X)2341 3831 y Fl(j)t Fm(=0)2491 3716 y Fo(j)p Fq(A)2592 3731 y Fl(i;j)2672 3716 y Fo(j)891 b Fu(\(4.31\))0 3979 y(Then)34 b(the)f(\()p Fq(n)519 3994 y Fl(\025)564 3979 y Fu(\()p Fq(\034)11 b Fu(\))23 b(+)f(1\))g Fo(\002)g Fu(\()p Fq(n)1118 3994 y Fl(\025)1164 3979 y Fu(\()p Fq(\034)11 b Fu(\))22 b(+)g(1\))32 b(matrix)1502 4137 y Fq(A)1575 4152 y Fl(k)r(;j)1698 4137 y Fu(=)27 b Fq(C)7 b(\025)1935 4096 y Fm(2)1974 4137 y Fq(T)31 b Fv(1)2118 4152 y Fl(j)t()e Fu(0)33 b(and)f Fq(C)j(<)27 b Fo(1)33 b Fu(so)f(that,)h(for)f(an)m(y)h Fq(')p Fu(,)1370 1650 y Fp(\015)1370 1710 y(\015)1370 1770 y(\015)1426 1765 y Fq(g)1477 1714 y Fm(\()p Fl(n;)p Fn(?)p Fm(\))1473 1787 y Fl(t)1652 1765 y Fq(')1716 1650 y Fp(\015)1716 1710 y(\015)1716 1770 y(\015)1772 1834 y Fn(1)1874 1765 y Fo(\024)28 b Fq(C)7 b(e)2101 1724 y Fn(\000)p Fl(\013t)2231 1765 y Fo(k)p Fq(')p Fo(k)2395 1780 y Fn(1)3591 1765 y Fu(\(4.33\))0 1993 y(Let)33 b(also)1082 2175 y Fq(z)1131 2134 y Fm(\()p Fl(n;)p Fn(?)p Fm(\))1308 2175 y Fu(\()p Fq(t)p Fu(\))28 b(:=)g Fq(z)1627 2134 y Fm(\()p Fl(n)p Fm(\))1729 2175 y Fu(\()p Fq(t)p Fu(\))22 b Fo(\000)1972 2107 y Fu(3)p 1972 2152 V 1972 2243 a(4)2031 2175 y Fo(h)c Fu(\026)-67 b Fq(m)2155 2134 y Fn(0)2155 2199 y Fl(x)2195 2207 y Fg(n)2241 2175 y Fq(;)17 b(z)2334 2134 y Fm(\()p Fl(n)p Fm(\))2436 2175 y Fu(\()p Fq(t)p Fu(\))p Fo(i)i Fu(\026)-68 b Fq(m)2671 2134 y Fn(0)2671 2199 y Fl(x)2711 2207 y Fg(n)3591 2175 y Fu(\(4.34\))0 2399 y(b)s(e)33 b(the)g(comp)s(onen)m(t)f(of)g Fq(z)959 2363 y Fm(\()p Fl(n)p Fm(\))1062 2399 y Fu(\()p Fq(t)p Fu(\))h(orthogonal)d(to)51 b(\026)-67 b Fq(m)1898 2363 y Fn(0)1898 2424 y Fl(x)1938 2432 y Fg(n)2017 2399 y Fu(and)936 2697 y Fo(B)1004 2646 y Fm(\(2\))1001 2725 y Fl(\025;\034)1133 2697 y Fu(:=)1264 2527 y Fp(\()1408 2697 y Fu(sup)1344 2783 y Fl(n)p Fn(\024)p Fl(n)1485 2795 y Fg(\025)1525 2783 y Fm(\()p Fl(\034)8 b Fm(\))1750 2697 y Fu(sup)1636 2783 y Fl(t)p Fn(2)p Fm([)p Fl(T)1769 2791 y Fg(n)1811 2783 y Fl(;T)1872 2792 y Fg(n)p Fi(+1)1992 2783 y Fm(])2028 2697 y Fo(k)p Fq(z)2127 2656 y Fm(\()p Fl(n;)p Fn(?)p Fm(\))2304 2697 y Fu(\()p Fq(t)p Fu(\))p Fo(k)2465 2712 y Fn(1)2567 2697 y Fo(\024)28 b Fq(\025)2729 2656 y Fn(\000)p Fl(\020)2824 2527 y Fp(\))3591 2697 y Fu(\(4.35\))0 2987 y(In)j(App)s(endix)f(B)h(w)m(e)g(will)d(pro)m(v)m (e)j(that)f(for)g(eac)m(h)h Fq(\034)6 b(;)17 b(\020)8 b(;)17 b(q)30 b(>)e Fu(0)i(there)h(exists)g(a)e(constan)m(t)j Fq(C)i Fu(=)28 b Fq(C)7 b Fu(\()p Fq(\034)f(;)17 b(\020)8 b(;)17 b(q)t Fu(\))0 3103 y(suc)m(h)34 b(that)e(for)g(an)m(y)i Fq(\025)27 b(>)h Fu(0)1499 3287 y Fj(P)1575 3176 y Fp(\020)1636 3287 y Fo(B)1704 3236 y Fm(\(2\))1701 3314 y Fl(\025;\034)1806 3176 y Fp(\021)1893 3287 y Fo(\025)g Fu(1)22 b Fo(\000)h Fq(C)7 b(\025)2303 3245 y Fl(q)3591 3287 y Fu(\(4.36\))100 3491 y(De\014ne)33 b(no)m(w)14 3713 y Fq(V)93 3671 y Fn(?)71 3737 y(\003)152 3713 y Fu(\()p Fq(\034)11 b Fu(\))28 b(:=)92 b(sup)440 3798 y Fl(n)p Fn(\024)p Fl(n)581 3810 y Fg(\025)621 3798 y Fm(\()p Fl(\034)8 b Fm(\))731 3628 y Fp(\015)731 3688 y(\015)787 3713 y Fq(v)838 3671 y Fm(\()p Fl(n;)p Fn(?)p Fm(\))1014 3713 y Fu(\()p Fq(T)1109 3728 y Fl(n)p Fm(+1)1246 3713 y Fu(\))1284 3628 y Fp(\015)1284 3688 y(\015)1339 3752 y Fn(1)1431 3713 y Fq(;)211 b(v)1720 3671 y Fm(\()p Fl(n;)p Fn(?)p Fm(\))1897 3713 y Fu(\()p Fq(T)1992 3728 y Fl(n)p Fm(+1)2129 3713 y Fu(\))27 b(:=)h Fq(v)2376 3671 y Fm(\()p Fl(n)p Fm(\))2478 3713 y Fu(\()p Fq(T)2573 3728 y Fl(n)p Fm(+1)2710 3713 y Fu(\))22 b Fo(\000)2880 3645 y Fu(3)p 2880 3690 V 2880 3781 a(4)2938 3713 y Fo(h)c Fu(\026)-67 b Fq(m)3062 3671 y Fn(0)3062 3737 y Fl(x)3102 3745 y Fg(n)3149 3713 y Fq(;)17 b(v)3244 3671 y Fm(\()p Fl(n)p Fm(\))3345 3713 y Fu(\()p Fq(T)3440 3728 y Fl(n)p Fm(+1)3577 3713 y Fu(\))p Fo(i)h Fu(\026)-67 b Fq(m)3739 3671 y Fn(0)3739 3737 y Fl(x)3779 3745 y Fg(n)3591 3908 y Fu(\(4.37\))0 4260 y Fv(Prop)s(osition)36 b(4.2.)41 b Ft(R)-5 b(e)g(c)g(al)5 b(ling)43 b Fu(\(4.5\))34 b Ft(and)45 b Fu(\(4.35\))o Ft(,)35 b(set)1507 4477 y Fo(B)1575 4426 y Fm(\(1)p Fl(;)p Fm(2\))1572 4504 y Fl(\025;\034)1753 4477 y Fu(:=)27 b Fo(B)1951 4426 y Fm(\(1\))1948 4504 y Fl(\025;\034)2075 4477 y Fo(\\)22 b(B)2231 4426 y Fm(\(2\))2228 4504 y Fl(\025;\034)3591 4477 y Fu(\(4.38\))0 4682 y Ft(Then,)34 b(for)h(e)-5 b(ach)34 b Fq(\034)6 b(;)17 b(\020)34 b(>)28 b Fu(0)35 b Ft(ther)-5 b(e)34 b(is)h(a)g(c)-5 b(onstant)34 b Fq(C)h Fu(=)27 b Fq(C)7 b Fu(\()p Fq(\034)f(;)17 b(\020)8 b Fu(\))34 b Ft(such)h(that,)g(for)f(any)h Fq(\025)28 b(>)f Fu(0)p Ft(,)1134 4898 y Fq(V)1213 4857 y Fn(?)1191 4923 y(\003)1272 4898 y Fu(\()p Fq(\034)11 b Fu(\))28 b Fo(\024)g Fq(C)7 b(\025)1668 4857 y Fm(1)p Fn(\000)p Fm(2)p Fl(\020)2032 4898 y Fu(on)33 b(the)g(set)g Fo(B)2556 4847 y Fm(\(1)p Fl(;)p Fm(2\))2553 4926 y Fl(\025;\034)3591 4898 y Fu(\(4.39\))0 5104 y Ft(and,)h(r)-5 b(e)g(c)g(al)5 b(ling)43 b Fu(\(4.1\))34 b Ft(for)h(the)g(de\014nition)f(of)g Fq(\030)1759 5119 y Fl(n)1806 5104 y Ft(,)949 5320 y Fu(sup)840 5406 y Fl(n)p Fn(\024)p Fl(n)981 5418 y Fg(\025)1022 5406 y Fm(\()p Fl(\034)8 b Fm(\)+1)1222 5320 y Fo(j)p Fq(\030)1293 5335 y Fl(n)1362 5320 y Fo(\000)22 b Fq(x)1516 5335 y Fl(n)1564 5320 y Fo(j)27 b(\024)h Fq(C)7 b(T)1872 5279 y Fn(\000)p Fm(1)p Fl(=)p Fm(2+)p Fl(\020)2326 5320 y Fu(on)33 b(the)g(set)g Fo(B)2850 5269 y Fm(\(1)p Fl(;)p Fm(2\))2847 5348 y Fl(\025;\034)3591 5320 y Fu(\(4.40\))1882 5532 y Fk(18)p eop %%Page: 19 20 19 19 bop 0 266 a Ft(Pr)-5 b(o)g(of.)43 b Fu(Let)634 483 y(\003)702 433 y Fm(\()p Fl(n;)p Fn(?)p Fm(\))702 508 y(3)878 483 y Fu(\()p Fq(T)973 498 y Fl(n)p Fm(+1)1110 483 y Fu(\))83 b(:=)g(\003)1485 433 y Fm(\()p Fl(n)p Fm(\))1485 508 y(3)1587 483 y Fu(\()p Fq(T)1682 498 y Fl(n)p Fm(+1)1819 483 y Fu(\))22 b Fo(\000)1989 416 y Fu(3)p 1989 461 49 4 v 1989 552 a(4)2048 483 y Fo(h)c Fu(\026)-67 b Fq(m)2172 442 y Fn(0)2172 508 y Fl(x)2212 516 y Fg(n)2258 483 y Fq(;)17 b Fu(\003)2370 433 y Fm(\()p Fl(n)p Fm(\))2370 508 y(3)2471 483 y Fu(\()p Fq(T)2566 498 y Fl(n)p Fm(+1)2704 483 y Fu(\))p Fo(i)h Fu(\026)-67 b Fq(m)2866 442 y Fn(0)2866 508 y Fl(x)2906 516 y Fg(n)1245 752 y Fu(=)96 b Fo(\000)p Fq(\025)1619 628 y Fl(n)1568 658 y Fp(X)1576 870 y Fl(k)r Fm(=1)1729 617 y Fp(Z)1829 643 y Fl(T)1870 652 y Fg(n)p Fi(+1)1784 842 y Fl(T)1825 850 y Fg(n)2010 752 y Fq(dt)17 b(g)2164 701 y Fm(\()p Fl(n;)p Fn(?)p Fm(\))2160 779 y Fl(T)2201 788 y Fg(n)p Fi(+1)2320 779 y Fn(\000)p Fl(t)2404 752 y Fq(p)2453 767 y Fl(t)p Fn(\000)p Fl(T)2574 779 y Fg(k)2617 752 y Fu([)h(\026)-67 b Fq(m)2729 767 y Fl(x)2769 779 y Fg(k)2833 752 y Fo(\000)41 b Fu(\026)-67 b Fq(m)3018 767 y Fl(x)3058 779 y Fg(k)q Fh(\000)p Fi(1)3179 752 y Fu(])385 b(\(4.41\))0 1021 y(where)34 b(w)m(e)f(used)h(\(4.32\))o(.)44 b(By)33 b(using)h(\(4.14\))o(,)f(\(4.33\))o(,)f Fo(k)19 b Fu(\026)-68 b Fq(m)2130 985 y Fn(0)2154 1021 y Fo(k)2204 1036 y Fn(1)2306 1021 y Fo(\024)28 b Fu(1,)33 b(and)f(recalling)g(\(4.3\))g(w)m(e)h(ha)m (v)m(e)599 1162 y Fp(\015)599 1221 y(\015)599 1281 y(\015)654 1276 y Fu(\003)722 1225 y Fm(\()p Fl(n;)p Fn(?)p Fm(\))722 1301 y(3)898 1276 y Fu(\()p Fq(T)993 1291 y Fl(n)p Fm(+1)1131 1276 y Fu(\))1169 1162 y Fp(\015)1169 1221 y(\015)1169 1281 y(\015)1224 1345 y Fn(1)1382 1276 y Fo(\024)83 b Fq(C)7 b(\016)1666 1235 y Fn(\003)1706 1276 y Fu(\()p Fq(\034)k Fu(\))p Fq(\025)1959 1152 y Fl(n)1908 1182 y Fp(X)1916 1394 y Fl(k)r Fm(=1)2069 1141 y Fp(Z)2169 1167 y Fl(T)2210 1176 y Fg(n)p Fi(+1)2124 1366 y Fl(T)2165 1374 y Fg(n)2333 1276 y Fq(dt)17 b(e)2481 1235 y Fn(\000)p Fl(\013)p Fm(\()p Fl(T)2649 1244 y Fg(n)p Fi(+1)2769 1235 y Fn(\000)p Fl(t)p Fm(\))3037 1209 y Fu(1)p 2891 1253 340 4 v 2891 1273 a Fo(p)p 2974 1273 257 4 v 77 x Fq(t)23 b Fo(\000)f Fq(T)3188 1365 y Fl(k)1382 1578 y Fo(\024)83 b Fq(C)7 b(\016)1666 1536 y Fn(\003)1706 1578 y Fu(\()p Fq(\034)k Fu(\))1893 1510 y Fq(\025)p 1845 1555 154 4 v 1845 1575 a Fo(p)p 1928 1575 71 4 v 84 x Fq(T)2009 1488 y Fp(p)p 2108 1488 233 4 v 2108 1578 a Fq(n)2166 1593 y Fl(\025)2212 1578 y Fu(\()p Fq(\034)g Fu(\))28 b Fo(\024)g Fq(C)7 b(\025)2608 1536 y Fm(1)p Fn(\000)p Fl(\020)2738 1578 y Fq(T)2809 1536 y Fn(\000)p Fm(1)p Fl(=)p Fm(2)3591 1578 y Fu(\(4.42\))0 1838 y(the)33 b(last)f(inequalit)m(y)f(b)s(eing)h(true,)h(b)m(y)i(\(4.7\),)d(on)h (the)g(set)g Fo(B)2222 1787 y Fm(\(1\))2219 1866 y Fl(\025;\034)2323 1838 y Fu(.)100 1988 y(Shorthanding)e(b)m(y)i Fq(R)916 1951 y Fm(\()p Fl(n)p Fm(\))1050 1988 y Fu(the)g(sum)f(of)g(all)e(the)j (terms)f(on)g(the)h(r.h.s.)g(of)38 b(\(3.7\),)32 b(except)i(for)e(\000) 3549 1937 y Fm(\()p Fl(n)p Fm(\))3549 2012 y(2)3683 1988 y Fu(and)0 2121 y(\003)68 2071 y Fm(\()p Fl(n)p Fm(\))68 2146 y(3)170 2121 y Fu(,)g(and)h(calling)d Fq(R)803 2085 y Fm(\()p Fl(n;)p Fn(?)p Fm(\))1012 2121 y Fu(its)i(orthogonal)e(pro)5 b(jection,)826 2313 y Fq(v)877 2272 y Fm(\()p Fl(n;)p Fn(?)p Fm(\))1053 2313 y Fu(\()p Fq(T)1148 2328 y Fl(n)p Fm(+1)1286 2313 y Fu(\))27 b(=)h Fq(\025z)1561 2272 y Fm(\()p Fl(n;)p Fn(?)p Fm(\))1738 2313 y Fu(\()p Fq(T)1833 2328 y Fl(n)p Fm(+1)1970 2313 y Fu(\))22 b(+)g(\003)2196 2263 y Fm(\()p Fl(n;)p Fn(?)p Fm(\))2196 2338 y(3)2372 2313 y Fu(\()p Fq(T)2467 2328 y Fl(n)p Fm(+1)2604 2313 y Fu(\))g(+)g Fq(R)2837 2272 y Fm(\()p Fl(n;)p Fn(?)p Fm(\))3591 2313 y Fu(\(4.43\))0 2488 y(The)34 b(last)d(term)h(is)g(b)s (ounded)i(b)m(y)927 2665 y Fo(j)p Fq(R)1030 2624 y Fm(\()p Fl(n;)p Fn(?)p Fm(\))1206 2665 y Fo(j)27 b(\024)i Fq(C)1460 2585 y Fp(\002)1502 2665 y Fq(e)1547 2624 y Fn(\000)p Fl(\013T)1724 2665 y Fu(+)22 b Fq(\025T)1950 2624 y Fm(2)2011 2665 y Fu(+)g Fq(\025T)31 b Fu(log)16 b Fq(n)2454 2680 y Fl(\025)2500 2665 y Fu(\()p Fq(\034)11 b Fu(\))2629 2585 y Fp(\003)2687 2665 y Fq(V)2744 2680 y Fn(\003)2784 2665 y Fu(\()p Fq(\034)g Fu(\))678 b(\(4.44\))0 2842 y(as)31 b(it)f(follo)m(ws)g(from)g(\(4.12\))g(and)h(\(4.18\))o ({\(4.20\),)g(and)g(\(4.23\))o(.)43 b(The)32 b(b)s(ound)f(\(4.39\))g (no)m(w)g(follo)m(ws)f(from)0 2958 y(the)j(de\014nition)f(\(4.35\))o(,) g(and)h(equations)g(\(4.42\))o({\(4.44\))o(.)100 3074 y(It)f(remains)g(to)g(pro)m(v)m(e)i(\(4.40\))o(.)44 b(Recalling)30 b(that)i Fq(x)1961 3089 y Fm(0)2029 3074 y Fu(=)27 b(0,)33 b(from)e(\(2.5\))h(and)h(\(4.1\))o(,)53 3343 y Fq(x)108 3358 y Fl(n)183 3343 y Fu(=)292 3219 y Fl(n)p Fn(\000)p Fm(1)287 3249 y Fp(X)295 3461 y Fl(k)r Fm(=0)431 3343 y Fu(\()p Fq(x)524 3358 y Fl(k)r Fm(+1)679 3343 y Fo(\000)23 b Fq(x)834 3358 y Fl(k)877 3343 y Fu(\))28 b(=)f Fq(\030)1089 3358 y Fl(n)1158 3343 y Fo(\000)1292 3276 y Fu(9)p 1267 3320 98 4 v 1267 3412 a(16)1397 3219 y Fl(n)p Fn(\000)p Fm(1)1392 3249 y Fp(X)1399 3461 y Fl(k)r Fm(=0)1535 3343 y Fo(h)18 b Fu(\026)-67 b Fq(m)1659 3302 y Fn(0)1659 3368 y Fl(x)1699 3380 y Fg(k)1742 3343 y Fq(;)17 b(v)1837 3302 y Fm(\()p Fl(k)r Fm(\))1934 3343 y Fu(\()p Fq(T)2029 3358 y Fl(k)r Fm(+1)2161 3343 y Fu(\))p Fo(ih)h Fu(\026)-67 b Fq(m)2362 3302 y Fn(00)2362 3368 y Fl(x)2402 3380 y Fg(k)2444 3343 y Fq(;)17 b(v)2539 3302 y Fm(\()p Fl(k)r Fm(\))2636 3343 y Fu(\()p Fq(T)2731 3358 y Fl(k)r Fm(+1)2864 3343 y Fu(\))p Fo(i)22 b Fu(+)3066 3219 y Fl(n)p Fn(\000)p Fm(1)3061 3249 y Fp(X)3069 3461 y Fl(k)r Fm(=0)3221 3343 y Fq(R)q Fu(\()p Fq(v)3385 3302 y Fm(\()p Fl(k)r Fm(\))3483 3343 y Fu(\()p Fq(T)3578 3358 y Fl(k)r Fm(+1)3711 3343 y Fu(\)\))3591 3554 y(\(4.45\))0 3670 y(then,)33 b(for)f Fq(n)c Fo(\024)g Fq(n)647 3685 y Fl(\025)693 3670 y Fu(\()p Fq(\034)11 b Fu(\),)33 b(since)g Fo(k)18 b Fu(\026)-67 b Fq(m)1256 3634 y Fn(0)1279 3670 y Fo(k)1329 3685 y Fm(1)1396 3670 y Fu(=)28 b(2,)73 3936 y Fo(j)p Fq(\030)144 3951 y Fl(n)212 3936 y Fo(\000)23 b Fq(x)367 3951 y Fl(n)414 3936 y Fo(j)k(\024)i Fq(n)633 3951 y Fl(\025)678 3936 y Fu(\()p Fq(\034)11 b Fu(\))824 3766 y Fp( )903 3936 y Fu(2)p Fq(V)1009 3951 y Fn(\003)1048 3936 y Fu(\()p Fq(\034)g Fu(\))81 b(sup)1194 4022 y Fl(n)p Fn(\024)p Fl(n)1335 4034 y Fg(\025)1375 4022 y Fm(\()p Fl(\034)8 b Fm(\))1486 3852 y Fp(\014)1486 3911 y(\014)1519 3936 y Fo(h)18 b Fu(\026)-67 b Fq(m)1643 3895 y Fn(00)1643 3961 y Fl(x)1683 3969 y Fg(n)1730 3936 y Fq(;)17 b(v)1825 3895 y Fm(\()p Fl(n)p Fm(\))1926 3936 y Fu(\()p Fq(T)2021 3951 y Fl(n)p Fm(+1)2158 3936 y Fu(\))p Fo(i)2235 3852 y Fp(\014)2235 3911 y(\014)2290 3936 y Fu(+)86 b(sup)2388 4022 y Fl(n)p Fn(\024)p Fl(n)2529 4034 y Fg(\025)2569 4022 y Fm(\()p Fl(\034)8 b Fm(\))2680 3852 y Fp(\015)2680 3911 y(\015)2735 3936 y Fq(R)q Fu(\()p Fq(v)2899 3895 y Fm(\()p Fl(n)p Fm(\))3001 3936 y Fu(\()p Fq(T)3096 3951 y Fl(n)p Fm(+1)3233 3936 y Fu(\)\))3309 3852 y Fp(\015)3309 3911 y(\015)3364 3975 y Fn(1)3439 3766 y Fp(!)3591 3936 y Fu(\(4.46\))0 4202 y(By)34 b(\(2.5\))1221 4344 y(sup)1157 4430 y Fl(n)p Fn(\024)p Fl(n)1298 4442 y Fg(\025)1338 4430 y Fm(\()p Fl(\034)8 b Fm(\))1449 4260 y Fp(\015)1449 4319 y(\015)1504 4344 y Fq(R)q Fu(\()p Fq(v)1668 4303 y Fm(\()p Fl(n)p Fm(\))1770 4344 y Fu(\()p Fq(T)1865 4359 y Fl(n)p Fm(+1)2002 4344 y Fu(\)\))2078 4260 y Fp(\015)2078 4319 y(\015)2133 4383 y Fn(1)2236 4344 y Fo(\024)28 b Fq(C)7 b(V)2475 4359 y Fn(\003)2514 4344 y Fu(\()p Fq(\034)k Fu(\))2643 4303 y Fm(3)3591 4344 y Fu(\(4.47\))0 4589 y(Since)33 b Fo(h)18 b Fu(\026)-67 b Fq(m)379 4553 y Fn(00)421 4589 y Fq(;)35 b Fu(\026)-67 b Fq(m)550 4553 y Fn(0)574 4589 y Fo(i)27 b Fu(=)h(0,)33 b Fo(jh)18 b Fu(\026)-67 b Fq(m)1005 4553 y Fn(00)1005 4614 y Fl(x)1045 4622 y Fg(n)1091 4589 y Fq(;)17 b(v)1186 4553 y Fm(\()p Fl(n)p Fm(\))1287 4589 y Fu(\()p Fq(T)1382 4604 y Fl(n)p Fm(+1)1520 4589 y Fu(\))p Fo(ij)27 b(\024)h Fq(C)7 b(V)1913 4553 y Fn(?)1891 4614 y(\003)1972 4589 y Fu(\()p Fq(\034)k Fu(\),)33 b(\(4.40\))f(follo)m(ws)g(from)f(\(4.7\),)h(\(4.39\),)h (\(4.46\))o(,)0 4705 y(and)g(\(4.47\))o(.)3297 b Fd(\003)100 5061 y Fu(The)36 b(b)s(ound)g(in)f(\(4.39\))g(holds)h(also)e(for)h Fq(v)1669 5025 y Fm(\()p Fl(n)p Fm(\))1771 5061 y Fu(\()p Fq(T)1866 5076 y Fl(n)1913 5061 y Fu(\).)53 b(Indeed,)38 b(since)e Fo(h)18 b Fu(\026)-67 b Fq(m)2745 5025 y Fn(0)2745 5085 y Fl(x)2785 5093 y Fg(n)2832 5061 y Fq(;)17 b(v)2927 5025 y Fm(\()p Fl(n)p Fm(\))3028 5061 y Fu(\()p Fq(T)3123 5076 y Fl(n)3170 5061 y Fu(\))p Fo(i)33 b Fu(=)g(0,)j(w)m(e)h(ha)m(v)m (e)0 5177 y(the)c(follo)m(wing)d(prop)s(osition.)1882 5532 y Fk(19)p eop %%Page: 20 21 20 20 bop 0 283 a Fv(Prop)s(osition)42 b(4.3.)i Ft(L)-5 b(et)40 b Fo(B)1071 232 y Fm(\(1)p Fl(;)p Fm(2\))1068 311 y Fl(\025;\034)1261 283 y Ft(b)-5 b(e)39 b(as)h(in)46 b Fu(\(4.38\))p Ft(.)59 b(Then,)40 b(for)g(e)-5 b(ach)39 b Fq(\034)6 b(;)17 b(\020)43 b(>)37 b Fu(0)j Ft(ther)-5 b(e)39 b(is)h(a)f(c)-5 b(onstant)0 407 y Fq(C)35 b Fu(=)27 b Fq(C)7 b Fu(\()p Fq(\034)f(;)17 b(\020)8 b Fu(\))34 b Ft(such)g(that,)i(for)e(any)h Fq(\025)28 b(>)f Fu(0)927 582 y(sup)863 667 y Fl(n)p Fn(\024)p Fl(n)1004 679 y Fg(\025)1044 667 y Fm(\()p Fl(\034)8 b Fm(\))1155 497 y Fp(\015)1155 557 y(\015)1210 582 y Fq(v)1261 541 y Fm(\()p Fl(n)p Fm(\))1363 582 y Fu(\()p Fq(T)1458 597 y Fl(n)1505 582 y Fu(\))1543 497 y Fp(\015)1543 557 y(\015)1598 621 y Fn(1)1700 582 y Fo(\024)28 b Fq(C)7 b(\025)1939 541 y Fm(1)p Fn(\000)p Fm(2)p Fl(\020)2304 582 y Fu(on)32 b(the)h(set)h Fo(B)2828 531 y Fm(\(1)p Fl(;)p Fm(2\))2825 609 y Fl(\025;\034)3591 582 y Fu(\(4.48\))0 1054 y Ft(Pr)-5 b(o)g(of.)43 b Fu(By)34 b(\(3.2\))1145 1178 y Fq(v)1196 1137 y Fm(\()p Fl(n)p Fm(\))1297 1178 y Fu(\()p Fq(T)1392 1193 y Fl(n)1440 1178 y Fu(\))27 b(=)h Fq(v)1660 1137 y Fm(\()p Fl(n)p Fn(\000)p Fm(1\))1851 1178 y Fu(\()p Fq(T)1946 1193 y Fl(n)1994 1178 y Fu(\))22 b(+)40 b(\026)-67 b Fq(m)2237 1193 y Fl(x)2277 1202 y Fg(n)p Fh(\000)p Fi(1)2424 1178 y Fo(\000)41 b Fu(\026)-68 b Fq(m)2608 1193 y Fl(x)2648 1201 y Fg(n)3591 1178 y Fu(\(4.49\))0 1315 y(By)33 b(T)-8 b(a)m(ylor)32 b(expansion)h(up)g(to)f(fourth)h (order,)g(w)m(e)g(ha)m(v)m(e)639 1515 y(\026)-67 b Fq(m)706 1530 y Fl(x)746 1538 y Fg(n)814 1515 y Fo(\000)41 b Fu(\026)-67 b Fq(m)999 1530 y Fl(x)1039 1539 y Fg(n)p Fh(\000)p Fi(1)1247 1515 y Fu(=)83 b Fo(\000)41 b Fu(\026)-67 b Fq(m)1591 1474 y Fn(0)1591 1539 y Fl(x)1631 1548 y Fg(n)p Fh(\000)p Fi(1)1756 1515 y Fu(\()p Fq(x)1849 1530 y Fl(n)1918 1515 y Fo(\000)23 b Fq(x)2073 1530 y Fl(n)p Fn(\000)p Fm(1)2210 1515 y Fu(\))g(+)2379 1447 y(1)p 2379 1492 49 4 v 2379 1583 a(2)2456 1515 y(\026)-68 b Fq(m)2522 1474 y Fn(00)2522 1539 y Fl(x)2562 1548 y Fg(n)p Fh(\000)p Fi(1)2687 1515 y Fu(\()p Fq(x)2780 1530 y Fl(n)2850 1515 y Fo(\000)22 b Fq(x)3004 1530 y Fl(n)p Fn(\000)p Fm(1)3142 1515 y Fu(\))3180 1474 y Fm(2)3591 1515 y Fu(\(4.50\))1406 1752 y Fo(\000)1516 1685 y Fu(1)p 1516 1729 V 1516 1820 a(6)1593 1752 y(\026)-68 b Fq(m)1659 1711 y Fn(000)1659 1777 y Fl(x)1699 1786 y Fg(n)p Fh(\000)p Fi(1)1824 1752 y Fu(\()p Fq(x)1917 1767 y Fl(n)1987 1752 y Fo(\000)23 b Fq(x)2142 1767 y Fl(n)p Fn(\000)p Fm(1)2279 1752 y Fu(\))2317 1711 y Fm(3)2379 1752 y Fu(+)f Fq(a)2528 1767 y Fl(n)p Fn(\000)p Fm(1)2665 1752 y Fu(\()p Fq(x)2758 1767 y Fl(n)2828 1752 y Fo(\000)g Fq(x)2982 1767 y Fl(n)p Fn(\000)p Fm(1)3120 1752 y Fu(\))3158 1711 y Fm(4)0 1938 y Fu(where)34 b Fq(a)333 1953 y Fl(n)p Fn(\000)p Fm(1)503 1938 y Fu(is)e(b)s(ounded.)44 b(Th)m(us,)34 b(b)m(y)g(Prop)s(osition)d(2.1,)h(w)m(e)h(ha)m(v)m(e)930 2141 y(\026)-67 b Fq(m)997 2156 y Fl(x)1037 2164 y Fg(n)1105 2141 y Fo(\000)41 b Fu(\026)-67 b Fq(m)1290 2156 y Fl(x)1330 2165 y Fg(n)p Fh(\000)p Fi(1)1483 2141 y Fu(=)1596 2074 y(3)p 1596 2118 V 1596 2209 a(4)1655 2141 y Fo(h)18 b Fu(\026)-67 b Fq(m)1779 2100 y Fn(0)1779 2166 y Fl(x)1819 2175 y Fg(n)p Fh(\000)p Fi(1)1944 2141 y Fq(;)17 b(v)2039 2100 y Fm(\()p Fl(n)p Fn(\000)p Fm(1\))2230 2141 y Fu(\()p Fq(T)2325 2156 y Fl(n)2372 2141 y Fu(\))p Fo(i)h Fu(\026)-67 b Fq(m)2534 2100 y Fn(0)2534 2166 y Fl(x)2574 2175 y Fg(n)p Fh(\000)p Fi(1)2721 2141 y Fu(+)22 b Fq(\013)2881 2156 y Fl(n)3591 2141 y Fu(\(4.51\))0 2327 y(where)193 2508 y Fq(\013)255 2523 y Fl(n)385 2508 y Fu(=)554 2441 y(1)p 554 2485 V 554 2576 a(2)631 2508 y(\026)-67 b Fq(m)698 2467 y Fn(00)698 2533 y Fl(x)738 2542 y Fg(n)p Fh(\000)p Fi(1)863 2508 y Fu(\()p Fq(x)956 2523 y Fl(n)1025 2508 y Fo(\000)23 b Fq(x)1180 2523 y Fl(n)p Fn(\000)p Fm(1)1318 2508 y Fu(\))1356 2467 y Fm(2)1417 2508 y Fo(\000)1527 2441 y Fu(1)p 1527 2485 V 1527 2576 a(6)1604 2508 y(\026)-67 b Fq(m)1671 2467 y Fn(000)1671 2533 y Fl(x)1711 2542 y Fg(n)p Fh(\000)p Fi(1)1836 2508 y Fu(\()p Fq(x)1929 2523 y Fl(n)1998 2508 y Fo(\000)23 b Fq(x)2153 2523 y Fl(n)p Fn(\000)p Fm(1)2290 2508 y Fu(\))2328 2467 y Fm(3)2390 2508 y Fu(+)f Fq(a)2539 2523 y Fl(n)p Fn(\000)p Fm(1)2676 2508 y Fu(\()p Fq(x)2769 2523 y Fl(n)2839 2508 y Fo(\000)g Fq(x)2993 2523 y Fl(n)p Fn(\000)p Fm(1)3131 2508 y Fu(\))3169 2467 y Fm(4)544 2758 y Fu(+)40 b(\026)-67 b Fq(m)727 2717 y Fn(0)727 2783 y Fl(x)767 2792 y Fg(n)p Fh(\000)p Fi(1)909 2618 y Fp(\032)1018 2691 y Fu(9)p 994 2735 98 4 v 994 2826 a(16)1101 2758 y Fo(h)18 b Fu(\026)-67 b Fq(m)1225 2717 y Fn(00)1225 2783 y Fl(x)1265 2792 y Fg(n)p Fh(\000)p Fi(1)1390 2758 y Fq(;)17 b(v)1485 2717 y Fm(\()p Fl(n)p Fn(\000)p Fm(1\))1676 2758 y Fu(\()p Fq(T)1771 2773 y Fl(n)1819 2758 y Fu(\))p Fo(ih)g Fu(\026)-66 b Fq(m)2020 2717 y Fn(0)2020 2783 y Fl(x)2060 2792 y Fg(n)p Fh(\000)p Fi(1)2184 2758 y Fq(;)17 b(v)2279 2717 y Fm(\()p Fl(n)p Fn(\000)p Fm(1\))2470 2758 y Fu(\()p Fq(T)2565 2773 y Fl(n)2613 2758 y Fu(\))p Fo(i)k(\000)i Fq(R)q Fu(\()p Fq(v)2975 2717 y Fm(\()p Fl(n)p Fn(\000)p Fm(1\))3167 2758 y Fu(\()p Fq(T)3262 2773 y Fl(n)3309 2758 y Fu(\)\))3385 2618 y Fp(\033)3591 2758 y Fu(\(4.52\))0 2977 y(F)-8 b(rom)31 b(\(4.49\))h(and)h(\(4.51\))e(w)m(e)j(get)1324 3141 y Fq(v)1375 3100 y Fm(\()p Fl(n)p Fm(\))1477 3141 y Fu(\()p Fq(T)1572 3156 y Fl(n)1619 3141 y Fu(\))27 b(=)h Fq(v)1839 3100 y Fm(\()p Fl(n)p Fn(\000)p Fm(1)p Fl(;)p Fn(?)p Fm(\))2105 3141 y Fu(\()p Fq(T)2200 3156 y Fl(n)2247 3141 y Fu(\))22 b Fo(\000)h Fq(\013)2469 3156 y Fl(n)3591 3141 y Fu(\(4.53\))0 3316 y(Note)35 b(that,)h(from)e(\(4.7\))o(,)i(in)e Fo(B)1160 3265 y Fm(\(1)p Fl(;)p Fm(2\))1157 3344 y Fl(\025;\034)1345 3316 y Fu(w)m(e)i(ha)m(v)m(e)h(sup)1865 3339 y Fl(n)p Fn(\024)p Fl(n)2006 3351 y Fg(\025)2047 3339 y Fm(\()p Fl(\034)8 b Fm(\))2161 3316 y Fo(j)p Fq(x)2244 3331 y Fl(n)2315 3316 y Fo(\000)25 b Fq(x)2472 3331 y Fl(n)p Fn(\000)p Fm(1)2609 3316 y Fo(j)32 b(\024)g Fq(C)7 b(\025)2912 3280 y Fm(1)p Fn(\000)p Fl(\020)3042 3232 y Fo(p)p 3125 3232 71 4 v 84 x Fq(T)14 b Fu(;)36 b(b)m(y)i(\(4.39\))c(and)0 3447 y(\(4.47\))e(w)m(e)i(ha)m(v)m(e)g Fo(j)p Fq(\013)741 3462 y Fl(n)787 3447 y Fo(j)28 b(\024)g Fq(C)7 b(\025)1082 3411 y Fm(2)p Fn(\000)p Fm(2)p Fl(\020)1247 3447 y Fq(T)47 b Fu(\(a)32 b(b)s(etter)h(b)s(ound)g(is)f(pro)m(v)m(ed)i(in)e(Section)h (6\).)43 b(The)34 b(b)s(ound)f(\(4.48\))0 3563 y(follo)m(ws.)3447 b Fd(\003)1882 5532 y Fk(20)p eop %%Page: 21 22 21 21 bop 0 282 a Fs(P)l(art)54 b(I)t(I)t(I.)h(Limit)f(motion)1197 753 y Fu(5.)86 b Fr(A)38 b(new)g(integral)f(equa)-7 b(tion)100 934 y Fu(Unless)32 b(otherwise)g(stated,)g(w)m(e)h(will)c(w)m(ork)k(in) e(the)h(set)g Fo(B)2221 883 y Fm(\(1)p Fl(;)p Fm(2\))2218 962 y Fl(\025;\034)2402 934 y Fu(whic)m(h)g(app)s(ears)g(in)f(Prop)s (ositions)f(4.2)0 1058 y(and)36 b(4.3.)54 b(In)37 b(particular)d(w)m(e) j(can)g(use)g(the)g(in)m(tegral)d(represen)m(tation)j(\(3.7\))f(for)g (all)e Fq(n)g Fo(\024)g Fq(n)3441 1073 y Fl(\025)3487 1058 y Fu(\()p Fq(\034)11 b Fu(\))36 b(\(if)f Fq(\025)0 1174 y Fu(is)c(small)d(enough\))k(and)f(the)g(b)s(ounds)h(of)e(Section) h(4.)43 b(It)31 b(is)g(no)m(w)g(con)m(v)m(enien)m(t)i(to)e(decomp)s (ose)g(the)g(term)0 1308 y(\003)68 1257 y Fm(\()p Fl(n)p Fm(\))68 1333 y(3)170 1308 y Fu(\()p Fq(t)p Fu(\))h(in)m(to)g(the)h (sum)f(of)h(three)g(new)g(terms;)g(w)m(e)h(th)m(us)f(use)h(\(4.51\))e (to)g(write)1366 1486 y(\003)1434 1435 y Fm(\()p Fl(n)p Fm(\))1434 1511 y(3)1536 1486 y Fu(\()p Fq(t)p Fu(\))c(=)f(\000)1839 1435 y Fm(\()p Fl(n)p Fm(\))1839 1511 y(3+8)1969 1486 y Fu(\()p Fq(t)p Fu(\))22 b(+)g(\000)2261 1435 y Fm(\()p Fl(n)p Fm(\))2261 1511 y(7)2363 1486 y Fu(\()p Fq(t)p Fu(\))1165 b(\(5.1\))0 1647 y(where)611 1862 y(\000)672 1811 y Fm(\()p Fl(n)p Fm(\))672 1886 y(3+8)802 1862 y Fu(\()p Fq(t)p Fu(\))83 b(:=)g Fo(\000)1269 1795 y Fu(3)p Fq(\025)p 1269 1839 106 4 v 1298 1930 a Fu(4)1452 1738 y Fl(n)1402 1767 y Fp(X)1409 1980 y Fl(k)r Fm(=1)1545 1862 y Fo(h)18 b Fu(\026)-67 b Fq(m)1669 1821 y Fn(0)1669 1887 y Fl(x)1709 1899 y Fg(k)q Fh(\000)p Fi(1)1830 1862 y Fq(;)17 b(v)1925 1821 y Fm(\()p Fl(k)r Fn(\000)p Fm(1\))2112 1862 y Fu(\()p Fq(T)2207 1877 y Fl(k)2250 1862 y Fu(\))p Fo(i)2344 1726 y Fp(Z)2443 1753 y Fl(t)2398 1952 y(T)2439 1960 y Fg(n)2486 1862 y Fq(ds)g(g)2651 1811 y Fm(\()p Fl(n)p Fm(\))2647 1885 y Fl(t)p Fn(\000)p Fl(s)2763 1862 y Fq(p)2812 1877 y Fl(s)p Fn(\000)p Fl(T)2941 1889 y Fg(k)3001 1862 y Fu(\026)-67 b Fq(m)3068 1821 y Fn(0)3068 1887 y Fl(x)3108 1899 y Fg(k)q Fh(\000)p Fi(1)3639 1862 y Fu(\(5.2\))639 2190 y(\000)700 2139 y Fm(\()p Fl(n)p Fm(\))700 2214 y(7)802 2190 y Fu(\()p Fq(t)p Fu(\))83 b(:=)g Fo(\000)p Fq(\025)1383 2065 y Fl(n)1333 2095 y Fp(X)1340 2308 y Fl(k)r Fm(=1)1493 2054 y Fp(Z)1593 2081 y Fl(t)1549 2280 y(T)1590 2288 y Fg(n)1636 2190 y Fq(ds)17 b(g)1801 2139 y Fm(\()p Fl(n)p Fm(\))1797 2212 y Fl(t)p Fn(\000)p Fl(s)1914 2190 y Fq(p)1963 2205 y Fl(s)p Fn(\000)p Fl(T)2092 2217 y Fg(k)2133 2190 y Fq(\013)2195 2205 y Fl(k)3639 2190 y Fu(\(5.3\))0 2471 y(and)33 b(next)g(w)m(e)h(decomp)s (ose)f(\000)1105 2420 y Fm(\()p Fl(n)p Fm(\))1105 2495 y(3+8)1234 2471 y Fu(\()p Fq(t)p Fu(\))28 b(=)g(\000)1538 2420 y Fm(\()p Fl(n)p Fm(\))1538 2495 y(3)1639 2471 y Fu(\()p Fq(t)p Fu(\))23 b(+)f(\000)1932 2420 y Fm(\()p Fl(n)p Fm(\))1932 2495 y(8)2034 2471 y Fu(\()p Fq(t)p Fu(\))32 b(where)124 2732 y(\000)185 2681 y Fm(\()p Fl(n)p Fm(\))185 2756 y(3)287 2732 y Fu(\()p Fq(t)p Fu(\))83 b(:=)g Fo(\000)754 2664 y Fu(3)p Fq(\025)p 754 2709 V 783 2800 a Fu(4)892 2607 y Fl(n)p Fn(\000)p Fm(1)887 2637 y Fp(X)894 2849 y Fl(k)r Fm(=0)1030 2732 y Fo(h)18 b Fu(\026)-67 b Fq(m)1154 2690 y Fn(0)1154 2756 y Fl(x)1194 2768 y Fg(k)1236 2732 y Fq(;)17 b(v)1331 2690 y Fm(\()p Fl(k)r Fm(\))1428 2732 y Fu(\()p Fq(T)1523 2747 y Fl(k)r Fm(+1)1656 2732 y Fu(\))p Fo(i)1750 2596 y Fp(Z)1849 2622 y Fl(t)1805 2822 y(T)1846 2830 y Fg(n)1893 2732 y Fq(ds)g(g)2058 2681 y Fm(\()p Fl(n)p Fm(\))2054 2754 y Fl(t)p Fn(\000)p Fl(s)2476 2664 y Fu(2)p 2180 2709 641 4 v 2180 2729 a Fp(p)p 2279 2729 542 4 v 2279 2814 a Fu(2)p Fq(\031)t Fu(\()p Fq(s)22 b Fo(\000)h Fq(T)2650 2829 y Fl(k)r Fm(+1)2783 2814 y Fu(\))3639 2732 y(\(5.4\))124 3074 y(\000)185 3023 y Fm(\()p Fl(n)p Fm(\))185 3098 y(8)287 3074 y Fu(\()p Fq(t)p Fu(\))83 b(:=)g Fo(\000)754 3006 y Fu(3)p Fq(\025)p 754 3051 106 4 v 783 3142 a Fu(4)892 2949 y Fl(n)p Fn(\000)p Fm(1)887 2979 y Fp(X)894 3191 y Fl(k)r Fm(=0)1030 3074 y Fo(h)18 b Fu(\026)-67 b Fq(m)1154 3033 y Fn(0)1154 3098 y Fl(x)1194 3110 y Fg(k)1236 3074 y Fq(;)17 b(v)1331 3033 y Fm(\()p Fl(k)r Fm(\))1428 3074 y Fu(\()p Fq(T)1523 3089 y Fl(k)r Fm(+1)1656 3074 y Fu(\))p Fo(i)1750 2938 y Fp(Z)1849 2964 y Fl(t)1805 3164 y(T)1846 3172 y Fg(n)1893 3074 y Fq(ds)g(g)2058 3023 y Fm(\()p Fl(n)p Fm(\))2054 3096 y Fl(t)p Fn(\000)p Fl(s)2186 2903 y Fp(")2245 3074 y Fq(p)2294 3089 y Fl(s)p Fn(\000)p Fl(T)2423 3101 y Fg(k)q Fi(+1)2560 3074 y Fu(\026)-68 b Fq(m)2626 3033 y Fn(0)2626 3098 y Fl(x)2666 3110 y Fg(k)2731 3074 y Fo(\000)3137 3006 y Fu(2)p 2841 3051 641 4 v 2841 3071 a Fp(p)p 2940 3071 542 4 v 2940 3156 a Fu(2)p Fq(\031)t Fu(\()p Fq(s)22 b Fo(\000)g Fq(T)3310 3171 y Fl(k)r Fm(+1)3443 3156 y Fu(\))3491 2903 y Fp(#)3639 3074 y Fu(\(5.5\))0 3354 y(Note)53 b(that)f(\000)548 3303 y Fm(\()p Fl(n)p Fm(\))548 3379 y(3)650 3354 y Fu(\()p Fq(t)p Fu(\))g(is)g(obtained)h(from)e(\000)1664 3303 y Fm(\()p Fl(n)p Fm(\))1664 3379 y(3+8)1793 3354 y Fu(\()p Fq(t)p Fu(\))i(replacing)e Fq(p)2443 3369 y Fl(t)2473 3354 y Fu(\()p Fq(x;)17 b(y)t Fu(\))51 b(b)m(y)j(\(2)p Fq(\031)t(t)p Fu(\))3126 3318 y Fn(\000)p Fm(1)p Fl(=)p Fm(2)3343 3354 y Fu(\(recall)d(that)0 3394 y Fp(R)66 3474 y Fq(dx)35 b Fu(\026)-67 b Fq(m)274 3438 y Fn(0)298 3474 y Fu(\()p Fq(x)p Fu(\))28 b(=)f(2\).)43 b(In)33 b(conclusion)f(w)m(e)i(ha)m(v)m(e)1270 3658 y Fq(v)1321 3617 y Fm(\()p Fl(n)p Fm(\))1423 3658 y Fu(\()p Fq(t)p Fu(\))28 b(=)f(\000)1726 3607 y Fm(\()p Fl(n)p Fm(\))1726 3682 y(1)1828 3658 y Fu(\()p Fq(t)p Fu(\))22 b(+)g Fo(\001)17 b(\001)g(\001)k Fu(+)h(\000)2357 3607 y Fm(\()p Fl(n)p Fm(\))2357 3682 y(10)2459 3658 y Fu(\()p Fq(t)p Fu(\))1069 b(\(5.6\))100 3818 y(Let)32 b(us)i(de\014ne)1355 3961 y Fq( )1418 3976 y Fl(n)1493 3961 y Fu(:=)27 b Fo(\000)1710 3894 y Fu(3)p 1710 3939 49 4 v 1710 4030 a(4)1770 3961 y Fo(h)18 b Fu(\026)-67 b Fq(m)1894 3920 y Fn(0)1894 3986 y Fl(x)1934 3994 y Fg(n)1980 3961 y Fq(;)17 b(v)2075 3920 y Fm(\()p Fl(n)p Fm(\))2176 3961 y Fu(\()p Fq(T)2271 3976 y Fl(n)p Fm(+1)2409 3961 y Fu(\))p Fo(i)1153 b Fu(\(5.7\))0 4137 y(so)33 b(that,)f(recalling)g(\(4.1\))o(,)1676 4350 y Fq(\030)1719 4365 y Fl(n)1794 4350 y Fu(=)1903 4226 y Fl(n)p Fn(\000)p Fm(1)1897 4256 y Fp(X)1905 4468 y Fl(k)r Fm(=0)2058 4350 y Fq( )2121 4365 y Fl(k)3639 4350 y Fu(\(5.8\))0 4585 y(W)-8 b(e)33 b(then)g(set)g Fq(t)28 b Fu(=)g Fq(T)766 4600 y Fl(n)p Fm(+1)936 4585 y Fu(in)k(\(5.6\))g(and) g(pro)5 b(ject)34 b(it)d(on)51 b(\026)-67 b Fq(m)2124 4549 y Fn(0)2124 4609 y Fl(x)2164 4617 y Fg(n)2210 4585 y Fu(,)33 b(getting)1455 4847 y Fq( )1518 4862 y Fl(n)1593 4847 y Fu(=)27 b Fq(\021)1744 4862 y Fl(n)1813 4847 y Fo(\000)1918 4723 y Fl(n)p Fn(\000)p Fm(1)1913 4753 y Fp(X)1921 4965 y Fl(k)r Fm(=0)2074 4847 y Fq(A)2147 4862 y Fl(n;k)2279 4847 y Fq( )2342 4862 y Fl(k)3639 4847 y Fu(\(5.9\))0 5096 y(where)571 5285 y Fq(\021)619 5300 y Fl(n)694 5285 y Fu(:=)889 5161 y Fm(10)852 5191 y Fp(X)875 5390 y Fg(i)p Fi(=1)875 5446 y Fg(i)p Fh(6)p Fi(=3)1013 5285 y Fq(\021)1061 5300 y Fl(n)1108 5285 y Fu(\()p Fq(i)p Fu(\))p Fq(;)212 b(\021)1504 5300 y Fl(n)1551 5285 y Fu(\()p Fq(i)p Fu(\))28 b(=)f Fo(\000)1878 5218 y Fu(3)p 1878 5262 V 1878 5354 a(4)1937 5285 y Fo(h)18 b Fu(\026)-67 b Fq(m)2061 5244 y Fn(0)2061 5310 y Fl(x)2101 5318 y Fg(n)2148 5285 y Fq(;)17 b Fu(\000)2253 5234 y Fm(\()p Fl(n)p Fm(\))2253 5311 y Fl(i)2354 5285 y Fu(\()p Fq(T)2449 5300 y Fl(n)p Fm(+1)2587 5285 y Fu(\))p Fo(i)p Fq(;)49 b(i)27 b Fu(=)h(1)p Fq(;)17 b(:)g(:)g(:)f(;)h Fu(10)321 b(\(5.10\))1882 5532 y Fk(21)p eop %%Page: 22 23 22 22 bop 0 266 a Fu(and)1039 455 y Fq(A)1112 470 y Fl(n;k)1245 455 y Fu(:=)28 b Fq(\025)1513 388 y Fu(3)p 1443 432 191 4 v 1443 452 a Fo(p)p 1525 452 108 4 v 1525 535 a Fu(2)p Fq(\031)2014 388 y Fu(2)2063 303 y Fo(p)p 2146 303 71 4 v 85 x Fq(T)p 1653 432 925 4 v 1653 452 a Fo(p)p 1736 452 235 4 v 81 x Fq(n)22 b Fo(\000)h Fq(k)i Fu(+)2090 452 y Fo(p)p 2173 452 405 4 v 81 x Fq(n)e Fo(\000)f Fq(k)k Fo(\000)c Fu(1)2605 455 y Fv(1)2661 470 y Fl(k)r()28 b Fu(0)34 b Ft(ther)-5 b(e)35 b(is)g(a)g(c)-5 b(onstant)34 b Fq(C)h Fu(=)27 b Fq(C)7 b Fu(\()p Fq(\034)k Fu(\))35 b Ft(such)g(that)g(for)g(any)g Fq(\025)27 b(>)h Fu(0)1571 2902 y(sup)1507 2987 y Fl(n)p Fn(\024)p Fl(n)1648 2999 y Fg(\025)1689 2987 y Fm(\()p Fl(\034)8 b Fm(\))1799 2817 y Fp(\015)1799 2877 y(\015)1854 2902 y Fq(A)1927 2860 y Fl(j)1964 2817 y Fp(\015)1964 2877 y(\015)2019 2941 y Fl(n)2094 2902 y Fo(\024)2209 2834 y Fq(C)2286 2798 y Fl(j)p 2209 2879 114 4 v 2229 2970 a Fq(j)e Fu(!)3591 2902 y(\(5.14\))0 3159 y Ft(Henc)-5 b(e,)34 b(fr)-5 b(om)42 b Fu(\(5.13\))p Ft(,)637 3418 y Fq( )31 b Fu(=)872 3293 y Fn(1)835 3323 y Fp(X)846 3533 y Fl(j)t Fm(=0)979 3418 y Fu(\()p Fo(\000)p Fq(A)p Fu(\))1205 3377 y Fl(j)1242 3418 y Fq(\021)g Fu(=)d(\(1)22 b(+)g Fq(A)p Fu(\))1743 3377 y Fn(\000)p Fm(1)1837 3418 y Fq(\021)t(;)280 b Fu(sup)2132 3503 y Fl(n)p Fn(\024)p Fl(n)2273 3515 y Fg(\025)2313 3503 y Fm(\()p Fl(\034)8 b Fm(\))2424 3333 y Fp(\015)2424 3393 y(\015)2479 3418 y Fu(\(1)22 b(+)g Fq(A)p Fu(\))2797 3377 y Fn(\000)p Fm(1)2891 3333 y Fp(\015)2891 3393 y(\015)2946 3457 y Fl(n)3021 3418 y Fo(\024)28 b Fq(C)395 b Fu(\(5.15\))100 3937 y(It)32 b(will)f(b)s(e)h(con)m(v)m(enien)m(t)j(to)d(consider)h (also)e(one)i(iteration)e(of)39 b(\(5.13\))o(,)33 b(i.e.)1522 4117 y Fq( )f Fu(=)27 b Fq(\021)f Fo(\000)d Fq(A\021)j Fu(+)c Fq(A)2212 4076 y Fm(2)2251 4117 y Fq( )1277 b Fu(\(5.16\))0 4295 y(Explicitly)31 b(w)m(e)i(ha)m(v)m(e)1213 4522 y(\()p Fq(A)1324 4480 y Fm(2)1363 4522 y Fu(\))1401 4537 y Fl(n;k)1534 4522 y Fu(=)1679 4397 y Fl(n)p Fn(\000)p Fm(1)1674 4427 y Fp(X)1638 4639 y Fl(j)t Fm(=)p Fl(k)r Fm(+1)1870 4522 y Fq(A)1943 4537 y Fl(n;j)2059 4522 y Fq(A)2132 4537 y Fl(j;k)2250 4522 y Fv(1)2306 4537 y Fm(0)p Fn(\024)p Fl(k)r()27 b Fu(0)35 b Ft(ther)-5 b(e)35 b(is)f(a)h(c)-5 b(onstant)35 b Fq(C)f Fu(=)28 b Fq(C)7 b Fu(\()p Fq(\034)k Fu(\))35 b Ft(such)g(that)1462 5354 y Fu(sup)1306 5439 y Fm(0)p Fn(\024)p Fl(k)r()27 b Fu(0)p Ft(,)916 486 y Fu(lim)928 549 y Fl(\025)p Fn(#)p Fm(0)1201 486 y Fu(sup)1106 557 y Fi(0)p Fh(\024)p Fg(\034)1213 572 y Fi(2)1248 557 y Fg(<\034)1325 572 y Fi(1)1360 557 y Fh(\024)p Fg(\034)1148 616 y(\034)1177 631 y Fi(1)1211 616 y Fh(\000)p Fg(\034)1288 631 y Fi(2)1323 616 y Fg(>\016)1496 342 y Fp(\014)1496 402 y(\014)1496 461 y(\014)1496 521 y(\014)1530 486 y Fu(\()p Fq(\025)1625 445 y Fm(2)1664 486 y Fq(T)14 b Fu(\))1773 445 y Fn(\000)p Fm(1)1884 406 y Fp(\000)1929 486 y Fq(A)2002 445 y Fm(2)2042 406 y Fp(\001)2087 525 y Fl(n)2130 537 y Fg(\025)2171 525 y Fm(\()p Fl(\034)2229 534 y Fi(1)2264 525 y Fm(\))p Fl(;n)2354 537 y Fg(\025)2395 525 y Fm(\()p Fl(\034)2453 534 y Fi(2)2488 525 y Fm(\))2542 486 y Fo(\000)2652 419 y Fu(9)p 2652 463 49 4 v 2652 555 a(2)2711 342 y Fp(\014)2711 402 y(\014)2711 461 y(\014)2711 521 y(\014)2771 486 y Fu(=)28 b(0)667 b(\(5.19\))0 1008 y Ft(Pr)-5 b(o)g(of.)43 b Fu(By)34 b(\(5.11\))1102 1112 y Fp(\014)1102 1172 y(\014)1102 1232 y(\014)1136 1146 y(\000)1181 1227 y Fq(A)1254 1185 y Fm(2)1294 1146 y Fp(\001)1340 1266 y Fl(n;k)1445 1112 y Fp(\014)1445 1172 y(\014)1445 1232 y(\014)1506 1227 y Fo(\024)28 b Fq(C)7 b(\025)1745 1185 y Fm(2)1784 1227 y Fq(T)1913 1102 y Fl(n)p Fn(\000)p Fm(1)1907 1132 y Fp(X)1871 1344 y Fl(j)t Fm(=)p Fl(k)r Fm(+1)2396 1159 y Fu(1)p 2114 1204 614 4 v 2114 1225 a Fo(p)p 2197 1225 226 4 v 74 x Fq(n)22 b Fo(\000)h Fq(j)2423 1224 y Fo(p)p 2506 1224 222 4 v 75 x Fq(j)28 b Fo(\000)23 b Fq(k)0 1478 y Fu(whic)m(h)33 b(pro)m(v)m(es)h(\(5.18\).)100 1594 y(By)g(\(5.11\))e(w)m(e)h(ha)m(v)m(e)174 1808 y(\()p Fq(\025)269 1766 y Fm(2)309 1808 y Fq(T)14 b Fu(\))418 1766 y Fn(\000)p Fm(1)528 1727 y Fp(\000)574 1808 y Fq(A)647 1766 y Fm(2)687 1727 y Fp(\001)732 1847 y Fl(n)775 1859 y Fg(\025)816 1847 y Fm(\()p Fl(\034)874 1856 y Fi(1)909 1847 y Fm(\))p Fl(;n)999 1859 y Fg(\025)1040 1847 y Fm(\()p Fl(\034)1098 1856 y Fi(2)1133 1847 y Fm(\))1187 1808 y Fo(\000)1297 1740 y Fu(9)p 1297 1785 49 4 v 1297 1876 a(2)1383 1808 y(=)1526 1740 y(9)p 1496 1785 108 4 v 1496 1876 a(2)p Fq(\031)207 2117 y Fo(\002)301 1913 y Fp(8)301 2003 y(<)301 2182 y(:)433 1985 y Fl(n)476 1997 y Fg(\025)517 1985 y Fm(\()p Fl(\034)575 1994 y Fi(1)610 1985 y Fm(\))p Fn(\000)p Fm(1)509 2023 y Fp(X)390 2239 y Fl(j)t Fm(=)p Fl(n)521 2251 y Fg(\025)561 2239 y Fm(\()p Fl(\034)619 2248 y Fi(2)654 2239 y Fm(\)+1)1448 2050 y Fu(2)p 798 2095 1348 4 v 798 2114 a Fp(p)p 898 2114 429 4 v 86 x Fq(n)956 2215 y Fl(\025)1001 2200 y Fu(\()p Fq(\034)1081 2215 y Fm(1)1121 2200 y Fu(\))22 b Fo(\000)h Fq(j)28 b Fu(+)1447 2114 y Fp(p)p 1546 2114 600 4 v 1546 2200 a Fq(n)1604 2215 y Fl(\025)1650 2200 y Fu(\()p Fq(\034)1730 2215 y Fm(1)1770 2200 y Fu(\))22 b Fo(\000)g Fq(j)28 b Fo(\000)23 b Fu(1)2837 2050 y(2)p 2188 2095 1348 4 v 2188 2114 a Fp(p)p 2288 2114 429 4 v 86 x Fq(j)28 b Fo(\000)22 b Fq(n)2513 2215 y Fl(\025)2559 2200 y Fu(\()p Fq(\034)2639 2215 y Fm(2)2679 2200 y Fu(\))g(+)2837 2114 y Fp(p)p 2936 2114 600 4 v 2936 2200 a Fq(j)28 b Fo(\000)23 b Fq(n)3162 2215 y Fl(\025)3208 2200 y Fu(\()p Fq(\034)3288 2215 y Fm(2)3327 2200 y Fu(\))f Fo(\000)h Fu(1)3568 2117 y Fo(\000)g Fq(\031)3727 1913 y Fp(9)3727 2003 y(=)3727 2182 y(;)3591 2384 y Fu(\(5.20\))0 2562 y(Since)31 b Fq(\031)g Fu(=)443 2482 y Fp(R)509 2508 y Fm(1)490 2597 y(0)549 2562 y Fq(dx)17 b Fu([)p Fq(x)p Fu(\(1)h Fo(\000)h Fq(x)p Fu(\)])1075 2526 y Fn(\000)p Fm(1)p Fl(=)p Fm(2)1240 2562 y Fu(,)31 b(b)m(y)h(letting)d Fq(\033)i Fu(=)d Fq(\034)1971 2577 y Fm(1)2029 2562 y Fo(\000)18 b Fq(\034)2166 2577 y Fm(2)2237 2562 y Fu(and)31 b(c)m(hanging)f(the)h(index)g(of)f (summation)0 2680 y(the)j(r.h.s.)g(of)39 b(\(5.20\))32 b(equals)144 2898 y(9)p 115 2943 108 4 v 115 3034 a(2)p Fq(\031)249 2762 y Fp(8)249 2851 y(<)249 3031 y(:)338 2834 y Fl(n)381 2846 y Fg(\025)421 2834 y Fm(\()p Fl(\033)r Fm(\))p Fn(\000)p Fm(2)401 2871 y Fp(X)412 3081 y Fl(j)t Fm(=0)1262 2898 y Fu(2)p 636 2943 1303 4 v 636 2963 a Fp(p)p 735 2963 407 4 v 735 3048 a Fq(n)793 3063 y Fl(\025)839 3048 y Fu(\()p Fq(\033)t Fu(\))22 b Fo(\000)g Fq(j)29 b Fu(+)1262 2963 y Fp(p)p 1361 2963 577 4 v 1361 3048 a Fq(n)1419 3063 y Fl(\025)1465 3048 y Fu(\()p Fq(\033)t Fu(\))22 b Fo(\000)g Fq(j)28 b Fo(\000)23 b Fu(1)2229 2898 y(2)p 1980 2943 548 4 v 1980 2963 a Fo(p)p 2063 2963 46 4 v 73 x Fq(j)28 b Fu(+)2229 2963 y Fo(p)p 2312 2963 215 4 v 73 x Fq(j)g Fu(+)22 b(1)2559 2966 y Fo(\000)2659 2830 y Fp(Z)2758 2857 y Fm(1)2714 3056 y(0)2798 2966 y Fq(dx)3135 2898 y Fu(1)p 2930 2943 457 4 v 2930 2963 a Fp(p)p 3030 2963 358 4 v 85 x Fq(x)p Fu(\(1)g Fo(\000)h Fq(x)p Fu(\))3397 2762 y Fp(9)3397 2851 y(=)3397 3031 y(;)3591 2966 y Fu(\(5.21\))0 3252 y(whic)m(h)i(con)m(v)m(erges,)j(as)c Fq(\025)j Fo(#)h Fu(0,)e(to)d(0)h(uniformly)e(for)h Fq(\033)32 b Fo(2)c Fu([)p Fq(\016)n(;)17 b(\034)11 b Fu(],)27 b(as)d(can)g(b)s(e) h(easily)e(c)m(hec)m(k)m(ed.)44 b(Prop)s(osition)0 3368 y(5.2)32 b(is)g(pro)m(v)m(ed.)3198 b Fd(\003)1547 3957 y Fu(6.)49 b Fr(Bounds)37 b(on)h Fq(\021)100 4131 y Fu(Recalling)i (\(5.8\))o({\(5.10\))o(,)k(a)d(term)g Fq(\021)t Fu(\()p Fq(i)p Fu(\))g(\(as)h(in)e(the)i(previous)g(section,)i(w)m(e)e(are)g (here)g(using)f(v)m(ec-)0 4247 y(torial)47 b(notation\))g(con)m (tributing)h(to)h Fq( )j Fu(whose)f(seminorm)c(is)h Fo(j)p Fq(\021)t Fu(\()p Fq(i)p Fu(\))p Fo(j)2641 4262 y Fl(n)2743 4247 y Fu(=)56 b Fq(o)p Fu(\()p Fq(\025)3017 4211 y Fm(2)3056 4247 y Fq(T)14 b Fu(\))48 b(\(i.e.)h(suc)m(h)h(that)0 4363 y(\()p Fq(\025)95 4327 y Fm(2)134 4363 y Fq(T)14 b Fu(\))243 4327 y Fn(\000)p Fm(1)337 4363 y Fo(j)p Fq(\021)t Fu(\()p Fq(i)p Fu(\))p Fo(j)554 4378 y Fl(n)635 4363 y Fu(v)-5 b(anishes)35 b(as)g Fq(\025)c Fo(#)g Fu(0\))k(do)s(es)g(not)f (con)m(tribute)h(to)f(the)i(limiting)30 b(equation)k(for)g Fq(\030)3552 4378 y Fl(n)3634 4363 y Fu(since)0 4480 y Fq(n)28 b Fo(\024)g Fu(\()p Fq(\025)286 4444 y Fm(2)325 4480 y Fq(T)14 b Fu(\))434 4444 y Fn(\000)p Fm(1)528 4480 y Fq(\034)d Fu(.)44 b(This)31 b(is)f(the)h(case)h(for)e(some)g(of) h(the)g Fq(\021)t Fu(\()p Fq(i)p Fu(\)'s,)g(i.e.)g Fq(\021)t Fu(\(1\),)f Fq(\021)t Fu(\(7\),)h(and)f Fq(\021)t Fu(\(10\),)h(as)g(w)m (e)g(shall)0 4596 y(see.)100 4712 y(Clearly)c Fq(\021)482 4727 y Fl(n)529 4712 y Fu(\(2\))h(is)f(not)h(negligible)d(b)s(ecause)k (its)f(t)m(ypical)f(magnitude)g(is)h Fq(\025)2805 4628 y Fo(p)p 2887 4628 71 4 v 2887 4712 a Fq(T)14 b Fu(.)42 b(It)28 b(will)e(b)s(e)i(examined)0 4828 y(in)33 b(the)i(next)g (section,)g(where)h(w)m(e)f(shall)e(see)i(that)f(,)h(summed)f(o)m(v)m (er)h Fq(n)f Fu(it)f(giv)m(es)i(a)f(\014nite)g(con)m(tribution)0 4945 y(b)s(ecause)k(of)f(cancellations)f(related)g(to)h(its)g (martingale)d(nature.The)39 b(other)e(terms,)h(i.e.)f Fq(\021)3429 4960 y Fl(n)3476 4945 y Fu(\()p Fq(i)p Fu(\),)h(with)0 5061 y Fq(i)44 b Fu(from)e(4)i(to)f(9,)j(liv)m(e)d(on)g(an)h(in)m (termediate)e(stage:)66 b(they)45 b(are)e(smaller)f(than)i(the)g Fe(a)f(priori)f Fu(b)s(ound)0 5181 y Fq(\025)57 5145 y Fm(1)p Fn(\000)p Fl(\020)187 5096 y Fo(p)p 270 5096 V 85 x Fq(T)14 b Fu(,)30 b(y)m(et)g(not)f(small)e(enough)j(to)f(b)s(e)h (directly)f(negligible.)39 b(W)-8 b(e)30 b(shall)e(rewrite)h(the)h (factors)f Fq(v)3627 5145 y Fm(\()p Fl(n)p Fm(\))3729 5181 y Fu(\()p Fq(t)p Fu(\))0 5297 y(via)j(\(5.6\))o(,)g(with)g(the)h (idea)e(that)h(if)f(w)m(e)j(get)e(t)m(w)m(o)h Fq(\021)t Fu(\()p Fq(i)p Fu(\),)f Fq(i)c Fu(=)f(4)p Fq(;)17 b(:)g(:)g(:)f(;)h Fu(10)31 b(then)i(the)g(corresp)s(onding)f(terms)0 5413 y(b)s(ecome)h(negligible.)40 b(W)-8 b(e)33 b(will)d(use)k(the)f(follo)m (wing)d(notation.)1882 5532 y Fk(23)p eop %%Page: 24 25 24 24 bop 0 266 a Fv(De\014nition)39 b(6.1.)k Ft(L)-5 b(et)37 b Fq(\021)961 281 y Fl(n)1008 266 y Fu(\()p Fq(i)1079 281 y Fm(1)1119 266 y Fq(;)17 b(:)g(:)g(:)f(;)h(i)1371 281 y Fl(k)1413 266 y Fu(\))p Ft(,)38 b Fq(k)c(>)e Fu(1)p Ft(,)37 b Fq(i)1861 281 y Fl(j)1929 266 y Fo(2)32 b(f)p Fu(3)p Fq(;)17 b Fu(4)p Fq(;)g Fu(5)p Fq(;)g Fu(6)p Fq(;)g Fu(8)p Fo(g)35 b Ft(when)h Fq(j)h(<)32 b(k)s Ft(,)37 b Fq(i)3173 281 y Fl(k)3248 266 y Fo(2)32 b(f)p Fu(1)p Fq(;)17 b(:)g(:)g(:)e(;)i Fu(10)p Fo(g)p Ft(,)0 388 y(b)-5 b(e)33 b(the)h(term)f(which)g(is)g(obtaine)-5 b(d)33 b(fr)-5 b(om)33 b Fq(\021)1548 403 y Fl(n)1595 388 y Fu(\()p Fq(i)1666 403 y Fm(1)1706 388 y Fu(\))g Ft(by)h(r)-5 b(eplacing)33 b(the)g Fq(v)2528 352 y Fm(\()p Fl(`)p Fm(\))2616 388 y Fu(\()p Fo(\001)p Fu(\))p Ft({function)f(in)i(its)f (expr)-5 b(ession)0 523 y(by)38 b Fu(\000)191 472 y Fm(\()p Fl(`)p Fm(\))191 549 y Fl(i)215 558 y Fi(2)279 523 y Fu(\()p Fo(\001)p Fu(\))p Ft(;)i(then,)e(the)h(new)e Fq(v)1122 487 y Fm(\()p Fl(`)1178 463 y Fh(0)1201 487 y Fm(\))1232 523 y Fu(\()p Fo(\001)p Fu(\))h Ft(function)g(which)f(app) -5 b(e)g(ars)38 b(is)g(r)-5 b(eplac)g(e)g(d)37 b(by)h Fu(\000)3064 472 y Fm(\()p Fl(`)3120 449 y Fh(0)3143 472 y Fm(\))3064 549 y Fl(i)3088 558 y Fi(3)3175 523 y Fu(\()p Fo(\001)p Fu(\))f Ft(and)h(so)g(forth)0 639 y(til)5 b(l)43 b(the)g(last)f(one,)i(which)e(is)h(not)g(change)-5 b(d.)67 b(Inste)-5 b(ad)42 b(of)g(an)h(index)e Fq(i)2666 654 y Fl(j)2746 639 y Ft(it)i(may)g(app)-5 b(e)g(ar)42 b(the)h(symb)-5 b(ol)0 761 y Fu(3)23 b(+)f(8)36 b Ft(or)g Fu(4)22 b(+)h(5)p Ft(,)36 b(it)g(me)-5 b(ans)35 b(that,)h(at)g(that)g (stage)g(of)f(the)h(iter)-5 b(ation,)36 b(we)f(r)-5 b(eplac)g(e)35 b(the)h Fq(v)3250 725 y Fm(\()p Fl(`)p Fm(\))3338 761 y Fu(\()p Fo(\001)p Fu(\))p Ft({function)0 895 y(by)f Fu(\000)188 844 y Fm(\()p Fl(`)p Fm(\))188 920 y(3)276 895 y Fu(\()p Fo(\001)p Fu(\))22 b(+)g(\000)561 844 y Fm(\()p Fl(`)p Fm(\))561 920 y(8)648 895 y Fu(\()p Fo(\001)p Fu(\))p Ft(,)35 b(or)f(r)-5 b(esp)g(e)g(ctively)35 b(by)g Fu(\000)1640 844 y Fm(\()p Fl(`)p Fm(\))1640 920 y(4)1728 895 y Fu(\()p Fo(\001)p Fu(\))21 b(+)h(\000)2012 844 y Fm(\()p Fl(`)p Fm(\))2012 920 y(5)2100 895 y Fu(\()p Fo(\001)p Fu(\))p Ft(.)100 1012 y(We)34 b(wil)5 b(l)33 b(also)h(c)-5 b(onsider)33 b Fq(i)1075 1027 y Fm(1)1142 1012 y Fu(=)28 b(9)p Ft(,)34 b(in)f(which)g(c)-5 b(ase)34 b(ther)-5 b(e)34 b(is)g(a)f(pr)-5 b(o)g(duct)35 b(of)e(two)h Fq(v)t Ft({functions.)44 b(Then,)0 1128 y Fq(\021)48 1143 y Fl(n)95 1128 y Fu(\(9)p Fo(j)p Fq(i)243 1143 y Fm(2)282 1128 y Fq(;)17 b(:)g(:)g(:)f(;)h(i)534 1143 y Fl(k)577 1128 y Fu(;)g Fq(i)654 1092 y Fn(0)654 1152 y Fm(2)693 1128 y Fq(;)g(:)g(:)g(:)f(;)h(i)945 1092 y Fn(0)945 1154 y Fl(h)990 1128 y Fu(\))26 b Ft(is)g(de\014ne)-5 b(d)25 b(by)i(doing)e(the)h(br)-5 b(anching)26 b Fq(i)2467 1143 y Fm(2)2506 1128 y Fq(;)17 b(:)g(:)g(:)f(;)h(i)2758 1143 y Fl(k)2827 1128 y Ft(\(with)26 b(the)g(pr)-5 b(evious)26 b(rules\))0 1244 y(for)j(the)h(\014rst)f Fq(v)34 b Ft(function)29 b(and)g(the)g(br)-5 b(anching)28 b Fq(i)1775 1208 y Fn(0)1775 1269 y Fm(2)1815 1244 y Fq(;)17 b(:)g(:)g(:)f(;)h(i)2067 1208 y Fn(0)2067 1270 y Fl(h)2141 1244 y Ft(for)30 b(the)f(se)-5 b(c)g(ond)29 b(one)f(\(the)i(se)-5 b(c)g(ond)28 b(br)-5 b(anching)0 1360 y(may)35 b(b)-5 b(e)34 b(absent\).)100 1739 y Fu(W)-8 b(e)33 b(de\014ne)909 1866 y Fq(\033)964 1881 y Fl(n)1039 1866 y Fu(:=)27 b Fq(\021)1217 1881 y Fl(n)1264 1866 y Fu(\(2\))22 b(+)g Fq(\021)1557 1881 y Fl(n)1604 1866 y Fu(\(4)g(+)g(5)p Fq(;)17 b Fu(2\))k(+)h Fq(\021)2158 1881 y Fl(n)2206 1866 y Fu(\(6)p Fq(;)17 b Fu(2\))k(+)h Fq(\021)2591 1881 y Fl(n)2638 1866 y Fu(\(9)p Fo(j)p Fu(2;)17 b(2\))706 b(\(6.1\))0 2014 y(whic)m(h)33 b(will)c(b)s(e)k(studied)f(in)f(the)i(next)g(section)f(where)h(w)m(e)g (will)d(b)s(ound)i(its)g(sum)g(o)m(v)m(er)h Fq(n)f Fu(using)g(proba-)0 2131 y(bilistic)e(argumen)m(ts,)k Fq(\033)873 2146 y Fl(n)953 2131 y Fu(is)e(in)h(fact)f(a)h(truly)g(sto)s(c)m(hastic)g (term.)44 b(The)34 b(di\013erence)f(b)s(et)m(w)m(een)j Fq(\021)3468 2146 y Fl(n)3548 2131 y Fu(and)d Fq(\033)3793 2146 y Fl(n)0 2247 y Fu(is)f(negligible,)e(this)i(b)s(eing)g(the)h (main)e(result)h(in)g(this)g(section.)44 b(Let)890 2501 y Fo(B)958 2450 y Fm(\(3\))955 2529 y Fl(\025;\034)1142 2501 y Fu(:=)1328 2331 y Fp(\()1473 2501 y Fu(sup)1408 2587 y Fl(n)p Fn(\024)p Fl(n)1549 2599 y Fg(\025)1590 2587 y Fm(\()p Fl(\034)8 b Fm(\))1700 2327 y Fp(\014)1700 2386 y(\014)1700 2446 y(\014)1700 2506 y(\014)1700 2566 y(\014)1733 2406 y(X)1737 2619 y Fl(k)r()e Fu(0)k(there)h(is)g Fq(C)h Fu(=)28 b Fq(C)7 b Fu(\()p Fq(\034)f(;)17 b(\020)8 b(;)17 b(q)t Fu(\))30 b(so)j(that)1359 3427 y Fj(P)1435 3317 y Fp(\020)1496 3427 y Fo(B)1564 3376 y Fm(\(3\))1561 3455 y Fl(\025;\034)1688 3427 y Fo(\\)22 b(B)1844 3376 y Fm(\(4\))1841 3455 y Fl(\025;\034)1946 3317 y Fp(\021)2033 3427 y Fo(\025)28 b Fu(1)22 b Fo(\000)h Fq(C)7 b(\025)2443 3386 y Fl(q)3639 3427 y Fu(\(6.4\))0 3622 y(Finally)30 b(de\014ne)1158 3749 y Fq(x)1213 3764 y Fl(n;)p Fn(\003)1343 3749 y Fu(:=)e(sup)1479 3831 y Fl(k)r Fn(\024)p Fl(n)1638 3749 y Fo(j)p Fq(x)1721 3764 y Fl(n)1768 3749 y Fo(j)p Fq(;)211 b(x)2089 3764 y Fn(\003)2129 3749 y Fu(\()p Fq(\034)11 b Fu(\))28 b(=)g Fq(x)2445 3765 y Fl(n)2488 3777 y Fg(\025)2529 3765 y Fm(\()p Fl(\034)8 b Fm(\))p Fl(;)p Fn(\003)3639 3749 y Fu(\(6.5\))0 4204 y Fv(Prop)s(osition)36 b(6.2.)41 b Ft(R)-5 b(e)g(c)g(al)5 b(ling)43 b Fu(\(4.38\))o Ft(,)35 b(let)1367 4390 y Fo(B)1432 4405 y Fl(\025;\034)1564 4390 y Fu(:=)28 b Fo(B)1763 4339 y Fm(\(1)p Fl(;)p Fm(2\))1760 4417 y Fl(\025;\034)1935 4390 y Fo(\\)22 b(B)2091 4339 y Fm(\(3\))2088 4417 y Fl(\025;\034)2215 4390 y Fo(\\)g(B)2371 4339 y Fm(\(4\))2368 4417 y Fl(\025;\034)3639 4390 y Fu(\(6.6\))0 4565 y Ft(Then)42 b(for)g(e)-5 b(ach)42 b Fq(\034)53 b(>)41 b Fu(0)p Ft(,)j(ther)-5 b(e)43 b(is)f(a)g(c)-5 b(onstant)42 b Fq(C)49 b Fu(=)41 b Fq(C)7 b Fu(\()p Fq(\034)k Fu(\))43 b Ft(so)f(that)h(for)f(any)g Fq(\025)g(>)f Fu(0)h Ft(and)g Fq(\020)50 b Ft(smal)5 b(l)0 4682 y(enough)831 4812 y Fo(j)p Fq(\021)907 4827 y Fl(n)976 4812 y Fo(\000)22 b Fq(\033)1130 4827 y Fl(n)1178 4812 y Fo(j)27 b(\024)h Fq(C)7 b(\025)1472 4771 y Fm(2)1512 4812 y Fq(T)1583 4771 y Fm(3)p Fl(=)p Fm(4)1692 4812 y Fu(\(1)22 b(+)g Fq(x)1954 4771 y Fm(2)1954 4836 y Fl(n)p Fm(+1)p Fl(;)p Fn(\003)2147 4812 y Fu(\))199 b(on)32 b(the)h(set)h Fo(B)2905 4827 y Fl(\025;\034)3639 4812 y Fu(\(6.7\))0 5200 y Ft(Pr)-5 b(o)g(of.)44 b Fu(W)-8 b(e)34 b(will)c(call)i(negligible)e(a)i(term)h (whic)m(h)h(is)e(b)s(ounded)i(b)m(y)g(the)f(r.h.s.)h(of)40 b(\(6.7\))o(.)45 b(W)-8 b(e)33 b(will)e(next)0 5316 y(examine)k(one)g (b)m(y)h(one)f(all)e(the)j(terms)f(whic)m(h)g(con)m(tribute)g(to)g Fq(\021)2390 5331 y Fl(n)2461 5316 y Fo(\000)24 b Fq(\033)2617 5331 y Fl(n)2700 5316 y Fu(and)35 b(sho)m(w)h(that)f(they)h(are)f(all)0 5433 y(negligible,)30 b(th)m(us)j(pro)m(ving)g(Prop)s(osition)d(6.2.) 1882 5532 y Fk(24)p eop %%Page: 25 26 25 25 bop 0 283 a Ft(Step)34 b(1.)44 b(The)33 b(terms)g Fq(\021)857 298 y Fl(n)904 283 y Fu(\(1\))h Ft(and)f Fq(\021)1299 298 y Fl(n)1346 283 y Fu(\(10\))p Ft(.)42 b Fu(W)-8 b(e)32 b(ha)m(v)m(e)g(that)g Fq(\021)2241 298 y Fl(n)2288 283 y Fu(\(1\))27 b(=)h(0;)j(indeed)h(b)m(y)h(\(3.8\),)e (since)h Fq(g)3642 232 y Fm(\()p Fl(n)p Fm(\))3638 306 y Fl(t)3774 283 y Fu(is)0 417 y(self{adjoin)m(t)f(and)i Fq(g)759 366 y Fm(\()p Fl(n)p Fm(\))755 440 y Fl(t)878 417 y Fu(\026)-67 b Fq(m)945 381 y Fn(0)945 442 y Fl(x)985 450 y Fg(n)1059 417 y Fu(=)46 b(\026)-67 b Fq(m)1248 381 y Fn(0)1248 442 y Fl(x)1288 450 y Fg(n)1335 417 y Fu(,)741 672 y Fq(\021)789 687 y Fl(n)836 672 y Fu(\(1\))28 b(=)f Fo(\000)1179 604 y Fu(3)p 1179 649 49 4 v 1179 740 a(4)1238 672 y Fo(h)18 b Fu(\026)-67 b Fq(m)1362 631 y Fn(0)1362 696 y Fl(x)1402 704 y Fg(n)1449 672 y Fq(;)17 b(g)1544 621 y Fm(\()p Fl(n)p Fm(\))1540 699 y Fl(T)1645 672 y Fq(v)1696 631 y Fm(\()p Fl(n)p Fm(\))1797 672 y Fu(\()p Fq(T)1892 687 y Fl(n)1939 672 y Fu(\))p Fo(i)28 b Fu(=)f Fo(\000)2234 604 y Fu(3)p 2234 649 V 2234 740 a(4)2293 672 y Fo(h)18 b Fu(\026)-67 b Fq(m)2417 631 y Fn(0)2417 696 y Fl(x)2457 704 y Fg(n)2504 672 y Fq(;)17 b(v)2599 631 y Fm(\()p Fl(n)p Fm(\))2700 672 y Fu(\()p Fq(T)2795 687 y Fl(n)2842 672 y Fu(\))p Fo(i)28 b Fu(=)f(0)540 b(\(6.8\))0 905 y(b)m(y)35 b(\(3.2\))o(.)100 1021 y(T)-8 b(o)39 b(sho)m(w)h Fq(\021)546 1036 y Fl(n)593 1021 y Fu(\(10\))e(is)h(negligible)d(w)m(e)k(use)f(the)h(b)s(ound)f (\(4.22\))o(;)j(since)e(w)m(e)g(are)f(in)f(the)h(set)h Fo(B)3557 1036 y Fl(\025;\034)3661 1021 y Fu(,)h(w)m(e)0 1137 y(get,)33 b(from)e(\(4.7\))525 1353 y(sup)411 1438 y Fl(t)p Fn(2)p Fm([)p Fl(T)544 1446 y Fg(n)587 1438 y Fl(;T)648 1447 y Fg(n)p Fi(+1)767 1438 y Fm(])803 1353 y Fo(k)p Fu(\000)914 1302 y Fm(\()p Fl(n)p Fm(\))914 1377 y(10)1016 1353 y Fu(\()p Fq(t)p Fu(\))p Fo(k)1177 1368 y Fn(1)1279 1353 y Fo(\024)d Fq(C)7 b(T)1532 1311 y Fm(5)p Fl(=)p Fm(2)1642 1353 y Fq(\025)1699 1311 y Fm(3)p Fn(\000)p Fm(3)p Fl(\020)2059 1353 y Fu(hence)197 b Fo(j)p Fq(\021)2569 1368 y Fl(n)2615 1353 y Fu(\(10\))p Fo(j)27 b(\024)h Fq(C)7 b(T)3097 1311 y Fm(5)p Fl(=)p Fm(2)3207 1353 y Fq(\025)3264 1311 y Fm(3)p Fn(\000)p Fm(3)p Fl(\020)3639 1353 y Fu(\(6.9\))0 1711 y Ft(Step)35 b(2.)44 b(The)34 b(term)h Fq(\021)821 1726 y Fl(n)868 1711 y Fu(\(7\))p Ft(.)43 b Fu(By)35 b(\(5.3\))d(and)g(\(4.52\),)1372 1926 y(\000)1433 1876 y Fm(\()p Fl(n)p Fm(\))1433 1951 y(7)1535 1926 y Fu(\()p Fq(t)p Fu(\))c(=)f Fq(I)1820 1941 y Fm(1)1882 1926 y Fu(+)22 b Fq(I)2023 1941 y Fm(2)2085 1926 y Fu(+)g Fq(I)2226 1941 y Fm(3)2287 1926 y Fu(+)g Fq(I)2428 1941 y Fm(4)3591 1926 y Fu(\(6.10\))0 2124 y(where)330 2377 y Fq(I)373 2392 y Fm(1)496 2377 y Fu(:=)83 b Fo(\000)769 2310 y Fq(\025)p 769 2354 57 4 v 773 2445 a Fu(2)903 2253 y Fl(n)853 2282 y Fp(X)860 2495 y Fl(k)r Fm(=1)997 2377 y Fu(\()p Fq(x)1090 2392 y Fl(k)1155 2377 y Fo(\000)22 b Fq(x)1309 2392 y Fl(k)r Fn(\000)p Fm(1)1443 2377 y Fu(\))1481 2336 y Fm(2)1537 2242 y Fp(Z)1636 2268 y Fl(t)1592 2467 y(T)1633 2475 y Fg(n)1680 2377 y Fq(ds)17 b(g)1845 2326 y Fm(\()p Fl(n)p Fm(\))1841 2400 y Fl(t)p Fn(\000)p Fl(s)1957 2377 y Fq(p)2006 2392 y Fl(s)p Fn(\000)p Fl(T)2135 2404 y Fg(k)2195 2377 y Fu(\026)-67 b Fq(m)2262 2336 y Fn(00)2262 2402 y Fl(x)2302 2414 y Fg(k)q Fh(\000)p Fi(1)3591 2377 y Fu(\(6.11\))330 2705 y Fq(I)373 2720 y Fm(2)496 2705 y Fu(:=)83 b Fo(\000)p Fq(\025)883 2581 y Fl(n)833 2610 y Fp(X)840 2823 y Fl(k)r Fm(=1)993 2565 y Fp(\024)1056 2638 y Fu(\()p Fq(x)1149 2653 y Fl(k)1214 2638 y Fo(\000)23 b Fq(x)1369 2653 y Fl(k)r Fn(\000)p Fm(1)1502 2638 y Fu(\))1540 2602 y Fm(3)p 1056 2682 524 4 v 1293 2773 a Fu(6)1606 2569 y Fp(Z)1705 2596 y Fl(t)1661 2795 y(T)1702 2803 y Fg(n)1749 2705 y Fq(ds)17 b(g)1914 2654 y Fm(\()p Fl(n)p Fm(\))1910 2728 y Fl(t)p Fn(\000)p Fl(s)2026 2705 y Fq(p)2075 2720 y Fl(s)p Fn(\000)p Fl(T)2204 2732 y Fg(k)2264 2705 y Fu(\026)-67 b Fq(m)2331 2664 y Fn(000)2331 2730 y Fl(x)2371 2742 y Fg(k)q Fh(\000)p Fi(1)692 3020 y Fu(+)22 b(\()p Fq(x)883 3035 y Fl(k)948 3020 y Fo(\000)g Fq(x)1102 3035 y Fl(k)r Fn(\000)p Fm(1)1236 3020 y Fu(\))1274 2979 y Fm(4)1330 2884 y Fp(Z)1429 2911 y Fl(t)1385 3110 y(T)1426 3118 y Fg(n)1473 3020 y Fq(ds)17 b(g)1638 2969 y Fm(\()p Fl(n)p Fm(\))1634 3043 y Fl(t)p Fn(\000)p Fl(s)1750 3020 y Fq(p)1799 3035 y Fl(s)p Fn(\000)p Fl(T)1928 3047 y Fg(k)1970 3020 y Fq(a)2021 3035 y Fl(k)r Fn(\000)p Fm(1)2154 2880 y Fp(\025)3591 3020 y Fu(\(6.12\))330 3319 y Fq(I)373 3334 y Fm(3)496 3319 y Fu(:=)83 b Fo(\000)769 3251 y Fu(9)p Fq(\025)p 769 3296 106 4 v 773 3387 a Fu(16)952 3194 y Fl(n)901 3224 y Fp(X)909 3436 y Fl(k)r Fm(=1)1045 3319 y Fo(h)18 b Fu(\026)-67 b Fq(m)1169 3278 y Fn(00)1169 3343 y Fl(x)1209 3355 y Fg(k)q Fh(\000)p Fi(1)1330 3319 y Fq(;)17 b(v)1425 3278 y Fm(\()p Fl(k)r Fn(\000)p Fm(1\))1612 3319 y Fu(\()p Fq(T)1707 3334 y Fl(k)1750 3319 y Fu(\))p Fo(ih)g Fu(\026)-66 b Fq(m)1951 3278 y Fn(0)1951 3343 y Fl(x)1991 3355 y Fg(k)q Fh(\000)p Fi(1)2111 3319 y Fq(;)17 b(v)2206 3278 y Fm(\()p Fl(k)r Fn(\000)p Fm(1\))2393 3319 y Fu(\()p Fq(T)2488 3334 y Fl(k)2531 3319 y Fu(\))p Fo(i)2625 3183 y Fp(Z)2724 3209 y Fl(t)2679 3409 y(T)2720 3417 y Fg(n)2767 3319 y Fq(ds)g(g)2932 3268 y Fm(\()p Fl(n)p Fm(\))2928 3341 y Fl(t)p Fn(\000)p Fl(s)3044 3319 y Fq(p)3093 3334 y Fl(s)p Fn(\000)p Fl(T)3222 3346 y Fg(k)3282 3319 y Fu(\026)-67 b Fq(m)3349 3278 y Fn(0)3349 3343 y Fl(x)3389 3355 y Fg(k)q Fh(\000)p Fi(1)3591 3319 y Fu(\(6.13\))330 3647 y Fq(I)373 3662 y Fm(4)496 3647 y Fu(:=)83 b Fq(\025)806 3522 y Fl(n)755 3552 y Fp(X)763 3764 y Fl(k)r Fm(=1)916 3511 y Fp(Z)1015 3537 y Fl(t)971 3737 y(T)1012 3745 y Fg(n)1059 3647 y Fq(ds)17 b(g)1224 3596 y Fm(\()p Fl(n)p Fm(\))1220 3669 y Fl(t)p Fn(\000)p Fl(s)1336 3647 y Fq(p)1385 3662 y Fl(s)p Fn(\000)p Fl(T)1514 3674 y Fg(k)1572 3536 y Fp(h)1638 3647 y Fu(\026)-68 b Fq(m)1704 3606 y Fn(0)1704 3671 y Fl(x)1744 3683 y Fg(k)q Fh(\000)p Fi(1)1865 3647 y Fq(R)q Fu(\()p Fq(v)2029 3606 y Fm(\()p Fl(k)r Fn(\000)p Fm(1\))2217 3647 y Fu(\()p Fq(T)2312 3662 y Fl(k)2354 3647 y Fu(\)\))2430 3536 y Fp(i)3591 3647 y Fu(\(6.14\))0 3933 y(T)-8 b(o)33 b(b)s(ound)f Fq(I)487 3948 y Fm(1)559 3933 y Fu(w)m(e)i(\014rst)f(consider)g(the)g (term)f(with)g Fq(k)f Fu(=)c Fq(n)p Fu(.)44 b(W)-8 b(e)33 b(write,)g(for)f Fq(t)27 b(>)h(T)3038 3948 y Fl(n)3107 3933 y Fu(+)22 b(1,)630 4062 y Fp(Z)730 4088 y Fl(t)686 4287 y(T)727 4295 y Fg(n)773 4197 y Fq(ds)17 b(g)938 4146 y Fm(\()p Fl(n)p Fm(\))934 4220 y Fl(t)p Fn(\000)p Fl(s)1050 4197 y Fq(p)1099 4212 y Fl(s)p Fn(\000)p Fl(T)1228 4220 y Fg(n)1293 4197 y Fu(\026)-67 b Fq(m)1360 4156 y Fn(00)1360 4222 y Fl(x)1400 4231 y Fg(n)p Fh(\000)p Fi(1)1552 4197 y Fu(=)1656 4057 y Fp(\032)1731 4062 y(Z)1830 4088 y Fl(T)1871 4096 y Fg(n)1914 4088 y Fm(+1)1786 4287 y Fl(T)1827 4295 y Fg(n)2025 4197 y Fu(+)2118 4062 y Fp(Z)2217 4088 y Fl(t)2173 4287 y(T)2214 4295 y Fg(n)2256 4287 y Fm(+1)2350 4057 y Fp(\033)2458 4197 y Fq(ds)17 b(g)2623 4146 y Fm(\()p Fl(n)p Fm(\))2619 4220 y Fl(t)p Fn(\000)p Fl(s)2736 4197 y Fq(p)2785 4212 y Fl(s)p Fn(\000)p Fl(T)2914 4220 y Fg(n)2978 4197 y Fu(\026)-67 b Fq(m)3045 4156 y Fn(00)3045 4222 y Fl(x)3085 4231 y Fg(n)p Fh(\000)p Fi(1)0 4454 y Fu(The)26 b(\014rst)f(in)m(tegral)e(is)h(b)s(ounded)h(b)m (y)h(a)e(constan)m(t.)42 b(In)25 b(the)g(second)h(in)m(tegral,)f(as)f (w)m(ell)g(as)h(in)f(the)h(in)m(tegrals)0 4571 y(in)32 b Fq(I)157 4586 y Fm(1)229 4571 y Fu(when)h Fq(k)e(<)d(n)p Fu(,)33 b(w)m(e)g(write)1326 4816 y Fq(p)1375 4831 y Fl(s)p Fn(\000)p Fl(T)1504 4843 y Fg(k)1564 4816 y Fu(\026)-67 b Fq(m)1631 4775 y Fn(00)1631 4841 y Fl(x)1671 4853 y Fg(k)q Fh(\000)p Fi(1)1819 4816 y Fu(=)28 b(2)1982 4749 y Fq(@)5 b(p)2087 4764 y Fl(s)p Fn(\000)p Fl(T)2216 4776 y Fg(k)p 1982 4793 277 4 v 2068 4885 a Fq(@)g(s)2287 4816 y Fu(\026)-68 b Fq(m)2353 4831 y Fl(x)2393 4843 y Fg(k)q Fh(\000)p Fi(1)0 5049 y Fu(and)33 b(then)g(use)g(\(4.17\).)43 b(W)-8 b(e)33 b(th)m(us)h(get)e(recalling)f(the)i(de\014nition)e (\(4.3\))h(of)g Fq(\016)2828 5013 y Fn(\003)2868 5049 y Fu(\()p Fq(\034)11 b Fu(\),)33 b(and)f(using)i(\(4.10\))o(,)432 5338 y Fo(k)p Fq(I)525 5353 y Fm(1)564 5338 y Fo(k)614 5353 y Fn(1)716 5338 y Fo(\024)29 b Fq(C)7 b(\025\016)1003 5297 y Fn(\003)1042 5338 y Fu(\()p Fq(\034)k Fu(\))1171 5297 y Fm(2)1227 5168 y Fp(\()1308 5338 y Fu(log)16 b Fq(T)36 b Fu(+)1692 5214 y Fl(n)1641 5244 y Fp(X)1649 5456 y Fl(k)r Fm(=1)1989 5271 y Fu(1)p 1811 5315 403 4 v 1811 5407 a Fq(n)23 b Fo(\000)f Fq(k)k Fu(+)c(1)2224 5168 y Fp(\))2332 5338 y Fo(\024)29 b Fq(C)2531 5198 y Fp(\022)2614 5271 y Fq(\016)2661 5235 y Fn(\003)2701 5271 y Fu(\()p Fq(\034)11 b Fu(\))p 2614 5315 216 4 v 2617 5420 a Fq(\025)2674 5335 y Fo(p)p 2757 5335 71 4 v 85 x Fq(T)2840 5198 y Fp(\023)2913 5220 y Fm(2)2969 5338 y Fq(\025)3026 5297 y Fm(3)3066 5338 y Fq(T)j Fo(j)j Fu(log)e Fq(\025)p Fo(j)1882 5532 y Fk(25)p eop %%Page: 26 27 26 26 bop 0 269 a Fu(Since)51 b(\026)-67 b Fq(m)340 232 y Fn(000)340 293 y Fl(x)380 305 y Fg(k)q Fh(\000)p Fi(1)528 269 y Fo(2)28 b Fq(L)688 284 y Fm(1)728 269 y Fu(\()p Fj(R)5 b Fq(;)17 b(dx)p Fu(\),)38 b Fo(k)p Fq(p)1184 284 y Fl(t)1232 269 y Fu(\026)-67 b Fq(m)1299 232 y Fn(000)1299 293 y Fl(x)1339 305 y Fg(k)q Fh(\000)p Fi(1)1459 269 y Fo(k)1509 284 y Fn(1)1611 269 y Fo(\024)29 b Fq(C)7 b(t)1829 232 y Fn(\000)p Fm(1)p Fl(=)p Fm(2)2026 269 y Fu(and)33 b(as)g Fq(a)2387 284 y Fl(k)r Fn(\000)p Fm(1)2552 269 y Fu(is)f(b)s(ounded,)h(using)g(again)e(\(4.10\))501 532 y Fo(k)p Fq(I)594 547 y Fm(2)633 532 y Fo(k)683 547 y Fn(1)841 532 y Fo(\024)83 b Fq(C)7 b(\025\016)1182 491 y Fn(\003)1222 532 y Fu(\()p Fq(\034)k Fu(\))1351 491 y Fm(3)1458 407 y Fl(n)1407 437 y Fp(X)1415 649 y Fl(k)r Fm(=1)1568 396 y Fp(Z)1667 422 y Fl(t)1623 622 y(T)1664 630 y Fg(n)1711 532 y Fq(ds)17 b Fu(\()p Fq(s)k Fo(\000)i Fq(T)2087 547 y Fl(k)2129 532 y Fu(\))2167 491 y Fn(\000)p Fm(1)p Fl(=)p Fm(2)2354 532 y Fu(+)f Fq(C)7 b(\025\016)2633 491 y Fn(\003)2673 532 y Fu(\()p Fq(\034)k Fu(\))2802 491 y Fm(4)2909 407 y Fl(n)2858 437 y Fp(X)2866 649 y Fl(k)r Fm(=1)3002 532 y Fu(\()p Fq(t)22 b Fo(\000)h Fq(T)3254 547 y Fl(n)3301 532 y Fu(\))841 857 y Fo(\024)83 b Fq(C)1095 717 y Fp(\022)1178 790 y Fq(\016)1225 754 y Fn(\003)1265 790 y Fu(\()p Fq(\034)11 b Fu(\))p 1178 835 216 4 v 1181 939 a Fq(\025)1238 855 y Fo(p)p 1321 855 71 4 v 84 x Fq(T)1404 717 y Fp(\023)1477 739 y Fm(3)1533 857 y Fq(\025)1590 816 y Fm(3)1630 857 y Fq(T)1701 816 y Fm(3)p Fl(=)p Fm(2)1833 857 y Fu(+)22 b Fq(C)2024 717 y Fp(\022)2107 790 y Fq(\016)2154 754 y Fn(\003)2194 790 y Fu(\()p Fq(\034)11 b Fu(\))p 2107 835 216 4 v 2110 939 a Fq(\025)2167 855 y Fo(p)p 2250 855 71 4 v 84 x Fq(T)2333 717 y Fp(\023)2406 739 y Fm(4)2463 857 y Fq(\025)2520 816 y Fm(3)2559 857 y Fq(T)2630 816 y Fm(2)0 1091 y Fu(Recalling)30 b(the)j(de\014nition)f(\(4.2\))g(of)g Fq(V)1427 1106 y Fn(\003)1466 1091 y Fu(\()p Fq(\034)11 b Fu(\))33 b(and)g(the)g(de\014nition)e(\(4.37\))h(of)g Fq(V)2891 1055 y Fn(?)2869 1116 y(\003)2950 1091 y Fu(\()p Fq(\034)11 b Fu(\),)33 b(w)m(e)g(get)634 1322 y Fo(k)p Fq(I)727 1337 y Fm(3)766 1322 y Fo(k)816 1337 y Fn(1)919 1322 y Fo(\024)28 b Fq(C)7 b(\025V)1215 1337 y Fn(\003)1254 1322 y Fu(\()p Fq(\034)k Fu(\))p Fq(V)1462 1281 y Fn(?)1440 1347 y(\003)1521 1322 y Fu(\()p Fq(\034)g Fu(\))1650 1232 y Fp(p)p 1750 1232 304 4 v 90 x Fq(T)j(n)1879 1337 y Fl(\025)1924 1322 y Fu(\()p Fq(\034)d Fu(\))28 b Fo(\024)g Fq(C)2280 1182 y Fp(\022)2363 1255 y Fq(V)2420 1270 y Fn(\003)2460 1255 y Fu(\()p Fq(\034)11 b Fu(\))p Fq(V)2667 1219 y Fn(?)2646 1280 y(\003)2726 1255 y Fu(\()p Fq(\034)g Fu(\))p 2363 1300 493 4 v 2484 1404 a Fq(\025)2541 1375 y Fm(2)2581 1319 y Fo(p)p 2664 1319 71 4 v 85 x Fq(T)2866 1182 y Fp(\023)2956 1322 y Fq(\025)3013 1281 y Fm(2)3052 1233 y Fo(p)p 3135 1233 V 89 x Fq(T)0 1548 y Fu(Finally)-8 b(,)30 b(recalling)h(\(2.5\),)848 1791 y Fo(k)p Fq(I)941 1806 y Fm(4)980 1791 y Fo(k)1030 1806 y Fn(1)1133 1791 y Fo(\024)d Fq(C)7 b(\025)1372 1701 y Fp(p)p 1471 1701 304 4 v 1471 1791 a Fq(T)14 b(n)1600 1806 y Fl(\025)1646 1791 y Fu(\()p Fq(\034)d Fu(\))p Fq(V)1832 1806 y Fn(\003)1871 1791 y Fu(\()p Fq(\034)g Fu(\))2000 1750 y Fm(3)2068 1791 y Fo(\024)28 b Fq(C)2266 1651 y Fp(\022)2350 1724 y Fq(V)2407 1739 y Fn(\003)2446 1724 y Fu(\()p Fq(\034)11 b Fu(\))p 2350 1768 226 4 v 2358 1872 a Fq(\025)2415 1788 y Fo(p)p 2497 1788 71 4 v 2497 1872 a Fq(T)2585 1651 y Fp(\023)2659 1673 y Fm(3)2715 1791 y Fq(\025)2772 1750 y Fm(3)2811 1791 y Fq(T)2882 1750 y Fm(3)p Fl(=)p Fm(2)0 2035 y Fu(Since)45 b(w)m(e)h(are)f(restricting)f(our)h (considerations)g(to)f(the)i(set)f(where)i Fq(\016)2729 1999 y Fn(\003)2768 2035 y Fu(\()p Fq(\034)11 b Fu(\))49 b Fo(\024)g Fq(\025)3129 1999 y Fm(1)p Fn(\000)p Fl(\020)3259 1950 y Fo(p)p 3342 1950 V 85 x Fq(T)14 b Fu(,)48 b Fq(V)3545 2050 y Fn(\003)3584 2035 y Fu(\()p Fq(\034)11 b Fu(\))50 b Fo(\024)0 2160 y Fq(\025)57 2124 y Fm(1)p Fn(\000)p Fl(\020)187 2076 y Fo(p)p 270 2076 V 84 x Fq(T)c Fu(and)33 b Fq(V)641 2124 y Fn(?)620 2185 y(\003)700 2160 y Fu(\()p Fq(\034)11 b Fu(\))28 b Fo(\024)h Fq(\025)1020 2124 y Fm(1)p Fn(\000)p Fl(\020)1149 2160 y Fu(,)k(w)m(e)h(get)452 2343 y(sup)338 2428 y Fl(t)p Fn(2)p Fm([)p Fl(T)471 2436 y Fg(n)513 2428 y Fl(;T)574 2437 y Fg(n)p Fi(+1)694 2428 y Fm(])730 2343 y Fo(k)p Fu(\000)841 2292 y Fm(\()p Fl(n)p Fm(\))841 2367 y(7)942 2343 y Fu(\()p Fq(t)p Fu(\))p Fo(k)1103 2358 y Fn(1)1206 2343 y Fo(\024)28 b Fq(C)7 b(\025)1445 2302 y Fm(2)p Fn(\000)p Fm(2)p Fl(\020)1610 2253 y Fo(p)p 1693 2253 V 90 x Fq(T)209 b Fu(hence)196 b Fo(j)p Fq(\021)2468 2358 y Fl(n)2515 2343 y Fu(\(7\))p Fo(j)27 b(\024)h Fq(C)7 b(\025)2934 2302 y Fm(2)p Fn(\000)p Fm(2)p Fl(\020)3099 2253 y Fo(p)p 3182 2253 V 90 x Fq(T)352 b Fu(\(6.15\))0 2635 y Ft(Step)35 b(3.)44 b(The)34 b(term)h Fq(\021)821 2650 y Fl(n)868 2635 y Fu(\(8\))p Ft(.)43 b Fu(In)33 b(order)g(to)f(sho)m(w)h Fq(\021)1851 2650 y Fl(n)1899 2635 y Fu(\(8\))f(is)g(negligible)d(\(as)k(sp)s(eci\014ed)g (at)f(the)h(b)s(eginning)0 2751 y(of)d(the)h(presen)m(t)h(pro)s(of)7 b(\))29 b(w)m(e)i(cannot)g(use)g(directly)f(the)g Fe(a)g(priori)f Fu(b)s(ounds,)i(but)g(w)m(e)g(use)g(equation)f(\(5.6\))0 2867 y(to)i(get)1203 3080 y Fq(\021)1251 3095 y Fl(n)1298 3080 y Fu(\(8\))27 b(=)1603 2955 y Fm(10)1566 2985 y Fp(X)1589 3184 y Fg(i)p Fi(=1)1564 3241 y Fg(i)p Fh(6)p Fi(=3)p Fg(;)p Fi(8)1739 3080 y Fq(\021)1787 3095 y Fl(n)1834 3080 y Fu(\(8)p Fq(;)17 b(i)p Fu(\))22 b(+)g Fq(\021)2204 3095 y Fl(n)2251 3080 y Fu(\(8)p Fq(;)17 b Fu(3)k(+)h(8\))954 b(\(6.16\))0 3366 y(and)33 b(sho)m(w)g(eac)m(h)h(of)e(the)h(terms)f(on) h(the)g(r.h.s.)g(ab)s(o)m(v)m(e)h(are)e(negligible.)100 3482 y(W)-8 b(e)33 b(ha)m(v)m(e)1522 3676 y Fq(\021)1570 3691 y Fl(n)1617 3676 y Fu(\(8\))28 b(=)1879 3551 y Fl(n)p Fn(\000)p Fm(1)1873 3581 y Fp(X)1881 3793 y Fl(k)r Fm(=0)2034 3676 y Fq(A)2107 3625 y Fm(\(1\))2107 3703 y Fl(n;k)2212 3676 y Fq( )2275 3691 y Fl(k)3591 3676 y Fu(\(6.17\))0 3914 y(where,)34 b(recalling)c(2)e(=)f Fo(h)p Fu(1)p Fq(;)35 b Fu(\026)-67 b Fq(m)1096 3878 y Fn(0)1096 3939 y Fl(x)1136 3951 y Fg(k)1178 3914 y Fo(i)p Fu(,)132 4182 y Fq(A)205 4131 y Fm(\(1\))205 4209 y Fl(n;k)338 4182 y Fu(:=)28 b Fo(\000)556 4114 y Fu(3)p Fq(\025)p 556 4159 106 4 v 585 4250 a Fu(4)688 4046 y Fp(Z)788 4072 y Fl(T)829 4081 y Fg(n)p Fi(+1)744 4272 y Fl(T)785 4280 y Fg(n)953 4182 y Fq(ds)1083 4011 y Fp(\()1163 4182 y Fo(h)18 b Fu(\026)-67 b Fq(m)1287 4140 y Fn(0)1287 4206 y Fl(x)1327 4214 y Fg(n)1373 4182 y Fq(;)17 b(p)1466 4197 y Fl(s)p Fn(\000)p Fl(T)1595 4209 y Fg(k)q Fi(+1)1732 4182 y Fu(\026)-67 b Fq(m)1799 4140 y Fn(0)1799 4206 y Fl(x)1839 4218 y Fg(k)1881 4182 y Fo(i)39 b(\000)g(h)18 b Fu(\026)-67 b Fq(m)2199 4140 y Fn(0)2199 4206 y Fl(x)2239 4214 y Fg(n)2285 4182 y Fq(;)17 b Fu(1)p Fo(i)2723 4114 y Fu(1)p 2427 4159 641 4 v 2427 4179 a Fp(p)p 2526 4179 542 4 v 2526 4264 a Fu(2)p Fq(\031)t Fu(\()p Fq(s)22 b Fo(\000)g Fq(T)2896 4279 y Fl(k)r Fm(+1)3029 4264 y Fu(\))3077 4182 y Fo(h)p Fu(1)p Fq(;)35 b Fu(\026)-67 b Fq(m)3294 4140 y Fn(0)3294 4206 y Fl(x)3334 4218 y Fg(k)3376 4182 y Fo(i)3415 4011 y Fp(\))3511 4182 y Fv(1)3568 4197 y Fl(k)r()28 b Fu(0\))138 4761 y Fq(B)217 4710 y Fm(\(1\))212 4789 y Fl(n;k)345 4761 y Fu(:=)f Fq(\025)549 4626 y Fp(Z)648 4652 y Fl(T)689 4661 y Fg(n)p Fi(+1)604 4851 y Fl(T)645 4859 y Fg(n)813 4761 y Fq(ds)927 4626 y Fp(Z)1026 4761 y Fq(dx)1149 4626 y Fp(Z)1248 4761 y Fq(dy)44 b Fu(\026)-68 b Fq(m)1457 4720 y Fn(0)1457 4786 y Fl(x)1497 4794 y Fg(n)1544 4761 y Fu(\()p Fq(x)p Fu(\))1692 4587 y Fp(\014)1692 4647 y(\014)1692 4707 y(\014)1692 4766 y(\014)1692 4826 y(\014)1725 4761 y Fq(p)1774 4776 y Fl(s)p Fn(\000)p Fl(T)1903 4788 y Fg(k)q Fi(+1)2022 4761 y Fu(\()p Fq(x)23 b Fo(\000)f Fq(y)t Fu(\))f Fo(\000)2754 4694 y Fu(1)p 2458 4738 641 4 v 2458 4758 a Fp(p)p 2558 4758 542 4 v 86 x Fu(2)p Fq(\031)t Fu(\()p Fq(s)g Fo(\000)i Fq(T)2928 4859 y Fl(k)r Fm(+1)3061 4844 y Fu(\))3109 4587 y Fp(\014)3109 4647 y(\014)3109 4707 y(\014)3109 4766 y(\014)3109 4826 y(\014)3177 4761 y Fu(\026)-67 b Fq(m)3244 4720 y Fn(0)3244 4786 y Fl(x)3284 4798 y Fg(k)3326 4761 y Fu(\()p Fq(y)t Fu(\))137 b(\(6.19\))0 5017 y(W)-8 b(e)35 b(are)g(going)f(to)g(pro)m(v)m(e)i(that)f(for)f(an)m (y)i Fq(\034)43 b(>)32 b Fu(0)i(there)i(is)e(a)h(constan)m(t)h Fq(C)i Fu(=)32 b Fq(C)7 b Fu(\()p Fq(\034)k Fu(\))35 b(suc)m(h)h(that)f(for)f(an)m(y)0 5134 y(0)27 b Fo(\024)i Fq(k)h(<)e(n)g Fo(\024)g Fq(n)616 5149 y Fl(\025)662 5134 y Fu(\()p Fq(\034)11 b Fu(\))629 5245 y Fp(\014)629 5305 y(\014)629 5365 y(\014)662 5360 y Fq(A)735 5309 y Fm(\(1\))735 5387 y Fl(n;k)840 5245 y Fp(\014)840 5305 y(\014)840 5365 y(\014)901 5360 y Fo(\024)28 b Fq(C)7 b(B)1162 5309 y Fm(\(1\))1157 5387 y Fl(n;k)1290 5360 y Fo(\024)28 b Fq(C)7 b(\025)1546 5279 y Fp(\002)1587 5360 y Fu(1)22 b(+)g Fq(x)1811 5318 y Fm(2)1811 5384 y Fl(n)p Fm(+1)p Fl(;)p Fn(\003)2003 5279 y Fp(\003)2062 5219 y(\032)2409 5292 y Fu(1)p 2146 5337 574 4 v 2146 5357 a Fo(p)p 2229 5357 71 4 v 84 x Fq(T)14 b Fu(\()p Fq(n)22 b Fo(\000)h Fq(k)s Fu(\))2610 5412 y Fm(3)p Fl(=)p Fm(2)2752 5360 y Fu(+)f Fv(1)2906 5375 y Fl(k)r Fm(=)p Fl(n)p Fn(\000)p Fm(1)3137 5219 y Fp(\033)3591 5360 y Fu(\(6.20\))1882 5532 y Fk(26)p eop %%Page: 27 28 27 27 bop 0 266 a Ft(Pr)-5 b(o)g(of)35 b(of)55 b Fu(\(6.20\))p Ft(.)43 b Fu(Let)33 b Fq(x)950 281 y Fl(n;k)1083 266 y Fu(:=)27 b Fq(x)1268 281 y Fl(n)1338 266 y Fo(\000)c Fq(x)1493 281 y Fl(k)1568 266 y Fu(and)33 b(de\014ne,)h(for)e(an)m(y)h Fq(\022)e Fo(2)d Fj(R)43 b Fu(and)33 b Fq(t)28 b(>)g Fu(0,)870 526 y Fq(f)11 b Fu(\()p Fq(t;)17 b(\022)s Fu(\))28 b(=)1263 391 y Fp(Z)1363 526 y Fq(dx)1486 391 y Fp(Z)1585 526 y Fq(dy)43 b Fu(\026)-67 b Fq(m)1794 485 y Fn(0)1818 526 y Fu(\()p Fq(x)p Fu(\))18 b(\026)-67 b Fq(m)2034 485 y Fn(0)2058 526 y Fu(\()p Fq(y)t Fu(\))2203 352 y Fp(\014)2203 412 y(\014)2203 472 y(\014)2203 531 y(\014)2203 591 y(\014)2245 459 y Fq(e)2290 423 y Fn(\000)p Fm(\()p Fl(y)r Fn(\000)p Fl(x)p Fn(\000)p Fl(\022)r Fm(\))2621 399 y Fi(2)2656 423 y Fl(=)p Fm(2)p Fl(t)2778 459 y Fo(\000)23 b Fu(1)p 2245 504 682 4 v 2473 523 a Fo(p)p 2556 523 143 4 v 83 x Fu(2)p Fq(\031)t(t)2936 352 y Fp(\014)2936 412 y(\014)2936 472 y(\014)2936 531 y(\014)2936 591 y(\014)0 809 y Fu(Then,)50 b(after)45 b(a)g(c)m(hange)h(of)f(v)-5 b(ariables)44 b(in)g(\(6.19\),)k(its)d(r.h.s.)h(b)s(ecomes)g(equal)f (to)g Fq(\025)3245 729 y Fp(R)3311 755 y Fl(T)3352 764 y Fg(n)p Fi(+1)3292 843 y Fl(T)3333 851 y Fg(n)3476 809 y Fq(ds)17 b(f)11 b Fu(\()p Fq(s)30 b Fo(\000)0 935 y Fq(T)57 950 y Fl(k)r Fm(+1)190 935 y Fq(;)17 b(x)289 950 y Fl(n;k)394 935 y Fu(\).)0 1101 y Ft(The)34 b(c)-5 b(ase)34 b Fq(k)d(<)d(n)22 b Fo(\000)h Fu(1)p Ft(.)43 b Fu(After)32 b(c)m(hanging)h(v)-5 b(ariables)31 b(in)h(the)h(time)e (in)m(tegral,)g(w)m(e)j(get,)e(for)g Fq(k)f(<)d(n)22 b Fo(\000)h Fu(1,)1028 1331 y Fq(B)1107 1280 y Fm(\(1\))1102 1359 y Fl(n;k)1234 1331 y Fu(=)28 b Fq(\025T)1482 1196 y Fp(Z)1582 1222 y Fm(1)1538 1421 y(0)1621 1331 y Fq(dt)17 b(f)1783 1221 y Fp(\020)1842 1331 y Fq(T)d Fu(\()p Fq(t)22 b Fu(+)g Fq(n)g Fo(\000)h Fu(1)f Fo(\000)h Fq(k)s Fu(\))p Fq(;)17 b(x)2648 1346 y Fl(n;k)2753 1221 y Fp(\021)3591 1331 y Fu(\(6.21\))0 1566 y(Since)33 b Fo(j)p Fq(e)328 1530 y Fn(\000j)p Fl(\030)s Fn(j)482 1566 y Fo(\000)22 b Fu(1)p Fo(j)27 b(\024)i(j)p Fq(\030)5 b Fo(j)31 b Fu(and)51 b(\026)-68 b Fq(m)1200 1530 y Fn(0)1224 1566 y Fu(\()p Fq(x)p Fu(\))33 b(deca)m(ys)h(exp)s(onen)m(tially)e(to)g(0)g(as)h Fo(j)p Fq(x)p Fo(j)28 b(!)f(1)p Fu(,)32 b(w)m(e)i(get)1538 1796 y Fq(f)11 b Fu(\()p Fq(t;)17 b(\022)s Fu(\))27 b Fo(\024)h Fq(C)2036 1728 y Fu(1)22 b(+)g Fq(\022)2253 1692 y Fm(2)p 2036 1773 257 4 v 2091 1864 a Fq(t)2126 1835 y Fm(3)p Fl(=)p Fm(2)3591 1796 y Fu(\(6.22\))0 1997 y(and,)33 b(from)e(\(6.21\))o(,)884 2233 y Fq(B)963 2182 y Fm(\(1\))958 2261 y Fl(n;k)1091 2233 y Fo(\024)d Fq(C)7 b(\025T)1401 2192 y Fn(\000)p Fm(1)p Fl(=)p Fm(2)1565 2233 y Fu([1)22 b(+)g Fq(x)1816 2192 y Fm(2)1816 2257 y Fl(n;k)1922 2233 y Fu(])1966 2097 y Fp(Z)2065 2124 y Fm(1)2021 2323 y(0)2105 2233 y Fq(dt)2549 2165 y Fu(1)p 2201 2210 746 4 v 2201 2301 a(\()p Fq(t)g Fu(+)g Fq(n)g Fo(\000)h Fu(1)f Fo(\000)g Fq(k)s Fu(\))2836 2272 y Fm(3)p Fl(=)p Fm(2)0 2459 y Fu(whic)m(h)33 b(pro)m(v)m(es)h(\(6.20\))e(when)i Fq(k)d(<)c(n)c Fo(\000)f Fu(1.)0 2626 y Ft(The)34 b(c)-5 b(ase)34 b Fq(k)d Fu(=)d Fq(n)22 b Fo(\000)h Fu(1)p Ft(.)43 b Fu(W)-8 b(e)33 b(ha)m(v)m(e,)h(after)e(a)g(c)m(hange)i(of)e(v)-5 b(ariables)31 b(in)h(the)h(time)e(in)m(tegral,)1318 2858 y Fq(B)1397 2808 y Fm(\(1\))1392 2883 y Fl(n;n)p Fn(\000)p Fm(1)1619 2858 y Fu(=)d Fq(\025)1797 2723 y Fp(Z)1896 2749 y Fl(T)1852 2948 y Fm(0)1951 2858 y Fq(dt)17 b(f)11 b Fu(\()p Fq(t;)17 b(x)2285 2873 y Fl(n;n)p Fn(\000)p Fm(1)2484 2858 y Fu(\))0 3093 y(Using)32 b(the)h(inequalit)m(y)f Fq(f)11 b Fu(\()p Fq(t;)17 b(\022)s Fu(\))27 b Fo(\024)h Fu(4\(2)p Fq(\031)t(t)p Fu(\))1557 3057 y Fn(\000)p Fm(1)p Fl(=)p Fm(2)1754 3093 y Fu(when)34 b Fq(t)28 b Fo(\024)g Fu(1)k(and)h(\(6.22\))f(when)h Fq(t)28 b(>)g Fu(1,)k(w)m(e)i(get)616 3332 y Fq(B)695 3281 y Fm(\(1\))690 3356 y Fl(n;n)p Fn(\000)p Fm(1)917 3332 y Fo(\024)29 b Fq(\025)1097 3196 y Fp(Z)1196 3222 y Fm(1)1151 3422 y(0)1235 3332 y Fq(dt)1436 3264 y Fu(4)p 1348 3309 226 4 v 1348 3329 a Fo(p)p 1431 3329 143 4 v 82 x Fu(2)p Fq(\031)t(t)1606 3332 y Fu(+)22 b Fq(C)7 b(\025)1855 3196 y Fp(Z)1954 3222 y Fl(T)1909 3422 y Fm(1)2009 3332 y Fq(dt)2121 3255 y Fu(1)22 b(+)g Fq(x)2345 3219 y Fm(2)2345 3280 y Fl(n;n)p Fn(\000)p Fm(1)p 2121 3309 424 4 v 2261 3400 a Fq(t)2296 3371 y Fm(3)p Fl(=)p Fm(2)2583 3332 y Fo(\024)28 b Fq(C)7 b(\025)p Fu(\(1)22 b(+)g Fq(x)3084 3291 y Fm(2)3084 3356 y Fl(n;)p Fn(\003)3186 3332 y Fu(\))0 3558 y(The)34 b(pro)s(of)d(of)39 b(\(6.20\))32 b(is)g(concluded.)100 3724 y(Recall)f(no)m(w)i(that)408 3960 y Fq(\021)456 3975 y Fl(n)503 3960 y Fu(\(8)p Fq(;)17 b(i)p Fu(\))27 b(=)h Fo(\000)923 3893 y Fu(3)p 923 3937 49 4 v 923 4028 a(4)1004 3836 y Fl(n)p Fn(\000)p Fm(1)999 3865 y Fp(X)1006 4078 y Fl(k)r Fm(=0)1159 3960 y Fq(A)1232 3909 y Fm(\(1\))1232 3988 y Fl(n;k)1337 3960 y Fo(h)18 b Fu(\026)-67 b Fq(m)1461 3919 y Fn(0)1461 3985 y Fl(x)1501 3997 y Fg(k)1543 3960 y Fq(;)17 b Fu(\000)1648 3919 y Fl(k)1648 3985 y(i)1691 3960 y Fu(\()p Fq(T)1786 3975 y Fl(k)r Fm(+1)1919 3960 y Fu(\))p Fo(i)p Fq(;)211 b(i)28 b Fu(=)g(1)p Fq(;)17 b Fu(2)p Fq(;)g Fu(3)k(+)h(8)p Fq(;)17 b Fu(4)p Fq(;)g Fu(5)p Fq(;)g Fu(6)p Fq(;)g Fu(7)p Fq(;)g Fu(9)p Fq(;)g Fu(10)p Fq(:)100 4221 y Fu(The)43 b(term)f Fq(\021)601 4236 y Fl(n)648 4221 y Fu(\(8)p Fq(;)17 b Fu(1\))41 b(in)h(\(6.16\))f(v)-5 b(anishes)43 b(b)m(y)h(the)e(same)h (argumen)m(t)f(as)g(in)g(\(6.8\))o(.)73 b(Since)43 b(w)m(e)g(are)0 4355 y(w)m(orking)33 b(in)e(the)i(set)h Fo(B)871 4304 y Fm(\(1\))868 4383 y Fl(\025;\034)972 4355 y Fu(,)f(see)g(\(4.38\),)f (w)m(e)i(ha)m(v)m(e)152 4631 y Fo(j)p Fq(\021)228 4646 y Fl(n)274 4631 y Fu(\(8)p Fq(;)17 b Fu(2\))p Fo(j)27 b(\024)h Fq(C)7 b(\025)803 4551 y Fp(\000)848 4631 y Fu(1)22 b(+)g Fq(x)1072 4590 y Fm(2)1072 4656 y Fl(n)p Fm(+1)p Fl(;)p Fn(\003)1265 4551 y Fp(\001)1327 4461 y(\()1412 4507 y Fl(n)p Fn(\000)p Fm(2)1407 4537 y Fp(X)1415 4749 y Fl(k)r Fm(=0)1840 4564 y Fu(1)p 1578 4608 574 4 v 1578 4628 a Fo(p)p 1661 4628 71 4 v 85 x Fq(T)13 b Fu(\()p Fq(n)23 b Fo(\000)f Fq(k)s Fu(\))2041 4684 y Fm(3)p Fl(=)p Fm(2)2183 4631 y Fu(+)g(1)2330 4461 y Fp(\))2427 4631 y Fq(\025)2484 4590 y Fm(1)p Fn(\000)p Fl(\020)2614 4542 y Fo(p)p 2697 4542 V 89 x Fq(T)41 b Fo(\024)28 b Fq(C)7 b(\025)3034 4590 y Fm(2)p Fn(\000)p Fl(\020)3181 4551 y Fp(\000)3226 4631 y Fu(1)22 b(+)g Fq(x)3450 4590 y Fm(2)3450 4656 y Fl(n)p Fm(+1)p Fl(;)p Fn(\003)3643 4551 y Fp(\001)3591 4842 y Fu(\(6.23\))0 4958 y(whic)m(h)33 b(is)f(negligible)e(for)i Fq(\020)39 b Fu(small.)100 5074 y(T)-8 b(o)32 b(b)s(ound)h Fq(\021)592 5089 y Fl(n)639 5074 y Fu(\(8)p Fq(;)17 b Fu(3)k(+)h(8\))33 b(use)g(\(6.17\),)f(\(5.2\),)g(and)h(\(5.7\))f(to)g(write)197 5338 y Fq(\021)245 5353 y Fl(n)292 5338 y Fu(\(8)p Fq(;)17 b Fu(3)k(+)h(8\))28 b(=)f Fq(\025)888 5214 y Fl(n)p Fn(\000)p Fm(1)883 5244 y Fp(X)890 5456 y Fl(k)r Fm(=0)1043 5338 y Fq(A)1116 5287 y Fm(\(1\))1116 5366 y Fl(n;k)1246 5214 y(k)r Fn(\000)p Fm(1)1238 5244 y Fp(X)1245 5456 y Fl(h)p Fm(=0)1399 5338 y Fq(B)1473 5353 y Fl(k)r(;h)1576 5338 y Fq( )1639 5353 y Fl(h)1684 5338 y Fq(;)212 b(B)1997 5353 y Fl(k)r(;h)2127 5338 y Fu(:=)2258 5203 y Fp(Z)2358 5229 y Fl(T)2399 5241 y Fg(k)q Fi(+1)2313 5428 y Fl(T)2354 5440 y Fg(k)2518 5338 y Fq(ds)22 b Fo(h)c Fu(\026)-67 b Fq(m)2761 5297 y Fn(0)2761 5363 y Fl(x)2801 5375 y Fg(k)2843 5338 y Fq(;)17 b(p)2936 5353 y Fl(s)p Fn(\000)p Fl(T)3065 5365 y Fg(h)p Fi(+1)3204 5338 y Fu(\026)-67 b Fq(m)3271 5297 y Fn(0)3271 5363 y Fl(x)3311 5375 y Fg(h)3355 5338 y Fo(i)197 b Fu(\(6.24\))1882 5532 y Fk(27)p eop %%Page: 28 29 28 28 bop 0 275 a Fu(W)-8 b(e)42 b(note)f(that,)j(b)m(y)f(\(5.7\),)g Fo(j)p Fq( )t Fo(j)1191 290 y Fl(n)1280 275 y Fo(\024)g Fq(C)7 b(V)1534 290 y Fl(n;)p Fn(\003)1636 275 y Fu(;)46 b(it)40 b(is)h(also)g(easy)h(to)f(v)m(erify)h Fq(B)2832 290 y Fl(k)r(;h)2977 275 y Fo(\024)h Fq(C)3174 191 y Fo(p)p 3257 191 71 4 v 84 x Fq(T)14 b Fu(\()p Fq(k)31 b Fo(\000)e Fq(h)p Fu(\))3648 239 y Fn(\000)p Fm(1)p Fl(=)p Fm(2)3813 275 y Fu(.)0 391 y(Plugging)i(this)h(b)s(ounds,)h (together)g(with)f(\(6.20\),)g(in)m(to)g(\(6.24\))o(,)h(w)m(e)h(get)183 665 y Fo(j)o Fq(\021)258 680 y Fl(n)305 665 y Fu(\(8)p Fq(;)17 b Fu(3)22 b(+)g(8\))p Fo(j)27 b(\024)h Fq(C)7 b(\025)986 623 y Fm(2)1025 575 y Fo(p)p 1108 575 V 90 x Fq(T)14 b(V)1236 680 y Fl(n;)p Fn(\003)1354 584 y Fp(\002)1396 665 y Fu(1)22 b(+)g Fq(x)1620 623 y Fm(2)1620 689 y Fl(n)p Fm(+1)p Fl(;)p Fn(\003)1812 584 y Fp(\003)1876 540 y Fl(n)p Fn(\000)p Fm(1)1870 570 y Fp(X)1878 782 y Fl(k)r Fm(=0)2031 524 y Fp(\032)2378 597 y Fu(1)p 2116 642 574 4 v 2116 662 a Fo(p)p 2199 662 71 4 v 84 x Fq(T)13 b Fu(\()p Fq(n)23 b Fo(\000)f Fq(k)s Fu(\))2579 717 y Fm(3)p Fl(=)p Fm(2)2721 665 y Fu(+)g Fv(1)2875 680 y Fl(k)r Fm(=)p Fl(n)p Fn(\000)p Fm(1)3106 524 y Fp(\033)3205 540 y Fl(k)r Fn(\000)p Fm(1)3197 570 y Fp(X)3204 782 y Fl(h)p Fm(=0)3501 597 y Fu(1)p 3368 642 316 4 v 3368 662 a Fo(p)p 3451 662 232 4 v 80 x Fq(k)j Fo(\000)e Fq(h)183 1007 y Fo(\024)28 b Fq(C)7 b(\025V)479 1022 y Fl(n;)p Fn(\003)597 926 y Fp(\002)639 1007 y Fu(1)21 b(+)h Fq(x)862 966 y Fm(2)862 1031 y Fl(n)p Fm(+1)p Fl(;)p Fn(\003)1055 926 y Fp(\003)1119 882 y Fl(n)p Fn(\000)p Fm(1)1113 912 y Fp(X)1121 1124 y Fl(k)r Fm(=0)1274 866 y Fp(\032)1621 939 y Fu(1)p 1358 984 574 4 v 1358 1004 a Fo(p)p 1441 1004 71 4 v 84 x Fq(T)14 b Fu(\()p Fq(n)22 b Fo(\000)h Fq(k)s Fu(\))1822 1059 y Fm(3)p Fl(=)p Fm(2)1964 1007 y Fu(+)f Fv(1)2118 1022 y Fl(k)r Fm(=)p Fl(n)p Fn(\000)p Fm(1)2349 866 y Fp(\033)2451 1007 y Fo(\024)28 b Fq(C)7 b(\025V)2747 1022 y Fl(n;)p Fn(\003)2865 926 y Fp(\002)2907 1007 y Fu(1)22 b(+)g Fq(x)3131 966 y Fm(2)3131 1031 y Fl(n)p Fm(+1)p Fl(;)p Fn(\003)3323 926 y Fp(\003)3591 1007 y Fu(\(6.25\))0 1277 y(whic)m(h)33 b(sho)m(ws)h Fq(\021)607 1292 y Fl(n)654 1277 y Fu(\(8)p Fq(;)17 b Fu(3)22 b(+)g(8\))32 b(is)g(negligible.)100 1393 y(It)j(remains)f(to)h (b)s(ound)g Fq(\021)1047 1408 y Fl(n)1094 1393 y Fu(\(8)p Fq(;)17 b(i)p Fu(\),)36 b Fq(i)c Fu(=)g(4)p Fq(;)17 b Fu(5)p Fq(;)g Fu(6)p Fq(;)g Fu(7)p Fq(;)g Fu(9)p Fq(;)g Fu(10.)48 b(The)36 b(negligibilit)m(y)31 b(of)j(those)i(terms)f(follo)m (ws)0 1509 y(directly)d(from)f(\(6.17\),)h(\(6.20\),)g(and)h(the)g(b)s (ounds)g(\(4.18\))o({\(4.22\))o(,)g(\(6.15\))o(.)44 b(W)-8 b(e)33 b(th)m(us)g(conclude)1249 1699 y Fo(j)p Fq(\021)1325 1714 y Fl(n)1372 1699 y Fu(\(8\))p Fo(j)27 b(\024)h Fq(C)7 b(\025)1791 1658 y Fm(2)p Fn(\000)p Fl(\020)1921 1610 y Fo(p)p 2004 1610 V 89 x Fq(T)2091 1619 y Fp(\002)2133 1699 y Fu(1)22 b(+)g Fq(x)2357 1658 y Fm(2)2357 1724 y Fl(n)p Fm(+1)p Fl(;)p Fn(\003)2549 1619 y Fp(\003)3591 1699 y Fu(\(6.26\))0 1939 y Ft(Step)35 b(4.)44 b(The)34 b(term)h Fq(\021)821 1954 y Fl(n)868 1939 y Fu(\(4)22 b(+)g(5\))p Ft(.)43 b Fu(W)-8 b(e)33 b(claim)966 2186 y Fq(\021)1014 2201 y Fl(n)1061 2186 y Fu(\(4)22 b(+)g(5\))27 b(=)1496 2119 y(3)p Fq(\025)p 1496 2164 106 4 v 1524 2255 a Fu(4)1628 2051 y Fp(Z)1728 2077 y Fl(T)1769 2086 y Fg(n)p Fi(+1)1683 2276 y Fl(T)1724 2284 y Fg(n)1892 2186 y Fq(ds)17 b Fo(h)h Fu(\026)-67 b Fq(m)2130 2145 y Fn(0)2130 2211 y Fl(x)2170 2219 y Fg(n)2216 2186 y Fq(;)17 b(p)2309 2201 y Fl(T)2350 2210 y Fg(n)p Fi(+1)2469 2201 y Fn(\000)p Fl(s)2561 2186 y Fq(v)2612 2145 y Fm(\()p Fl(n)p Fm(\))2714 2186 y Fu(\()p Fq(s)p Fu(\))p Fo(i)716 b Fu(\(6.27\))0 2422 y(indeed)790 2644 y Fq(\021)838 2659 y Fl(n)885 2644 y Fu(\(4\))83 b(=)f Fo(\000)1338 2577 y Fu(3)p Fq(\025)p 1338 2621 V 1367 2713 a Fu(4)1471 2509 y Fp(Z)1571 2535 y Fl(T)1612 2544 y Fg(n)p Fi(+1)1526 2734 y Fl(T)1567 2742 y Fg(n)1735 2644 y Fq(ds)1849 2509 y Fp(Z)1948 2535 y Fl(s)1904 2734 y(T)1945 2742 y Fg(n)1992 2644 y Fq(ds)2089 2603 y Fn(0)2128 2644 y Fo(h)18 b Fu(\026)-67 b Fq(m)2252 2603 y Fn(0)2252 2669 y Fl(x)2292 2677 y Fg(n)2339 2644 y Fq(;)2392 2577 y(@)5 b(p)2497 2592 y Fl(s)p Fn(\000)p Fl(s)2618 2573 y Fh(0)p 2392 2621 253 4 v 2456 2713 a Fq(@)g(s)2558 2684 y Fn(0)2655 2644 y Fq(v)2706 2603 y Fm(\()p Fl(n)p Fm(\))2808 2644 y Fu(\()p Fq(s)2892 2603 y Fn(0)2915 2644 y Fu(\))p Fo(i)1093 2935 y Fu(=)1261 2867 y(3)p Fq(\025)p 1261 2912 106 4 v 1290 3003 a Fu(4)1394 2799 y Fp(Z)1493 2825 y Fl(T)1534 2834 y Fg(n)p Fi(+1)1449 3025 y Fl(T)1490 3033 y Fg(n)1658 2935 y Fq(ds)1755 2894 y Fn(0)1795 2799 y Fp(Z)1894 2825 y Fl(T)1935 2834 y Fg(n)p Fi(+1)1850 3025 y Fl(s)1883 3006 y Fh(0)2059 2935 y Fq(ds)17 b Fo(h)g Fu(\026)-66 b Fq(m)2297 2894 y Fn(0)2297 2959 y Fl(x)2337 2967 y Fg(n)2383 2935 y Fq(;)2436 2867 y(@)5 b(p)2541 2882 y Fl(s)p Fn(\000)p Fl(s)2662 2863 y Fh(0)p 2436 2912 253 4 v 2511 3003 a Fq(@)g(s)2699 2935 y(v)2750 2894 y Fm(\()p Fl(n)p Fm(\))2851 2935 y Fu(\()p Fq(s)2935 2894 y Fn(0)2959 2935 y Fu(\))p Fo(i)1093 3225 y Fu(=)1261 3158 y(3)p Fq(\025)p 1261 3202 106 4 v 1290 3293 a Fu(4)1394 3089 y Fp(Z)1493 3116 y Fl(T)1534 3125 y Fg(n)p Fi(+1)1449 3315 y Fl(T)1490 3323 y Fg(n)1658 3225 y Fq(ds)1755 3184 y Fn(0)1795 3225 y Fo(h)18 b Fu(\026)-67 b Fq(m)1919 3184 y Fn(0)1919 3250 y Fl(x)1959 3258 y Fg(n)2005 3225 y Fq(;)17 b(p)2098 3240 y Fl(T)2139 3249 y Fg(n)p Fi(+1)2258 3240 y Fn(\000)p Fl(s)2346 3221 y Fh(0)2372 3225 y Fq(v)2423 3184 y Fm(\()p Fl(n)p Fm(\))2525 3225 y Fu(\()p Fq(s)2609 3184 y Fn(0)2632 3225 y Fu(\))p Fo(i)22 b(\000)g Fq(\021)2878 3240 y Fl(n)2926 3225 y Fu(\(5\))0 3466 y(whic)m(h)33 b(pro)m(v)m(es)h(\(6.27\).)100 3582 y(In)f(order)f(to)g(b)s(ound)h Fq(\021)944 3597 y Fl(n)991 3582 y Fu(\(4)22 b(+)g(5\),)33 b(w)m(e)g(use)h(equation)e(\(5.6\))g(to)g(get)670 3758 y Fq(\021)718 3773 y Fl(n)765 3758 y Fu(\(4)22 b(+)g(5\))83 b(=)g Fq(\021)1349 3773 y Fl(n)1396 3758 y Fu(\(4)22 b(+)g(5)p Fq(;)17 b Fu(2\))k(+)h Fq(\021)1950 3773 y Fl(n)1997 3758 y Fu(\(4)g(+)g(5)p Fq(;)17 b Fu(3)p Fq(;)g Fu(2\))k(+)h Fq(\021)2644 3773 y Fl(n)2691 3758 y Fu(\(4)g(+)g(5)p Fq(;)17 b Fu(3)p Fq(;)g Fu(3\))1301 3997 y(+)1442 3873 y Fm(10)1406 3903 y Fp(X)1428 4102 y Fg(i)p Fi(=1)1403 4158 y Fg(i)p Fh(6)p Fi(=2)p Fg(;)p Fi(3)1579 3997 y Fu([)p Fq(\021)1654 4012 y Fl(n)1701 3997 y Fu(\(4)22 b(+)g(5)p Fq(;)17 b(i)p Fu(\))k(+)h Fq(\021)2239 4012 y Fl(n)2287 3997 y Fu(\(4)f(+)h(5)p Fq(;)17 b Fu(3)p Fq(;)g(i)p Fu(\)])814 b(\(6.28\))0 4320 y(The)37 b(term)f Fq(\021)489 4335 y Fl(n)536 4320 y Fu(\(4)24 b(+)h(5)p Fq(;)17 b Fu(2\))35 b(is)h(in)f Fq(\033)1237 4335 y Fl(n)1285 4320 y Fu(.)54 b(P)m(ostp)s(oning)36 b(the)h(analysis)e(of)h Fq(\021)2586 4335 y Fl(n)2633 4320 y Fu(\(4)24 b(+)h(5)p Fq(;)17 b Fu(3)p Fq(;)g Fu(2\))35 b(let)g(us)i(\014rst)g(sho)m(w)0 4437 y(that)32 b Fq(\021)259 4452 y Fl(n)306 4437 y Fu(\(4)22 b(+)g(5)p Fq(;)17 b(i)p Fu(\),)33 b Fq(i)28 b Fo(6)p Fu(=)f(2)p Fq(;)17 b Fu(3)32 b(are)g(negligible.)0 4613 y Ft(The)i(terms)h Fq(\021)520 4628 y Fl(n)567 4613 y Fu(\(4)22 b(+)g(5)p Fq(;)17 b(i)p Fu(\))p Ft(,)34 b Fq(i)28 b Fo(6)p Fu(=)g(2)p Fq(;)17 b Fu(3)p Ft(.)42 b Fu(W)-8 b(e)33 b(ha)m(v)m(e)780 4860 y Fq(\021)828 4875 y Fl(n)875 4860 y Fu(\(4)22 b(+)g(5)p Fq(;)17 b Fu(1\))26 b(=)1402 4793 y(3)p Fq(\025)p 1402 4837 V 1430 4928 a Fu(4)1534 4724 y Fp(Z)1634 4751 y Fl(T)1675 4760 y Fg(n)p Fi(+1)1589 4950 y Fl(T)1630 4958 y Fg(n)1799 4860 y Fq(ds)17 b Fo(h)g Fu(\026)-66 b Fq(m)2037 4819 y Fn(0)2037 4885 y Fl(x)2077 4893 y Fg(n)2122 4860 y Fq(;)17 b(p)2215 4875 y Fl(T)2256 4884 y Fg(n)p Fi(+1)2375 4875 y Fn(\000)p Fl(s)2467 4860 y Fq(g)2518 4809 y Fm(\()p Fl(n)p Fm(\))2514 4887 y Fl(s)p Fn(\000)p Fl(T)2643 4895 y Fg(n)2689 4860 y Fq(v)2740 4819 y Fm(\()p Fl(n)p Fm(\))2842 4860 y Fu(\()p Fq(T)2937 4875 y Fl(n)2984 4860 y Fu(\))p Fo(i)0 5112 y Fu(since)33 b Fq(v)290 5076 y Fm(\()p Fl(n)p Fm(\))392 5112 y Fu(\()p Fq(T)487 5127 y Fl(n)534 5112 y Fu(\))f(is)g(orthogonal)f(to)50 b(\026)-67 b Fq(m)1394 5076 y Fn(0)1394 5137 y Fl(x)1434 5145 y Fg(n)1481 5112 y Fu(,)32 b(see)i(\(3.2\),)e(w)m(e)i(can)f(apply) g(\(4.11\))f(to)g(deduce)j(that)1022 5368 y Fo(j)p Fq(\021)1098 5383 y Fl(n)1145 5368 y Fu(\(4)22 b(+)g(5)p Fq(;)17 b Fu(1\))p Fo(j)27 b(\024)h Fq(C)7 b(\025)1843 5232 y Fp(Z)1942 5258 y Fl(T)1983 5267 y Fg(n)p Fi(+1)1898 5458 y Fl(T)1939 5466 y Fg(n)2107 5368 y Fq(ds)17 b(e)2266 5326 y Fn(\000)p Fl(\013)p Fm(\()p Fl(s)p Fn(\000)p Fl(T)2522 5334 y Fg(n)2564 5326 y Fm(\))2595 5368 y Fq(\025)2652 5326 y Fm(1)p Fn(\000)p Fm(2)p Fl(\020)1882 5532 y Fk(28)p eop %%Page: 29 30 29 29 bop 0 269 a Fu(ha)m(ving)30 b(used)h(that)g Fo(k)p Fq(v)843 232 y Fm(\()p Fl(n)p Fm(\))944 269 y Fu(\()p Fq(T)1039 284 y Fl(n)1086 269 y Fu(\))p Fo(k)1174 284 y Fn(1)1276 269 y Fo(\024)d Fq(C)7 b(\025)1515 232 y Fm(1)p Fn(\000)p Fm(2)p Fl(\020)1680 269 y Fu(,)31 b(see)h(\(4.48\))o (.)43 b(Th)m(us)31 b Fo(j)p Fq(\021)2533 284 y Fl(n)2580 269 y Fu(\(4)17 b(+)h(5)p Fq(;)f Fu(1\))p Fo(j)26 b(\024)i Fq(C)7 b(\025)3251 232 y Fm(2)p Fn(\000)p Fm(2)p Fl(\020)3447 269 y Fu(whic)m(h)30 b(for)0 385 y Fq(\020)40 b Fu(small)30 b(is)i(negligible.)100 501 y(W)-8 b(e)33 b(next)g(sho)m(w)h Fq(\021)777 516 y Fl(n)824 501 y Fu(\(4)22 b(+)g(5)p Fq(;)17 b Fu(8\))31 b(is)i(negligible.)40 b(W)-8 b(e)33 b(ha)m(v)m(e)302 775 y Fq(\021)350 790 y Fl(n)397 775 y Fu(\(4)22 b(+)g(5)p Fq(;)17 b Fu(8\))82 b(=)1035 707 y(3)p 1035 752 49 4 v 1035 843 a(4)1094 775 y Fq(\025)1151 734 y Fm(2)1212 650 y Fl(n)p Fn(\000)p Fm(1)1207 680 y Fp(X)1215 892 y Fl(k)r Fm(=0)1367 775 y Fq( )1430 790 y Fl(k)1490 639 y Fp(Z)1590 666 y Fl(T)1631 675 y Fg(n)p Fi(+1)1546 865 y Fl(T)1587 873 y Fg(n)1755 775 y Fq(dt)1858 639 y Fp(Z)1957 666 y Fl(t)1912 865 y(T)1953 873 y Fg(n)2000 775 y Fq(ds)1225 1114 y Fo(\002)1324 944 y Fp(*)1423 1114 y Fu(\026)-67 b Fq(m)1490 1073 y Fn(0)1490 1139 y Fl(x)1530 1147 y Fg(n)1576 1114 y Fq(;)17 b(p)1669 1129 y Fl(T)1710 1138 y Fg(n)p Fi(+1)1830 1129 y Fn(\000)p Fl(t)1914 1114 y Fq(g)1965 1063 y Fm(\()p Fl(n)p Fm(\))1961 1137 y Fl(t)p Fn(\000)p Fl(s)2094 944 y Fp(")2153 1114 y Fq(p)2202 1129 y Fl(s)p Fn(\000)p Fl(T)2331 1141 y Fg(k)q Fi(+1)2468 1114 y Fu(\026)-68 b Fq(m)2534 1073 y Fn(0)2534 1139 y Fl(x)2574 1151 y Fg(k)2639 1114 y Fo(\000)3045 1047 y Fu(2)p 2748 1091 641 4 v 2748 1111 a Fp(p)p 2848 1111 542 4 v 85 x Fu(2)p Fq(\031)t Fu(\()p Fq(s)22 b Fo(\000)g Fq(T)3218 1211 y Fl(k)r Fm(+1)3351 1196 y Fu(\))3399 944 y Fp(#+)3591 1114 y Fu(\(6.29\))100 1411 y(W)-8 b(e)28 b(no)m(w)h(use)g(the)f(decomp)s(osition)e(\(4.32\))h (for)h Fq(g)1900 1360 y Fm(\()p Fl(n)p Fm(\))1896 1433 y Fl(t)p Fn(\000)p Fl(s)2041 1411 y Fu(ab)s(o)m(v)m(e.)43 b(Recalling)26 b(\(6.19\),)j(the)f(term)f(obtained)0 1549 y(b)m(y)33 b(replacing)f Fq(g)604 1498 y Fm(\()p Fl(n)p Fm(\))600 1572 y Fl(t)p Fn(\000)p Fl(s)717 1549 y Fu(\()p Fq(x;)17 b(y)t Fu(\))31 b(b)m(y)j(\(3)p Fq(=)p Fu(4\))17 b(\026)-66 b Fq(m)1419 1513 y Fn(0)1419 1574 y Fl(x)1459 1582 y Fg(n)1505 1549 y Fu(\()p Fq(x)p Fu(\))18 b(\026)-67 b Fq(m)1721 1513 y Fn(0)1721 1574 y Fl(x)1761 1582 y Fg(n)1808 1549 y Fu(\()p Fq(y)t Fu(\))31 b(in)h(\(6.29\))g(is)g (b)s(ounded)h(b)m(y)219 1914 y Fq(C)7 b(\025)375 1790 y Fl(n)p Fn(\000)p Fm(1)369 1820 y Fp(X)377 2032 y Fl(k)r Fm(=0)530 1914 y Fq(B)609 1864 y Fm(\(1\))604 1942 y Fl(n;k)709 1914 y Fo(j)p Fq( )800 1929 y Fl(k)843 1914 y Fo(j)888 1779 y Fp(Z)987 1805 y Fl(T)1028 1814 y Fg(n)p Fi(+1)942 2004 y Fl(T)983 2012 y Fg(n)1151 1914 y Fq(dt)17 b Fo(h)h Fu(\026)-67 b Fq(m)1378 1873 y Fn(0)1378 1939 y Fl(x)1418 1947 y Fg(n)1464 1914 y Fq(;)17 b(p)1557 1929 y Fl(T)1598 1938 y Fg(n)p Fi(+1)1718 1929 y Fn(\000)p Fl(t)1820 1914 y Fu(\026)-67 b Fq(m)1887 1873 y Fn(0)1887 1939 y Fl(x)1927 1947 y Fg(n)1974 1914 y Fo(i)219 2257 y(\024)28 b Fq(C)401 2167 y Fo(p)p 484 2167 71 4 v 90 x Fq(T)14 b(\025)612 2216 y Fm(2)651 2257 y Fq(V)708 2272 y Fl(n;)p Fn(\003)826 2176 y Fp(\002)868 2257 y Fu(1)22 b(+)g Fq(x)1092 2216 y Fm(2)1092 2281 y Fl(n)p Fm(+1)p Fl(;)p Fn(\003)1284 2176 y Fp(\003)1348 2132 y Fl(n)p Fn(\000)p Fm(1)1342 2162 y Fp(X)1350 2374 y Fl(k)r Fm(=0)1503 2116 y Fp(\032)1850 2189 y Fu(1)p 1588 2234 574 4 v 1588 2254 a Fo(p)p 1671 2254 71 4 v 84 x Fq(T)13 b Fu(\()p Fq(n)23 b Fo(\000)f Fq(k)s Fu(\))2051 2309 y Fm(3)p Fl(=)p Fm(2)2193 2257 y Fu(+)g Fv(1)2347 2272 y Fl(k)r Fm(=)p Fl(n)p Fn(\000)p Fm(1)2578 2116 y Fp(\033)2680 2257 y Fo(\024)28 b Fq(C)2862 2167 y Fo(p)p 2945 2167 V 90 x Fq(T)14 b(\025)3073 2216 y Fm(2)3113 2257 y Fq(V)3170 2272 y Fl(n;)p Fn(\003)3288 2176 y Fp(\002)3329 2257 y Fu(1)22 b(+)g Fq(x)3553 2216 y Fm(2)3553 2281 y Fl(n)p Fm(+1)p Fl(;)p Fn(\003)3746 2176 y Fp(\003)0 2556 y Fu(where)35 b(w)m(e)g(used)g(\(6.20\))o(.)47 b(W)-8 b(e)34 b(next)g(consider)g(the)g(case)h(when)g(w)m(e)g(replace)e Fq(g)2905 2506 y Fm(\()p Fl(n)p Fm(\))2901 2579 y Fl(t)p Fn(\000)p Fl(s)3052 2556 y Fu(in)g(\(6.29\))g(b)m(y)h Fq(g)3637 2506 y Fm(\()p Fl(n;)p Fn(?)p Fm(\))3633 2579 y Fl(t)p Fn(\000)p Fl(s)3813 2556 y Fu(.)0 2673 y(Since,)f(b)m(y)i (\(4.14\))c(and)i(\(4.33\))o(,)252 2786 y Fp(\015)252 2846 y(\015)252 2906 y(\015)307 2901 y Fq(p)356 2916 y Fl(T)397 2925 y Fg(n)p Fi(+1)517 2916 y Fn(\000)p Fl(t)601 2901 y Fq(g)652 2850 y Fm(\()p Fl(n;)p Fn(?)p Fm(\))648 2923 y Fl(t)p Fn(\000)p Fl(s)828 2901 y Fq(p)877 2916 y Fl(s)p Fn(\000)p Fl(T)1006 2928 y Fg(k)q Fi(+1)1143 2901 y Fu(\026)-67 b Fq(m)1210 2860 y Fn(0)1210 2925 y Fl(x)1250 2937 y Fg(k)1292 2786 y Fp(\015)1292 2846 y(\015)1292 2906 y(\015)1348 2970 y Fn(1)1450 2901 y Fo(\024)28 b Fq(C)7 b(e)1677 2860 y Fn(\000)p Fl(\013)p Fm(\()p Fl(t)p Fn(\000)p Fl(s)p Fm(\))1966 2816 y Fp(\015)1966 2876 y(\015)2021 2901 y Fq(p)2070 2916 y Fl(s)p Fn(\000)p Fl(t)2183 2928 y Fg(k)q Fi(+1)2321 2901 y Fu(\026)-67 b Fq(m)2388 2860 y Fn(0)2388 2925 y Fl(x)2428 2937 y Fg(k)2470 2816 y Fp(\015)2470 2876 y(\015)2525 2941 y Fn(1)2628 2901 y Fo(\024)28 b Fq(C)7 b(e)2855 2860 y Fn(\000)p Fl(\013)p Fm(\()p Fl(t)p Fn(\000)p Fl(s)p Fm(\))3333 2833 y Fu(1)p 3137 2878 441 4 v 3137 2898 a Fo(p)p 3220 2898 358 4 v 73 x Fq(s)22 b Fo(\000)h Fq(T)3445 2986 y Fl(k)r Fm(+1)0 3166 y Fu(w)m(e)34 b(can)e(b)s(ound)h(\(6.29\))f (\(with)g Fq(g)1216 3115 y Fm(\()p Fl(n)p Fm(\))1212 3188 y Fl(t)p Fn(\000)p Fl(s)1362 3166 y Fu(replaced)g(b)m(y)i Fq(g)1933 3115 y Fm(\()p Fl(n;)p Fn(?)p Fm(\))1929 3188 y Fl(t)p Fn(\000)p Fl(s)2108 3166 y Fu(\))f(as)653 3531 y Fq(C)7 b(\025)787 3490 y Fm(2)848 3406 y Fl(n)p Fn(\000)p Fm(1)843 3436 y Fp(X)851 3648 y Fl(k)r Fm(=0)1003 3531 y Fo(j)p Fq( )1094 3546 y Fl(k)1137 3531 y Fo(j)1182 3395 y Fp(Z)1281 3422 y Fl(T)1322 3431 y Fg(n)p Fi(+1)1237 3621 y Fl(T)1278 3629 y Fg(n)1446 3531 y Fq(dt)1549 3395 y Fp(Z)1648 3422 y Fl(t)1604 3621 y(T)1645 3629 y Fg(n)1691 3531 y Fq(ds)22 b(e)1855 3490 y Fn(\000)p Fl(\013)p Fm(\()p Fl(t)p Fn(\000)p Fl(s)p Fm(\))2334 3464 y Fu(1)p 2138 3508 441 4 v 2138 3528 a Fo(p)p 2221 3528 358 4 v 73 x Fq(s)g Fo(\000)g Fq(T)2445 3616 y Fl(k)r Fm(+1)653 3873 y Fo(\024)28 b Fq(C)835 3784 y Fo(p)p 918 3784 71 4 v 89 x Fq(T)14 b(\025)1046 3832 y Fm(2)1085 3873 y Fq(V)1142 3888 y Fl(n;)p Fn(\003)1266 3749 y Fl(n)p Fn(\000)p Fm(1)1261 3778 y Fp(X)1268 3991 y Fl(k)r Fm(=0)1421 3733 y Fp(\032)1496 3873 y Fv(1)1552 3888 y Fl(k)r Fm(=)p Fl(n)p Fn(\000)p Fm(1)1805 3873 y Fu(+)2047 3806 y(1)p 1913 3850 318 4 v 1913 3870 a Fo(p)p 1996 3870 235 4 v 81 x Fq(n)22 b Fo(\000)h Fq(k)2240 3733 y Fp(\033)2342 3873 y Fo(\024)28 b Fq(C)2541 3762 y Fp(h)2588 3873 y Fq(\025)2645 3832 y Fm(3)p Fn(\000)p Fl(\020)2775 3873 y Fq(T)35 b Fu(+)22 b Fq(\025)3022 3832 y Fm(2)p Fn(\000)p Fl(\020)3152 3784 y Fo(p)p 3235 3784 71 4 v 89 x Fq(T)3306 3762 y Fp(i)0 4141 y Fu(whic)m(h)33 b(is)f(negligible)e(for)i Fq(\020)39 b Fu(small)31 b(enough.)100 4258 y(It)41 b(remains)f(to)g (sho)m(w)i Fq(\021)1011 4273 y Fl(n)1058 4258 y Fu(\(4)28 b(+)f(5)p Fq(;)17 b(i)p Fu(\),)43 b Fq(i)f Fu(=)g(4)p Fq(;)17 b Fu(5)p Fq(;)g Fu(6)p Fq(;)g Fu(7)p Fq(;)g Fu(9)p Fq(;)g Fu(10)37 b(is)k(negligible.)65 b(This)41 b(follo)m(ws)f(from)g (the)0 4374 y(b)s(ounds)33 b(\(4.18\){\(4.22\))o(,)f(\(6.15\),)g(and)h (\(6.27\))o(.)0 4553 y Ft(The)h(terms)h Fq(\021)520 4568 y Fl(n)567 4553 y Fu(\(4)22 b(+)g(5)p Fq(;)17 b Fu(3)p Fq(;)g(i)p Fu(\))p Ft(.)42 b Fu(W)-8 b(e)33 b(write)1364 4827 y Fq(\021)1412 4842 y Fl(n)1459 4827 y Fu(\(4)22 b(+)g(5)p Fq(;)17 b Fu(3\))27 b(=)1982 4703 y Fl(n)p Fn(\000)p Fm(1)1976 4732 y Fp(X)1984 4945 y Fl(k)r Fm(=0)2164 4827 y Fq(A)2237 4776 y Fm(\(2\))2237 4855 y Fl(n;k)2370 4827 y Fq( )2433 4842 y Fl(k)3591 4827 y Fu(\(6.30\))0 5095 y(where)206 5341 y Fq(A)279 5290 y Fm(\(2\))279 5369 y Fl(n;k)412 5341 y Fu(=)526 5274 y(3)p Fq(\025)632 5238 y Fm(2)p 526 5318 146 4 v 574 5410 a Fu(4)697 5206 y Fp(Z)797 5232 y Fl(T)838 5241 y Fg(n)p Fi(+1)753 5431 y Fl(T)794 5439 y Fg(n)962 5341 y Fq(ds)1059 5300 y Fn(00)1117 5206 y Fp(Z)1217 5232 y Fl(T)1258 5241 y Fg(n)p Fi(+1)1173 5431 y Fl(s)1206 5412 y Fh(00)1382 5341 y Fq(ds)1479 5300 y Fn(0)1535 5171 y Fp(*)1633 5341 y Fu(\026)-67 b Fq(m)1700 5300 y Fn(0)1700 5366 y Fl(x)1740 5374 y Fg(n)1787 5341 y Fq(;)17 b(p)1880 5356 y Fl(T)1921 5365 y Fg(n)p Fi(+1)2040 5356 y Fn(\000)p Fl(s)2128 5337 y Fh(0)2154 5341 y Fq(g)2205 5290 y Fm(\()p Fl(n)p Fm(\))2201 5370 y Fl(s)2234 5351 y Fh(0)2256 5370 y Fn(\000)p Fl(s)2344 5351 y Fh(00)2715 5274 y Fu(2)p 2398 5318 684 4 v 2398 5338 a Fp(p)p 2498 5338 584 4 v 86 x Fu(2)p Fq(\031)t Fu(\()p Fq(s)2690 5395 y Fn(00)2754 5424 y Fo(\000)22 b Fq(T)2910 5439 y Fl(k)r Fm(+1)3043 5424 y Fu(\))3091 5171 y Fp(+)3188 5341 y Fv(1)3244 5356 y Fl(k)r()f Fu(0,)33 b(and)f Fq(g)720 1013 y Fm(\()p Fl(n)p Fm(\))716 1086 y Fl(t)821 1063 y Fu(\()p Fq(x;)17 b(y)t Fu(\))27 b Fo(\025)h Fu(0\),)33 b(w)m(e)g(ha)m(v)m(e)183 1315 y Fo(j)p Fq(\021)259 1330 y Fl(n)306 1315 y Fu(\(9)p Fo(j)p Fu(8\))p Fo(j)82 b(\024)h Fq(\025C)7 b(V)969 1330 y Fl(n;)p Fn(\003)1093 1191 y Fl(n)p Fn(\000)p Fm(1)1087 1221 y Fp(X)1095 1433 y Fl(k)r Fm(=0)1248 1315 y Fo(j)p Fq( )1339 1330 y Fl(k)1382 1315 y Fo(j)1427 1180 y Fp(Z)1525 1206 y Fl(T)1566 1215 y Fg(n)p Fi(+1)1481 1405 y Fl(T)1522 1413 y Fg(n)1690 1315 y Fq(dt)1793 1180 y Fp(Z)1892 1206 y Fl(t)1848 1405 y(T)1889 1413 y Fg(n)1936 1315 y Fq(ds)2255 1248 y Fu(1)p 2059 1292 441 4 v 2059 1312 a Fo(p)p 2142 1312 358 4 v 74 x Fq(s)22 b Fo(\000)h Fq(T)2367 1401 y Fl(k)r Fm(+1)977 1616 y Fo(\002)1077 1480 y Fp(Z)1177 1616 y Fq(dx)17 b(dy)i(dz)45 b Fu(\026)-68 b Fq(m)1625 1575 y Fn(0)1625 1640 y Fl(x)1665 1648 y Fg(n)1712 1616 y Fu(\()p Fq(x)p Fu(\))p Fq(g)1894 1565 y Fm(\()p Fl(n)p Fm(\))1890 1638 y Fl(t)p Fn(\000)p Fl(s)2007 1616 y Fu(\()p Fq(x;)17 b(y)t Fu(\))2251 1501 y Fp(\014)2251 1561 y(\014)2251 1621 y(\014)2283 1616 y Fq(e)2328 1575 y Fn(\000)p Fm(\()p Fl(y)r Fn(\000)p Fl(z)s Fm(\))2565 1551 y Fi(2)2601 1575 y Fl(=)p Fm([2\()p Fl(s)p Fn(\000)p Fl(T)2847 1587 y Fg(k)q Fi(+1)2962 1575 y Fm(\)])3035 1616 y Fo(\000)23 b Fu(1)3184 1501 y Fp(\014)3184 1561 y(\014)3184 1621 y(\014)3252 1616 y Fu(\026)-67 b Fq(m)3319 1575 y Fn(0)3319 1640 y Fl(x)3359 1652 y Fg(k)3401 1616 y Fu(\()p Fq(z)t Fu(\))65 b(\(6.58\))0 1852 y(Since)33 b Fq(g)306 1801 y Fm(\()p Fl(n)p Fm(\))302 1874 y Fl(t)440 1852 y Fu(is)f(self{adjoin)m (t)g(in)g Fq(L)1237 1867 y Fm(2)1276 1852 y Fu(\()p Fj(R)5 b Fq(;)17 b(dx)p Fu(\))39 b(w)m(e)34 b(ha)m(v)m(e)1976 1772 y Fp(R)2042 1852 y Fq(dx)h Fu(\026)-67 b Fq(m)2250 1816 y Fn(0)2250 1876 y Fl(x)2290 1884 y Fg(n)2337 1852 y Fu(\()p Fq(x)p Fu(\))p Fq(g)2519 1801 y Fm(\()p Fl(n)p Fm(\))2515 1874 y Fl(t)p Fn(\000)p Fl(s)2632 1852 y Fu(\()p Fq(x;)17 b(y)t Fu(\))27 b(=)46 b(\026)-67 b Fq(m)3075 1816 y Fn(0)3075 1876 y Fl(x)3115 1884 y Fg(n)3162 1852 y Fu(\()p Fq(y)t Fu(\).)43 b(Then,)34 b(from)0 1968 y(\(6.19\))e(and)g (\(6.20\))g(it)g(follo)m(ws)f(that)669 2220 y Fo(j)p Fq(\021)745 2235 y Fl(n)792 2220 y Fu(\(9)p Fo(j)p Fu(8\))p Fo(j)26 b(\024)i Fq(C)7 b(V)1287 2235 y Fn(\003)1326 2220 y Fu(\()p Fq(\034)k Fu(\))p Fq(T)1549 2095 y Fl(n)p Fn(\000)p Fm(1)1543 2125 y Fp(X)1551 2337 y Fl(k)r Fm(=0)1704 2220 y Fq(B)1783 2169 y Fm(\(1\))1778 2248 y Fl(n;k)1883 2220 y Fo(j)p Fq( )1974 2235 y Fl(k)2017 2220 y Fo(j)27 b(\024)h Fq(C)7 b(\025V)2368 2235 y Fn(\003)2407 2220 y Fu(\()p Fq(\034)k Fu(\))2536 2179 y Fm(2)2576 2220 y Fq(T)2663 2139 y Fp(\000)2709 2220 y Fu(1)22 b(+)g Fq(x)2933 2179 y Fm(2)2933 2244 y Fl(n)p Fm(+1)p Fl(;)p Fn(\003)3125 2139 y Fp(\001)0 2472 y Fu(whic)m(h)33 b(is)f(negligible)e (b)s(ecause)k(of)39 b(\(4.7\))o(.)100 2588 y(W)-8 b(e)33 b(next)g(consider)g Fq(\021)t Fu(\(9)p Fo(j)p Fu(3)p Fq(;)17 b(i)p Fu(\),)31 b Fq(i)d Fo(6)p Fu(=)g(2)p Fq(;)17 b Fu(3,)140 2840 y Fq(\021)188 2855 y Fl(n)235 2840 y Fu(\(9)p Fo(j)p Fu(3)p Fq(;)g(i)p Fu(\))27 b(=)655 2772 y(9)p Fq(\025)p 655 2817 106 4 v 683 2908 a Fu(4)792 2715 y Fl(n)p Fn(\000)p Fm(1)787 2745 y Fp(X)794 2957 y Fl(k)r Fm(=0)947 2840 y Fq(\021)995 2855 y Fl(k)1038 2840 y Fu(\()p Fq(i)p Fu(\))1164 2704 y Fp(Z)1264 2730 y Fl(T)1305 2739 y Fg(n)p Fi(+1)1219 2930 y Fl(T)1260 2938 y Fg(n)1428 2840 y Fq(dt)1531 2704 y Fp(Z)1630 2730 y Fl(t)1586 2930 y(T)1627 2938 y Fg(n)1674 2840 y Fq(ds)1804 2669 y Fp(*)1902 2840 y Fu(\026)-67 b Fq(m)1969 2799 y Fn(0)1969 2864 y Fl(x)2009 2872 y Fg(n)2056 2840 y Fq(;)35 b Fu(\026)-67 b Fq(m)2185 2855 y Fl(x)2225 2863 y Fg(n)2271 2840 y Fq(v)2322 2799 y Fm(\()p Fl(n)p Fm(\))2424 2840 y Fu(\()p Fq(s)p Fu(\))p Fq(g)2597 2789 y Fm(\()p Fl(n)p Fm(\))2593 2862 y Fl(t)p Fn(\000)p Fl(s)3015 2772 y Fu(2)p 2719 2817 641 4 v 2719 2837 a Fp(p)p 2819 2837 542 4 v 85 x Fu(2)p Fq(\031)t Fu(\()p Fq(s)22 b Fo(\000)g Fq(T)3189 2937 y Fl(k)r Fm(+1)3322 2922 y Fu(\))3370 2669 y Fp(+)3591 2840 y Fu(\(6.59\))0 3091 y(Note)33 b(that,)f(from)f(\(6.8\))i Fq(\021)t Fu(\(9)p Fo(j)p Fu(3)p Fq(;)17 b Fu(1\))25 b(=)j(0)k(and)h(that,)f(for)g Fq(i)c Fo(\025)h Fu(4,)j(from)f(\(5.11\),)77 3343 y Fo(j)p Fq(\021)153 3358 y Fl(n)200 3343 y Fu(\(9)p Fo(j)p Fu(3)p Fq(;)17 b(i)p Fu(\))p Fo(j)26 b(\024)i Fq(C)7 b(\025V)829 3358 y Fl(n;)p Fn(\003)953 3219 y Fl(n)p Fn(\000)p Fm(1)948 3249 y Fp(X)955 3461 y Fl(k)r Fm(=0)1108 3343 y Fo(j)p Fq(\021)1184 3358 y Fl(k)1227 3343 y Fu(\()p Fq(i)p Fu(\))p Fo(j)1381 3208 y Fp(Z)1480 3234 y Fl(T)1521 3243 y Fg(n)p Fi(+1)1435 3433 y Fl(T)1476 3441 y Fg(n)1644 3343 y Fq(dt)1747 3208 y Fp(Z)1846 3234 y Fl(t)1802 3433 y(T)1843 3441 y Fg(n)1890 3343 y Fq(ds)2209 3276 y Fu(1)p 2013 3320 441 4 v 2013 3340 a Fo(p)p 2096 3340 358 4 v 74 x Fq(s)22 b Fo(\000)h Fq(T)2321 3429 y Fl(k)r Fm(+1)2492 3343 y Fo(\024)28 b Fq(T)14 b(C)7 b(V)2802 3358 y Fl(n;)p Fn(\003)2925 3219 y Fl(n)p Fn(\000)p Fm(1)2920 3249 y Fp(X)2927 3461 y Fl(k)r Fm(=0)3080 3343 y Fq(A)3153 3358 y Fl(n;k)3258 3343 y Fo(j)p Fq(\021)3334 3358 y Fl(k)3377 3343 y Fu(\()p Fq(i)p Fu(\))p Fo(j)77 b Fu(\(6.60\))0 3595 y(Since)37 b Fo(k)p Fq(A)p Fo(k)432 3610 y Fl(n)515 3595 y Fo(\024)f Fq(C)7 b Fu(,)38 b(b)m(y)g(using)f(again)f(\(4.7\))o(,)j(\(4.18\))o(,)f (\(4.19\),)g(and)g(\(4.20\))o({\(6.9\))o(,)h(w)m(e)f(sho)m(w)g Fq(\021)t Fu(\(9)p Fo(j)p Fu(3)p Fq(;)17 b(i)p Fu(\),)0 3711 y Fq(i)28 b Fo(\025)g Fu(4)k(is)g(negligible.)41 b(Next,)34 b(using)f(\(5.12\))o(,)g(\(5.15\))f(and)h(\(6.59\))o(,)5 3963 y Fq(\021)t Fu(\(9)p Fo(j)p Fu(3)p Fq(;)17 b Fu(3\))25 b(=)j Fo(\000)568 3896 y Fu(9)p Fq(\025)p 568 3940 106 4 v 597 4031 a Fu(4)706 3839 y Fl(n)p Fn(\000)p Fm(1)701 3868 y Fp(X)708 4081 y Fl(k)r Fm(=0)861 3882 y Fp(\000)907 3963 y Fu(\(1)22 b(+)g Fq(A)p Fu(\))1225 3922 y Fn(\000)p Fm(1)1319 3963 y Fq(A\021)1444 3882 y Fp(\001)1490 4002 y Fl(k)1549 3827 y Fp(Z)1649 3854 y Fl(T)1690 3863 y Fg(n)p Fi(+1)1604 4053 y Fl(T)1645 4061 y Fg(n)1813 3963 y Fq(dt)1916 3827 y Fp(Z)2015 3854 y Fl(t)1971 4053 y(T)2012 4061 y Fg(n)2059 3963 y Fq(ds)2189 3793 y Fp(*)2287 3963 y Fu(\026)-67 b Fq(m)2354 3922 y Fn(0)2354 3988 y Fl(x)2394 3996 y Fg(n)2441 3963 y Fq(;)35 b Fu(\026)-67 b Fq(m)2570 3978 y Fl(x)2610 3986 y Fg(n)2656 3963 y Fq(v)2707 3922 y Fm(\()p Fl(n)p Fm(\))2809 3963 y Fu(\()p Fq(s)p Fu(\))p Fq(g)2982 3912 y Fm(\()p Fl(n)p Fm(\))2978 3986 y Fl(t)p Fn(\000)p Fl(s)3400 3896 y Fu(2)p 3104 3940 641 4 v 3104 3960 a Fp(p)p 3204 3960 542 4 v 85 x Fu(2)p Fq(\031)t Fu(\()p Fq(s)22 b Fo(\000)g Fq(T)3574 4060 y Fl(k)r Fm(+1)3707 4045 y Fu(\))3755 3793 y Fp(+)3591 4174 y Fu(\(6.61\))0 4290 y(whence,)35 b(recalling)30 b Fo(k)p Fu(\(1)21 b(+)h Fq(A)p Fu(\))1126 4254 y Fn(\000)p Fm(1)1221 4290 y Fo(k)1271 4305 y Fl(n)1345 4290 y Fo(\024)28 b Fq(C)7 b Fu(,)33 b Fo(k)p Fq(A)p Fo(k)1760 4305 y Fl(n)1834 4290 y Fo(\024)c Fq(C)7 b Fu(,)32 b Fq(\021)2124 4305 y Fl(n)2171 4290 y Fu(\(1\))27 b(=)h(0,)475 4542 y Fo(j)p Fq(\021)t Fu(\(9)p Fo(j)p Fu(3)p Fq(;)17 b Fu(3\))p Fo(j)876 4571 y Fl(n)1006 4542 y Fo(\024)83 b Fq(C)7 b(\025T)1387 4371 y Fp( )1466 4542 y Fu(sup)1471 4624 y Fl(k)r Fn(\024)p Fl(n)1630 4542 y Fo(j)o Fu(\()p Fq(A\021)t Fu(\(2\)\))1983 4571 y Fl(k)2025 4542 y Fo(j)22 b Fu(+)2210 4417 y Fm(10)2173 4447 y Fp(X)2188 4657 y Fl(i)p Fm(=4)2334 4542 y Fo(j)o Fq(\021)t Fu(\()p Fq(i)p Fu(\))p Fo(j)2550 4571 y Fl(n)2597 4371 y Fp(!)2692 4542 y Fq(V)2749 4557 y Fl(n;)p Fn(\003)2873 4417 y Fl(n)p Fn(\000)p Fm(1)2868 4447 y Fp(X)2875 4659 y Fl(k)r Fm(=0)3120 4390 y Fo(p)p 3203 4390 71 4 v 84 x Fq(T)p 3038 4519 318 4 v 3038 4539 a Fo(p)p 3121 4539 235 4 v 80 x Fq(n)g Fo(\000)h Fq(k)1006 4794 y Fo(\024)83 b Fq(C)7 b(T)1330 4713 y Fp(\002)1372 4794 y Fq(\025)1429 4753 y Fm(2)p Fn(\000)p Fl(\020)1559 4794 y Fq(T)35 b Fu(+)22 b Fq(\025)1806 4753 y Fm(2)p Fn(\000)p Fm(2)p Fl(\020)1972 4794 y Fq(T)2043 4753 y Fm(2)2098 4713 y Fp(\000)2144 4794 y Fu(1)g(+)g Fq(x)2368 4753 y Fm(2)2368 4818 y Fl(n)p Fm(+1)p Fl(;)p Fn(\003)2560 4713 y Fp(\001\003)2664 4794 y Fq(V)2721 4809 y Fl(n;)p Fn(\003)0 4956 y Fu(where)37 b(w)m(e)h(used)f(the)g(b)s(ounds)g(\(6.8\))o(,)g(\(6.15\),)g(\(6.26\))o (,)g(\(6.9\),)g(and)f(\(4.18\))o({\(4.21\))g(to)f(estimate)h Fo(j)o Fq(\021)t Fu(\()p Fq(i)p Fu(\))p Fo(j)0 5072 y Fu(together)f(with)f(the)h(fact)f(that)g(w)m(e)i(are)f(on)f(the)h(set)g Fo(B)2025 5087 y Fl(\025;\034)2164 5072 y Fu(\(see)h(\(6.2\))d(and)i (\(6.6\)\).)49 b(Hence)36 b Fq(\021)t Fu(\(9)p Fo(j)p Fu(3)p Fq(;)17 b Fu(3\))32 b(is)0 5188 y(negligible.)100 5316 y(Since)45 b(on)h(the)g(set)g Fo(B)927 5331 y Fl(\025;\034)1077 5316 y Fu(\000)1138 5265 y Fm(\()p Fl(n)p Fm(\))1138 5341 y(2)1240 5316 y Fu(\()p Fq(t)p Fu(\))f(has)h(the)g(same)f(order)h (as)g Fq(v)2473 5280 y Fm(\()p Fl(n)p Fm(\))2574 5316 y Fu(\()p Fq(t)p Fu(\))g(\(see)g(\(4.5\)\))f(w)m(e)i(can)f(b)s(ound)0 5433 y Fq(\021)t Fu(\(9)p Fo(j)p Fu(2;)17 b Fq(i)p Fu(\),)31 b Fq(\021)t Fu(\(9)p Fo(j)p Fu(2;)17 b(3)p Fq(;)g(i)p Fu(\),)31 b Fq(i)d Fo(6)p Fu(=)g(2)p Fq(;)17 b Fu(3,)32 b(and)g Fq(\021)t Fu(\(9)p Fo(j)p Fu(2;)17 b(3)p Fq(;)g Fu(3\))31 b(as)h(ab)s(o)m(v)m(e.)1882 5532 y Fk(34)p eop %%Page: 35 36 35 35 bop 100 283 a Fu(W)-8 b(e)22 b(are)g(left)f(with)h(the)h(b)s (ounds)f(of)g Fq(\021)t Fu(\(9)p Fo(j)p Fu(3)p Fq(;)17 b Fu(2\))j(and)j Fq(\021)t Fu(\(9)p Fo(j)p Fu(2;)17 b(3)p Fq(;)g Fu(2\).)37 b(W)-8 b(e)23 b(ha)m(v)m(e)g Fq(\021)2839 298 y Fl(n)2886 283 y Fu(\(9)p Fo(j)p Fu(3)p Fq(;)17 b Fu(2\))27 b(=)g Fq(\021)3363 232 y Fm(\()p Fl(L)p Fm(\))3359 294 y Fl(n)3470 283 y Fu(\(9)p Fo(j)p Fu(3)p Fq(;)17 b Fu(2\))q(+)0 417 y Fq(\021)52 366 y Fm(\()p Fl(R)p Fm(\))48 428 y Fl(n)164 417 y Fu(\(9)p Fo(j)p Fu(3)p Fq(;)g Fu(2\))31 b(where)64 723 y Fq(\021)116 682 y Fm(\()p Fl(L)p Fm(\))112 748 y Fl(n)223 723 y Fu(\(9)p Fo(j)p Fu(3)p Fq(;)17 b Fu(2\))81 b(:=)795 656 y(9)p Fq(\025)p 795 701 106 4 v 824 792 a Fu(4)933 599 y Fl(n)p Fn(\000)p Fm(1)928 629 y Fp(X)935 841 y Fl(k)r Fm(=0)1468 656 y Fu(2)p 1098 701 790 4 v 1098 720 a Fp(p)p 1198 720 690 4 v 86 x Fu(2)p Fq(\031)t Fu(\()p Fq(T)1401 821 y Fl(n)p Fm(+1)1560 806 y Fo(\000)22 b Fq(T)1716 821 y Fl(k)r Fm(+1)1849 806 y Fu(\))1897 723 y Fq(\021)1945 738 y Fl(k)1988 723 y Fu(\(2\))2130 588 y Fp(Z)2229 614 y Fl(T)2270 623 y Fg(n)p Fi(+1)2185 813 y Fl(T)2226 821 y Fg(n)2394 723 y Fq(dt)2497 588 y Fp(Z)2596 614 y Fl(t)2552 813 y(T)2593 821 y Fg(n)2639 723 y Fq(ds)2769 613 y Fp(D)2848 723 y Fu(\026)-67 b Fq(m)2915 682 y Fn(0)2915 748 y Fl(x)2955 756 y Fg(n)3002 723 y Fq(;)35 b Fu(\026)-67 b Fq(m)3131 738 y Fl(x)3171 746 y Fg(n)3217 723 y Fq(v)3268 682 y Fm(\()p Fl(n)p Fm(\))3370 723 y Fu(\()p Fq(s)p Fu(\))p Fq(g)3543 673 y Fm(\()p Fl(n)p Fm(\))3539 746 y Fl(t)p Fn(\000)p Fl(s)3655 723 y Fu(1)3704 613 y Fp(E)58 1066 y Fq(\021)110 1024 y Fm(\()p Fl(R)p Fm(\))106 1090 y Fl(n)223 1066 y Fu(\(9)p Fo(j)p Fu(3)p Fq(;)17 b Fu(2\))81 b(:=)795 998 y(9)p Fq(\025)p 795 1043 106 4 v 824 1134 a Fu(4)933 941 y Fl(n)p Fn(\000)p Fm(1)928 971 y Fp(X)935 1183 y Fl(k)r Fm(=0)1088 1066 y Fq(\021)1136 1081 y Fl(k)1179 1066 y Fu(\(2\))1321 930 y Fp(Z)1420 956 y Fl(T)1461 965 y Fg(n)p Fi(+1)1375 1156 y Fl(T)1416 1164 y Fg(n)1584 1066 y Fq(dt)1687 930 y Fp(Z)1787 956 y Fl(t)1742 1156 y(T)1783 1164 y Fg(n)1830 1066 y Fq(ds)1960 895 y Fp(")2324 998 y Fu(2)p 2028 1043 641 4 v 2028 1063 a Fp(p)p 2128 1063 542 4 v 85 x Fu(2)p Fq(\031)t Fu(\()p Fq(s)21 b Fo(\000)i Fq(T)2498 1163 y Fl(k)r Fm(+1)2631 1148 y Fu(\))2701 1066 y Fo(\000)3181 998 y Fu(2)p 2811 1043 790 4 v 2811 1063 a Fp(p)p 2910 1063 690 4 v 2910 1148 a Fu(2)p Fq(\031)t Fu(\()p Fq(T)3113 1163 y Fl(n)p Fm(+1)3272 1148 y Fo(\000)g Fq(T)3429 1163 y Fl(k)r Fm(+1)3562 1148 y Fu(\))3610 895 y Fp(#)985 1345 y Fo(\002)1084 1234 y Fp(D)1163 1345 y Fu(\026)-67 b Fq(m)1230 1304 y Fn(0)1230 1370 y Fl(x)1270 1378 y Fg(n)1317 1345 y Fq(;)35 b Fu(\026)-67 b Fq(m)1446 1360 y Fl(x)1486 1368 y Fg(n)1532 1345 y Fq(v)1583 1304 y Fm(\()p Fl(n)p Fm(\))1685 1345 y Fu(\()p Fq(s)p Fu(\))p Fq(g)1858 1294 y Fm(\()p Fl(n)p Fm(\))1854 1367 y Fl(t)p Fn(\000)p Fl(s)1971 1345 y Fu(1)2020 1234 y Fp(E)3591 1345 y Fu(\(6.62\))100 1588 y(Since,)32 b(b)m(y)j(\(4.7\))d(and)h (\(4.10\))o(,)780 1727 y Fp(\014)780 1787 y(\014)780 1847 y(\014)780 1907 y(\014)813 1736 y(Z)913 1763 y Fl(T)954 1772 y Fg(n)p Fi(+1)869 1962 y Fl(T)910 1970 y Fg(n)1078 1872 y Fq(dt)1181 1736 y Fp(Z)1280 1763 y Fl(t)1236 1962 y(T)1277 1970 y Fg(n)1323 1872 y Fq(ds)1453 1761 y Fp(D)1532 1872 y Fu(\026)-67 b Fq(m)1599 1831 y Fn(0)1599 1897 y Fl(x)1639 1905 y Fg(n)1686 1872 y Fq(;)35 b Fu(\026)-67 b Fq(m)1815 1887 y Fl(x)1855 1895 y Fg(n)1901 1872 y Fq(v)1952 1831 y Fm(\()p Fl(n)p Fm(\))2054 1872 y Fu(\()p Fq(s)p Fu(\))p Fq(g)2227 1821 y Fm(\()p Fl(n)p Fm(\))2223 1894 y Fl(t)p Fn(\000)p Fl(s)2340 1872 y Fu(1)2389 1761 y Fp(E)2449 1727 y(\014)2449 1787 y(\014)2449 1847 y(\014)2449 1907 y(\014)2510 1872 y Fo(\024)28 b Fq(C)7 b(\025)2749 1831 y Fm(1)p Fn(\000)p Fl(\020)2879 1872 y Fq(T)2950 1831 y Fm(5)p Fl(=)p Fm(2)0 2143 y Fu(w)m(e)34 b(ha)m(v)m(e)974 2299 y Fp(\014)974 2359 y(\014)1007 2384 y Fq(\021)1059 2343 y Fm(\()p Fl(L)p Fm(\))1055 2408 y Fl(n)1166 2384 y Fu(\(9)p Fo(j)p Fu(3)p Fq(;)17 b Fu(2\))1461 2299 y Fp(\014)1461 2359 y(\014)1520 2384 y Fo(\024)29 b Fq(C)7 b(\025)1760 2343 y Fm(1)p Fn(\000)p Fl(\020)1889 2384 y Fq(T)1960 2343 y Fm(3)p Fl(=)p Fm(2)2087 2209 y Fp(\014)2087 2269 y(\014)2087 2329 y(\014)2087 2389 y(\014)2087 2449 y(\014)2125 2259 y Fl(n)p Fn(\000)p Fm(1)2120 2289 y Fp(X)2127 2501 y Fl(k)r Fm(=0)2344 2316 y Fq(\025)2401 2232 y Fo(p)p 2483 2232 71 4 v 2483 2316 a Fq(T)p 2290 2361 318 4 v 2290 2381 a Fo(p)p 2373 2381 235 4 v 81 x Fq(n)23 b Fo(\000)f Fq(k)2617 2384 y(\021)2665 2399 y Fl(k)2708 2384 y Fu(\(2\))2833 2209 y Fp(\014)2833 2269 y(\014)2833 2329 y(\014)2833 2389 y(\014)2833 2449 y(\014)3591 2384 y Fu(\(6.63\))0 2696 y(whic)m(h)29 b(is)g(negligible)d (since)j(w)m(e)h(are)f(in)f(the)i(set)f Fo(B)1825 2645 y Fm(\(4\))1822 2723 y Fl(\025;\034)1927 2696 y Fu(,)g(see)i(\(6.3\))o (.)42 b(On)29 b(the)h(other)f(hand,)h(again)d(b)m(y)k(\(4.7\))0 2819 y(and)i(\(4.10\))o(,)281 2948 y Fp(\014)281 3008 y(\014)281 3068 y(\014)281 3127 y(\014)281 3187 y(\014)315 2987 y(Z)414 3013 y Fl(T)455 3022 y Fg(n)p Fi(+1)370 3212 y Fl(T)411 3220 y Fg(n)579 3122 y Fq(dt)682 2987 y Fp(Z)781 3013 y Fl(t)737 3212 y(T)778 3220 y Fg(n)825 3122 y Fq(ds)954 2952 y Fp(")1319 3055 y Fu(2)p 1022 3100 641 4 v 1022 3119 a Fp(p)p 1122 3119 542 4 v 86 x Fu(2)p Fq(\031)t Fu(\()p Fq(s)22 b Fo(\000)g Fq(T)1492 3220 y Fl(k)r Fm(+1)1625 3205 y Fu(\))1695 3122 y Fo(\000)2175 3055 y Fu(2)p 1805 3100 790 4 v 1805 3119 a Fp(p)p 1905 3119 690 4 v 86 x Fu(2)p Fq(\031)t Fu(\()p Fq(T)2108 3220 y Fl(n)p Fm(+1)2267 3205 y Fo(\000)g Fq(T)2423 3220 y Fl(k)r Fm(+1)2556 3205 y Fu(\))2604 2952 y Fp(#)2679 3012 y(D)2758 3122 y Fu(\026)-67 b Fq(m)2825 3081 y Fn(0)2825 3147 y Fl(x)2865 3155 y Fg(n)2912 3122 y Fq(;)35 b Fu(\026)-67 b Fq(m)3041 3137 y Fl(x)3081 3145 y Fg(n)3127 3122 y Fq(v)3178 3081 y Fm(\()p Fl(n)p Fm(\))3280 3122 y Fu(\()p Fq(s)p Fu(\))p Fq(g)3453 3072 y Fm(\()p Fl(n)p Fm(\))3449 3145 y Fl(t)p Fn(\000)p Fl(s)3565 3122 y Fu(1)3614 3012 y Fp(E)3675 2948 y(\014)3675 3008 y(\014)3675 3068 y(\014)3675 3127 y(\014)3675 3187 y(\014)687 3429 y Fo(\024)28 b Fq(C)7 b(\025)926 3388 y Fm(1)p Fn(\000)p Fl(\020)1056 3429 y Fq(T)1127 3388 y Fm(2)1183 3288 y Fp(\024)1235 3429 y Fv(1)1292 3444 y Fl(k)r Fm(=)p Fl(n)p Fn(\000)p Fm(1)1544 3429 y Fu(+)1923 3361 y(1)p 1652 3406 591 4 v 1652 3497 a(\()p Fq(n)22 b Fo(\000)h Fq(k)i Fo(\000)e Fu(1\))2133 3468 y Fm(3)p Fl(=)p Fm(2)2253 3288 y Fp(\025)0 3702 y Fu(whence,)35 b(b)m(y)f(\(4.13\))e(and)h(\(4.5\))191 3923 y Fp(\014)191 3983 y(\014)225 4008 y Fq(\021)277 3967 y Fm(\()p Fl(R)p Fm(\))273 4033 y Fl(n)389 4008 y Fu(\(9)p Fo(j)p Fu(3)p Fq(;)17 b Fu(2\))684 3923 y Fp(\014)684 3983 y(\014)743 4008 y Fo(\024)28 b Fq(C)7 b(\025)982 3967 y Fm(2)p Fn(\000)p Fl(\020)1112 4008 y Fq(T)1183 3967 y Fm(2)1239 3838 y Fp(")1303 3884 y Fl(n)p Fn(\000)p Fm(2)1297 3914 y Fp(X)1305 4126 y Fl(k)r Fm(=0)1738 3941 y Fu(1)p 1468 3985 V 1468 4077 a(\()p Fq(n)22 b Fo(\000)h Fq(k)i Fo(\000)d Fu(1\))1948 4048 y Fm(3)p Fl(=)p Fm(2)2085 4008 y Fo(j)o Fq(\021)2160 4023 y Fl(k)2203 4008 y Fu(\(2\))p Fo(j)g Fu(+)g Fo(j)o Fq(\021)2551 4023 y Fl(n)p Fn(\000)p Fm(1)2689 4008 y Fu(\(2\))p Fo(j)2841 3838 y Fp(#)2927 4008 y Fo(\024)28 b Fq(\025)3089 3967 y Fm(3)p Fn(\000)p Fl(\020)3219 4008 y Fq(T)3290 3967 y Fm(5)p Fl(=)p Fm(2)3591 4008 y Fu(\(6.64\))0 4309 y(whic)m(h)33 b(is)f(negligible.)100 4437 y(Since)27 b(\000)410 4386 y Fm(\(2\))410 4447 y Fl(n)504 4437 y Fu(\()p Fq(t)p Fu(\))h(has)f(the)h(same)f(order)h(as)f Fq(v)1628 4401 y Fm(\()p Fl(n)p Fm(\))1730 4437 y Fu(\()p Fq(t)p Fu(\),)i(it)d(follo)m(ws)g(from)g(\(6.56\))h(that)g(the)g(term)g Fq(\021)t Fu(\(9)p Fo(j)p Fu(2;)17 b(3)p Fq(;)g Fu(2\))0 4553 y(can)33 b(b)s(e)g(analyzed)f(as)h Fq(\021)t Fu(\(9)p Fo(j)p Fu(3)p Fq(;)17 b Fu(2\).)42 b(Prop)s(osition)31 b(6.2)h(is)g(th)m(us)i(pro)m(v)m(ed.)1207 b Fd(\003)1029 5142 y Fu(7.)49 b Fr(Bounds)37 b(on)h(the)g(stochastic)f(terms)100 5316 y Fu(The)e(sto)s(c)m(hastic)f(terms)h(in)e(the)i(title)d(are)j (the)f(ones)h(left)f(from)f(the)h(previous)h(section,)g(i.e.)f Fq(\033)3606 5331 y Fl(n)3653 5316 y Fu(,)h(see)0 5433 y(\(6.1\).)41 b(As)29 b(already)e(men)m(tioned)g(w)m(e)i(will)c(obtain) i(b)s(ounds)h(for)f(the)h(sums)g(of)f(the)h(comp)s(onen)m(ts)h(of)e (these)1882 5532 y Fk(35)p eop %%Page: 36 37 36 36 bop 0 266 a Fu(v)m(ectors)34 b(rather)f(than)f(for)g(the)h(comp)s (onen)m(ts)g(themselv)m(es.)45 b(W)-8 b(e)33 b(de\014ne)579 543 y Fq(S)639 558 y Fl(n)686 543 y Fu(\()p Fq(`)p Fu(\))83 b(:=)1077 419 y Fl(n)p Fn(\000)p Fm(1)1072 448 y Fp(X)1079 661 y Fl(k)r Fm(=0)1232 543 y Fq(\021)1280 558 y Fl(k)1323 543 y Fu(\()p Fq(`)p Fu(\))p Fq(;)212 b(`)27 b Fo(2)h(L)g Fu(:=)f Fo(f)p Fu(2)p Fq(;)17 b Fu(\(4)k(+)h(5)p Fq(;)17 b Fu(2\))p Fq(;)g Fu(\(6)p Fq(;)g Fu(2\))p Fq(;)g Fu(\(9)p Fo(j)p Fu(2;)g(2\))p Fo(g)696 885 y Fq(S)756 900 y Fl(n)886 885 y Fu(:=)1072 791 y Fp(X)1081 1003 y Fl(`)p Fn(2L)1232 885 y Fq(S)1292 900 y Fl(n)1339 885 y Fu(\()p Fq(`)p Fu(\))28 b(=)1593 761 y Fl(n)p Fn(\000)p Fm(1)1587 791 y Fp(X)1595 1003 y Fl(k)r Fm(=0)1748 885 y Fq(\033)1803 900 y Fl(k)3639 885 y Fu(\(7.1\))0 1167 y(and)36 b(call)f Fq(S)42 b Fu(the)36 b(v)m(ector)i(whose)f(comp)s(onen)m(ts)g(are)f Fq(S)2004 1182 y Fl(n)2051 1167 y Fu(.)54 b(Analogously)35 b Fq(\030)41 b Fu(is)36 b(the)g(v)m(ector)i(whose)f(com-)0 1283 y(p)s(onen)m(ts)d(are)e Fq(\030)569 1298 y Fl(n)616 1283 y Fu(,)g(see)i(\(5.8\).)0 1636 y Fv(Prop)s(osition)i(7.1.)41 b Ft(R)-5 b(e)g(c)g(al)5 b(l)35 b Fq(A)g Ft(is)f(the)h(matrix)g(whose)f (entries)g(ar)-5 b(e)35 b(de\014ne)-5 b(d)34 b(in)42 b Fu(\(5.11\))o Ft(.)j(Then)1018 1825 y Fq(\030)32 b Fu(=)27 b Fq(S)h Fo(\000)23 b Fq(AS)28 b Fu(+)22 b Fq(R)h Fu(+)f Fq(A)1911 1784 y Fm(2)1951 1825 y Fq(\030)203 b Fu(on)33 b(the)g(set)g Fo(B)2718 1840 y Fl(\025;\034)3639 1825 y Fu(\(7.2\))0 2014 y Ft(wher)-5 b(e)1126 2231 y Fq(R)1200 2246 y Fl(n)1275 2231 y Fu(:=)1411 2107 y Fl(n)p Fn(\000)p Fm(1)1406 2137 y Fp(X)1414 2349 y Fl(k)r Fm(=0)1566 2231 y Fu(\()p Fq(\021)1652 2246 y Fl(k)1717 2231 y Fo(\000)23 b Fq(\033)1872 2246 y Fl(k)1915 2231 y Fu(\))f Fo(\000)2080 2107 y Fl(n)p Fn(\000)p Fm(1)2075 2137 y Fp(X)2082 2349 y Fl(k)r Fm(=0)2235 2231 y Fu(\()p Fq(A)p Fu([)p Fq(\021)k Fo(\000)d Fq(\033)t Fu(]\))2671 2260 y Fl(k)3639 2231 y Fu(\(7.3\))0 2494 y Ft(Mor)-5 b(e)g(over,)41 b(for)e(e)-5 b(ach)39 b Fq(\034)48 b(>)37 b Fu(0)i Ft(ther)-5 b(e)40 b(is)f(a)h(c)-5 b(onstant)39 b Fq(C)44 b Fu(=)36 b Fq(C)7 b Fu(\()p Fq(\034)k Fu(\))40 b Ft(such)g(that)g(for)g(any)f Fq(\025)h Ft(and)f Fq(\020)47 b Ft(smal)5 b(l)0 2610 y(enough)930 2761 y Fo(j)p Fq(R)q Fo(j)1061 2776 y Fl(n)1135 2761 y Fo(\024)28 b Fq(C)7 b(T)1388 2720 y Fn(\000)p Fm(1)p Fl(=)p Fm(4)1570 2680 y Fp(\002)1611 2761 y Fu(1)22 b(+)g Fq(x)1835 2720 y Fm(2)1835 2785 y Fl(n)p Fm(+1)p Fl(;)p Fn(\003)2027 2680 y Fp(\003)2285 2761 y Fu(on)32 b(the)h(set)h Fo(B)2806 2776 y Fl(\025;\034)3639 2761 y Fu(\(7.4\))100 3165 y(The)f(pro)s(of)f(is)g(based)h(on)g(the)g(follo) m(wing)d(simple)h(lemma.)0 3518 y Fv(Lemma)26 b(7.2.)35 b Ft(L)-5 b(et)26 b Fq(M)37 b Ft(b)-5 b(e)26 b(a)g(lower)f(triangular)h (matrix)g(such)g(that)h Fq(M)2572 3533 y Fl(n;k)2705 3518 y Fu(=)g Fq(M)2902 3533 y Fl(n)2945 3514 y Fh(0)2968 3533 y Fl(;k)3027 3514 y Fh(0)3079 3518 y Ft(whenever)e Fq(n)3553 3482 y Fn(0)3579 3518 y Fo(\000)s Fq(k)3713 3482 y Fn(0)3764 3518 y Fu(=)0 3645 y Fq(n)c Fo(\000)g Fq(k)s Ft(.)44 b(L)-5 b(et)35 b(also)e Fq(u)28 b Fu(=)f Fo(f)p Fq(u)964 3660 y Fl(h)1008 3645 y Fu(;)17 b Fq(h)28 b Fu(=)f(0)p Fq(;)17 b(:)g(:)g(:)f(;)h(n)p Fo(g)34 b Ft(b)-5 b(e)34 b(a)g(ve)-5 b(ctor)34 b(and)f(de\014ne)g Fq(v)2657 3660 y Fl(n)2732 3645 y Fu(=)2836 3570 y Fp(P)2941 3597 y Fl(n)p Fn(\000)p Fm(1)2941 3674 y Fl(h)p Fm(=0)3095 3645 y Fq(u)3151 3660 y Fl(h)3229 3645 y Ft(\(r)-5 b(esp.)34 b Fq(v)3551 3660 y Fm(0)3618 3645 y Fu(=)27 b(0)p Ft(\).)0 3761 y(Then)1508 3854 y Fl(n)p Fn(\000)p Fm(1)1503 3884 y Fp(X)1509 4096 y Fl(h)p Fm(=0)1647 3979 y Fu(\()p Fq(M)10 b(u)p Fu(\))1883 3994 y Fl(h)1955 3979 y Fu(=)28 b(\()p Fq(M)10 b(v)t Fu(\))2290 3994 y Fl(n)3639 3979 y Fu(\(7.5\))0 4478 y Ft(Pr)-5 b(o)g(of.)43 b Fu(The)34 b(l.h.s.)e(of)39 b(\(7.5\))32 b(can)h(b)s(e)g(written)f(as)611 4639 y Fl(n)p Fn(\000)p Fm(1)606 4669 y Fp(X)612 4881 y Fl(h)p Fm(=0)773 4639 y Fl(h)p Fn(\000)p Fm(1)766 4669 y Fp(X)774 4881 y Fl(k)r Fm(=0)927 4764 y Fq(M)1021 4779 y Fl(h;k)1152 4764 y Fu(\()p Fq(v)1237 4779 y Fl(k)r Fm(+1)1392 4764 y Fo(\000)23 b Fq(v)1539 4779 y Fl(k)1581 4764 y Fu(\))28 b(=)1756 4639 y Fl(n)p Fn(\000)p Fm(1)1751 4669 y Fp(X)1757 4881 y Fl(h)p Fm(=0)1918 4639 y Fl(h)p Fn(\000)p Fm(1)1911 4669 y Fp(X)1919 4881 y Fl(k)r Fm(=0)2072 4764 y Fq(M)2166 4779 y Fl(h;k)2297 4764 y Fq(v)2344 4779 y Fl(k)r Fm(+1)2499 4764 y Fo(\000)2604 4639 y Fl(n)p Fn(\000)p Fm(1)2598 4669 y Fp(X)2605 4881 y Fl(h)p Fm(=0)2765 4639 y Fl(h)p Fn(\000)p Fm(1)2759 4669 y Fp(X)2766 4881 y Fl(k)r Fm(=0)2919 4764 y Fq(M)3013 4779 y Fl(h;k)3144 4764 y Fq(v)3191 4779 y Fl(k)0 5045 y Fu(W)-8 b(e)33 b(call)e Fq(k)401 5009 y Fn(0)452 5045 y Fu(=)c Fq(k)f Fu(+)c(1)32 b(in)g(the)h(\014rst)g (sum)f(and)h Fq(h)1746 5009 y Fn(0)1797 5045 y Fu(=)28 b Fq(h)22 b Fo(\000)h Fu(1)32 b(in)g(the)h(second)h(one,)e(getting)791 5206 y Fl(n)p Fn(\000)p Fm(1)786 5236 y Fp(X)792 5449 y Fl(h)p Fm(=0)930 5331 y Fu(\()p Fq(M)10 b(u)p Fu(\))1166 5346 y Fl(h)1238 5331 y Fu(=)1347 5206 y Fl(n)p Fn(\000)p Fm(1)1342 5236 y Fp(X)1348 5449 y Fl(h)p Fm(=0)1558 5206 y Fl(h)1506 5236 y Fp(X)1502 5449 y Fl(k)1541 5430 y Fh(0)1563 5449 y Fm(=1)1670 5331 y Fq(M)1764 5346 y Fl(h;k)1864 5327 y Fh(0)1886 5346 y Fn(\000)p Fm(1)2008 5331 y Fq(v)2055 5346 y Fl(k)2094 5327 y Fh(0)2142 5331 y Fo(\000)2279 5206 y Fl(n)p Fn(\000)p Fm(2)2274 5236 y Fp(X)2242 5449 y Fl(h)2283 5430 y Fh(0)2305 5449 y Fm(=)p Fn(\000)p Fm(1)2507 5206 y Fl(h)2548 5183 y Fh(0)2466 5236 y Fp(X)2474 5449 y Fl(k)r Fm(=0)2627 5331 y Fq(M)2721 5346 y Fl(h)2762 5327 y Fh(0)2784 5346 y Fm(+1)p Fl(;k)2964 5331 y Fq(v)3011 5346 y Fl(k)1882 5532 y Fk(36)p eop %%Page: 37 38 37 37 bop 0 266 a Fu(W)-8 b(e)33 b(ma)m(y)f(as)h(w)m(ell)f(extend)i (the)f(\014rst)g(sum)f(from)g Fq(k)1874 230 y Fn(0)1925 266 y Fu(=)27 b(0)33 b(b)s(ecause)h Fq(v)2518 281 y Fm(0)2585 266 y Fu(=)27 b(0.)44 b(W)-8 b(e)33 b(then)g(get)655 392 y Fl(n)p Fn(\000)p Fm(1)650 422 y Fp(X)656 634 y Fl(h)p Fm(=0)794 517 y Fu(\()p Fq(M)10 b(u)p Fu(\))1030 532 y Fl(h)1103 517 y Fu(=)1212 392 y Fl(n)p Fn(\000)p Fm(1)1206 422 y Fp(X)1214 634 y Fl(k)r Fm(=1)1367 517 y Fq(M)1461 532 y Fl(n)p Fn(\000)p Fm(1)p Fl(;k)r Fn(\000)p Fm(1)1774 517 y Fq(v)1821 532 y Fl(k)1886 517 y Fu(+)1989 392 y Fl(n)p Fn(\000)p Fm(2)1984 422 y Fp(X)1991 634 y Fl(h)p Fm(=0)2196 392 y Fl(h)2145 422 y Fp(X)2152 634 y Fl(k)r Fm(=0)2305 517 y Fu([)p Fq(M)2426 532 y Fl(h;k)r Fn(\000)p Fm(1)2642 517 y Fo(\000)22 b Fq(M)2835 532 y Fl(h)p Fm(+1)p Fl(;k)3029 517 y Fu(])44 b Fq(v)3147 532 y Fl(k)0 765 y Fu(Note)37 b(that)g(the)g(second)h(term)f(ab)s(o)m (v)m(e)g(v)-5 b(anishes)38 b(b)m(y)g(the)f(assumptions)g(on)g Fq(M)10 b Fu(,)39 b(and)e(this)f(completes)0 881 y(the)d(pro)s(of)f(of) 39 b(\(7.5\))o(.)2995 b Fd(\003)0 1040 y Ft(Pr)-5 b(o)g(of)35 b(of)f(Pr)-5 b(op)g(osition)34 b(7.1.)43 b Fu(F)-8 b(rom)31 b(\(5.8\))h(and)h(\(5.16\))o(,)1390 1293 y Fq(\030)1433 1308 y Fl(n)1507 1293 y Fu(=)1616 1169 y Fl(n)p Fn(\000)p Fm(1)1611 1199 y Fp(X)1619 1411 y Fl(k)r Fm(=0)1755 1293 y Fu([)p Fq(\021)26 b Fo(\000)c Fq(A\021)k Fu(+)c Fq(A)2273 1252 y Fm(2)2313 1293 y Fq( )t Fu(])2407 1308 y Fl(k)0 1551 y Fu(Next,)30 b(apply)f(Lemma)e(7.2)i(for)f Fq(A)h Fu(and)g Fq(A)1531 1515 y Fm(2)1570 1551 y Fu(.)43 b(F)-8 b(rom)27 b(\(7.1\))h(and)h(\(5.8\),)g(w)m(e)h(get)f(\(7.2\))o(.)42 b(Since)29 b Fo(k)p Fq(A)p Fo(k)3556 1566 y Fl(n)3631 1551 y Fo(\024)f Fq(C)7 b Fu(,)0 1667 y(the)33 b(b)s(ound)g(\(7.4\))f (follo)m(ws)f(from)g(Prop)s(osition)g(6.2.)1834 b Fd(\003)100 1826 y Fu(Our)32 b(next)i(task)f(is)f(to)g(con)m(trol)g(the)h(sto)s(c)m (hastic)g(term)f Fq(S)2186 1841 y Fl(n)2233 1826 y Fu(.)0 2179 y Fv(Prop)s(osition)k(7.3.)41 b Ft(F)-7 b(or)34 b(any)h Fq(\034)k(>)28 b Fu(0)34 b Ft(we)h(have)1161 2423 y Fu(lim)1134 2485 y Fl(L)p Fn(!1)p 1351 2340 136 3 v 1351 2423 a Fu(lim)1363 2486 y Fl(\025)p Fn(#)p Fm(0)1514 2423 y Fj(P)1590 2253 y Fp( )1735 2423 y Fu(sup)1671 2509 y Fl(n)p Fn(\024)p Fl(n)1812 2521 y Fg(\025)1852 2509 y Fm(\()p Fl(\034)8 b Fm(\))1962 2423 y Fo(j)p Fq(S)2050 2438 y Fl(n)2097 2423 y Fu(\(2\))p Fo(j)27 b Fq(>)h(L)2447 2253 y Fp(!)2553 2423 y Fu(=)g(0)933 b(\(7.6\))776 2776 y(lim)787 2838 y Fl(\025)p Fn(#)p Fm(0)928 2776 y Fj(P)1004 2605 y Fp( )1148 2776 y Fu(sup)1084 2861 y Fl(n)p Fn(\024)p Fl(n)1225 2873 y Fg(\025)1265 2861 y Fm(\()p Fl(\034)8 b Fm(\))1409 2708 y Fo(j)p Fq(S)1497 2723 y Fl(n)1544 2708 y Fu(\()p Fq(`)p Fu(\))p Fo(j)p 1386 2753 327 4 v 1386 2844 a Fu(1)22 b(+)g Fq(x)1610 2859 y Fl(n;)p Fn(\003)1750 2776 y Fq(>)27 b(T)1924 2734 y Fn(\000)p Fm(1)p Fl(=)p Fm(4)2089 2605 y Fp(!)2195 2776 y Fu(=)h(0)p Fq(;)216 b(`)27 b Fo(2)h(L)22 b(n)g(f)p Fu(2)p Fo(g)574 b Fu(\(7.7\))0 3242 y Ft(Pr)-5 b(o)g(of)35 b(of)55 b Fu(\(7.6\))p Ft(.)43 b Fu(By)34 b(\(3.9\),)f(\(5.10\))o(,)f(and)h (\(7.1\),)1206 3495 y Fq(S)1266 3510 y Fl(n)1313 3495 y Fu(\(2\))27 b(=)h Fo(\000)1656 3428 y Fu(3)p Fq(\025)p 1656 3472 106 4 v 1685 3564 a Fu(4)1794 3371 y Fl(n)p Fn(\000)p Fm(1)1788 3401 y Fp(X)1796 3613 y Fl(k)r Fm(=0)1932 3495 y Fo(h)18 b Fu(\026)-67 b Fq(m)2056 3454 y Fn(0)2056 3520 y Fl(x)2096 3532 y Fg(k)2138 3495 y Fq(;)17 b(z)2231 3454 y Fm(\()p Fl(k)r Fm(\))2329 3495 y Fu(\()p Fq(T)2424 3510 y Fl(k)r Fm(+1)2557 3495 y Fu(\))p Fo(i)1005 b Fu(\(7.8\))0 3749 y(whic)m(h)23 b(is)f(an)h Fo(F)555 3764 y Fl(T)596 3772 y Fg(n)642 3749 y Fu({martingale.)37 b(In)23 b(App)s(endix)g(B)f (it)g(is)g(pro)m(v)m(ed)i(there)f(is)f Fq(C)30 b Fu(so)22 b(that,)j(for)d(an)m(y)h Fq(n)28 b Fo(\024)g Fq(n)3665 3764 y Fl(\025)3711 3749 y Fu(\()p Fq(\034)11 b Fu(\))0 3865 y(and)33 b Fq(t)28 b Fo(2)g Fu([)p Fq(T)431 3880 y Fl(n)500 3865 y Fu(+)22 b(1)p Fq(;)17 b(T)748 3880 y Fl(n)p Fm(+1)885 3865 y Fu(],)889 4055 y(sup)895 4137 y Fl(x)p Fn(2)p Ff(R)1074 4055 y Fj(E)1152 3945 y Fp(\020)1217 3975 y(\000)1263 4055 y Fq(z)1312 4014 y Fm(\()p Fl(n)p Fm(\))1414 4055 y Fu(\()p Fq(t;)g(x)p Fu(\))1624 3975 y Fp(\001)1670 3991 y Fm(2)p Fl(p)1745 3945 y Fp(\021)1832 4055 y Fo(\024)28 b Fq(C)7 b Fu(\()p Fq(t)23 b Fo(\000)f Fq(T)2266 4070 y Fl(n)2313 4055 y Fu(\))2351 4014 y Fl(p)2391 4055 y Fq(;)212 b(p)27 b Fu(=)h(1)p Fq(;)17 b Fu(2)687 b(\(7.9\))0 4272 y(Then,)34 b(b)m(y)f(the)g(Do)s(ob's)f(martingale)e (inequalit)m(y)-8 b(,)32 b(w)m(e)h(get,)g(for)f(some)g(constan)m(t)i Fq(C)7 b Fu(,)680 4516 y Fj(P)756 4346 y Fp( )901 4516 y Fu(sup)836 4602 y Fl(n)p Fn(\024)p Fl(n)977 4614 y Fg(\025)1018 4602 y Fm(\()p Fl(\034)h Fm(\))1128 4516 y Fo(j)p Fq(S)1216 4531 y Fl(n)1263 4516 y Fu(\(2\))p Fo(j)27 b Fq(>)g(L)1612 4346 y Fp(!)1719 4516 y Fo(\024)1863 4449 y Fu(1)p 1834 4494 V 1834 4585 a Fq(L)1900 4556 y Fm(2)1950 4516 y Fj(E)2028 4436 y Fp(\000)2079 4516 y Fq(S)2139 4532 y Fl(n)2182 4544 y Fg(\025)2223 4532 y Fm(\()p Fl(\034)8 b Fm(\))2321 4516 y Fu(\(2\))2446 4475 y Fm(2)2485 4436 y Fp(\001)2558 4516 y Fo(\024)2673 4449 y Fq(C)f(n)2808 4464 y Fl(\025)2854 4449 y Fu(\()p Fq(\034)k Fu(\))p Fq(\025)3040 4413 y Fm(2)3079 4449 y Fq(T)p 2673 4494 477 4 v 2859 4585 a(L)2925 4556 y Fm(2)3591 4516 y Fu(\(7.10\))0 4767 y(whic)m(h)33 b(pro)m(v)m(es)h (\(7.6\).)0 4926 y Ft(Pr)-5 b(o)g(of)35 b(of)55 b Fu(\(7.7\))p Ft(.)0 5085 y(The)34 b(c)-5 b(ase)34 b Fq(`)28 b Fu(=)g(\(4)21 b(+)h(5)p Fq(;)17 b Fu(2\))p Ft(.)43 b Fu(W)-8 b(e)33 b(ha)m(v)m(e)h(\(recall)d(\(6.27\))o(\))819 5338 y Fq(S)879 5353 y Fl(n)926 5338 y Fu(\(4)21 b(+)h(5)p Fq(;)17 b Fu(2\))27 b(=)1453 5271 y(3)p Fq(\025)1559 5235 y Fm(2)p 1453 5315 146 4 v 1501 5407 a Fu(4)1630 5214 y Fl(n)p Fn(\000)p Fm(1)1624 5244 y Fp(X)1631 5456 y Fl(h)p Fm(=0)1785 5203 y Fp(Z)1885 5229 y Fl(T)1926 5241 y Fg(h)p Fi(+1)1840 5428 y Fl(T)1881 5440 y Fg(h)2047 5338 y Fq(ds)17 b Fo(h)g Fu(\026)-66 b Fq(m)2285 5297 y Fn(0)2285 5363 y Fl(x)2325 5375 y Fg(h)2369 5338 y Fq(;)17 b(p)2462 5353 y Fl(T)2503 5365 y Fg(h)p Fi(+1)2620 5353 y Fn(\000)p Fl(s)2711 5338 y Fq(z)2760 5297 y Fm(\()p Fl(h)p Fm(\))2861 5338 y Fu(\()p Fq(s)p Fu(\))p Fo(i)1882 5532 y Fk(37)p eop %%Page: 38 39 38 38 bop 0 266 a Fu(This)32 b(is)f(again)f(a)i Fo(F)731 281 y Fl(T)772 289 y Fg(n)818 266 y Fu({martingale,)d(so)j(that,)g (using)f(again)g(the)h(Do)s(ob's)f(inequalit)m(y)-8 b(,)31 b(and)h(the)g(b)s(ound)968 567 y Fj(E)1046 396 y Fp( )1130 422 y(\014)1130 482 y(\014)1130 542 y(\014)1130 602 y(\014)1164 431 y(Z)1263 457 y Fl(T)1304 469 y Fg(h)p Fi(+1)1219 657 y Fl(T)1260 669 y Fg(h)1426 567 y Fq(ds)17 b Fo(h)g Fu(\026)-66 b Fq(m)1664 526 y Fn(0)1664 591 y Fl(x)1704 603 y Fg(h)1747 567 y Fq(;)17 b(p)1840 582 y Fl(T)1881 594 y Fg(h)p Fi(+1)1998 582 y Fn(\000)p Fl(s)2090 567 y Fq(z)2139 526 y Fm(\()p Fl(h)p Fm(\))2239 567 y Fu(\()p Fq(s)p Fu(\))p Fo(i)2400 422 y Fp(\014)2400 482 y(\014)2400 542 y(\014)2400 602 y(\014)2433 438 y Fm(2)2473 396 y Fp(!)2579 567 y Fo(\024)28 b Fq(C)7 b(T)2832 526 y Fm(3)0 870 y Fu(\(whic)m(h)32 b(follo)m(ws)f(from)f(\(7.9\)\))i(w)m(e)g (conclude)h(that)e(for)g(eac)m(h)i Fq(\020)i(>)28 b Fu(0)j(there)i(is)e (a)g(constan)m(t)i Fq(C)i Fu(=)27 b Fq(C)7 b Fu(\()p Fq(\034)f(;)17 b(\020)8 b Fu(\))0 986 y(suc)m(h)34 b(that)554 1264 y Fj(P)630 1093 y Fp( )775 1264 y Fu(sup)710 1349 y Fl(n)p Fn(\024)p Fl(n)851 1361 y Fg(\025)892 1349 y Fm(\()p Fl(\034)8 b Fm(\))1002 1264 y Fo(j)p Fq(S)1090 1279 y Fl(n)1137 1264 y Fu(\(4)22 b(+)g(5)p Fq(;)17 b Fu(2\))p Fo(j)26 b Fq(>)i(T)1753 1222 y Fn(\000)p Fm(1)p Fl(=)p Fm(4)1917 1093 y Fp(!)2024 1264 y Fo(\024)g Fq(C)7 b(n)2264 1279 y Fl(\025)2310 1264 y Fu(\()p Fq(\034)k Fu(\))p Fq(\025)2496 1222 y Fm(4)2535 1264 y Fq(T)2606 1222 y Fm(3)2645 1174 y Fo(p)p 2728 1174 71 4 v 90 x Fq(T)42 b Fo(\024)28 b Fq(C)7 b(\025)3066 1222 y Fm(2)3105 1264 y Fq(T)3176 1222 y Fm(5)p Fl(=)p Fm(2)3591 1264 y Fu(\(7.11\))0 1662 y Ft(The)34 b(c)-5 b(ase)34 b Fq(`)28 b Fu(=)g(\(6)p Fq(;)17 b Fu(2\))p Ft(.)42 b Fu(By)34 b(\(3.13\))e(and)h(\(3.9\))543 1971 y Fq(S)603 1986 y Fl(n)650 1971 y Fu(\(6)p Fq(;)17 b Fu(2\))26 b(=)1008 1903 y(3)p Fq(\025)1114 1867 y Fm(2)p 1008 1948 146 4 v 1056 2039 a Fu(4)1185 1846 y Fl(n)p Fn(\000)p Fm(1)1180 1876 y Fp(X)1186 2088 y Fl(h)p Fm(=1)1392 1846 y Fl(h)1340 1876 y Fp(X)1348 2088 y Fl(k)r Fm(=1)1501 1835 y Fp(Z)1600 1861 y Fl(T)1641 1873 y Fg(h)p Fi(+1)1556 2061 y Fl(T)1597 2073 y Fg(h)1763 1971 y Fq(ds)1877 1835 y Fp(Z)1976 1861 y Fl(T)2017 1873 y Fg(k)1931 2061 y Fl(T)1972 2073 y Fg(k)q Fh(\000)p Fi(1)2110 1971 y Fq(ds)2207 1929 y Fn(0)2280 1830 y Fp(\034)2373 1971 y Fu(\026)-68 b Fq(m)2439 1929 y Fn(0)2439 1995 y Fl(x)2479 2007 y Fg(h)2524 1971 y Fq(;)2577 1903 y(@)5 b(p)2682 1918 y Fl(s)p Fn(\000)p Fl(s)2803 1899 y Fh(0)p 2577 1948 253 4 v 2652 2039 a Fq(@)g(s)2840 1971 y(z)2889 1929 y Fm(\()p Fl(k)r Fn(\000)p Fm(1\))3077 1971 y Fu(\()p Fq(s)3161 1929 y Fn(0)3185 1971 y Fu(\))3223 1830 y Fp(\035)3591 1971 y Fu(\(7.12\))0 2276 y(F)-8 b(or)32 b(eac)m(h)h Fq(h)g Fu(and)g Fq(k)i Fu(ab)s(o)m(v)m(e,)f(w)m(e)f(write,)g(recalling)d(that)51 b(\026)-68 b Fq(m)2169 2291 y Fl(\030)2207 2276 y Fu(\()p Fq(x)p Fu(\))28 b(=)46 b(\026)-67 b Fq(m)p Fu(\()p Fq(x)23 b Fo(\000)f Fq(\030)5 b Fu(\),)1170 2546 y(\026)-67 b Fq(m)1237 2504 y Fn(0)1237 2570 y Fl(x)1277 2582 y Fg(h)1321 2546 y Fu(\()p Fq(x)p Fu(\))28 b(=)46 b(\026)-67 b Fq(m)1669 2504 y Fn(0)1669 2570 y Fl(x)1709 2582 y Fg(k)q Fh(\000)p Fi(1)1829 2546 y Fu(\()p Fq(x)p Fu(\))23 b Fo(\000)2082 2410 y Fp(Z)2182 2436 y Fl(x)2222 2448 y Fg(h)2138 2636 y Fl(x)2178 2648 y Fg(k)q Fh(\000)p Fi(1)2315 2546 y Fq(d\030)39 b Fu(\026)-68 b Fq(m)2514 2504 y Fn(00)2514 2570 y Fl(\030)2557 2546 y Fu(\()p Fq(x)p Fu(\))0 2832 y(th)m(us)34 b(obtaining,)c(b)m(y)k(exc)m(hanging)f(the)g(sums)g(in)f (\(7.12\))o(,)1258 3138 y Fq(S)1318 3153 y Fl(n)1365 3138 y Fu(\(6)p Fq(;)17 b Fu(2\))27 b(=)h Fo(\000)1808 3113 y Fu(~)1791 3138 y Fq(S)1851 3153 y Fl(n)1898 3138 y Fu(\(6)p Fq(;)17 b Fu(2\))k(+)2241 3014 y Fl(n)p Fn(\000)p Fm(1)2235 3043 y Fp(X)2243 3256 y Fl(k)r Fm(=1)2396 3138 y Fq(X)2477 3153 y Fl(n;k)3591 3138 y Fu(\(7.13\))0 3438 y(where)390 3697 y(~)374 3722 y Fq(S)434 3737 y Fl(n)481 3722 y Fu(\(6)p Fq(;)c Fu(2\))82 b(:=)977 3655 y(3)p Fq(\025)1083 3619 y Fm(2)p 977 3700 146 4 v 1025 3791 a Fu(4)1154 3598 y Fl(n)p Fn(\000)p Fm(1)1148 3628 y Fp(X)1155 3840 y Fl(h)p Fm(=1)1361 3598 y Fl(h)1309 3628 y Fp(X)1317 3840 y Fl(k)r Fm(=1)1470 3587 y Fp(Z)1569 3613 y Fl(T)1610 3625 y Fg(h)p Fi(+1)1525 3812 y Fl(T)1566 3824 y Fg(h)1732 3722 y Fq(ds)1846 3587 y Fp(Z)1945 3613 y Fl(T)1986 3625 y Fg(k)1900 3812 y Fl(T)1941 3824 y Fg(k)q Fh(\000)p Fi(1)2079 3722 y Fq(ds)2176 3681 y Fn(0)2215 3587 y Fp(Z)2315 3613 y Fl(x)2355 3625 y Fg(h)2271 3812 y Fl(x)2311 3824 y Fg(k)q Fh(\000)p Fi(1)2448 3722 y Fq(d\030)20 b Fo(h)e Fu(\026)-67 b Fq(m)2686 3681 y Fn(00)2686 3747 y Fl(\030)2729 3722 y Fq(;)2782 3655 y(@)5 b(p)2887 3670 y Fl(s)p Fn(\000)p Fl(s)3008 3651 y Fh(0)p 2782 3700 253 4 v 2857 3791 a Fq(@)g(s)3045 3722 y(z)3094 3681 y Fm(\()p Fl(k)r Fn(\000)p Fm(1\))3282 3722 y Fu(\()p Fq(s)3366 3681 y Fn(0)3390 3722 y Fu(\))p Fo(i)124 b Fu(\(7.14\))512 4042 y Fq(X)593 4057 y Fl(n;k)781 4042 y Fu(:=)977 3975 y(3)p Fq(\025)1083 3938 y Fm(2)p 977 4019 146 4 v 1025 4110 a Fu(4)1148 3906 y Fp(Z)1248 3933 y Fl(T)1289 3941 y Fg(n)1204 4132 y Fl(T)1245 4144 y Fg(k)1336 4042 y Fq(ds)1450 3906 y Fp(Z)1549 3933 y Fl(T)1590 3945 y Fg(k)1504 4132 y Fl(T)1545 4144 y Fg(k)q Fh(\000)p Fi(1)1683 4042 y Fq(ds)1780 4001 y Fn(0)1819 4042 y Fo(h)18 b Fu(\026)-67 b Fq(m)1943 4001 y Fn(0)1943 4067 y Fl(x)1983 4079 y Fg(k)q Fh(\000)p Fi(1)2104 4042 y Fq(;)2157 3975 y(@)5 b(p)2262 3990 y Fl(s)p Fn(\000)p Fl(s)2383 3971 y Fh(0)p 2157 4019 253 4 v 2232 4110 a Fq(@)g(s)2420 4042 y(z)2469 4001 y Fm(\()p Fl(k)r Fn(\000)p Fm(1\))2657 4042 y Fu(\()p Fq(s)2741 4001 y Fn(0)2765 4042 y Fu(\))p Fo(i)749 b Fu(\(7.15\))100 4323 y(W)-8 b(e)33 b(de\014ne)1188 4567 y Fq(M)1292 4516 y Fm(\()p Fl(n)p Fm(\))1282 4594 y Fl(k)1423 4567 y Fu(:=)1561 4442 y Fl(k)r Fn(\000)p Fm(1)1553 4472 y Fp(X)1560 4684 y Fl(h)p Fm(=1)1714 4567 y Fq(X)1795 4582 y Fl(n;h)1902 4567 y Fq(;)212 b(k)31 b Fu(=)c(1)p Fq(;)17 b(:)g(:)g(:)f(;)h(n)0 4878 y Fu(with)48 b Fq(M)342 4827 y Fm(\()p Fl(n)p Fm(\))332 4902 y(1)500 4878 y Fu(=)56 b(0.)92 b(F)-8 b(or)48 b(eac)m(h)h(\014xed)h Fq(n)p Fu(,)j Fo(f)p Fq(M)1770 4827 y Fm(\()p Fl(n)p Fm(\))1760 4906 y Fl(k)1873 4878 y Fu(;)17 b Fq(k)58 b Fu(=)d(1)p Fq(;)17 b(:)g(:)g(:)f(;)h(n)p Fo(g)48 b Fu(is)h(an)f Fo(F)2919 4893 y Fl(T)2960 4905 y Fg(k)q Fh(\000)p Fi(1)3081 4878 y Fu({martingale)e(since)0 5021 y Fj(E)12 b Fu(\()p Fq(X)179 5036 y Fl(n;h)293 5021 y Fo(jF)393 5036 y Fl(T)434 5048 y Fg(h)p Fh(\000)p Fi(1)556 5021 y Fu(\))27 b(=)h(0.)43 b(The)34 b(quadratic)e(v)-5 b(ariation)30 b(of)i Fq(M)2105 4970 y Fm(\()p Fl(n)p Fm(\))2095 5048 y Fl(k)2240 5021 y Fu(is)1010 5258 y Fp(\002)1051 5338 y Fq(M)1155 5297 y Fm(\()p Fl(n)p Fm(\))1258 5258 y Fp(\003)1300 5377 y Fl(k)1370 5338 y Fu(=)1526 5214 y Fl(k)1473 5244 y Fp(X)1480 5456 y Fl(h)p Fm(=2)1634 5228 y Fp(\020)1693 5338 y Fq(M)1797 5287 y Fm(\()p Fl(n)p Fm(\))1787 5366 y Fl(h)1922 5338 y Fo(\000)23 b Fq(M)2126 5287 y Fm(\()p Fl(n)p Fm(\))2116 5366 y Fl(h)p Fn(\000)p Fm(1)2251 5228 y Fp(\021)2311 5250 y Fm(2)2378 5338 y Fu(=)2489 5214 y Fl(k)r Fn(\000)p Fm(1)2481 5244 y Fp(X)2488 5456 y Fl(h)p Fm(=1)2642 5338 y Fq(X)2731 5297 y Fm(2)2723 5363 y Fl(n;h)1882 5532 y Fk(38)p eop %%Page: 39 40 39 39 bop 100 266 a Fu(W)-8 b(e)33 b(ha)m(v)m(e)423 477 y Fj(E)501 396 y Fp(\000)562 477 y Fq(X)651 436 y Fm(2)643 502 y Fl(n;k)748 392 y Fp(\014)748 452 y(\014)798 477 y Fo(F)870 492 y Fl(T)911 504 y Fg(k)q Fh(\000)p Fi(1)1031 396 y Fp(\001)1104 477 y Fu(=)1218 410 y(9)p Fq(\025)1324 374 y Fm(4)p 1218 454 146 4 v 1242 545 a Fu(16)1390 341 y Fp(Z)1489 368 y Fl(T)1530 376 y Fg(n)1445 567 y Fl(T)1486 579 y Fg(k)1560 477 y Fq(ds)1674 341 y Fp(Z)1773 368 y Fl(T)1814 380 y Fg(k)1729 567 y Fl(T)1770 579 y Fg(k)q Fh(\000)p Fi(1)1874 477 y Fq(ds)1971 436 y Fn(0)1994 341 y Fp(Z)2094 368 y Fl(T)2135 376 y Fg(n)2049 567 y Fl(T)2090 579 y Fg(k)2165 477 y Fq(du)2272 341 y Fp(Z)2371 368 y Fl(T)2412 380 y Fg(k)2326 567 y Fl(T)2367 579 y Fg(k)q Fh(\000)p Fi(1)2472 477 y Fq(du)2579 436 y Fn(0)2601 341 y Fp(Z)2701 368 y Fl(s)2734 344 y Fh(0)2756 368 y Fn(^)p Fl(u)2844 344 y Fh(0)2656 567 y Fl(T)2697 579 y Fg(k)q Fh(\000)p Fi(1)2853 477 y Fq(ds)2950 436 y Fn(00)2993 341 y Fp(Z)3092 477 y Fq(dx)17 b(dx)3321 436 y Fn(0)3361 477 y Fq(dy)423 767 y Fo(\002)517 627 y Fp(\022)601 700 y Fq(@)5 b(p)706 715 y Fl(s)p Fn(\000)p Fl(s)827 696 y Fh(0)p 601 745 253 4 v 675 836 a Fq(@)g(s)881 767 y Fu(\026)-67 b Fq(m)948 726 y Fn(0)972 627 y Fp(\023)1045 767 y Fu(\()p Fq(x)p Fu(\))1193 627 y Fp(\022)1276 700 y Fq(@)5 b(p)1381 715 y Fl(u)p Fn(\000)p Fl(u)1518 696 y Fh(0)p 1276 745 269 4 v 1355 836 a Fq(@)g(u)1574 767 y Fu(\026)-68 b Fq(m)1640 726 y Fn(0)1664 627 y Fp(\023)1737 767 y Fu(\()p Fq(x)1830 726 y Fn(0)1854 767 y Fu(\))17 b Fq(g)1956 782 y Fl(s)1989 763 y Fh(0)2010 782 y Fn(\000)p Fl(s)2098 763 y Fh(00)2142 767 y Fu(\()p Fq(x;)g(y)t Fu(\))g Fq(g)2433 782 y Fl(u)2474 763 y Fh(0)2495 782 y Fn(\000)p Fl(s)2583 763 y Fh(00)2627 767 y Fu(\()p Fq(x)2720 726 y Fn(0)2744 767 y Fq(;)g(y)t Fu(\))p Fq(a)p Fu(\()p Fq(\025)3024 726 y Fl(\014)3070 767 y Fu(\()p Fq(y)25 b Fu(+)d Fq(x)3334 782 y Fl(k)r Fn(\000)p Fm(1)3467 767 y Fu(\)\))3543 726 y Fm(2)456 1060 y Fo(\024)28 b Fq(C)7 b Fo(k)p Fq(a)p Fo(k)789 1019 y Fm(2)789 1085 y Fn(1)863 1060 y Fq(\025)920 1019 y Fm(4)976 924 y Fp(Z)1076 951 y Fl(T)1117 959 y Fg(n)1032 1150 y Fl(T)1073 1162 y Fg(k)1147 1060 y Fq(ds)1261 924 y Fp(Z)1360 951 y Fl(T)1401 963 y Fg(k)1316 1150 y Fl(T)1357 1162 y Fg(k)q Fh(\000)p Fi(1)1461 1060 y Fq(ds)1558 1019 y Fn(0)1581 924 y Fp(Z)1680 951 y Fl(T)1721 959 y Fg(n)1636 1150 y Fl(T)1677 1162 y Fg(k)1752 1060 y Fq(du)1859 924 y Fp(Z)1957 951 y Fl(T)1998 963 y Fg(k)1913 1150 y Fl(T)1954 1162 y Fg(k)q Fh(\000)p Fi(1)2058 1060 y Fq(du)2165 1019 y Fn(0)2188 924 y Fp(Z)2287 951 y Fl(s)2320 927 y Fh(0)2342 951 y Fn(^)p Fl(u)2430 927 y Fh(0)2243 1150 y Fl(T)2284 1162 y Fg(k)q Fh(\000)p Fi(1)2440 1060 y Fq(ds)2537 1019 y Fn(00)655 1350 y Fo(\002)755 1206 y Fp(\015)755 1266 y(\015)755 1325 y(\015)755 1385 y(\015)810 1350 y Fq(g)857 1365 y Fl(s)890 1346 y Fh(0)912 1365 y Fn(\000)p Fl(s)1000 1346 y Fh(00)1054 1283 y Fq(@)e(p)1159 1298 y Fl(s)p Fn(\000)p Fl(s)1280 1279 y Fh(0)p 1054 1327 253 4 v 1129 1419 a Fq(@)g(s)1335 1350 y Fu(\026)-67 b Fq(m)1402 1309 y Fn(0)1425 1206 y Fp(\015)1425 1266 y(\015)1425 1325 y(\015)1425 1385 y(\015)1480 1449 y Fn(1)1594 1206 y Fp(\015)1594 1266 y(\015)1594 1325 y(\015)1594 1385 y(\015)1659 1283 y Fq(@)5 b(p)1764 1298 y Fl(u)p Fn(\000)p Fl(u)1901 1279 y Fh(0)p 1659 1327 269 4 v 1738 1419 a Fq(@)g(u)1957 1350 y Fu(\026)-68 b Fq(m)2023 1309 y Fn(0)2047 1206 y Fp(\015)2047 1266 y(\015)2047 1325 y(\015)2047 1385 y(\015)2102 1449 y Fm(1)2180 1350 y Fo(k)p Fq(g)2277 1365 y Fl(u)2318 1346 y Fh(0)2340 1365 y Fn(\000)p Fl(s)2428 1346 y Fh(00)2472 1350 y Fu(1)p Fo(k)2571 1379 y Fn(1)456 1647 y Fo(\024)28 b Fq(C)7 b Fo(k)p Fq(a)p Fo(k)789 1606 y Fm(2)789 1672 y Fn(1)863 1647 y Fq(\025)920 1606 y Fm(4)976 1511 y Fp(Z)1076 1538 y Fl(T)1117 1546 y Fg(n)1032 1737 y Fl(T)1073 1749 y Fg(k)1147 1647 y Fq(ds)1261 1511 y Fp(Z)1360 1538 y Fl(T)1401 1550 y Fg(k)1316 1737 y Fl(T)1357 1749 y Fg(k)q Fh(\000)p Fi(1)1461 1647 y Fq(ds)1558 1606 y Fn(0)1581 1511 y Fp(Z)1680 1538 y Fl(T)1721 1546 y Fg(n)1636 1737 y Fl(T)1677 1749 y Fg(k)1752 1647 y Fq(du)1859 1511 y Fp(Z)1957 1538 y Fl(T)1998 1550 y Fg(k)1913 1737 y Fl(T)1954 1749 y Fg(k)q Fh(\000)p Fi(1)2058 1647 y Fq(du)2165 1606 y Fn(0)2188 1511 y Fp(Z)2287 1538 y Fl(s)2320 1514 y Fh(0)2342 1538 y Fn(^)p Fl(u)2430 1514 y Fh(0)2243 1737 y Fl(T)2284 1749 y Fg(k)q Fh(\000)p Fi(1)2440 1647 y Fq(ds)2537 1606 y Fn(00)2760 1579 y Fu(1)p 2573 1624 423 4 v 2573 1715 a(\()p Fq(s)22 b Fo(\000)g Fq(s)2824 1686 y Fn(0)2848 1715 y Fu(\))2886 1686 y Fm(3)p Fl(=)p Fm(2)3119 1579 y Fu(1)p 3015 1624 257 4 v 3015 1715 a Fq(u)g Fo(\000)h Fq(u)3249 1686 y Fn(0)456 1884 y Fo(\024)28 b Fq(C)7 b Fo(k)p Fq(a)p Fo(k)789 1843 y Fm(2)789 1908 y Fn(1)863 1884 y Fq(\025)920 1843 y Fm(4)960 1884 y Fo(j)17 b Fu(log)f Fq(\025)p Fo(j)p Fq(T)1303 1843 y Fm(5)p Fl(=)p Fm(2)100 2043 y Fu(Since,)32 b(conditionally)e(on)j Fo(F)1172 2058 y Fl(T)1213 2070 y Fg(k)q Fh(\000)p Fi(1)1333 2043 y Fu(,)f Fq(z)1441 2007 y Fl(k)r Fn(\000)p Fm(1)1608 2043 y Fu(is)g(Gaussian)g(w)m(e)h(also)f(ha)m(v)m(e)1117 2212 y Fj(E)1194 2131 y Fp(\000)1256 2212 y Fq(X)1345 2171 y Fm(4)1337 2237 y Fl(n;k)1442 2127 y Fp(\014)1442 2187 y(\014)1491 2212 y Fo(F)1563 2227 y Fl(T)1604 2239 y Fg(k)q Fh(\000)p Fi(1)1725 2131 y Fp(\001)1798 2212 y Fo(\024)c Fq(C)7 b Fo(k)p Fq(a)p Fo(k)2131 2171 y Fm(4)2131 2237 y Fn(1)2206 2212 y Fq(\025)2263 2171 y Fm(8)2302 2212 y Fo(j)17 b Fu(log)f Fq(\025)p Fo(j)2574 2171 y Fm(2)2613 2212 y Fq(T)2684 2171 y Fm(5)0 2375 y Fu(By)33 b(the)g(BDG)f(inequalit)m(y)f(\(see)j([16,)e Fo(x)p Fu(6,)h(E.)g (4.1]\))f(and)h(the)g(ab)s(o)m(v)m(e)g(b)s(ounds)g(it)f(follo)m(ws)416 2645 y Fj(E)493 2534 y Fp(\020)559 2564 y(\000)604 2645 y Fq(M)708 2604 y Fm(\()p Fl(n)p Fm(\))698 2669 y Fl(n)811 2564 y Fp(\001)856 2580 y Fm(4)896 2534 y Fp(\021)1038 2645 y Fo(\024)84 b Fq(C)24 b Fj(E)1370 2534 y Fp(\020)1435 2564 y(\002)1477 2645 y Fq(M)1581 2604 y Fm(\()p Fl(n)p Fm(\))1683 2564 y Fp(\003)1725 2580 y Fm(2)1725 2684 y Fl(n)1772 2534 y Fp(\021)1859 2645 y Fu(=)j Fq(C)d Fj(E)2133 2474 y Fp( )2223 2520 y Fl(n)p Fn(\000)p Fm(1)2218 2550 y Fp(X)2224 2762 y Fl(h)p Fm(=1)2378 2645 y Fq(X)2467 2604 y Fm(2)2459 2669 y Fl(n;h)2567 2474 y Fp(!)2645 2494 y Fm(2)1038 2990 y Fo(\024)84 b Fq(C)1309 2820 y Fp(\()1395 2866 y Fl(n)p Fn(\000)p Fm(1)1389 2896 y Fp(X)1396 3108 y Fl(h)p Fm(=1)1550 2990 y Fj(E)12 b Fq(X)1699 2949 y Fm(4)1691 3015 y Fl(n;h)1827 2990 y Fu(+)22 b(2)1995 2866 y Fl(n)p Fn(\000)p Fm(1)1990 2896 y Fp(X)1998 3108 y Fl(k)r Fm(=1)2158 2866 y Fl(k)r Fn(\000)p Fm(1)2151 2896 y Fp(X)2157 3108 y Fl(h)p Fm(=1)2311 2990 y Fj(E)2388 2910 y Fp(\000)2440 2990 y Fq(X)2529 2949 y Fm(2)2521 3015 y Fl(n;h)2628 2990 y Fj(E)2705 2910 y Fp(\000)2767 2990 y Fq(X)2856 2949 y Fm(2)2848 3015 y Fl(n;k)2953 2906 y Fp(\014)2953 2965 y(\014)3002 2990 y Fo(F)3074 3005 y Fl(T)3115 3017 y Fg(k)q Fh(\000)p Fi(1)3236 2910 y Fp(\001)o(\001)3327 2820 y(\))1038 3239 y Fo(\024)84 b Fq(C)24 b(n)1351 3198 y Fm(2)1390 3239 y Fq(\025)1447 3198 y Fm(8)1487 3239 y Fo(j)17 b Fu(log)e Fq(\025)p Fo(j)1758 3198 y Fm(2)1797 3239 y Fq(T)1868 3198 y Fm(5)3591 3239 y Fu(\(7.16\))0 3398 y(W)-8 b(e)33 b(next)g(use)h(the)f(follo)m (wing)d(corollary)h(of)h(the)h(Cheb)m(yshev)j(inequalit)m(y)-8 b(,)651 3643 y Fj(P)727 3502 y Fp(\022)808 3643 y Fu(sup)801 3724 y Fl(n)p Fn(\024)p Fl(N)979 3643 y Fo(j)p Fq(!)1068 3658 y Fl(n)1114 3643 y Fo(j)28 b Fq(>)f(\016)1320 3502 y Fp(\023)1421 3643 y Fo(\024)h Fj(P)1602 3472 y Fp( )1691 3548 y(X)1683 3759 y Fl(n)p Fn(\024)p Fl(N)1860 3643 y Fo(j)p Fq(!)1949 3658 y Fl(n)1995 3643 y Fo(j)2023 3601 y Fl(p)2090 3643 y Fq(>)g(\016)2241 3601 y Fl(p)2281 3472 y Fp(!)2387 3643 y Fo(\024)g Fq(\016)2539 3601 y Fn(\000)p Fl(p)2659 3548 y Fp(X)2650 3759 y Fl(n)p Fn(\024)p Fl(N)2828 3643 y Fj(E)h Fu(\()6 b Fo(j)p Fq(!)3038 3658 y Fl(n)3084 3643 y Fo(j)3112 3601 y Fl(p)3151 3643 y Fu(\))402 b(\(7.17\))0 3906 y(and)33 b(get,)f(b)m(y)j(\(7.16\))o(,)e(c) m(ho)s(osing)f Fq(p)c Fu(=)f(4,)505 4174 y Fj(P)581 4004 y Fp( )726 4174 y Fu(sup)662 4260 y Fl(n)p Fn(\024)p Fl(n)803 4272 y Fg(\025)843 4260 y Fm(\()p Fl(\034)8 b Fm(\))953 4000 y Fp(\014)953 4060 y(\014)953 4119 y(\014)953 4179 y(\014)953 4239 y(\014)992 4050 y Fl(n)p Fn(\000)p Fm(1)987 4080 y Fp(X)994 4292 y Fl(k)r Fm(=1)1147 4174 y Fq(X)1228 4189 y Fl(n;k)1333 4000 y Fp(\014)1333 4060 y(\014)1333 4119 y(\014)1333 4179 y(\014)1333 4239 y(\014)1394 4174 y Fq(>)28 b(T)1569 4133 y Fn(\000)p Fm(1)p Fl(=)p Fm(4)1733 4004 y Fp(!)1840 4174 y Fo(\024)g Fq(T)2032 4042 y Fl(n)2075 4054 y Fg(\025)2116 4042 y Fm(\()p Fl(\034)8 b Fm(\))2049 4080 y Fp(X)2055 4289 y Fl(n)p Fm(=1)2226 4174 y Fq(C)f(n)2361 4133 y Fm(2)2401 4174 y Fq(\025)2458 4133 y Fm(8)2497 4174 y Fo(j)17 b Fu(log)f Fq(\025)p Fo(j)2769 4133 y Fm(2)2808 4174 y Fq(T)2879 4133 y Fm(5)2946 4174 y Fo(\024)28 b Fq(C)7 b(\025)3185 4133 y Fm(2)3225 4174 y Fq(T)3296 4133 y Fm(4)100 4444 y Fu(It)42 b(remains)f(to)g(b)s (ound)1043 4419 y(~)1027 4444 y Fq(S)1087 4459 y Fl(n)1134 4444 y Fu(\(6)p Fq(;)17 b Fu(2\).)71 b(W)-8 b(e)42 b(split)f(it)g(in)m (to)g(t)m(w)m(o)i(terms)f(b)m(y)h(using)e(the)i(decomp)s(osition)0 4563 y(\(4.34\))32 b(for)g Fq(z)480 4527 y Fm(\()p Fl(k)r Fn(\000)p Fm(1\))668 4563 y Fu(.)44 b(W)-8 b(e)33 b(get)174 4794 y(~)157 4820 y Fq(S)223 4778 y Fn(?)217 4844 y Fl(n)282 4820 y Fu(\(6)p Fq(;)17 b Fu(2\))82 b(:=)778 4752 y(3)p Fq(\025)884 4716 y Fm(2)p 778 4797 146 4 v 826 4888 a Fu(4)955 4695 y Fl(n)p Fn(\000)p Fm(1)950 4725 y Fp(X)956 4937 y Fl(h)p Fm(=1)1162 4695 y Fl(h)1110 4725 y Fp(X)1118 4937 y Fl(k)r Fm(=1)1271 4684 y Fp(Z)1370 4710 y Fl(T)1411 4722 y Fg(h)p Fi(+1)1326 4910 y Fl(T)1367 4922 y Fg(h)1533 4820 y Fq(ds)1647 4684 y Fp(Z)1746 4710 y Fl(T)1787 4722 y Fg(k)1701 4910 y Fl(T)1742 4922 y Fg(k)q Fh(\000)p Fi(1)1880 4820 y Fq(ds)1977 4778 y Fn(0)2016 4684 y Fp(Z)2116 4710 y Fl(x)2156 4722 y Fg(h)2072 4910 y Fl(x)2112 4922 y Fg(k)q Fh(\000)p Fi(1)2249 4820 y Fq(d\030)20 b Fo(h)e Fu(\026)-67 b Fq(m)2487 4778 y Fn(00)2487 4844 y Fl(\030)2530 4820 y Fq(;)2584 4752 y(@)5 b(p)2689 4767 y Fl(s)p Fn(\000)p Fl(s)2810 4748 y Fh(0)p 2584 4797 253 4 v 2658 4888 a Fq(@)g(s)2846 4820 y(z)2895 4778 y Fm(\()p Fl(k)r Fn(\000)p Fm(1)p Fl(;)p Fn(?)p Fm(\))3158 4820 y Fu(\()p Fq(s)3242 4778 y Fn(0)3265 4820 y Fu(\))p Fo(i)192 5140 y Fu(~)175 5165 y Fq(S)241 5124 y Fn(k)235 5190 y Fl(n)282 5165 y Fu(\(6)p Fq(;)17 b Fu(2\))82 b(:=)778 5098 y(3)p Fq(\025)884 5062 y Fm(2)p 778 5142 146 4 v 826 5233 a Fu(4)955 5041 y Fl(n)p Fn(\000)p Fm(1)950 5070 y Fp(X)956 5283 y Fl(h)p Fm(=1)1162 5041 y Fl(h)1110 5070 y Fp(X)1118 5283 y Fl(k)r Fm(=1)1271 5029 y Fp(Z)1370 5056 y Fl(T)1411 5068 y Fg(h)p Fi(+1)1326 5255 y Fl(T)1367 5267 y Fg(h)1533 5165 y Fq(ds)1647 5029 y Fp(Z)1746 5056 y Fl(T)1787 5068 y Fg(k)1701 5255 y Fl(T)1742 5267 y Fg(k)q Fh(\000)p Fi(1)1880 5165 y Fq(ds)1977 5124 y Fn(0)2016 5029 y Fp(Z)2116 5056 y Fl(x)2156 5068 y Fg(h)2072 5255 y Fl(x)2112 5267 y Fg(k)q Fh(\000)p Fi(1)2249 5165 y Fq(d\030)20 b Fo(h)e Fu(\026)-67 b Fq(m)2487 5124 y Fn(00)2487 5190 y Fl(\030)2530 5165 y Fq(;)2584 5098 y(@)5 b(p)2689 5113 y Fl(s)p Fn(\000)p Fl(s)2810 5094 y Fh(0)p 2584 5142 253 4 v 2658 5233 a Fq(@)g(s)2864 5165 y Fu(\026)-67 b Fq(m)2931 5124 y Fn(0)2931 5190 y Fl(x)2971 5202 y Fg(k)q Fh(\000)p Fi(1)3092 5165 y Fo(i)17 b Fq(w)3221 5124 y Fm(\()p Fl(k)r Fn(\000)p Fm(1\))3407 5165 y Fu(\()p Fq(s)3491 5124 y Fn(0)3514 5165 y Fu(\))39 b(\(7.18\))0 5430 y(where)34 b(w)m(e)f(set)h Fq(w)651 5393 y Fm(\()p Fl(k)r Fm(\))748 5430 y Fu(\()p Fq(t)p Fu(\))27 b(:=)1027 5390 y Fm(3)p 1027 5407 36 4 v 1027 5464 a(4)1072 5430 y Fo(h)18 b Fu(\026)-67 b Fq(m)1196 5393 y Fn(0)1196 5454 y Fl(x)1236 5466 y Fg(k)1278 5430 y Fq(;)17 b(z)1371 5393 y Fm(\()p Fl(k)r Fm(\))1469 5430 y Fu(\()p Fq(t)p Fu(\))p Fo(i)p Fu(.)1882 5532 y Fk(39)p eop %%Page: 40 41 40 40 bop 100 266 a Fu(In)33 b(App)s(endix)g(A)f(w)m(e)i(pro)m(v)m(e)g (for)e(eac)m(h)h Fq(\015)g Fo(2)28 b Fu(\(0)p Fq(;)17 b Fu(1)p Fq(=)p Fu(2\))31 b(there)i(exists)h(a)e(constan)m(t)i Fq(C)39 b Fu(suc)m(h)34 b(that)1100 364 y Fp(\014)1100 424 y(\014)1100 484 y(\014)1100 544 y(\014)1133 368 y(\034)1226 509 y Fu(\026)-67 b Fq(m)1293 468 y Fn(00)1336 509 y Fq(;)1389 442 y(@)5 b(p)1494 457 y Fl(t)p 1389 486 136 4 v 1411 577 a Fq(@)g(t)1535 509 y(')1599 368 y Fp(\035)1674 364 y(\014)1674 424 y(\014)1674 484 y(\014)1674 544 y(\014)1734 509 y Fo(\024)1884 442 y Fq(C)p 1850 486 146 4 v 1850 577 a(t)1885 548 y Fm(3)p Fl(=)p Fm(2)2021 368 y Fp(\022)2095 509 y Fo(k)p Fq(')p Fo(k)2259 524 y Fn(1)2355 509 y Fo(^)2454 442 y(k)p Fq(')p Fo(k)2618 457 y Fm(1)p 2454 486 203 4 v 2515 577 a Fq(t)2550 548 y Fl(\015)2667 368 y Fp(\023)3591 509 y Fu(\(7.19\))100 747 y(W)-8 b(e)33 b(th)m(us)g(get)289 1032 y(sup)225 1118 y Fl(n)p Fn(\024)p Fl(n)366 1130 y Fg(\025)406 1118 y Fm(\()p Fl(\034)8 b Fm(\))527 965 y Fo(j)571 939 y Fu(~)555 965 y Fq(S)621 928 y Fn(?)615 989 y Fl(n)679 965 y Fu(\(6)p Fq(;)17 b Fu(2\))p Fo(j)p 527 1009 398 4 v 562 1100 a Fu(1)22 b(+)g Fq(x)786 1115 y Fl(n;)p Fn(\003)1017 1032 y Fo(\024)83 b Fq(C)7 b(\025)1311 991 y Fm(2)1367 900 y Fl(n)1410 912 y Fg(\025)1451 900 y Fm(\()p Fl(\034)h Fm(\))p Fn(\000)p Fm(1)1429 937 y Fp(X)1436 1149 y Fl(h)p Fm(=1)1703 907 y Fl(h)1651 937 y Fp(X)1659 1149 y Fl(k)r Fm(=1)1812 896 y Fp(Z)1912 923 y Fl(T)1953 935 y Fg(h)p Fi(+1)1867 1122 y Fl(T)1908 1134 y Fg(h)2074 1032 y Fq(ds)2188 896 y Fp(Z)2287 923 y Fl(T)2328 935 y Fg(k)2243 1122 y Fl(T)2284 1134 y Fg(k)q Fh(\000)p Fi(1)2405 1032 y Fq(ds)2502 991 y Fn(0)2738 965 y Fu(1)p 2551 1009 423 4 v 2551 1100 a(\()p Fq(s)22 b Fo(\000)h Fq(s)2803 1072 y Fn(0)2826 1100 y Fu(\))2864 1072 y Fm(3)p Fl(=)p Fm(2)2984 1032 y Fo(k)p Fq(z)3083 991 y Fm(\()p Fl(k)r Fn(\000)p Fm(1)p Fl(;)p Fn(?)p Fm(\))3346 1032 y Fu(\()p Fq(s)3430 991 y Fn(0)3453 1032 y Fu(\))p Fo(k)3541 1047 y Fn(1)1017 1389 y Fo(\024)83 b Fq(C)7 b(\025)1311 1348 y Fm(2)1367 1257 y Fl(n)1410 1269 y Fg(\025)1451 1257 y Fm(\()p Fl(\034)h Fm(\))p Fn(\000)p Fm(1)1429 1294 y Fp(X)1436 1506 y Fl(h)p Fm(=1)1703 1264 y Fl(h)1651 1294 y Fp(X)1659 1506 y Fl(k)r Fm(=1)2038 1237 y Fo(p)p 2122 1237 71 4 v 2122 1321 a Fq(T)p 1822 1366 587 4 v 1822 1457 a Fu(\()p Fq(h)22 b Fo(\000)h Fq(k)i Fu(+)d(1\))2299 1428 y Fm(3)p Fl(=)p Fm(2)2546 1389 y Fu(sup)2435 1475 y Fl(t)p Fn(2)p Fm([)p Fl(T)2568 1487 y Fg(k)q Fh(\000)p Fi(1)2685 1475 y Fl(;T)2746 1487 y Fg(k)2784 1475 y Fm(])2820 1389 y Fo(k)p Fq(z)2919 1348 y Fm(\()p Fl(k)r Fn(\000)p Fm(1)p Fl(;)p Fn(?)p Fm(\))3182 1389 y Fu(\()p Fq(t)p Fu(\))p Fo(k)3343 1404 y Fn(1)1017 1645 y Fo(\024)83 b Fq(C)24 b(T)1342 1604 y Fn(\000)p Fm(1)p Fl(=)p Fm(2)1587 1645 y Fu(sup)1523 1730 y Fl(n)p Fn(\024)p Fl(n)1664 1742 y Fg(\025)1704 1730 y Fm(\()p Fl(\034)8 b Fm(\))1926 1645 y Fu(sup)1815 1730 y Fl(t)p Fn(2)p Fm([)p Fl(T)1948 1742 y Fg(k)q Fh(\000)p Fi(1)2065 1730 y Fl(;T)2126 1742 y Fg(k)2163 1730 y Fm(])2200 1645 y Fo(k)p Fq(z)2299 1604 y Fm(\()p Fl(k)r Fn(\000)p Fm(1)p Fl(;)p Fn(?)p Fm(\))2561 1645 y Fu(\()p Fq(t)p Fu(\))p Fo(k)2722 1660 y Fn(1)3591 1645 y Fu(\(7.20\))0 1908 y(On)33 b(the)g(other)f(hand,)h(for)f(eac)m(h)i Fq(\015)e Fo(2)d Fu(\(0)p Fq(;)17 b Fu(1)p Fq(=)p Fu(2\),)31 b(since)i Fo(k)18 b Fu(\026)-67 b Fq(m)2151 1872 y Fn(0)2151 1932 y Fl(x)2191 1944 y Fg(k)q Fh(\000)p Fi(1)2311 1908 y Fo(k)2361 1923 y Fm(1)2428 1908 y Fo(\024)28 b Fq(C)7 b Fu(,)33 b(w)m(e)h(also)d(ha)m(v)m(e)336 2217 y(sup)272 2303 y Fl(n)p Fn(\024)p Fl(n)413 2315 y Fg(\025)453 2303 y Fm(\()p Fl(\034)8 b Fm(\))573 2149 y Fo(j)618 2124 y Fu(~)601 2149 y Fq(S)667 2099 y Fn(k)661 2160 y Fl(n)708 2149 y Fu(\(6)p Fq(;)17 b Fu(2\))p Fo(j)p 573 2194 380 4 v 600 2285 a Fu(1)22 b(+)g Fq(x)824 2300 y Fl(n;)p Fn(\003)1046 2217 y Fo(\024)83 b Fq(C)7 b(\025)1340 2176 y Fm(2)1396 2085 y Fl(n)1439 2097 y Fg(\025)1480 2085 y Fm(\()p Fl(\034)h Fm(\))p Fn(\000)p Fm(1)1458 2122 y Fp(X)1465 2334 y Fl(h)p Fm(=1)1732 2092 y Fl(h)1680 2122 y Fp(X)1688 2334 y Fl(k)r Fm(=1)1841 2081 y Fp(Z)1941 2108 y Fl(T)1982 2120 y Fg(h)p Fi(+1)1896 2307 y Fl(T)1937 2319 y Fg(h)2103 2217 y Fq(ds)2217 2081 y Fp(Z)2316 2108 y Fl(T)2357 2120 y Fg(k)2272 2307 y Fl(T)2313 2319 y Fg(k)q Fh(\000)p Fi(1)2433 2217 y Fq(ds)2530 2176 y Fn(0)2815 2149 y Fu(1)p 2580 2194 518 4 v 2580 2285 a(\()p Fq(s)22 b Fo(\000)h Fq(s)2832 2256 y Fn(0)2855 2285 y Fu(\))2893 2256 y Fm(3)p Fl(=)p Fm(2+)p Fl(\015)3108 2217 y Fo(j)p Fq(w)3209 2176 y Fm(\()p Fl(k)r Fn(\000)p Fm(1\))3396 2217 y Fu(\()p Fq(s)3480 2176 y Fn(0)3503 2217 y Fu(\))p Fo(j)1046 2574 y(\024)83 b Fq(C)7 b(\025)1340 2533 y Fm(2)1396 2441 y Fl(n)1439 2453 y Fg(\025)1480 2441 y Fm(\()p Fl(\034)h Fm(\))p Fn(\000)p Fm(1)1458 2479 y Fp(X)1465 2691 y Fl(h)p Fm(=1)1732 2449 y Fl(h)1680 2479 y Fp(X)1688 2691 y Fl(k)r Fm(=1)2054 2506 y Fq(T)2125 2470 y Fm(1)p Fl(=)p Fm(2)p Fn(\000)p Fl(\015)p 1851 2551 682 4 v 1851 2642 a Fu(\()p Fq(h)22 b Fo(\000)h Fq(k)i Fu(+)d(1\))2328 2613 y Fm(3)p Fl(=)p Fm(2+)p Fl(\015)2670 2574 y Fu(sup)2559 2660 y Fl(t)p Fn(2)p Fm([)p Fl(T)2692 2672 y Fg(k)q Fh(\000)p Fi(1)2809 2660 y Fl(;T)2870 2672 y Fg(k)2908 2660 y Fm(])2944 2574 y Fo(j)p Fq(w)3045 2533 y Fm(\()p Fl(k)r Fn(\000)p Fm(1\))3232 2574 y Fu(\()p Fq(t)p Fu(\))p Fo(j)1046 2892 y(\024)83 b Fq(C)24 b(T)1371 2851 y Fn(\000)p Fl(\015)1551 2892 y Fu(sup)1486 2978 y Fl(n)p Fn(\024)p Fl(n)1627 2990 y Fg(\025)1668 2978 y Fm(\()p Fl(\034)8 b Fm(\))1889 2892 y Fu(sup)1778 2978 y Fl(t)p Fn(2)p Fm([)p Fl(T)1911 2990 y Fg(k)q Fh(\000)p Fi(1)2028 2978 y Fl(;T)2089 2990 y Fg(k)2127 2978 y Fm(])2173 2825 y Fo(j)p Fq(w)2274 2788 y Fm(\()p Fl(k)r Fm(\))2371 2825 y Fu(\()p Fq(t)p Fu(\))p Fo(j)p 2173 2869 337 4 v 2264 2889 a(p)p 2347 2889 71 4 v 84 x Fq(T)3591 2892 y Fu(\(7.21\))100 3166 y(Since)35 b Fo(j)p Fq(w)458 3130 y Fm(\()p Fl(k)r Fm(\))555 3166 y Fu(\()p Fq(t)p Fu(\))p Fo(j)c(\024)i Fq(C)7 b Fo(k)p Fq(z)1011 3130 y Fm(\()p Fl(k)r Fn(\000)p Fm(1\))1199 3166 y Fu(\()p Fq(t)p Fu(\))p Fo(k)1360 3181 y Fn(1)1470 3166 y Fu(b)m(y)36 b(using)f(the)g(Gaussian) g(estimate)f(\(4.6\))g(\(resp.)j(\(4.36\))o(\))e(and)0 3291 y(\(7.21\))d(\(resp.)h(in)f(\(7.20\))o(\),)h(the)g(b)s(ound)g(in)f (\(7.7\))g(for)1995 3266 y(~)1979 3291 y Fq(S)2039 3306 y Fl(n)2086 3291 y Fu(\(6)p Fq(;)17 b Fu(2\))31 b(no)m(w)i(follo)m(ws.) 0 3481 y Ft(The)h(c)-5 b(ase)34 b Fq(`)28 b Fu(=)g(\(9)p Fo(j)p Fu(2;)17 b(2\))p Ft(.)41 b Fu(W)-8 b(e)33 b(ha)m(v)m(e)h Fq(S)1400 3496 y Fl(n)1447 3481 y Fu(\(9)p Fo(j)p Fu(2;)17 b(2\))27 b(=)1872 3406 y Fp(P)1977 3432 y Fl(n)p Fn(\000)p Fm(1)1977 3510 y Fl(k)r Fm(=0)2131 3481 y Fq(\021)2179 3496 y Fl(k)2222 3481 y Fu(\(9)p Fo(j)p Fu(2;)17 b(2\))31 b(where)1013 3738 y Fq(\021)1061 3753 y Fl(k)1104 3738 y Fu(\(9)p Fo(j)p Fu(2;)17 b(2\))26 b(=)1539 3671 y(9)p Fq(\025)1645 3635 y Fm(2)p 1539 3716 146 4 v 1587 3807 a Fu(4)1711 3603 y Fp(Z)1810 3629 y Fl(T)1851 3641 y Fg(k)q Fi(+1)1766 3828 y Fl(T)1807 3840 y Fg(k)1971 3738 y Fq(dt)17 b Fo(h)g Fu(\026)-66 b Fq(m)2198 3697 y Fn(0)2198 3763 y Fl(x)2238 3775 y Fg(k)2279 3738 y Fq(;)35 b Fu(\026)-67 b Fq(m)2408 3753 y Fl(x)2448 3765 y Fg(k)2490 3738 y Fq(z)2539 3697 y Fm(\()p Fl(k)r Fm(\))2637 3738 y Fu(\()p Fq(t)p Fu(\))2748 3697 y Fm(2)2788 3738 y Fo(i)764 b Fu(\(7.22\))0 3983 y(W)-8 b(e)33 b(de\014ne)1418 4124 y Fq(\015)1469 4139 y Fl(k)1539 4124 y Fu(:=)28 b Fj(E)h Fu(\()15 b Fq(\021)1848 4139 y Fl(k)1891 4124 y Fu(\(9)p Fo(j)p Fu(2;)i(2\))p Fo(j)e(F)2301 4139 y Fl(T)2342 4151 y Fg(k)2384 4124 y Fu(\))1169 b(\(7.23\))0 4286 y(and)33 b(then)g(decomp)s(ose)1396 4516 y Fq(S)1456 4531 y Fl(n)1503 4516 y Fu(\(9)p Fo(j)p Fu(2;)17 b(2\))27 b(=)g Fq(M)2022 4531 y Fl(n)2092 4516 y Fu(+)2195 4391 y Fl(n)p Fn(\000)p Fm(1)2190 4421 y Fp(X)2197 4633 y Fl(k)r Fm(=0)2350 4516 y Fq(\015)2401 4531 y Fl(k)3591 4516 y Fu(\(7.24\))0 4766 y(where)34 b Fq(M)376 4781 y Fl(n)456 4766 y Fu(is)e(an)g Fo(F)761 4781 y Fl(T)802 4789 y Fg(n)849 4766 y Fu({martingale)d(with)j (brac)m(k)m(et)1103 5040 y Fo(h)p Fq(M)10 b Fo(i)1285 5055 y Fl(n)1360 5040 y Fu(=)1469 4916 y Fl(n)p Fn(\000)p Fm(1)1463 4946 y Fp(X)1471 5158 y Fl(k)r Fm(=0)1624 4960 y Fp(\002)1665 5040 y Fj(E)1743 4960 y Fp(\000)1804 5040 y Fq(\021)1852 5055 y Fl(k)1895 5040 y Fu(\(9)p Fo(j)p Fu(2;)17 b(2\))2190 4999 y Fm(2)2228 4956 y Fp(\014)2228 5015 y(\014)2278 5040 y Fo(F)2350 5055 y Fl(T)2391 5067 y Fg(k)2433 4960 y Fp(\001)2500 5040 y Fo(\000)23 b Fq(\015)2656 4999 y Fm(2)2651 5065 y Fl(k)2696 4960 y Fp(\003)3591 5040 y Fu(\(7.25\))0 5316 y(Note)47 b(that)g(the)h(di\013erence)g(b)s (et)m(w)m(een)h(the)e(brac)m(k)m(et)i Fo(h\001i)2148 5331 y Fl(n)2241 5316 y Fu(and)f(the)f(quadratic)g(v)-5 b(ariation)45 b([)p Fo(\001)p Fu(])3585 5331 y Fl(n)3679 5316 y Fu(is)h(a)0 5433 y(martingale,)30 b(see)k([16)o(].)1882 5532 y Fk(40)p eop %%Page: 41 42 41 41 bop 100 266 a Fu(W)-8 b(e)33 b(ha)m(v)m(e)179 485 y Fq(\015)230 500 y Fl(k)300 485 y Fu(=)413 418 y(9)p Fq(\025)519 381 y Fm(2)p 413 462 146 4 v 461 553 a Fu(4)585 349 y Fp(Z)685 376 y Fl(T)726 388 y Fg(k)q Fi(+1)640 575 y Fl(T)681 587 y Fg(k)845 485 y Fq(dt)948 349 y Fp(Z)1047 376 y Fl(t)1003 575 y(T)1044 587 y Fg(k)1086 485 y Fq(ds)1200 349 y Fp(Z)1299 485 y Fq(dx)1422 349 y Fp(Z)1521 485 y Fq(dy)38 b Fu(\026)-67 b Fq(m)1725 444 y Fn(0)1748 485 y Fu(\()p Fq(x)p Fu(\))19 b(\026)-68 b Fq(m)q Fu(\()p Fq(x)p Fu(\))p Fq(g)2143 500 y Fl(t)p Fn(\000)p Fl(s)2260 485 y Fu(\()p Fq(x;)17 b(y)t Fu(\))2487 444 y Fm(2)2542 404 y Fp(\002)2584 485 y Fq(a)p Fu(\()p Fq(\025)2730 444 y Fl(\014)2777 485 y Fu(\()p Fq(y)25 b Fu(+)d Fq(x)3041 500 y Fl(k)3084 485 y Fu(\)\))3160 444 y Fm(2)3222 485 y Fo(\000)h Fu(1)3371 404 y Fp(\003)3591 485 y Fu(\(7.26\))0 721 y(where)34 b(w)m(e)f(exploited)f(the)h(iden)m(tit)m(y)519 805 y Fp(Z)619 940 y Fq(dx)742 805 y Fp(Z)841 940 y Fq(dy)38 b Fu(\026)-67 b Fq(m)1045 899 y Fn(0)1069 940 y Fu(\()p Fq(x)p Fu(\))18 b(\026)-67 b Fq(m)p Fu(\()p Fq(x)p Fu(\))p Fq(g)1463 955 y Fl(t)p Fn(\000)p Fl(s)1580 940 y Fu(\()p Fq(x;)17 b(y)t Fu(\))1807 899 y Fm(2)1874 940 y Fu(=)1977 805 y Fp(Z)2077 940 y Fq(dx)35 b Fu(\026)-67 b Fq(m)2285 899 y Fn(0)2308 940 y Fu(\()p Fq(x)p Fu(\))19 b(\026)-68 b Fq(m)q Fu(\()p Fq(x)p Fu(\))p Fq(g)2703 956 y Fm(2\()p Fl(t)p Fn(\000)p Fl(s)p Fm(\))2910 940 y Fu(\()p Fq(x;)17 b(x)p Fu(\))28 b(=)g(0)0 1165 y(whic)m(h)33 b(holds)f(b)s(ecause)i Fq(x)28 b Fo(7!)g Fq(g)1153 1180 y Fl(t)1182 1165 y Fu(\()p Fq(x;)17 b(x)p Fu(\))33 b(is)f(an)h(ev)m(en)h(function)e(of)g Fq(x)p Fu(.)100 1282 y(W)-8 b(e)33 b(claim)d(for)i(eac)m(h)h Fq(q)f(>)27 b Fu(log)1200 1305 y Fl(\025)1241 1286 y Fh(\000)p Fi(1)1345 1282 y Fq(T)41 b Fu(=)28 b(\(1)22 b Fo(^)g Fq(\014)6 b Fu(\))p Fq(=)p Fu(20)31 b(there)j(is)e Fq(C)j Fu(=)27 b Fq(C)7 b Fu(\()p Fq(q)t Fu(\))32 b(so)h(that)1041 1486 y Fo(j)p Fq(\015)1120 1501 y Fl(k)1161 1486 y Fo(j)28 b(\024)g Fq(C)7 b Fu(\()p Fq(\025T)14 b Fu(\))1603 1445 y Fm(2)1658 1375 y Fp(n)1725 1486 y Fq(\025)1782 1445 y Fl(\014)s Fn(\000)p Fl(q)1940 1486 y Fu(+)22 b Fq(\025)2095 1445 y Fl(\014)2142 1486 y Fq(x)2197 1501 y Fl(k)r(;)p Fn(\003)2317 1486 y Fu(+)g Fq(e)2460 1445 y Fn(\000)p Fl(\025)2556 1421 y Fh(\000)p Fg(q)2639 1445 y Fl(=C)2733 1375 y Fp(o)3591 1486 y Fu(\(7.27\))0 1684 y(By)33 b(T)-8 b(a)m(ylor)32 b(expansion)604 1772 y Fp(\014)604 1832 y(\014)637 1857 y Fq(a)p Fu(\()p Fq(\025)783 1816 y Fl(\014)830 1857 y Fu(\()p Fq(y)25 b Fu(+)d Fq(x)1094 1872 y Fl(k)1137 1857 y Fu(\)\))1213 1816 y Fm(2)1275 1857 y Fo(\000)g Fu(1)1423 1772 y Fp(\014)1423 1832 y(\014)1539 1857 y Fo(\024)84 b Fq(C)1793 1776 y Fp(\010)1852 1857 y Fq(\025)1909 1816 y Fl(\014)1956 1857 y Fu(\()p Fo(j)p Fq(y)t Fo(j)20 b Fu(+)i Fo(j)p Fq(x)2303 1872 y Fl(k)2346 1857 y Fo(j)p Fu(\))p Fv(1)2468 1873 y Fn(j)p Fl(y)r Fn(j\024)p Fl(\025)2641 1854 y Fh(\000)p Fg(q)2749 1857 y Fu(+)g Fv(1)2903 1873 y Fn(j)p Fl(y)r Fn(j)p Fl(>\025)3076 1854 y Fh(\000)p Fg(q)3162 1776 y Fp(\011)1539 2016 y Fo(\024)84 b Fq(C)1793 1936 y Fp(\010)1852 2016 y Fq(\025)1909 1975 y Fl(\014)s Fn(\000)p Fl(q)2066 2016 y Fu(+)22 b Fq(\025)2221 1975 y Fl(\014)2269 2016 y Fq(x)2324 2031 y Fl(k)r(;)p Fn(\003)2444 2016 y Fu(+)g Fv(1)2598 2033 y Fn(j)p Fl(y)r Fn(j)p Fl(>\025)2771 2014 y Fh(\000)p Fg(q)2857 1936 y Fp(\011)3591 2016 y Fu(\(7.28\))0 2191 y(W)-8 b(e)33 b(use)h(the)f(b)s(ound)f(\(7.28\))g (in)g(\(7.26\))o(.)44 b(There)33 b(is)g Fq(C)39 b Fu(so)33 b(that,)f(for)g(an)m(y)i Fq(t)27 b Fo(2)i Fu([0)p Fq(;)17 b(T)d Fu(])31 b(\()p Fq(T)42 b(>)27 b Fu(1\),)414 2295 y Fp(Z)514 2321 y Fl(t)470 2520 y Fm(0)544 2430 y Fq(ds)658 2295 y Fp(Z)757 2430 y Fq(dx)880 2295 y Fp(Z)979 2430 y Fq(dy)38 b Fu(\026)-68 b Fq(m)1182 2389 y Fn(0)1206 2430 y Fu(\()p Fq(x)p Fu(\))p Fo(j)18 b Fu(\026)-67 b Fq(m)p Fu(\()p Fq(x)p Fu(\))p Fo(j)p Fq(g)1656 2445 y Fl(s)1693 2430 y Fu(\()p Fq(x;)17 b(y)t Fu(\))1920 2389 y Fm(2)1986 2430 y Fo(\024)2091 2295 y Fp(Z)2191 2321 y Fl(t)2146 2520 y Fm(0)2220 2430 y Fq(ds)2334 2295 y Fp(Z)2433 2430 y Fq(dx)35 b Fu(\026)-67 b Fq(m)2641 2389 y Fn(0)2665 2430 y Fu(\()p Fq(x)p Fu(\))p Fq(g)2843 2445 y Fm(2)p Fl(s)2915 2430 y Fu(\()p Fq(x;)17 b(x)p Fu(\))28 b Fo(\024)g Fq(C)7 b(T)0 2660 y Fu(the)29 b(\014rst)g(t)m(w)m(o)g (terms)g(on)g(the)g(r.h.s.)g(of)35 b(\(7.28\))28 b(pro)s(duce)h(the)g (\014rst)h(t)m(w)m(o)f(terms)f(on)h(the)g(r.h.s.)h(of)35 b(\(7.27\))o(.)0 2776 y(W)-8 b(e)28 b(estimate)f(next)i(the)f(con)m (tribution)e(of)h(the)i(last)d(one.)43 b(Denoting)26 b(b)m(y)j Fj(E)2749 2791 y Fl(x;y)r(;t)2928 2776 y Fu(the)g(exp)s (ectation)e(w.r.t.)0 2892 y(a)32 b(Bro)m(wnian)h(bridge)f(from)f Fq(x)i Fu(to)f Fq(y)k Fu(in)c(time)f Fq(t)p Fu(,)i(b)m(y)g(the)g(F)-8 b(eynman{Kac)32 b(form)m(ula)f(w)m(e)j(get)602 3126 y Fq(g)649 3141 y Fl(t)678 3126 y Fu(\()p Fq(x;)17 b(y)t Fu(\))27 b(=)h Fq(p)1085 3141 y Fl(t)1114 3126 y Fu(\()p Fq(x;)17 b(y)t Fu(\))g Fj(E)1417 3141 y Fl(x;y)r(;t)1586 3126 y Fu(exp)1751 2986 y Fp(\022)1825 3126 y Fo(\000)1919 2991 y Fp(Z)2019 3017 y Fl(t)1974 3216 y Fm(0)2048 3126 y Fq(ds)g(V)2240 3085 y Fn(00)2282 3126 y Fu(\()i(\026)-68 b Fq(m)q Fu(\()p Fq(!)2505 3141 y Fl(s)2541 3126 y Fu(\)\))2617 2986 y Fp(\023)2718 3126 y Fo(\024)28 b Fq(e)2868 3085 y Fm(2)p Fl(t)2933 3126 y Fq(p)2982 3141 y Fl(t)3012 3126 y Fu(\()p Fq(x;)17 b(y)t Fu(\))352 b(\(7.29\))0 3355 y(whence)111 3443 y Fp(Z)210 3579 y Fq(dx)333 3443 y Fp(Z)433 3579 y Fq(dy)19 b Fv(1)607 3595 y Fn(j)p Fl(x)p Fn(j\024)p Fl(\025)783 3576 y Fh(\000)p Fg(q)864 3595 y Fl(=)p Fm(2)974 3579 y Fu(\026)-67 b Fq(m)1041 3538 y Fn(0)1064 3579 y Fu(\()p Fq(x)p Fu(\))17 b Fq(g)1259 3594 y Fl(t)1289 3579 y Fu(\()p Fq(x;)g(y)t Fu(\))1516 3538 y Fm(2)1571 3579 y Fv(1)1627 3595 y Fn(j)p Fl(y)r Fn(j)p Fl(>\025)1800 3576 y Fh(\000)p Fg(q)1969 3579 y Fo(\024)2139 3511 y Fq(e)2184 3475 y Fn(\000)p Fm([)p Fl(\025)2300 3452 y Fh(\000)p Fi(2)p Fg(q)2413 3475 y Fl(=)p Fm(\(8)p Fl(t)p Fm(\)]+2)p Fl(t)p 2139 3556 564 4 v 2308 3576 a Fo(p)p 2391 3576 143 4 v 82 x Fu(2)p Fq(\031)t(t)2730 3443 y Fp(Z)2830 3579 y Fq(dx)2953 3443 y Fp(Z)3052 3579 y Fq(dy)i(g)3217 3594 y Fl(t)3247 3579 y Fu(\()p Fq(x;)e(y)t Fu(\))34 b(\026)-66 b Fq(m)3576 3538 y Fn(0)3598 3579 y Fu(\()p Fq(x)p Fu(\))1970 3868 y(=)83 b(2)2205 3800 y Fq(e)2250 3764 y Fn(\000)p Fm([)p Fl(\025)2366 3741 y Fh(\000)p Fi(2)p Fg(q)2478 3764 y Fl(=)p Fm(\(8)p Fl(t)p Fm(\)]+2)p Fl(t)p 2205 3845 564 4 v 2374 3865 a Fo(p)p 2457 3865 143 4 v 82 x Fu(2)p Fq(\031)t(t)3591 3868 y Fu(\(7.30\))0 4100 y(On)33 b(the)g(other)f (hand,)h(since)52 b(\026)-68 b Fq(m)1180 4064 y Fn(0)1204 4100 y Fu(\()p Fq(x)p Fu(\))33 b(deca)m(ys)h(exp)s(onen)m(tially)e(as)g Fq(x)d Fo(!)e(1)p Fu(,)278 4189 y Fp(Z)377 4325 y Fq(dx)500 4189 y Fp(Z)600 4325 y Fq(dy)19 b Fv(1)774 4341 y Fn(j)p Fl(x)p Fn(j)p Fl(>\025)950 4323 y Fh(\000)p Fg(q)1032 4341 y Fl(=)p Fm(2)1125 4325 y Fu(\026)-68 b Fq(m)1191 4284 y Fn(0)1215 4325 y Fu(\()p Fq(x)p Fu(\))p Fq(g)1393 4340 y Fl(t)1423 4325 y Fu(\()p Fq(x;)17 b(y)t Fu(\))1650 4284 y Fm(2)1688 4325 y Fv(1)1744 4341 y Fn(j)p Fl(y)r Fn(j)p Fl(>\025)1917 4323 y Fh(\000)p Fg(q)2086 4325 y Fo(\024)84 b Fq(C)7 b(e)2369 4284 y Fn(\000)p Fl(\025)2465 4260 y Fh(\000)p Fg(q)2547 4284 y Fl(=C)2658 4189 y Fp(Z)2758 4325 y Fq(dx)17 b(g)2928 4340 y Fm(2)p Fl(t)2992 4325 y Fu(\()p Fq(x;)g(x)p Fu(\))3222 4235 y Fp(p)p 3322 4235 240 4 v 3341 4325 a Fu(\026)-68 b Fq(m)3407 4296 y Fn(0)3431 4325 y Fu(\()p Fq(x)p Fu(\))2086 4607 y Fo(\024)84 b Fq(C)2334 4540 y(e)2379 4503 y Fn(\000)p Fl(\025)2475 4480 y Fh(\000)p Fg(q)2557 4503 y Fl(=C)5 b Fm(+4)p Fl(t)p 2334 4584 434 4 v 2437 4604 a Fo(p)p 2520 4604 143 4 v 82 x Fu(4)p Fq(\031)t(t)3591 4607 y Fu(\(7.31\))0 4839 y(T)-8 b(aking)40 b Fq(q)45 b(>)40 b Fu(log)663 4863 y Fl(\025)704 4844 y Fh(\000)p Fi(1)808 4839 y Fq(T)14 b Fu(,)42 b(from)d(\(7.30\))h(and)g(\(7.31\))g(w)m(e)h(get)f(the)h (last)f(term)g(on)g(the)h(r.h.s.)g(of)47 b(\(7.28\))0 4955 y(yields)32 b(the)h(last)f(one)h(on)f(the)h(r.h.s.)h(of)39 b(\(7.27\))o(.)100 5071 y(F)-8 b(rom)31 b(\(7.27\))o(,)i(for)f(an)m(y)h Fq(\025)f Fu(small)f(enough,)1260 5338 y(sup)1196 5424 y Fl(n)p Fn(\024)p Fl(n)1337 5436 y Fg(\025)1377 5424 y Fm(\()p Fl(\034)8 b Fm(\))1636 5271 y Fu(1)p 1498 5315 327 4 v 1498 5407 a(1)22 b(+)g Fq(x)1722 5422 y Fl(n;)p Fn(\003)1851 5164 y Fp(\014)1851 5224 y(\014)1851 5284 y(\014)1851 5343 y(\014)1851 5403 y(\014)1889 5214 y Fl(n)p Fn(\000)p Fm(1)1884 5244 y Fp(X)1891 5456 y Fl(k)r Fm(=0)2044 5338 y Fq(\015)2095 5353 y Fl(k)2137 5164 y Fp(\014)2137 5224 y(\014)2137 5284 y(\014)2137 5343 y(\014)2137 5403 y(\014)2198 5338 y Fo(\024)29 b Fq(C)7 b(T)14 b(\025)2509 5297 y Fl(\014)s Fn(\000)p Fl(q)3591 5338 y Fu(\(7.32\))1882 5532 y Fk(41)p eop %%Page: 42 43 42 42 bop 0 266 a Fu(By)37 b(c)m(ho)s(osing)e Fq(q)j Fo(2)c Fu(\(log)901 289 y Fl(\025)942 270 y Fh(\000)p Fi(1)1046 266 y Fq(T)8 b(;)17 b(\014)30 b Fo(\000)25 b Fu(log)1468 289 y Fl(\025)1509 270 y Fh(\000)p Fi(1)1612 266 y Fq(T)14 b Fu(\))36 b(\(recall)f(w)m(e)i(\014xed)g(log)2571 289 y Fl(\025)2612 270 y Fh(\000)p Fi(1)2716 266 y Fq(T)47 b Fu(=)34 b(\(1)24 b Fo(^)h Fq(\014)6 b Fu(\))p Fq(=)p Fu(20\))35 b(it)g(follo)m(ws)0 382 y(the)e(l.h.s.)g(of)39 b(\(7.32\))31 b(v)-5 b(anishes)34 b(as)e Fq(\025)c Fo(#)f Fu(0.)100 498 y(W)-8 b(e)34 b(are)g(left)f(with)h(the)h(b)s(ound)f(on)g (the)g(martingale)e(part)i Fq(M)2404 513 y Fl(n)2451 498 y Fu(.)48 b(By)34 b(Do)s(ob's)g(inequalit)m(y)-8 b(,)33 b(recalling)0 614 y(\(7.25\))o(,)266 884 y Fj(P)342 713 y Fp( )486 884 y Fu(sup)422 970 y Fl(n)p Fn(\024)p Fl(n)563 982 y Fg(\025)603 970 y Fm(\()p Fl(\034)8 b Fm(\))714 884 y Fo(j)p Fq(M)836 899 y Fl(n)883 884 y Fo(j)27 b Fq(>)h(T)1113 843 y Fn(\000)p Fm(1)p Fl(=)p Fm(4)1277 713 y Fp(!)1439 884 y Fo(\024)84 b Fq(C)1677 795 y Fo(p)p 1760 795 71 4 v 89 x Fq(T)30 b Fj(E)1924 803 y Fp(\000)1976 884 y Fo(h)p Fq(M)10 b Fo(i)2158 899 y Fl(n)2201 911 y Fg(\025)2242 899 y Fm(\()p Fl(\034)e Fm(\))2340 803 y Fp(\001)2413 884 y Fo(\024)28 b Fq(C)2595 795 y Fo(p)p 2678 795 V 89 x Fq(T)2765 751 y Fl(n)2808 763 y Fg(\025)2849 751 y Fm(\()p Fl(\034)8 b 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y(the)f(stopping)f(time)1057 2894 y Fq(t)1092 2909 y Fl(\016)1158 2894 y Fu(:=)27 b(inf)6 b Fo(f)p Fq(t)28 b Fo(\025)g Fu(0)44 b(:)51 b(inf)1789 2956 y Fl(z)s Fn(2)p Ff(R)1936 2894 y Fo(k)p Fq(m)p Fu(\()p Fq(t)p Fu(\))22 b Fo(\000)41 b Fu(\026)-67 b Fq(m)2389 2909 y Fl(z)2429 2894 y Fo(k)2479 2909 y Fn(1)2581 2894 y Fo(\025)28 b Fq(\016)t Fo(g)856 b Fu(\(8.1\))0 3077 y(By)37 b(Prop)s(osition)e(2.1,)i(for)f(eac)m(h)i Fq(t)d Fo(2)g Fu([0)p Fq(;)17 b(t)1578 3092 y Fl(\016)1615 3077 y Fu(])37 b(the)g(pro\014le)f Fq(m)p Fu(\()p Fq(t)p Fu(\))h(has)g(a)f (w)m(ell)g(de\014ned)i(cen)m(ter)g Fq(x)p Fu(\()p Fq(m)p Fu(\()p Fq(t)p Fu(\)\).)0 3194 y(W)-8 b(e)33 b(pro)m(v)m(e)h(the)f (theorem)f(with)1452 3316 y Fq(x)1507 3275 y Fm(\()p Fl(\025)p Fm(\))1607 3316 y Fu(\()p Fq(t)p Fu(\))c(:=)g Fq(x)p Fu(\()p Fq(m)p Fu(\()p Fq(t)23 b Fo(^)f Fq(t)2274 3331 y Fl(\016)2312 3316 y Fu(\)\))0 3459 y(Observ)m(e)36 b(the)e(pro)s(cess)i Fq(x)942 3422 y Fm(\()p Fl(\025)p Fm(\))1042 3459 y Fu(\()p Fq(t)p 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Fg(n)p Fi(+1)1453 4118 y Fm(])1490 4032 y Fo(j)p Fq(x)1573 3991 y Fm(\()p Fl(\025)p Fm(\))1673 4032 y Fu(\()p Fq(t)p Fu(\))22 b Fo(\000)h Fq(x)1961 4047 y Fl(n)2008 4032 y Fo(j)28 b(\024)g Fq(C)7 b(\025)2303 3991 y Fm(1)p Fn(\000)p Fl(\020)2432 3943 y Fo(p)p 2515 3943 71 4 v 89 x Fq(T)209 b Fu(on)33 b(the)g(set)g Fo(B)3305 3982 y Fm(\(1\))3302 4060 y Fl(\025;\034)3639 4032 y Fu(\(8.2\))0 4272 y(On)g(the)g(other)f(hand,)h(b)m(y)i(\(3.1\),)d (\(4.2\),)g(and)h Fo(k)18 b Fu(\026)-67 b Fq(m)1838 4236 y Fn(0)1861 4272 y Fo(k)1911 4287 y Fn(1)2014 4272 y Fo(\024)28 b Fu(1,)k(w)m(e)i(ha)m(v)m(e)269 4440 y(sup)155 4526 y Fl(t)p Fn(2)p Fm([)p Fl(T)288 4534 y Fg(n)330 4526 y Fl(;T)391 4535 y Fg(n)p Fi(+1)510 4526 y Fm(])547 4356 y Fp(\015)547 4415 y(\015)602 4440 y Fq(m)p Fu(\()p Fq(t)p Fu(\))23 b Fo(\000)40 b Fu(\026)-67 b Fq(m)1005 4463 y Fl(x)1045 4443 y Fi(\()p Fg(\025)p Fi(\))1134 4463 y Fm(\()p Fl(t)p Fm(\))1219 4356 y Fp(\015)1219 4415 y(\015)1274 4484 y Fn(1)1376 4440 y Fo(\024)28 b 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b(Let)36 b Fq(\030)1496 4971 y Fl(\025)1541 4956 y Fu(\()p Fq(\034)11 b Fu(\),)37 b Fq(\034)45 b Fo(2)34 b Fj(R)1986 4971 y Fm(+)2051 4956 y Fu(,)j(b)s(e)f(the)g(\(con)m(tin)m(uous\))h(pro)s(cess)g(obtained)e (b)m(y)0 5073 y(linearly)h(in)m(terp)s(olating)f(the)k(v)-5 b(alues)38 b Fq(\030)1453 5088 y Fl(\025)1498 5073 y Fu(\()p Fq(\025)1593 5036 y Fm(2)1632 5073 y Fq(T)1689 5088 y Fl(n)1736 5073 y Fu(\))f(=)g Fq(\030)1967 5088 y Fl(n)2014 5073 y Fu(.)60 b(F)-8 b(rom)37 b(\(4.6\))o(,)j(\(4.36\))o (,)f(\(4.40\),)g(and)f(\(8.2\),)h(for)0 5189 y(an)m(y)33 b Fq(\034)44 b Fu(and)33 b Fq(")f Fu(p)s(ositiv)m(e,)1132 5364 y(lim)1144 5427 y Fl(\025)p Fn(#)p Fm(0)1284 5364 y Fj(P)1360 5224 y Fp(\022)1470 5364 y Fu(sup)1435 5445 y Fm(0)p Fn(\024)p Fl(s)p Fn(\024)p Fl(\034)1668 5364 y Fo(j)p Fq(\030)1739 5379 y Fl(\025)1784 5364 y Fu(\()p Fq(s)p Fu(\))22 b Fo(\000)g Fq(x)2082 5379 y Fl(\025)2128 5364 y Fu(\()p Fq(s)p Fu(\))p Fo(j)27 b Fq(>)h(")2455 5224 y Fp(\023)2556 5364 y Fu(=)f(0)1882 5532 y Fk(42)p eop %%Page: 43 44 43 43 bop 0 266 a Fu(It)33 b(is)f(therefore)h(enough)g(to)f(pro)m(v)m (e)h(the)g(con)m(v)m(ergence)j(of)c(the)g(pro)s(cess)i Fq(\030)2699 281 y Fl(\025)2744 266 y Fu(.)44 b(T)-8 b(o)32 b(this)g(end)h(w)m(e)h(shall)d(use)0 382 y(that)h Fq(\030)254 397 y Fl(n)333 382 y Fu(solv)m(es)i(the)f(equation)f (\(7.2\).)100 498 y(In)i(Sections)g(7)f(and)h(8)f(w)m(e)i(pro)m(v)m(ed) g(b)s(ounds)g(on)e Fq(R)i Fu(and)f Fq(S)6 b Fu(,)33 b(see)i(\(7.4\))e (and)h(Prop)s(osition)e(7.3)h(whic)m(h)0 614 y(hold)27 b(with)h(probabilit)m(y)d(going)i(to)g(1)h(as)g Fq(\025)g Fo(#)f Fu(0.)42 b(They)29 b(ho)m(w)m(ev)m(er)h(dep)s(end)f(on)f(the)h (unkno)m(wn)g(quan)m(tities)0 731 y Fq(x)55 746 y Fl(n;)p Fn(\003)157 731 y Fu(.)69 b(The)42 b Fe(a)e(priori)f Fu(b)s(ounds)j(of)e(Section)h(4)f(yield)h Fq(x)2031 746 y Fl(n;)p Fn(\003)2175 731 y Fo(\024)h Fq(n\016)2399 694 y Fn(\003)2439 731 y Fu(\()p Fq(\034)11 b Fu(\),)43 b(whic)m(h)e(go)s(es)g(lik)m(e)g Fq(\025)3393 694 y Fn(\000)p Fm(1)3487 731 y Fq(T)3558 694 y Fn(\000)p Fm(1)p Fl(=)p Fm(2+)p Fl(\020)3813 731 y Fu(,)0 847 y(according)f(to)g(\(4.7\))g(and) g(with)g Fq(n)h Fu(=)g Fq(n)1516 862 y Fl(\025)1562 847 y Fu(\()p Fq(\034)11 b Fu(\).)67 b(With)40 b(suc)m(h)i(a)e(b)s(ound,)j (our)d(estimates)g(on)g Fq(R)i Fu(and)e Fq(S)0 963 y Fu(b)s(ecome)i(v)m(ery)h(bad)f(and)g(in)g(an)m(y)g(case)h(inadequate)f (to)g(study)h(the)f(equation)g(\(7.2\),)i(for)d(whic)m(h)i(w)m(e)0 1079 y(could)32 b(only)g(tolerate)g(a)g(b)s(ound)h(on)f Fq(x)1404 1094 y Fl(n;)p Fn(\003)1539 1079 y Fu(whic)m(h)h(div)m(erges) h(v)m(ery)g(w)m(eakly)f(as)g Fq(\025)28 b Fo(#)f Fu(0.)100 1196 y(As)g(men)m(tioned,)g(the)g(b)s(ounds)g(of)f(Section)g(4)g(on)g Fq(\016)1920 1159 y Fn(\003)1959 1196 y Fu(\()p Fq(\034)11 b Fu(\))27 b(are)f(quasi)h(optimal,)e(y)m(et)i(they)g(yield)f(a)g(b)s (ound)0 1312 y(on)e Fq(x)182 1327 y Fl(n;)p Fn(\003)308 1312 y Fu(whic)m(h)h(is)e(far)h(from)f(correct.)41 b(The)25 b(p)s(oin)m(t)e(is)h(that)g Fq(x)2177 1327 y Fl(n)2248 1312 y Fu(is)g(the)g(sum)g(of)g(the)g(incremen)m(ts)h Fq(x)3495 1327 y Fl(k)3542 1312 y Fo(\000)5 b Fq(x)3679 1327 y Fl(k)r Fn(\000)p Fm(1)3813 1312 y Fu(,)0 1428 y(whose)36 b(size)g(has)f(indeed)h(the)f(order)g(of)g Fq(\016)1554 1392 y Fn(\003)1593 1428 y Fu(\()p Fq(\034)11 b Fu(\);)37 b(but)f(there)f(are)h(a)e(lot)g(of)h(cancellations,)f(whic) m(h)i(mak)m(e)0 1544 y(the)42 b(absolute)f(v)-5 b(alue)42 b(of)f(the)h(sum)f(m)m(uc)m(h)i(smaller)c(than)j(the)g(sum)f(of)h(the)g (absolute)f(v)-5 b(alues.)71 b(Suc)m(h)0 1660 y(cancellations)37 b(are)h(lost)f(using)h(the)h Fe(a)f(priori)e Fu(b)s(ounds)j(of)f (Section)g(4,)h(w)m(e)h(th)m(us)f(need)g(to)f(go)g(bac)m(k)h(to)0 1777 y(the)e(equation)e(\(7.2\))h(itself)f(\(recalling)e(that)j Fq(\030)1739 1792 y Fl(n)1822 1777 y Fu(and)g Fq(x)2070 1792 y Fl(n)2153 1777 y Fu(are)g(essen)m(tially)g(the)h(same,)f(see)i (\(4.40\))o(\).)54 b(It)0 1893 y(seems,)42 b(at)d(this)g(p)s(oin)m(t,)h (that)g(w)m(e)g(are)f(bac)m(k)i(to)e(a)g(non)g(linear)f(problem,)i (with)f(the)h(unkno)m(wn)h Fq(\030)3673 1908 y Fl(n)3759 1893 y Fu(in)0 2009 y(b)s(oth)32 b(the)i(\\kno)m(wn)f(terms")g Fq(R)h Fu(and)e Fq(S)39 b Fu(hidden)33 b(through)f Fq(x)2216 2024 y Fl(n)2264 2009 y Fu(.)44 b(Suc)m(h)34 b(a)e(non)h(linearit)m(y)e (is)h(ho)m(w)m(ev)m(er)j(not)0 2125 y(really)27 b(dangerous,)j(as)f(in) f(our)g(b)s(ounds)h(for)f Fq(R)h Fu(and)g Fq(S)34 b Fu(the)29 b(dep)s(endence)i(on)d Fq(x)2876 2140 y Fl(n;)p Fn(\003)3007 2125 y Fu(is)g(alw)m(a)m(ys)h(m)m(ultiplied)0 2242 y(b)m(y)39 b(a)e(function)g(of)g Fq(\025)h Fu(whic)m(h)g(v)-5 b(anishes)38 b(as)g Fq(\025)f Fo(#)f Fu(0.)58 b(As)39 b(w)m(e)f(will)e(see,)k(this)d (mak)m(es)i(easy)f(the)h(pro)s(of)d(of)0 2358 y(Prop)s(osition)31 b(8.1)h(b)s(elo)m(w.)100 2474 y(Let)1514 2596 y Fq(\030)1562 2555 y Fn(\003)1601 2596 y Fu(\()p Fq(\034)11 b Fu(\))28 b(:=)92 b(sup)1889 2681 y Fl(n)p Fn(\024)p Fl(n)2030 2693 y Fg(\025)2070 2681 y Fm(\()p Fl(\034)8 b Fm(\))2181 2596 y Fo(j)p Fq(\030)2252 2611 y Fl(n)2298 2596 y Fo(j)1313 b Fu(\(8.4\))0 2817 y(and)33 b Fq(x)245 2832 y Fn(\003)285 2817 y Fu(\()p Fq(\034)11 b Fu(\))32 b(b)s(e)h(as)g(in)f(\(6.5\))o(.)0 3169 y Fv(Prop)s(osition)k(8.1.)41 b Ft(F)-7 b(or)34 b(any)h Fq(\034)k Fo(2)28 b Fj(R)1431 3184 y Fm(+)1496 3169 y Ft(,)1383 3420 y Fu(lim)1356 3482 y Fl(L)p Fn(!1)p 1573 3337 136 3 v 1573 3420 a Fu(lim)1585 3483 y Fl(\025)p Fn(#)p Fm(0)1736 3420 y Fj(P)17 b Fu(\()r Fq(\030)1900 3379 y Fn(\003)1939 3420 y Fu(\()p Fq(\034)11 b Fu(\))28 b Fq(>)f(L)p Fu(\))h(=)g(0)1155 b(\(8.5\))1379 3740 y(lim)1352 3802 y Fl(L)p Fn(!1)p 1569 3656 V 1569 3740 a Fu(lim)1581 3802 y Fl(\025)p Fn(#)p Fm(0)1732 3740 y Fj(P)17 b Fu(\()r Fq(x)1903 3755 y Fn(\003)1943 3740 y Fu(\()p Fq(\034)11 b Fu(\))28 b Fq(>)f(L)p Fu(\))h(=)g(0)1151 b(\(8.6\))0 4170 y Ft(Pr)-5 b(o)g(of.)57 b Fu(Since)38 b(\(8.6\))e(follo)m(ws)h (from)f(\(8.5\))g(and)i(\(4.40\))o(,)h(w)m(e)f(only)f(need)h(to)f(pro)m (v)m(e)i(\(8.5\))o(.)58 b(Let)37 b(\(recall)0 4287 y(\(7.1\)\))1499 4430 y Fq(S)1565 4389 y Fm(\()p Fl(R)p Fm(\))1559 4454 y Fl(n)1705 4430 y Fu(:=)1896 4335 y Fp(X)1835 4551 y Fl(`)p Fn(2Lnf)p Fm(2)p Fn(g)2118 4430 y Fq(S)2178 4445 y Fl(n)2225 4430 y Fu(\()p Fq(`)p Fu(\))1297 b(\(8.7\))0 4712 y(and)33 b(denote)g(as)g(usual)f(b)m(y)i Fq(S)1081 4676 y Fm(\()p Fl(R)p Fm(\))1225 4712 y Fu(the)f(v)m(ector)h(whose)g (comp)s(onen)m(ts)f(are)g Fq(S)2740 4661 y Fm(\()p Fl(R)p Fm(\))2734 4723 y Fl(n)2852 4712 y Fu(.)100 4829 y(Giv)m(en)f Fq(\034)39 b(>)28 b Fu(0)k(and)h Fq(\020)i(>)27 b Fu(0)32 b(small)f(enough,)i(let)f Fo(B)1925 4844 y Fl(\025;\034)2062 4829 y Fu(b)s(e)g(as)h(in)f(\(6.6\))g(and)h(de\014ne)1519 4991 y Fo(G)1578 5006 y Fl(\025;L)1718 4991 y Fu(:=)28 b Fo(A)1929 5006 y Fl(\025;L)2064 4991 y Fo(\\)22 b(B)2217 5006 y Fl(\025;\034)3639 4991 y Fu(\(8.8\))0 5153 y(where)812 5341 y Fo(A)892 5356 y Fl(\025;L)1032 5341 y Fu(=)1136 5171 y Fp(\()1216 5341 y Fo(j)o Fq(S)6 b Fu(\(2\))p Fo(j)1462 5370 y Fl(n)1505 5382 y Fg(\025)1545 5370 y Fm(\()p Fl(\034)i Fm(\))1671 5341 y Fo(\024)28 b Fq(L)g Fu(;)109 b(sup)1941 5427 y Fl(n)p Fn(\024)p Fl(n)2082 5439 y Fg(\025)2123 5427 y Fm(\()p Fl(\034)8 b Fm(\))2260 5179 y Fp(\014)2260 5239 y(\014)2294 5263 y Fq(S)2360 5227 y Fm(\()p Fl(R)p Fm(\))2472 5179 y Fp(\014)2472 5239 y(\014)2505 5302 y Fl(n)p 2243 5318 327 4 v 2243 5410 a Fu(1)22 b(+)g Fq(x)2467 5425 y Fl(n;)p Fn(\003)2607 5341 y Fo(\024)28 b Fq(T)2783 5300 y Fn(\000)p Fm(1)p Fl(=)p Fm(4)2948 5171 y Fp(\))3639 5341 y Fu(\(8.9\))1882 5532 y Fk(43)p eop %%Page: 44 45 44 44 bop 0 266 a Fu(F)-8 b(rom)31 b(\(4.6\),)h(\(4.36\))o(,)h(\(6.6\)) f(\(6.4\),)g(and)h(Prop)s(osition)e(7.3)h(w)m(e)i(ha)m(v)m(e)1504 541 y(lim)1477 603 y Fl(L)p Fn(!1)p 1694 458 136 3 v 1694 541 a Fu(lim)1706 604 y Fl(\025)p Fn(#)p Fm(0)1857 541 y Fj(P)17 b Fu(\()r Fo(G)2032 556 y Fl(\025;L)2145 541 y Fu(\))27 b(=)h(1)1228 b(\(8.10\))100 772 y(W)-8 b(e)33 b(are)g(going)f(to)g(sho)m(w)j(that)d(giv)m(en)i Fq(L)f Fu(there)h(is)e(a)h(constan)m(t)h Fq(c)f Fu(so)g(that,)g(for)g (all)e Fq(\025)i Fu(small)d(enough,)0 888 y(in)i Fo(G)173 903 y Fl(\025;L)319 888 y Fu(w)m(e)h(ha)m(v)m(e)h Fo(j)p Fq(\030)758 903 y Fl(n)804 888 y Fo(j)28 b(\024)g Fq(c)k Fu(for)g(all)f Fq(n)d Fo(\024)g Fq(n)1573 903 y Fl(\025)1618 888 y Fu(\()p Fq(\034)11 b Fu(\))23 b(+)f(1.)100 1004 y(Giv)m(en)h(an)m(y)h Fq(L)614 1019 y Fm(1)682 1004 y Fq(>)j Fu(0,)e(let)e Fq(N)10 b Fu(\()p Fq(L)1210 1019 y Fm(1)1250 1004 y Fq(;)17 b(\025)p Fu(\))23 b(b)s(e)h(the)g(\014rst)g (index)g Fq(n)k Fo(\024)g Fq(n)2390 1019 y Fl(\025)2435 1004 y Fu(\()p Fq(\034)11 b Fu(\))24 b(suc)m(h)h(that)e Fo(j)p Fq(\030)3072 1019 y Fl(n)3119 1004 y Fo(j)k(\025)h Fq(L)3345 1019 y Fm(1)3385 1004 y Fu(;)e(otherwise)0 1120 y(w)m(e)38 b(set)g Fq(N)10 b Fu(\()p Fq(L)497 1135 y Fm(1)537 1120 y Fq(;)17 b(\025)p Fu(\))34 b(=)h Fq(n)879 1135 y Fl(\025)925 1120 y Fu(\()p Fq(\034)11 b Fu(\))25 b(+)g(1.)57 b(Then)38 b(what)f(w)m(e)h(ha)m(v)m(e)g(to)f(sho)m(w)h(is)e (that)h(there)h(is)e Fq(L)3307 1135 y Fm(1)3384 1120 y Fu(so)h(that)f(for)0 1237 y(all)d Fq(\025)h Fu(small)e(enough,)k Fq(N)10 b Fu(\()p Fq(L)1044 1252 y Fm(1)1084 1237 y Fq(;)17 b(\025)p Fu(\))31 b(=)g Fq(n)1419 1252 y Fl(\025)1465 1237 y Fu(\()p Fq(\034)11 b Fu(\))23 b(+)h(1)34 b(in)g Fo(G)1975 1252 y Fl(\025;L)2088 1237 y Fu(.)50 b(Ha)m(ving)34 b(\014xed)i Fq(L)2804 1252 y Fm(1)2844 1237 y Fu(,)f(there)g(is)f Fq(\025)3314 1252 y Fm(1)3388 1237 y Fu(so)h(that)f(for)0 1373 y Fq(\025)28 b Fo(\024)g Fq(\025)247 1388 y Fm(1)319 1373 y Fu(and)k Fq(n)c Fo(\024)g Fq(N)10 b Fu(\()p Fq(L)891 1388 y Fm(1)932 1373 y Fq(;)17 b(\025)p Fu(\),)32 b(b)m(y)j(\(4.40\))c (\(recall)g Fo(G)1905 1388 y Fl(\025;L)2046 1373 y Fo(\022)d(B)2219 1322 y Fm(\(1)p Fl(;)p Fm(2\))2216 1401 y Fl(\025;\034)2369 1373 y Fu(\))981 1565 y Fo(j)p Fq(x)1064 1580 y Fl(n)1111 1565 y Fo(j)f(\024)h(j)p Fq(\030)1342 1580 y Fl(n)1388 1565 y Fo(j)22 b Fu(+)g(1)28 b Fo(\024)g Fq(L)1784 1580 y Fm(1)1846 1565 y Fu(+)22 b(1)p Fq(;)211 b Fu(on)33 b(the)g(set)g Fo(G)2746 1580 y Fl(\025;L)0 1749 y Fu(Using)f(this,)g (\(7.4\))o(,)g(and)g(\(8.9\))g(w)m(e)h(then)g(get)f(there)h(is)e Fq(\025)2102 1764 y Fm(2)2169 1749 y Fu(=)d Fq(\025)2330 1764 y Fm(2)2369 1749 y Fu(\()p Fq(\025)2464 1764 y Fm(1)2503 1749 y Fq(;)17 b(L)2613 1764 y Fm(1)2653 1749 y Fu(\))28 b Fo(2)g Fu(\(0)p Fq(;)17 b(\025)3001 1764 y Fm(1)3039 1749 y Fu(\))32 b(suc)m(h)i(that)e(for)f(an)m(y)0 1865 y Fq(\025)d Fo(\024)g Fq(\025)247 1880 y Fm(2)319 1865 y Fu(and)k Fq(n)c Fo(\024)g Fq(N)10 b Fu(\()p Fq(L)891 1880 y Fm(1)932 1865 y Fq(;)17 b(\025)p Fu(\),)920 2049 y Fo(j)p Fq(R)q Fo(j)1051 2064 y Fl(n)1125 2049 y Fo(\024)28 b Fu(1)p Fq(;)211 b Fo(j)p Fq(S)6 b Fo(j)1639 2064 y Fl(n)1713 2049 y Fo(\024)28 b Fq(L)23 b Fu(+)f(1)p Fq(;)211 b Fu(on)33 b(the)g(set)g Fo(G)2807 2064 y Fl(\025;L)3591 2049 y Fu(\(8.11\))0 2233 y(Then,)h(b)m(y)h(\(7.2\))d(and)g(\(5.18\),)g (for)g Fq(n)c Fo(\024)g Fq(N)10 b Fu(\()p Fq(L)1682 2248 y Fm(1)1723 2233 y Fq(;)17 b(\025)p Fu(\),)802 2511 y Fo(j)p Fq(\030)873 2526 y Fl(n)919 2511 y Fo(j)27 b(\024)h Fq(C)7 b Fu(\()p Fq(L)23 b Fu(+)f(1\))g(+)g Fq(C)1687 2386 y Fl(n)p Fn(\000)p Fm(1)1681 2416 y Fp(X)1689 2628 y Fl(k)r Fm(=0)1842 2511 y Fq(\025)1899 2470 y Fm(2)1938 2511 y Fq(T)36 b Fo(j)p Fq(\030)2102 2526 y Fl(k)2144 2511 y Fo(j)p Fq(;)211 b Fu(on)33 b(the)g(set)g Fo(G)2925 2526 y Fl(\025;L)3591 2511 y Fu(\(8.12\))0 2789 y(By)44 b(solving)e(this)g(inequalit)m(y)-8 b(,)45 b(w)m(e)f(conclude)g(that)e (there)i(is)f Fq(c)i Fu(=)h Fq(c)p Fu(\()p Fq(C)r(;)17 b(L;)g(\034)11 b Fu(\),)45 b(indep)s(enden)m(t)f(of)f Fq(L)3773 2804 y Fm(1)3813 2789 y Fu(,)0 2905 y(so)g(that)g Fo(j)p Fq(\030)423 2920 y Fl(n)470 2905 y Fo(j)i(\024)h Fq(c)p Fu(,)g(for)d(all)e Fq(n)46 b Fo(\024)g Fq(N)10 b Fu(\()p Fq(L)1506 2920 y Fm(1)1546 2905 y Fq(;)17 b(\025)p Fu(\).)75 b(By)44 b(taking)e Fq(L)2328 2920 y Fm(1)2414 2905 y Fo(\025)k Fq(c)p Fu(,)g(w)m(e)e(ha)m(v)m(e)h Fq(N)10 b Fu(\()p Fq(L)3234 2920 y Fm(1)3274 2905 y Fq(;)17 b(\025)p Fu(\))45 b Fq(>)h(n)3638 2920 y Fl(\025)3684 2905 y Fu(\()p Fq(\034)11 b Fu(\).)0 3021 y(Prop)s(osition)31 b(8.1)h(is)g(pro)m(v)m (ed.)2673 b Fd(\003)100 3205 y Fu(Setting)1329 3437 y Fq(Y)1386 3452 y Fl(n)1461 3437 y Fu(:=)27 b Fq(S)1651 3452 y Fl(n)1698 3437 y Fu(\(2\))22 b Fo(\000)1950 3313 y Fl(n)p Fn(\000)p Fm(1)1945 3342 y Fp(X)1952 3555 y Fl(k)r Fm(=0)2105 3437 y Fq(A)2178 3452 y Fl(n;k)2283 3437 y Fq(S)2343 3452 y Fl(k)2386 3437 y Fu(\(2\))1080 b(\(8.13\))0 3695 y(from)31 b(\(7.4\),)h(\(8.6\),)h(\(8.10\))o(,)f(and) h Fo(k)p Fq(A)p Fo(k)1422 3710 y Fl(n)1497 3695 y Fo(\024)28 b Fq(C)7 b Fu(,)32 b(w)m(e)i(ha)m(v)m(e,)g(for)e(eac)m(h)h Fq(")28 b(>)f Fu(0,)990 3970 y(lim)1002 4033 y Fl(\025)p Fn(#)p Fm(0)1142 3970 y Fj(P)1218 3800 y Fp( )1363 3970 y Fu(sup)1298 4056 y Fl(n)p Fn(\024)p Fl(n)1439 4068 y Fg(\025)1480 4056 y Fm(\()p Fl(\034)8 b Fm(\))1590 3970 y Fo(j)p Fq(S)28 b Fo(\000)22 b Fq(AS)29 b Fu(+)22 b Fq(R)h Fo(\000)f Fq(Y)g Fo(j)2368 3985 y Fl(n)2442 3970 y Fq(>)28 b(")2592 3800 y Fp(!)2698 3970 y Fu(=)f(0)741 b(\(8.14\))0 4245 y(so)42 b(that)f(the)g(analysis)g(of)g(the)h (limiting)36 b(b)s(eha)m(vior)42 b(of)e(the)i(\\kno)m(wn)h(term")d Fq(S)34 b Fo(\000)29 b Fq(AS)34 b Fu(+)28 b Fq(R)42 b Fu(in)e(\(7.2\))0 4361 y(reduces)34 b(to)f(that)f(of)g Fq(Y)21 b Fu(,)33 b(whic)m(h)g(is)f(the)h(con)m(ten)m(t)h(of)e(the)h (follo)m(wing)d(prop)s(osition.)0 4714 y Fv(Prop)s(osition)37 b(8.2.)42 b Ft(L)-5 b(et)36 b Fq(Y)1049 4729 y Fl(\025)1094 4714 y Fu(\()p Fq(\034)11 b Fu(\))p Ft(,)36 b Fq(\034)41 b Fo(2)30 b Fj(R)1534 4729 y Fm(+)1599 4714 y Ft(,)36 b(b)-5 b(e)36 b(the)g(pr)-5 b(o)g(c)g(ess)35 b(obtaine)-5 b(d)35 b(by)g(line)-5 b(arly)36 b(interp)-5 b(olating)35 b(the)0 4830 y(values)f Fq(Y)348 4845 y Fl(\025)393 4830 y Fu(\()p Fq(\025)488 4794 y Fm(2)528 4830 y Fq(T)585 4845 y Fl(n)632 4830 y Fu(\))27 b(=)h Fq(Y)858 4845 y Fl(n)904 4830 y Ft(,)35 b Fq(n)28 b Fo(2)g Fj(N)9 b Ft(.)51 b(Then)1147 5073 y Fq(Y)1204 5088 y Fl(\025)1249 5073 y Fu(\()p Fq(\034)11 b Fu(\))1430 5013 y Fl(\025)p Fn(#)p Fm(0)1406 5073 y Fu(=)-17 b Fo(\))28 b Fq(b)p Fu(\()p Fq(\034)11 b Fu(\))23 b Fo(\000)1966 5006 y Fu(3)p 1895 5050 191 4 v 1895 5070 a Fo(p)p 1978 5070 108 4 v 83 x Fu(2)p Fq(\031)2113 4938 y Fp(Z)2212 4964 y Fl(\034)2168 5163 y Fm(0)2255 5073 y Fq(ds)2449 5006 y(b)p Fu(\()p Fq(s)p Fu(\))p 2379 5050 305 4 v 2379 5070 a Fo(p)p 2462 5070 222 4 v 75 x Fq(\034)33 b Fo(\000)23 b Fq(s)3591 5073 y Fu(\(8.15\))0 5316 y Ft(wher)-5 b(e)29 b(the)g(c)-5 b(onver)g(genc)g(e)28 b(is)h(in)h Fq(C)7 b Fu(\()p Fj(R)1349 5331 y Fm(+)1414 5316 y Fu(\))p Ft(.)43 b(A)n(b)-5 b(ove)29 b Fq(b)p Fu(\()p Fq(\034)11 b Fu(\))30 b Ft(is)f(a)h(one)f(dimensional) e(Br)-5 b(ownian)28 b(motion)h(with)0 5433 y(di\013usion)34 b(c)-5 b(o)g(e\016cient)34 b Fq(D)c Fu(=)e(3)p Fq(=)p Fu(4)p Ft(.)1882 5532 y Fk(44)p eop %%Page: 45 46 45 45 bop 0 266 a Ft(Pr)-5 b(o)g(of.)62 b Fu(W)-8 b(e)39 b(pro)m(v)m(e)h(\014rst)f(that)g(\()p Fq(S)6 b Fu(\(2\)\))1454 281 y Fl(\025)1498 266 y Fu(\()p Fq(\034)11 b Fu(\))39 b(\(obtained)f(b)m(y)i(linear)d(in)m(terp)s(olation)f(from)i Fq(S)3413 281 y Fl(n)3460 266 y Fu(\(2\)\))g(con-)0 382 y(v)m(erges)g(w)m(eakly)g(to)e(a)g(Bro)m(wnian)h(motion.)53 b(W)-8 b(e)37 b(\014rst)g(pro)m(v)m(e)h(tigh)m(tness.)56 b(Boundedness)39 b(has)e(already)0 498 y(b)s(een)43 b(pro)m(v)m(ed,)i (see)e(\(7.6\).)71 b(Since)42 b Fq(S)1385 513 y Fl(n)1432 498 y Fu(\(2\))f(is)h(a)f Fo(F)1868 513 y Fl(T)1909 521 y Fg(n)1955 498 y Fu({martingale,)h(b)m(y)g(Do)s(ob's)g(inequalit)m(y) -8 b(,)43 b(for)e(eac)m(h)0 614 y Fq(")27 b(>)h Fu(0,)k(w)m(e)i(ha)m(v) m(e)186 896 y(lim)202 959 y Fl(\016)r Fn(#)p Fm(0)p 350 813 136 3 v 350 896 a Fu(lim)362 959 y Fl(\025)p Fn(#)p Fm(0)513 896 y Fj(P)589 696 y Fp(0)589 875 y(@)745 896 y Fu(sup)687 967 y Fi(0)p Fh(\024)p Fg(s)f(")2214 696 y Fp(1)2214 875 y(A)2329 896 y Fo(\024)h Fu(lim)2449 959 y Fl(\016)r Fn(#)p Fm(0)p 2597 813 V 2597 896 a Fu(lim)2609 959 y Fl(\025)p Fn(#)p Fm(0)2770 829 y Fq(C)7 b Fu([)p Fq(n)2932 844 y Fl(\025)2978 829 y Fu(\()p Fq(\016)t Fu(\))22 b(+)g(1])p Fq(\025)3354 792 y Fm(2)3393 829 y Fq(T)p 2770 873 694 4 v 3074 964 a(")3120 936 y Fm(2)3501 896 y Fu(=)28 b(0)0 1183 y(T)-8 b(o)33 b(iden)m(tify)f(the)h(limit)c(w)m(e)k(compute)g(the)g(brac)m(k)m (et)h(of)e Fq(S)2122 1199 y Fl(n)2165 1211 y Fg(\025)2206 1199 y Fm(\()p Fl(\034)8 b Fm(\))2304 1183 y Fu(\(2\))32 b(as)h(follo)m(ws)224 1333 y Fl(n)267 1345 y Fg(\025)307 1333 y Fm(\()p Fl(\034)8 b Fm(\))241 1371 y Fp(X)248 1583 y Fl(k)r Fm(=1)418 1465 y Fj(E)495 1385 y Fp(\000)557 1465 y Fu(\()o Fq(S)654 1480 y Fl(k)697 1465 y Fu(\(2\))22 b Fo(\000)h Fq(S)1004 1480 y Fl(k)r Fn(\000)p Fm(1)1136 1465 y Fu(\(2\)\))1299 1417 y Fm(2)1338 1381 y Fp(\014)1338 1440 y(\014)1388 1465 y Fo(F)1460 1480 y Fl(T)1501 1492 y Fg(k)q Fh(\000)p Fi(1)1622 1385 y Fp(\001)1750 1465 y Fu(=)1919 1398 y(9)p Fq(\025)2025 1362 y Fm(2)p 1919 1442 146 4 v 1943 1534 a Fu(16)2091 1333 y Fl(n)2134 1345 y Fg(\025)2174 1333 y Fm(\()p Fl(\034)8 b Fm(\))2108 1371 y Fp(X)2115 1583 y Fl(k)r Fm(=1)2285 1465 y Fj(E)2362 1355 y Fp(\020)2437 1465 y Fo(h)18 b Fu(\026)-67 b Fq(m)2561 1424 y Fn(0)2561 1490 y Fl(x)2601 1502 y Fg(k)q Fh(\000)p Fi(1)2722 1465 y Fq(;)17 b(z)2815 1424 y Fm(\()p Fl(k)r Fn(\000)p Fm(1\))3003 1465 y Fu(\()p Fq(T)3098 1480 y Fl(k)3141 1465 y Fu(\))p Fo(i)3218 1424 y Fm(2)3257 1351 y Fp(\014)3257 1410 y(\014)3257 1470 y(\014)3307 1465 y Fo(F)3379 1480 y Fl(T)3420 1492 y Fg(k)q Fh(\000)p Fi(1)3540 1355 y Fp(\021)1750 1822 y Fu(=)1919 1755 y(9)p Fq(\025)2025 1719 y Fm(2)2064 1755 y Fq(T)p 1919 1799 216 4 v 1978 1890 a Fu(16)2162 1690 y Fl(n)2205 1702 y Fg(\025)2245 1690 y Fm(\()p Fl(\034)8 b Fm(\))2178 1728 y Fp(X)2186 1940 y Fl(k)r Fm(=1)2356 1687 y Fp(Z)2455 1822 y Fq(dx)35 b Fu(\026)-67 b Fq(m)2663 1781 y Fn(0)2687 1822 y Fu(\()p Fq(x)p Fu(\))2818 1781 y Fm(2)2857 1822 y Fq(a)p Fu(\()p Fq(\025)3003 1781 y Fl(\014)3051 1822 y Fu(\()p Fq(x)22 b Fu(+)g Fq(x)3319 1837 y Fl(k)r Fn(\000)p Fm(1)3452 1822 y Fu(\)\))3528 1781 y Fm(2)0 2082 y Fu(whic)m(h)41 b(con)m(v)m(erges)j(to)c Fq(D)s(\034)52 b Fu(on)41 b(the)h(set)f Fo(f)p Fq(x)1624 2046 y Fn(\003)1664 2082 y Fu(\()p Fq(\034)11 b Fu(\))42 b Fo(\024)h Fq(L)p Fo(g)e Fu(for)f(an)m(y)i Fq(L)g(>)g Fu(0.)69 b(F)-8 b(rom)39 b(\(8.6\))h(and)i(routine)0 2199 y(manipulation,)29 b(\()p Fq(S)6 b Fu(\(2\)\))890 2214 y Fl(\025)968 2199 y Fu(con)m(v)m(erges)35 b(w)m(eakly)e(to)g(a)f (Bro)m(wnian)g(motion)f(with)i(di\013usion)e Fq(D)36 b Fu(b)m(y)d(Levy's)0 2315 y(c)m(haracterization)f(theorem.)100 2431 y(T)-8 b(o)29 b(complete)g(the)h(pro)s(of,)f(w)m(e)h(in)m(tro)s (duce)g(the)g(family)d(of)i(linear)e(mappings)i Fq(J)2990 2446 y Fl(\025)3063 2431 y Fu(:)e Fq(C)7 b Fu(\()p Fj(R)3298 2446 y Fm(+)3363 2431 y Fu(\))28 b Fo(!)f Fq(C)7 b Fu(\()p Fj(R)3737 2446 y Fm(+)3802 2431 y Fu(\))0 2547 y(de\014ned)34 b(b)m(y)1079 2776 y Fq(J)1133 2791 y Fl(\025)1178 2776 y Fu(\()p Fq(')p Fu(\)\()p Fq(\034)11 b Fu(\))28 b(:=)1606 2644 y Fl(n)1649 2656 y Fg(\025)1689 2644 y Fm(\()p Fl(\034)8 b Fm(\))p Fn(\000)p Fm(1)1668 2681 y Fp(X)1675 2894 y Fl(k)r Fm(=0)1890 2776 y Fq(A)1963 2792 y Fl(n)2006 2804 y Fg(\025)2047 2792 y Fm(\()p Fl(\034)g Fm(\))p Fn(\000)p Fm(1)p Fl(;k)r Fn(\000)p Fm(1)2400 2776 y Fq(')p Fu(\()p Fq(\025)2559 2735 y Fm(2)2598 2776 y Fq(T)14 b(k)s Fu(\))0 3020 y(when)32 b(\()p Fq(\025)348 2984 y Fm(2)387 3020 y Fq(T)14 b Fu(\))496 2984 y Fn(\000)p Fm(1)590 3020 y Fq(\034)42 b Fu(is)30 b(an)g(in)m(teger)h(and)f(b)m(y)i(linear)d(in)m (terp)s(olation)f(otherwise.)43 b(It)30 b(is)h(easy)g(to)f(v)m(erify)h (that)937 3245 y(lim)949 3308 y Fl(\025)p Fn(#)p Fm(0)1106 3245 y Fq(J)1160 3260 y Fl(\025)1205 3245 y Fu(\()p Fq(')p Fu(\)\()p Fq(\034)11 b Fu(\))28 b(=)f Fq(J)9 b Fu(\()p Fq(')p Fu(\)\()p Fq(\034)i Fu(\))28 b(:=)2177 3178 y(3)p 2106 3222 191 4 v 2106 3242 a Fo(p)p 2189 3242 108 4 v 83 x Fu(2)p Fq(\031)2323 3110 y Fp(Z)2423 3136 y Fl(\034)2378 3335 y Fm(0)2466 3245 y Fq(ds)2648 3178 y(')p Fu(\()p Fq(s)p Fu(\))p 2589 3222 305 4 v 2589 3242 a Fo(p)p 2672 3242 222 4 v 75 x Fq(\034)34 b Fo(\000)22 b Fq(s)0 3471 y Fu(uniformly)34 b(for)i Fq(')h Fu(in)f(a)g(compact)g(set.)56 b(Since)37 b Fq(J)46 b Fu(is)36 b(con)m(tin)m(uous,)i(w)m(e)g(can)e (apply)h([6)o(,)h(Thm.)f(5.3])f(and)0 3587 y(get)740 3766 y Fq(J)794 3781 y Fl(\025)839 3766 y Fu(\(\()p Fq(S)6 b Fu(\(2\)\))1144 3781 y Fl(\025)1189 3766 y Fu(\)\()p Fq(\034)11 b Fu(\))1407 3705 y Fl(\025)p Fn(#)p Fm(0)1384 3766 y Fu(=)-17 b Fo(\))1651 3698 y Fu(3)p 1580 3743 191 4 v 1580 3763 a Fo(p)p 1663 3763 108 4 v 82 x Fu(2)p Fq(\031)1797 3630 y Fp(Z)1897 3656 y Fl(\034)1853 3855 y Fm(0)1940 3766 y Fq(ds)2134 3698 y(b)p Fu(\()p Fq(s)p Fu(\))p 2063 3743 305 4 v 2063 3763 a Fo(p)p 2146 3763 222 4 v 75 x Fq(\034)34 b Fo(\000)23 b Fq(s)2703 3766 y Fu(in)31 b Fq(C)7 b Fu(\()p Fj(R)2997 3781 y Fm(+)3062 3766 y Fu(\))0 3965 y(Prop)s(osition)31 b(8.2)h(is)g(pro)m(v)m(ed.)2673 b Fd(\003)0 4131 y Ft(Conclusion)35 b(of)g(the)h(pr)-5 b(o)g(of)36 b(of)f(The)-5 b(or)g(em)35 b(1.1.)47 b Fu(W)-8 b(e)34 b(\014rst)g(sho)m(w)h(that)f Fq(\030)2630 4146 y Fl(\025)2708 4131 y Fu(con)m(v)m(erges)i(b)m(y)f(subsequences)0 4247 y(to)e(a)g(con)m(tin)m(uous)i(pro)s(cess,)g(and)e(then)i(that)e (an)m(y)h(limit)c(p)s(oin)m(t)j(solv)m(es)h(the)g(in)m(tegral)e (equation)i(\(1.12\))o(.)0 4363 y(By)f(the)g(uniqueness)i(\(in)c(la)m (w\))h(of)h(the)g(latter)e(Theorem)i(1.1)f(follo)m(ws.)100 4480 y(The)e(b)s(oundedness)i(of)d Fq(\030)1018 4495 y Fl(\025)1092 4480 y Fu(follo)m(ws)g(from)f(\(8.5\))o(.)43 b(In)30 b(order)f(to)h(pro)m(v)m(e)g(equicon)m(tin)m(uit)m(y)-8 b(,)31 b(w)m(e)f(note)g(that)0 4596 y(from)h(\(7.2\))h(and)h(\(8.14\))f (w)m(e)h(ha)m(v)m(e,)h(for)f(eac)m(h)g Fq(")27 b(>)h Fu(0,)889 4856 y(lim)901 4919 y Fl(\025)p Fn(#)p Fm(0)1041 4856 y Fj(P)1117 4686 y Fp( )1262 4856 y Fu(sup)1198 4942 y Fl(n)p Fn(\024)p Fl(n)1339 4954 y Fg(\025)1379 4942 y Fm(\()p Fl(\034)8 b Fm(\))1489 4682 y Fp(\014)1489 4741 y(\014)1489 4801 y(\014)1489 4861 y(\014)1489 4921 y(\014)1523 4856 y Fq(\030)1566 4871 y Fl(n)1634 4856 y Fo(\000)23 b Fq(Y)1791 4871 y Fl(n)1860 4856 y Fo(\000)1965 4732 y Fl(n)p Fn(\000)p Fm(1)1959 4761 y Fp(X)1967 4974 y Fl(k)r Fm(=0)2103 4856 y Fu(\()p Fq(A)2214 4815 y Fm(2)2254 4856 y Fu(\))2292 4871 y Fl(n;k)2397 4856 y Fq(\030)2440 4871 y Fl(k)2482 4682 y Fp(\014)2482 4741 y(\014)2482 4801 y(\014)2482 4861 y(\014)2482 4921 y(\014)2543 4856 y Fq(>)k(")2692 4686 y Fp(!)2799 4856 y Fu(=)g(0)640 b(\(8.16\))0 5110 y(Setting)1332 5338 y(\004)1397 5353 y Fl(\025)1443 5338 y Fu(\()p Fq(s)p Fu(\))27 b(:=)1723 5206 y Fl(n)1766 5218 y Fg(\025)1806 5206 y Fm(\()p Fl(s)p Fm(\))p Fn(\000)p Fm(1)1781 5244 y Fp(X)1789 5456 y Fl(k)r Fm(=0)1984 5338 y Fu(\()p Fq(A)2095 5297 y Fm(2)2135 5338 y Fu(\))2173 5354 y Fl(n)2216 5366 y Fg(\025)2256 5354 y Fm(\()p Fl(s)p Fm(\))p Fl(;k)2423 5338 y Fq(\030)2466 5353 y Fl(k)3591 5338 y Fu(\(8.17\))1882 5532 y Fk(45)p eop %%Page: 46 47 46 46 bop 0 266 a Fu(w)m(e)34 b(p)s(ostp)s(one)e(the)h(pro)s(of)f (that,)h(for)f(eac)m(h)h Fq(")27 b(>)h Fu(0,)958 538 y(lim)974 601 y Fl(\016)r Fn(#)p Fm(0)p 1121 455 136 3 v 1121 538 a Fu(lim)1133 601 y Fl(\025)p Fn(#)p Fm(0)1284 538 y Fj(P)1360 338 y Fp(0)1360 517 y(@)1516 538 y Fu(sup)1459 616 y Fg(s;s)1536 602 y Fh(0)1558 616 y(2)p Fi([0)p Fg(;\034)6 b Fi(])1467 680 y Fh(j)p Fg(s)p Fh(\000)p Fg(s)1592 666 y Fh(0)1614 680 y(j\024)p Fg(\016)1773 538 y Fo(j)p Fu(\004)1866 553 y Fl(\025)1911 538 y Fu(\()p Fq(s)p Fu(\))22 b Fo(\000)h Fu(\004)2220 553 y Fl(\025)2265 538 y Fu(\()p Fq(s)2349 497 y Fn(0)2372 538 y Fu(\))p Fo(j)28 b Fq(>)f(")2615 338 y Fp(1)2615 517 y(A)2730 538 y Fu(=)g(0)709 b(\(8.18\))0 826 y(and)33 b(observ)m(e)h(that,)f(b)m(y)h(\(8.16\),)e(Prop)s(osition) f(8.2,)h(and)h(\(8.18\))o(,)g(for)f(eac)m(h)h Fq(")28 b(>)f Fu(0,)980 1104 y(lim)996 1167 y Fl(\016)r Fn(#)p Fm(0)p 1144 1021 V 1144 1104 a Fu(lim)1155 1167 y Fl(\025)p Fn(#)p Fm(0)1307 1104 y Fj(P)1383 904 y Fp(0)1383 1083 y(@)1538 1104 y Fu(sup)1481 1182 y Fg(s;s)1558 1168 y Fh(0)1581 1182 y(2)p Fi([0)p Fg(;\034)6 b Fi(])1489 1245 y Fh(j)p Fg(s)p Fh(\000)p Fg(s)1614 1231 y Fh(0)1637 1245 y(j\024)p Fg(\016)1796 1104 y Fo(j)o Fq(\030)1866 1119 y Fl(\025)1911 1104 y Fu(\()p Fq(s)p Fu(\))22 b Fo(\000)h Fq(\030)2198 1119 y Fl(\025)2243 1104 y Fu(\()p Fq(s)2327 1063 y Fn(0)2350 1104 y Fu(\))p Fo(j)k Fq(>)h(")2593 904 y Fp(1)2593 1083 y(A)2707 1104 y Fu(=)g(0)731 b(\(8.19\))0 1391 y(whic)m(h)33 b(sho)m(ws)h Fq(\030)602 1406 y Fl(\025)680 1391 y Fu(is)e(tigh)m(t)g(in)f Fq(C)7 b Fu(\()p Fj(R)1308 1406 y Fm(+)1373 1391 y Fu(\).)100 1508 y(It)32 b(remains)g(to)g(pro)m (v)m(e)i(\(8.18\))o(.)44 b(Let)32 b(us)i(\014x)f(0)27 b Fo(\024)h Fq(s)1936 1471 y Fn(0)1987 1508 y Fo(\024)h Fq(s)e Fo(\024)h Fq(\034)11 b Fu(,)33 b Fq(s)23 b Fo(\000)f Fq(s)2598 1471 y Fn(0)2649 1508 y Fo(\024)28 b Fq(\016)t Fu(.)44 b(W)-8 b(e)33 b(ha)m(v)m(e)196 1775 y(\004)261 1790 y Fl(\025)306 1775 y Fu(\()p Fq(s)p Fu(\))22 b Fo(\000)h Fu(\004)615 1790 y Fl(\025)660 1775 y Fu(\()p Fq(s)744 1734 y Fn(0)768 1775 y Fu(\))83 b(=)1047 1643 y Fl(n)1090 1655 y Fg(\025)1131 1643 y Fm(\()p Fl(s)1191 1619 y Fh(0)1213 1643 y Fn(\000)p Fl(\016)r Fm(\))p Fn(\000)p Fm(1)1162 1680 y Fp(X)1169 1892 y Fl(k)r Fm(=0)1436 1694 y Fp(\000)1482 1775 y Fu(\()p Fq(A)1593 1734 y Fm(2)1632 1775 y Fu(\))1670 1790 y Fl(n)1713 1802 y Fg(\025)1754 1790 y Fm(\()p Fl(s)p Fm(\))p Fl(;k)1926 1775 y Fo(\000)23 b Fu(\()p Fq(A)2137 1734 y Fm(2)2176 1775 y Fu(\))2214 1790 y Fl(n)2257 1802 y Fg(\025)2298 1790 y Fm(\()p Fl(s)2358 1771 y Fh(0)2380 1790 y Fm(\))p Fl(;k)2470 1694 y Fp(\001)2532 1775 y Fq(\030)2575 1790 y Fl(k)1047 2133 y Fu(+)1186 2001 y Fl(n)1229 2013 y Fg(\025)1269 2001 y Fm(\()p Fl(s)1329 1977 y Fh(0)1352 2001 y Fm(\))p Fn(\000)p Fm(1)1256 2038 y Fp(X)1140 2254 y Fl(k)r Fm(=)p Fl(n)1277 2266 y Fg(\025)1317 2254 y Fm(\()p Fl(s)1377 2235 y Fh(0)1399 2254 y Fn(\000)p Fl(\016)r Fm(\))1532 2052 y Fp(\000)1578 2133 y Fu(\()p Fq(A)1689 2092 y Fm(2)1728 2133 y Fu(\))1766 2148 y Fl(n)1809 2160 y Fg(\025)1850 2148 y Fm(\()p Fl(s)p Fm(\))p Fl(;k)2022 2133 y Fo(\000)f Fu(\()p Fq(A)2232 2092 y Fm(2)2272 2133 y Fu(\))2310 2148 y Fl(n)2353 2160 y Fg(\025)2393 2148 y Fm(\()p Fl(s)2453 2130 y Fh(0)2476 2148 y Fm(\))p Fl(;k)2566 2052 y Fp(\001)2628 2133 y Fq(\030)2671 2148 y Fl(k)2735 2133 y Fu(+)2846 2001 y Fl(n)2889 2013 y Fg(\025)2930 2001 y Fm(\()p Fl(s)p Fm(\))p Fn(\000)p Fm(1)2905 2038 y Fp(X)2833 2254 y Fl(k)r Fm(=)p Fl(n)2970 2266 y Fg(\025)3010 2254 y Fm(\()p Fl(s)3070 2235 y Fh(0)3093 2254 y Fm(\))3120 2133 y Fu(\()p Fq(A)3231 2092 y Fm(2)3271 2133 y Fu(\))3309 2148 y Fl(n)3352 2160 y Fg(\025)3392 2148 y Fm(\()p Fl(s)p Fm(\))p Fl(;k)3559 2133 y Fq(\030)3602 2148 y Fl(k)0 2399 y Fu(so)33 b(that)299 2592 y Fo(j)o Fu(\004)391 2607 y Fl(\025)437 2592 y Fu(\()p Fq(s)p Fu(\))22 b Fo(\000)g Fu(\004)745 2607 y Fl(\025)791 2592 y Fu(\()p Fq(s)875 2551 y Fn(0)898 2592 y Fu(\))p Fo(j)83 b(\024)g Fu(2)17 b Fq(\030)1321 2551 y Fn(\003)1359 2592 y Fu(\()p Fq(\034)11 b Fu(\))17 b Fq(n)1563 2607 y Fl(\025)1609 2592 y Fu(\()p Fq(\034)11 b Fu(\))17 b Fq(\025)1812 2551 y Fm(2)1851 2592 y Fq(T)120 b Fu(sup)1948 2666 y Fg(\034)1977 2681 y Fi(1)2012 2666 y Fg(;\034)2060 2681 y Fi(2)2094 2666 y Fh(2)p Fi([0)p Fg(;\034)6 b Fi(])1975 2724 y Fg(\034)2004 2739 y Fi(1)2039 2724 y Fh(\000)p Fg(\034)2116 2739 y Fi(2)2150 2724 y Fg(>\016)2309 2448 y Fp(\014)2309 2508 y(\014)2309 2567 y(\014)2309 2627 y(\014)2343 2592 y Fu(\()p Fq(\025)2438 2551 y Fm(2)2477 2592 y Fq(T)14 b Fu(\))2586 2551 y Fn(\000)p Fm(1)2680 2592 y Fu(\()p Fq(A)2791 2551 y Fm(2)2830 2592 y Fu(\))2868 2608 y Fl(n)2911 2620 y Fg(\025)2952 2608 y Fm(\()p Fl(\034)3010 2617 y Fi(1)3045 2608 y Fm(\))p Fl(;n)3135 2620 y Fg(\025)3176 2608 y Fm(\()p Fl(\034)3234 2617 y Fi(2)3269 2608 y Fm(\))3323 2592 y Fo(\000)3433 2525 y Fu(9)p 3433 2569 49 4 v 3433 2661 a(2)3491 2448 y Fp(\014)3491 2508 y(\014)3491 2567 y(\014)3491 2627 y(\014)1207 2871 y Fu(+)j(3)g Fq(\030)1414 2830 y Fn(\003)1452 2871 y Fu(\()p Fq(\034)11 b Fu(\))17 b Fq(n)1656 2886 y Fl(\025)1701 2871 y Fu(\()p Fq(\016)t Fu(\))173 b(sup)1841 2957 y Fm(0)p Fn(\024)p Fl(k)r()37 b Fu(0)0 5167 y(suc)m(h)d(that,)e(for)h(an)m (y)g Fq(n)28 b Fo(\024)g Fq(n)1041 5182 y Fl(\025)1086 5167 y Fu(\()p Fq(\034)11 b Fu(\),)1297 5354 y(sup)1182 5439 y Fl(t)p Fn(2)p Fm([)p Fl(T)1315 5447 y Fg(n)1358 5439 y Fl(;T)1419 5448 y Fg(n)p Fi(+1)1538 5439 y Fm(])1574 5354 y Fu(sup)1581 5435 y Fl(x)p Fn(2)p Ff(R)1755 5354 y Fj(E)1815 5369 y Fl(n)1885 5273 y Fp(\000)1930 5354 y Fq(z)1979 5312 y Fm(\()p Fl(n)p Fm(\))2082 5354 y Fu(\()p Fq(t;)17 b(x)p Fu(\))2292 5312 y Fm(2)2331 5273 y Fp(\001)2405 5354 y Fo(\024)28 b Fq(C)7 b(T)975 b Fu(\(B.2\))1882 5532 y Fk(47)p eop %%Page: 48 49 48 48 bop 0 266 a Fu(Pro)s(ceeding)37 b(as)g(in)f(the)h(pro)s(of)f(of)g ([7,)h(Thm.)g(2.3],)h(\(taking)d Fq(")g Fu(=)f(1)j(and)g Fq(\026)p Fu(\()p Fq(y)t Fu(\))c(=)i Fq(a)p Fu(\()p Fq(\025)3212 230 y Fl(\014)3259 266 y Fq(y)t Fu(\)\),)i(it)e(follo)m(ws)0 382 y(that)41 b(giv)m(en)h Fq(\013)i Fo(2)f Fu(\(0)p Fq(;)17 b Fu(1\))40 b(there)j(exists)f Fq(C)48 b Fu(suc)m(h)43 b(that,)h(for)d(an)m(y)h Fq(h;)17 b(\025)43 b Fo(2)g Fu(\(0)p Fq(;)17 b Fu(1],)43 b Fq(t)g Fo(2)h Fu([)p Fq(T)3300 397 y Fl(n)3347 382 y Fq(;)17 b(T)3448 397 y Fl(n)p Fm(+1)3585 382 y Fu(],)44 b(and)0 498 y Fq(x)28 b Fo(2)g Fj(R)5 b Fu(,)1026 641 y Fj(E)1087 656 y Fl(n)1156 530 y Fp(\020)1216 560 y(\000)1261 641 y Fq(z)1310 599 y Fm(\()p Fl(n)p Fm(\))1413 641 y Fu(\()p Fq(t;)17 b(x)22 b Fu(+)g Fq(h)p Fu(\))h Fo(\000)f Fq(z)1970 599 y Fm(\()p Fl(n)p Fm(\))2073 641 y Fu(\()p Fq(t;)17 b(x)p Fu(\))2283 560 y Fp(\001)2329 576 y Fm(2)2368 530 y Fp(\021)2455 641 y Fo(\024)29 b Fq(C)7 b(T)14 b(h)2765 599 y Fl(\013)3619 641 y Fu(\(B.3\))0 803 y(It)33 b(is)f(not)g(di\016cult)g(to)g(pro)m(v)m(e)i(also)e(H\177) -49 b(older)32 b(con)m(tin)m(uit)m(y)g(in)g Fq(t)p Fu(:)910 986 y Fj(E)971 1001 y Fl(n)1040 876 y Fp(\020)1100 906 y(\000)1145 986 y Fq(z)1194 945 y Fm(\()p Fl(n)p Fm(\))1297 986 y Fu(\()p Fq(t)22 b Fu(+)g Fq(h;)17 b(x)p Fu(\))23 b Fo(\000)f Fq(z)1854 945 y Fm(\()p Fl(n)p Fm(\))1957 986 y Fu(\()p Fq(t;)17 b(x)p Fu(\))2167 906 y Fp(\001)2213 922 y Fm(2)2252 876 y Fp(\021)2339 986 y Fo(\024)29 b Fq(C)7 b(T)2593 945 y Fm(2)2632 986 y Fq(\025)2689 945 y Fn(\000)p Fl(\014)2791 897 y Fo(p)p 2874 897 57 4 v 89 x Fq(h)689 b Fu(\(B.4\))0 1175 y(Indeed)34 b(recall)d(that,)i(from)e (\(1.3\))o(,)40 1399 y Fj(E)101 1414 y Fl(n)170 1289 y Fp(\020)230 1318 y(\000)276 1399 y Fq(z)325 1358 y Fm(\()p Fl(n)p Fm(\))427 1399 y Fu(\()p Fq(t)23 b Fu(+)f Fq(h;)17 b(x)p Fu(\))22 b Fo(\000)h Fq(z)985 1358 y Fm(\()p Fl(n)p Fm(\))1087 1399 y Fu(\()p Fq(t;)17 b(x)p Fu(\))1297 1318 y Fp(\001)1343 1334 y Fm(2)1382 1289 y Fp(\021)1525 1399 y Fu(=)1684 1264 y Fp(Z)1784 1290 y Fl(t)1739 1489 y(T)1780 1497 y Fg(n)1827 1399 y Fq(ds)1941 1264 y Fp(Z)2040 1399 y Fq(dy)2175 1289 y Fp(\020)2234 1399 y Fq(g)2285 1348 y Fm(\()p Fl(n)p Fm(\))2281 1427 y Fl(t)p Fn(\000)p Fl(s)p Fm(+)p Fl(h)2494 1399 y Fu(\()p Fq(x;)g(y)t Fu(\))k Fo(\000)i Fq(g)2893 1348 y Fm(\()p Fl(n)p Fm(\))2889 1422 y Fl(t)p Fn(\000)p Fl(s)3006 1399 y Fu(\()p Fq(x;)17 b(y)t Fu(\))3233 1289 y Fp(\021)3291 1311 y Fm(2)3347 1399 y Fq(a)p Fu(\()p Fq(\025)3493 1358 y Fl(\014)3540 1399 y Fq(y)t Fu(\))3630 1358 y Fm(2)1684 1690 y Fu(+)1793 1555 y Fp(Z)1893 1581 y Fl(h)1848 1780 y Fm(0)1937 1690 y Fq(ds)2051 1555 y Fp(Z)2150 1690 y Fq(dy)j(g)2320 1649 y Fm(\()p Fl(n)p Fm(\))2316 1715 y Fl(s)2421 1690 y Fu(\()p Fq(x;)d(y)t Fu(\))2648 1649 y Fm(2)2686 1690 y Fq(a)p Fu(\()p Fq(\025)2832 1649 y Fl(\014)2880 1690 y Fq(y)t Fu(\))2970 1649 y Fm(2)3619 1690 y Fu(\(B.5\))0 1904 y(Using)34 b(\(7.29\))d(it)h(is)g(easy)i(to)e(see)i(that)1138 1998 y Fp(Z)1238 2024 y Fl(h)1193 2223 y Fm(0)1282 2133 y Fq(ds)1396 1998 y Fp(Z)1495 2133 y Fq(dy)20 b(g)1665 2092 y Fm(\()p Fl(n)p Fm(\))1661 2158 y Fl(s)1766 2133 y Fu(\()p Fq(x;)d(y)t Fu(\))1993 2092 y Fm(2)2031 2133 y Fq(a)p Fu(\()p Fq(\025)2177 2092 y Fl(\014)2225 2133 y Fq(y)t Fu(\))2315 2092 y Fm(2)2381 2133 y Fo(\024)28 b Fq(C)2563 2044 y Fo(p)p 2646 2044 V 89 x Fq(h)0 2373 y Fu(T)-8 b(o)33 b(b)s(ound)f(the)h(\014rst)g(term)f(on)h(the)g(r.h.s.) g(of)39 b(\(B.5\),)33 b(w)m(e)g(use)h(the)f(follo)m(wing)d(form)m(ula)h (for)h Fq(g)3443 2322 y Fm(\()p Fl(n)p Fm(\))3439 2396 y Fl(t)3544 2373 y Fu(:)543 2597 y Fq(g)594 2546 y Fm(\()p Fl(n)p Fm(\))590 2620 y Fl(t)695 2597 y Fu(\()p Fq(x;)17 b(y)t Fu(\))26 b(=)i Fq(p)1101 2612 y Fl(t)1131 2597 y Fu(\()p Fq(x;)17 b(y)t Fu(\))k(+)1477 2462 y Fp(Z)1577 2488 y Fl(t)1532 2687 y Fm(0)1606 2597 y Fq(ds)1720 2462 y Fp(Z)1819 2597 y Fq(dz)g(p)1985 2612 y Fl(t)p Fn(\000)p Fl(s)2102 2597 y Fu(\()p Fq(x;)c(z)t Fu(\)[1)23 b Fo(\000)g Fu(3)18 b(\026)-67 b Fq(m)2659 2556 y Fm(2)2659 2622 y Fl(x)2699 2630 y Fg(n)2745 2597 y Fu(\()p Fq(z)t Fu(\)])p Fq(g)2948 2556 y Fm(\()p Fl(n)p Fm(\))2944 2622 y Fl(s)3050 2597 y Fu(\()p Fq(z)t(;)17 b(y)t Fu(\))p Fq(;)321 b Fu(\(B.6\))0 2837 y(F)-8 b(rom)22 b(the)i(prop)s(erties)g(of)f(the)h(heat)g(k)m (ernel)h Fq(p)1653 2852 y Fl(t)1682 2837 y Fu(,)h(the)e(estimate)f(sup) q Fo(f)p Fq(g)2524 2786 y Fm(\()p Fl(n)p Fm(\))2520 2860 y Fl(t)2625 2837 y Fu(\()p Fq(x;)17 b(y)t Fu(\);)32 b Fq(x;)17 b(y)31 b Fo(2)d Fj(R)5 b Fq(;)39 b(t)28 b Fo(\025)g Fu(1)p Fo(g)g Fq(<)f Fo(1)p Fu(,)0 2953 y(the)33 b(fact)f(that)h Fq(a)f Fu(has)h(compact)g(supp)s(ort)g(and)f(\(7.29\))g(for)g Fq(t)c Fo(2)g Fu(\(0)p Fq(;)17 b Fu(1],)32 b(one)h(gets)g(\(B.4\).)100 3070 y(Recall)e(that)h Fq(z)650 3085 y Fl(n)730 3070 y Fu(conditioned)f(on)i Fq(x)1446 3085 y Fl(n)1526 3070 y Fu(is)f(Gaussian,)g(and)h(let)f(us)h(no)m(w)g(de\014ne)1144 3229 y Fq(\033)1203 3188 y Fm(2)1199 3254 y Fl(n)1329 3229 y Fu(=)197 b(sup)1488 3315 y Fl(t)p Fn(2)p Fm([)p Fl(T)1621 3323 y Fg(n)1664 3315 y Fl(;T)1725 3324 y Fg(n)p Fi(+1)1844 3315 y Fm(])1880 3229 y Fu(sup)1886 3311 y Fl(x)p Fn(2)p Ff(R)2060 3229 y Fj(E)2121 3244 y Fl(n)2190 3148 y Fp(\000)2236 3229 y Fq(z)2285 3188 y Fm(\()p Fl(n)p Fm(\))2388 3229 y Fu(\()p Fq(t;)17 b(x)p Fu(\))2598 3188 y Fm(2)2637 3148 y Fp(\001)3619 3229 y Fu(\(B.7\))1141 3553 y Fq(\026)1200 3568 y Fl(n)1329 3553 y Fu(=)83 b Fj(E)1549 3568 y Fl(n)1618 3383 y Fp( )1811 3553 y Fu(sup)1697 3639 y Fl(t)p Fn(2)p Fm([)p Fl(T)1830 3647 y Fg(n)1873 3639 y Fl(;T)1934 3648 y Fg(n)p Fi(+1)2053 3639 y Fm(])2089 3553 y Fo(k)p Fq(z)2188 3512 y Fm(\()p Fl(n)p Fm(\))2290 3553 y Fu(\()p Fq(t)p Fu(\))p Fo(k)2451 3568 y Fn(1)2526 3383 y Fp(!)3619 3553 y Fu(\(B.8\))100 3819 y(Borell's)31 b(inequalit)m(y)h(\(see)h(for)f(instance)h([1,)g(Thm.)f(2.1]\))g(tells) g(us)h(that,)g(if)e Fq(\026)2970 3834 y Fl(n)3044 3819 y Fq(<)3148 3735 y Fo(p)p 3231 3735 71 4 v 84 x Fq(T)14 b(\025)3359 3783 y Fn(\000)p Fl(\020)3453 3819 y Fu(,)33 b(then)533 4068 y Fj(P)592 4083 y Fl(n)658 3897 y Fp( )851 4068 y Fu(sup)737 4153 y Fl(t)p Fn(2)p Fm([)p Fl(T)870 4161 y Fg(n)912 4153 y Fl(;T)973 4162 y Fg(n)p Fi(+1)1093 4153 y Fm(])1129 4068 y Fo(k)p Fq(z)1228 4026 y Fm(\()p Fl(n)p Fm(\))1330 4068 y Fu(\()p Fq(t)p Fu(\))p Fo(k)1491 4083 y Fn(1)1594 4068 y Fq(>)27 b(\025)1754 4026 y Fn(\000)p Fl(\020)1849 3978 y Fo(p)p 1932 3978 V 90 x Fq(T)2002 3897 y Fp(!)2109 4068 y Fo(\024)h Fu(4)17 b(exp)2445 3897 y Fp(")2503 4068 y Fo(\000)2590 4000 y Fu(\()p Fq(\025)2685 3964 y Fn(\000)p Fl(\020)2780 3916 y Fo(p)p 2863 3916 V 84 x Fq(T)36 b Fo(\000)23 b Fq(\026)3115 4015 y Fl(n)3161 4000 y Fu(\))3199 3964 y Fm(2)p 2590 4045 648 4 v 2839 4136 a Fu(2)p Fq(\033)2947 4107 y Fm(2)2943 4160 y Fl(n)3249 3897 y Fp(#)3619 4068 y Fu(\(B.9\))0 4316 y(T)-8 b(o)33 b(estimate)e Fq(\026)593 4331 y Fl(n)640 4316 y Fu(,)i(w)m(e)g(use)h(w) m(e)f(use)h(\(see)g([1)o(,)f(Cor.)44 b(4.15]\),)1363 4528 y Fq(\026)1422 4543 y Fl(n)1496 4528 y Fo(\024)29 b Fq(K)1708 4392 y Fp(Z)1808 4419 y Fn(1)1764 4618 y Fm(0)1883 4528 y Fq(dr)1997 4438 y Fp(p)p 2096 4438 354 4 v 2096 4528 a Fu(log)17 b Fq(N)10 b Fu(\()p Fq(r)s Fu(\))p Fq(;)1093 b Fu(\(B.10\))0 4742 y(where)28 b Fq(K)34 b Fu(is)26 b(a)h(univ)m(ersal)f(constan)m(t)i(and)f Fq(N)10 b Fu(\()p Fq(r)s Fu(\))26 b(is)h(the)g(minimal)c(n)m(um)m(b)s(er)k(of)f (balls)f(of)i(radius)f Fq(r)j Fu(needed)0 4858 y(to)j(co)m(v)m(er)i([)p Fq(T)455 4873 y Fl(n)502 4858 y Fq(;)17 b(T)603 4873 y Fl(n)p Fm(+1)740 4858 y Fu(])23 b Fo(\002)f Fj(R)44 b Fu(when)34 b(considering)e(in)f(that)i(set)g(the)g(metric)876 5090 y Fq(d)927 4979 y Fp(\020)987 5090 y Fu(\()p Fq(t;)17 b(x)p Fu(\))p Fq(;)g Fu(\()p Fq(s;)g(y)t Fu(\))1459 4979 y Fp(\021)1544 5090 y Fu(=)1648 4940 y Fp(r)p 1747 4940 1217 4 v 1747 5090 a Fj(E)1808 5105 y Fl(n)1877 4979 y Fp(\020)1937 5090 y Fu(\()p Fq(z)2024 5061 y Fm(\()p Fl(n)p Fm(\))2127 5090 y Fu(\()p Fq(t;)g(x)p Fu(\))22 b Fo(\000)h Fq(z)2508 5061 y Fm(\()p Fl(n)p Fm(\))2610 5090 y Fu(\()p Fq(s;)17 b(y)t Fu(\)\))2865 5037 y Fm(2)2904 4979 y Fp(\021)100 5316 y Fu(Let)31 b(us)h(then)f(estimate)g Fq(N)10 b Fu(\()p Fq(r)s Fu(\).)43 b(Recall)29 b(that,)i(from)f (\(B.2\),)i(w)m(e)g(only)e(need)j(to)d(consider)i Fq(r)e Fo(\024)3582 5232 y(p)p 3665 5232 148 4 v 84 x Fq(C)7 b(T)14 b Fu(,)0 5433 y(since)44 b(for)f(larger)f Fq(r)s Fu(,)j Fq(N)10 b Fu(\()p Fq(r)s Fu(\))46 b(=)g(1)d(and)h(the)g(in)m (tegrand)f(in)f(\(B.10\))h(equals)h(zero.)76 b(F)-8 b(or)42 b(eac)m(h)j Fq(r)j(>)e Fu(0,)1882 5532 y Fk(48)p eop %%Page: 49 50 49 49 bop 0 266 a Fu(consider)38 b(the)h(sets)g Fq(A)828 281 y Fm(1)904 266 y Fu(=)e([)p Fq(T)1101 281 y Fl(n)1148 266 y Fq(;)17 b(T)1249 281 y Fl(n)p Fm(+1)1386 266 y Fu(])26 b Fo(\002)h Fu([)p Fo(\000)p Fu(\()p Fq(\025)1742 230 y Fm(2)p Fl(\014)1824 266 y Fq(r)s Fu(\))1909 230 y Fn(\000)p Fm(1)2003 266 y Fq(;)17 b Fu(\()p Fq(\025)2142 230 y Fm(2)p Fl(\014)2224 266 y Fq(r)s Fu(\))2309 230 y Fn(\000)p Fm(1)2403 266 y Fu(])38 b(and)g Fq(A)2736 281 y Fm(2)2813 266 y Fu(=)f(\([)p Fq(T)3048 281 y Fl(n)3095 266 y Fq(;)17 b(T)3196 281 y Fl(n)p Fm(+1)3333 266 y Fu(])26 b Fo(\002)g Fj(R)5 b Fu(\))32 b Fo(n)25 b Fq(A)3773 281 y Fm(1)3813 266 y Fu(.)0 382 y(Let)33 b(us)g(estimate)f(\014rst)h (the)g(n)m(um)m(b)s(er)g(of)f(balls)f(needed)j(to)e(co)m(v)m(er)i Fq(A)2523 397 y Fm(2)2563 382 y Fu(.)100 498 y(F)-8 b(rom)35 b(no)m(w)i(on,)h(to)f(a)m(v)m(oid)f(in)m(tro)s(ducing)g(new)h(constan)m (ts)h(in)e(the)i(notation)d(w)m(e)j(supp)s(ose)g(that)f(the)0 614 y(supp)s(ort)d(of)f Fq(a)h Fu(is)f(included)g(in)g([)p Fo(\000)p Fu(1)p Fq(;)17 b Fu(1],)34 b(and)f(that)h Fq(\025)f Fu(is)g(su\016cien)m(tly)h(small.)45 b(F)-8 b(rom)32 b(\(7.29\))o(,)i(and)g(since)0 740 y Fq(T)57 755 y Fl(n)p Fm(+1)216 740 y Fo(\000)23 b Fq(T)373 755 y Fl(n)448 740 y Fu(=)k Fq(T)42 b Fo(\024)28 b Fq(\025)812 704 y Fn(\000)p Fl(\014)s(=)p Fm(20)1052 740 y Fu(and)33 b Fq(r)d Fo(\024)1422 656 y(p)p 1505 656 148 4 v 84 x Fq(C)7 b(T)13 b Fu(,)33 b(it)e(is)h(easy)i(to)e(see)i(that)e(if)g(\()p Fq(t;)17 b(x)p Fu(\))28 b Fo(2)g Fq(A)3105 755 y Fm(2)509 998 y Fj(E)570 1013 y Fl(n)639 917 y Fp(\000)685 998 y Fq(z)734 956 y Fm(\()p Fl(n)p Fm(\))837 998 y Fu(\()p Fq(t;)17 b(x)p Fu(\))1047 956 y Fm(2)1086 917 y Fp(\001)1215 998 y Fo(\024)1376 862 y Fp(Z)1475 888 y Fl(t)1431 1088 y(T)1472 1096 y Fg(n)1519 998 y Fq(ds)1633 862 y Fp(Z)1732 998 y Fq(dy)i(e)1895 956 y Fm(4\()p Fl(t)p Fn(\000)p Fl(s)p Fm(\))2103 998 y Fq(p)2152 1013 y Fl(t)p Fn(\000)p Fl(s)2269 998 y Fu(\()p Fq(x;)e(y)t Fu(\))2496 956 y Fm(2)2534 998 y Fq(a)p Fu(\()p Fq(\025)2680 956 y Fl(\014)2727 998 y Fq(y)t Fu(\))2817 956 y Fm(2)1215 1308 y Fo(\024)84 b Fq(C)7 b(e)1498 1267 y Fm(4)p Fl(T)1605 1173 y Fp(Z)1704 1199 y Fl(t)1660 1398 y(T)1701 1406 y Fg(n)1748 1308 y Fq(ds)1990 1241 y Fu(1)p 1871 1285 286 4 v 1871 1305 a Fo(p)p 1954 1305 203 4 v 77 x Fq(t)22 b Fo(\000)h Fq(s)2184 1308 y Fu(exp)2349 1138 y Fp(")2407 1308 y Fo(\000)2494 1154 y Fp(\000)2540 1234 y Fu(\()p Fq(\025)2635 1198 y Fm(2)p Fl(\014)2718 1234 y Fq(r)s Fu(\))2803 1198 y Fn(\000)p Fm(1)2919 1234 y Fo(\000)f Fq(\025)3075 1198 y Fn(\000)p Fl(\014)3177 1154 y Fp(\001)3223 1176 y Fm(2)p 2494 1285 768 4 v 2715 1376 a Fu(2\()p Fq(t)g Fo(\000)g Fq(s)p Fu(\))3272 1138 y Fp(#)1215 1616 y Fo(\024)84 b Fu(exp)17 b([)p Fo(\000)p Fq(\025)1702 1575 y Fn(\000)p Fl(\014)1805 1616 y Fq(=)p Fu(\(2)p Fq(r)1988 1575 y Fm(2)2026 1616 y Fu(\)])28 b Fo(\024)2234 1549 y Fq(r)2281 1513 y Fm(2)p 2234 1593 87 4 v 2253 1685 a Fu(4)3570 1616 y(\(B.11\))0 1842 y(F)-8 b(rom)31 b(\(B.11\),)i(it)f(follo)m(ws)g (that)g(w)m(e)i(ma)m(y)f(co)m(v)m(er)i Fq(A)1898 1857 y Fm(2)1970 1842 y Fu(with)e(just)g(one)g(ball.)43 b(F)-8 b(rom)31 b(\(B.3\))i(and)g(\(B.4\),)g(it)0 1958 y(follo)m(ws)28 b(that)i Fq(A)599 1973 y Fm(1)668 1958 y Fu(ma)m(y)g(b)s(e)g(co)m(v)m (ered)h(b)m(y)g(\()p Fq(\025)1581 1922 y Fm(7)p Fl(\014)1663 1958 y Fq(r)1710 1922 y Fm(9)1749 1958 y Fu(\))1787 1922 y Fn(\000)p Fm(1)1911 1958 y Fu(balls)e(of)g(radius)g Fq(r)s Fu(.)42 b(Then,)32 b(from)c(\(B.10\))i(it)e(follo)m(ws)0 2074 y(that)427 2329 y Fq(\026)486 2344 y Fl(n)616 2329 y Fo(\024)83 b Fq(K)883 2193 y Fp(Z)983 2219 y Fn(1)938 2419 y Fm(0)1057 2329 y Fq(dr)1171 2239 y Fp(p)p 1271 2239 354 4 v 90 x Fu(log)17 b Fq(N)10 b Fu(\()p Fq(r)s Fu(\))27 b Fo(\024)h Fq(K)1881 2193 y Fp(Z)1980 2163 y Fn(p)p 2039 2163 106 3 v 56 x Fl(C)5 b(T)1936 2419 y Fm(0)2149 2329 y Fq(dr)2263 2239 y Fp(p)p 2363 2239 784 4 v 90 x Fu(log)16 b(\(1)22 b(+)g(\()p Fq(\025)2807 2300 y Fm(7)p Fl(\014)2890 2329 y Fq(r)2937 2300 y Fm(9)2976 2329 y Fu(\))3014 2300 y Fn(\000)p Fm(1)3108 2329 y Fu(\))616 2618 y Fo(\024)83 b Fq(K)7 b(\025)923 2577 y Fn(\000)p Fm(7)p Fl(\014)s(=)p Fm(9)1148 2483 y Fp(Z)1247 2509 y Fn(1)1213 2681 y Fg(\025)1250 2663 y Fh(\000)p Fi(7)p Fg(\014)s(=)p Fi(9)p 1213 2693 219 3 v 1252 2701 a Fh(p)p 1301 2701 92 3 v 1301 2743 a Fg(C)s(T)1463 2618 y Fq(du)1595 2464 y Fp(p)p 1695 2464 483 4 v 85 x Fu(log)17 b(\(1)22 b(+)g Fq(u)2101 2520 y Fm(9)2140 2549 y Fu(\))p 1595 2595 582 4 v 1839 2687 a Fq(u)1895 2658 y Fm(2)2215 2618 y Fo(\024)28 b Fq(K)7 b(\025)2467 2577 y Fn(\000)p Fm(7)p Fl(\014)s(=)p Fm(9)2692 2483 y Fp(Z)2791 2509 y Fn(1)2757 2681 y Fg(\025)2794 2663 y Fh(\000)p Fi(7)p Fg(\014)s(=)p Fi(9)p 2757 2693 219 3 v 2796 2701 a Fh(p)p 2845 2701 92 3 v 2845 2743 a Fg(C)s(T)3007 2618 y Fq(du)3139 2551 y Fu(9)17 b(log)f Fq(u)p 3139 2595 264 4 v 3224 2687 a(u)3280 2658 y Fm(2)616 2883 y Fo(\024)83 b Fq(C)7 b Fu(\()p Fq(\014)f Fu(\))990 2794 y Fo(p)p 1073 2794 71 4 v 89 x Fq(T)13 b Fo(j)k Fu(log)f Fq(\025)p Fo(j)2155 b Fu(\(B.12\))0 3077 y(where)40 b(w)m(e)g(ha)m(v)m(e)g(done)g(the)f(c)m (hange)h(of)e(v)-5 b(ariables)38 b Fq(u)g Fu(=)g Fq(\025)2205 3040 y Fn(\000)p Fm(7)p Fl(\014)s(=)p Fm(9)2413 3077 y Fq(r)2460 3040 y Fn(\000)p Fm(1)2554 3077 y Fu(,)i(and)f Fq(C)7 b Fu(\()p Fq(\014)f Fu(\))38 b(is)h(a)f(constan)m(t)i(that)0 3193 y(dep)s(ends)34 b(on)f Fq(\014)6 b Fu(.)43 b(Then,)34 b(w)m(e)f(ma)m(y)g(apply)g(\(B.9\),)g(to)f(obtain)738 3475 y Fj(P)797 3490 y Fl(n)862 3304 y Fp( )1056 3475 y Fu(sup)941 3561 y Fl(t)p Fn(2)p Fm([)p Fl(T)1074 3569 y Fg(n)1117 3561 y Fl(;T)1178 3570 y Fg(n)p Fi(+1)1297 3561 y Fm(])1333 3475 y Fo(k)p Fq(z)1432 3434 y Fm(\()p Fl(n)p Fm(\))1535 3475 y Fu(\()p Fq(t)p Fu(\))p Fo(k)1696 3490 y Fn(1)1798 3475 y Fq(>)c(\025)1959 3434 y Fn(\000)p Fl(\020)2053 3386 y Fo(p)p 2136 3386 V 89 x Fq(T)2207 3304 y Fp(!)2313 3475 y Fo(\024)h Fu(4)17 b(exp)g(\()p Fo(\000)p Fq(C)7 b(\025)2899 3434 y Fn(\000)p Fl(\020)e(=)p Fm(2)3064 3475 y Fu(\))468 b(\(B.13\))0 3751 y(Finally)-8 b(,)30 b(recall)h(that,)h(giv)m(en)h Fq(x)1159 3766 y Fl(n)1207 3751 y Fu(,)f(the)h(random)f(v)-5 b(ariables)1220 3857 y Fp(\()1414 4027 y Fu(sup)1300 4113 y Fl(t)p Fn(2)p Fm([)p Fl(T)1433 4121 y Fg(n)1476 4113 y Fl(;T)1537 4122 y Fg(n)p Fi(+1)1656 4113 y Fm(])1692 4027 y Fo(k)p Fq(z)1791 3986 y Fm(\()p Fl(n)p Fm(\))1894 4027 y Fu(\()p Fq(t)p Fu(\))p Fo(k)2055 4042 y Fn(1)2129 4027 y Fu(;)50 b Fq(n)28 b Fo(\024)g Fq(n)2455 4042 y Fl(\025\034)2540 3857 y Fp(\))0 4309 y Fu(are)33 b(indep)s(enden)m(t,)h(and)e(the)i(estimate)d (\(B.13\))i(do)s(es)g(not)f(dep)s(end)i(on)f Fq(n)f Fu(\(in)g (particular,)f(it)h(do)s(es)h(not)0 4426 y(dep)s(end)h(on)e Fq(x)529 4441 y Fl(n)577 4426 y Fu(\).)43 b(Then)1107 4607 y Fj(P)1183 4496 y Fp(\020)1244 4607 y Fo(B)1312 4556 y Fm(\(1\))1309 4635 y Fl(\025;\034)1414 4496 y Fp(\021)1501 4607 y Fo(\025)1606 4526 y Fp(\000)1652 4607 y Fu(1)21 b Fo(\000)i Fu(4)17 b(exp)g(\()p Fo(\000)p Fq(C)7 b(\025)2302 4566 y Fn(\000)p Fl(\020)e(=)p Fm(2)2468 4607 y Fu(\))2505 4526 y Fp(\001)2551 4542 y Fl(n)2594 4554 y Fg(\025)2635 4542 y Fm(\()p Fl(\034)j Fm(\))3570 4607 y Fu(\(B.14\))0 4809 y(and)33 b(\(4.6\))f(follo)m(ws.)0 4999 y Ft(Pr)-5 b(o)g(of)40 b(of)61 b Fu(\(4.36\))o Ft(.)g Fu(The)40 b(estimate)e(\(4.36\))f(follo)m(ws)h(b)m(y)h(the)g(same)f (pro)s(cedure)i(used)f(to)g(pro)m(v)m(e)g(\(4.6\).)0 5115 y(W)-8 b(e)33 b(use)h(the)f(notation)e(in)m(tro)s(duced)h(in)g (that)h(pro)s(of.)42 b(Recall)31 b(that)1360 5368 y Fq(z)1409 5326 y Fl(n;)p Fn(?)1531 5368 y Fu(\()p Fq(t)p Fu(\))d(=)1774 5232 y Fp(Z)1873 5258 y Fl(t)1829 5458 y(T)1870 5466 y Fg(n)1917 5368 y Fq(g)1968 5320 y Fl(n;)p Fn(?)1964 5390 y Fl(t)p Fn(\000)p Fl(s)2105 5368 y Fq(dW)2262 5326 y Fm(\()p Fl(a)p Fm(\))2358 5368 y Fu(\()p Fq(s)p Fu(\))1090 b(\(B.15\))1882 5532 y Fk(49)p eop %%Page: 50 51 50 50 bop 0 266 a Fu(and)33 b(de\014ne)1042 456 y Fq(\033)1101 415 y Fm(2)1097 481 y(\()p Fl(n;)p Fn(?)p Fm(\))1357 456 y Fu(=)197 b(sup)1516 542 y Fl(t)p Fn(2)p Fm([)p Fl(T)1649 550 y Fg(n)1691 542 y Fl(;T)1752 551 y Fg(n)p Fi(+1)1871 542 y Fm(])1908 456 y Fu(sup)1914 538 y Fl(x)p Fn(2)p Ff(R)2088 456 y Fj(E)2148 471 y Fl(n)2218 376 y Fp(\000)2263 456 y Fq(z)2312 415 y Fm(\()p Fl(n;)p Fn(?)p Fm(\))2489 456 y Fu(\()p Fq(t;)17 b(x)p Fu(\))2699 415 y Fm(2)2739 376 y Fp(\001)3570 456 y Fu(\(B.16\))1039 780 y Fq(\026)1098 796 y Fm(\()p Fl(n;)p Fn(?)p Fm(\))1357 780 y Fu(=)83 b Fj(E)1576 795 y Fl(n)1646 610 y Fp( )1839 780 y Fu(sup)1724 866 y Fl(t)p Fn(2)p Fm([)p Fl(T)1857 874 y Fg(n)1900 866 y Fl(;T)1961 875 y Fg(n)p Fi(+1)2080 866 y Fm(])2117 780 y Fo(k)p Fq(z)2216 739 y Fm(\()p Fl(n;)p Fn(?)p Fm(\))2392 780 y Fu(\()p Fq(t)p Fu(\))p Fo(k)2553 795 y Fn(1)2628 610 y Fp(!)3570 780 y Fu(\(B.17\))0 1057 y(Then,)351 1320 y Fj(E)411 1335 y Fl(n)481 1239 y Fp(\000)527 1320 y Fq(z)576 1279 y Fm(\()p Fl(n;)p Fn(?)p Fm(\))753 1320 y Fu(\()p Fq(t;)17 b(x)p Fu(\))963 1279 y Fm(2)1002 1239 y Fp(\001)1132 1320 y Fu(=)1292 1185 y Fp(Z)1391 1211 y Fl(t)1347 1410 y(T)1388 1418 y Fg(n)1435 1320 y Fq(ds)1549 1185 y Fp(Z)1647 1320 y Fq(dy)1782 1180 y Fp(\022)1856 1320 y Fq(g)1907 1269 y Fm(\()p Fl(n)p Fm(\))1903 1343 y Fl(t)p Fn(\000)p Fl(s)2020 1320 y Fu(\()p Fq(x;)g(y)t Fu(\))k Fo(\000)2378 1253 y Fu(3)p 2378 1297 49 4 v 2378 1388 a(4)2455 1320 y(\026)-68 b Fq(m)2521 1279 y Fn(0)2521 1345 y Fl(x)2561 1353 y Fg(n)2608 1320 y Fu(\()p Fq(x)p Fu(\))19 b(\026)-68 b Fq(m)2824 1279 y Fn(0)2824 1345 y Fl(x)2864 1353 y Fg(n)2911 1320 y Fu(\()p Fq(y)t Fu(\))3039 1180 y Fp(\023)3111 1202 y Fm(2)3167 1320 y Fq(a)p Fu(\()p Fq(\025)3313 1279 y Fl(\014)3361 1320 y Fq(y)t Fu(\))3451 1279 y Fm(2)1131 1617 y Fo(\024)84 b(k)p Fq(a)p Fo(k)1443 1632 y Fn(1)1534 1481 y Fp(Z)1633 1507 y Fn(1)1589 1707 y Fm(0)1708 1617 y Fq(ds)1822 1481 y Fp(Z)1921 1617 y Fq(dy)2056 1476 y Fp(\022)2129 1617 y Fq(g)2176 1632 y Fl(t)p Fn(\000)p Fl(s)2293 1617 y Fu(\()p Fq(x;)17 b(y)t Fu(\))k Fo(\000)2651 1549 y Fu(3)p 2651 1594 V 2651 1685 a(4)2728 1617 y(\026)-67 b Fq(m)2795 1576 y Fn(0)2818 1617 y Fu(\()p Fq(x)p Fu(\))19 b(\026)-68 b Fq(m)3034 1576 y Fn(0)3058 1617 y Fu(\()p Fq(y)t Fu(\))3186 1476 y Fp(\023)3259 1499 y Fm(2)1132 1893 y Fu(=)84 b Fo(k)p Fq(a)p Fo(k)1443 1908 y Fn(1)1534 1758 y Fp(Z)1633 1784 y Fn(1)1589 1983 y Fm(0)1708 1893 y Fq(ds)1838 1753 y Fp(\022)1911 1893 y Fq(g)1958 1908 y Fm(2)p Fl(t)2023 1893 y Fu(\()p Fq(x;)17 b(x)p Fu(\))22 b Fo(\000)2385 1826 y Fu(3)p 2385 1870 V 2385 1962 a(4)2462 1893 y(\026)-67 b Fq(m)2529 1852 y Fn(0)2552 1893 y Fu(\()p Fq(x)p Fu(\))2683 1852 y Fm(2)2723 1753 y Fp(\023)2824 1893 y Fo(\024)28 b Fq(C)571 b Fu(\(B.18\))0 2145 y(\(see)40 b([8])e(for)g(the)h(last)f(inequalit)m(y\).)61 b(It)39 b(is)f(easy)h(to)g(see)h(that)e(the)h(estimates)f(\(B.3\))h(and)f (\(B.4\))h(also)0 2265 y(hold)32 b(for)g Fq(z)415 2228 y Fm(\()p Fl(n;)p Fn(?)p Fm(\))592 2265 y Fu(.)43 b(Moreo)m(v)m(er,)899 2458 y Fj(E)960 2473 y Fl(n)1029 2377 y Fp(\000)1075 2458 y Fq(z)1124 2417 y Fm(\()p Fl(n;)p Fn(?)p Fm(\))1301 2458 y Fu(\()p Fq(t;)17 b(x)p Fu(\))1511 2417 y Fm(2)1551 2377 y Fp(\001)1624 2458 y Fo(\024)28 b Fq(C)7 b Fj(E)1867 2473 y Fl(n)1936 2377 y Fp(\000)1982 2458 y Fq(z)2031 2417 y Fm(\()p Fl(n)p Fm(\))2134 2458 y Fu(\()p Fq(t;)17 b(x)p Fu(\))2344 2417 y Fm(2)2383 2377 y Fp(\001)2451 2458 y Fu(+)22 b Fq(C)7 b(t)18 b Fu(\026)-67 b Fq(m)2746 2417 y Fn(0)2770 2458 y Fu(\()p Fq(x)p Fu(\))2901 2417 y Fm(2)3570 2458 y Fu(\(B.19\))0 2650 y(\(see)33 b(\(4.34\)\),)f(hence) i(from)d(\(B.11\))h(and)g(recalling)e(that)51 b(\026)-68 b Fq(m)2220 2614 y Fn(0)2276 2650 y Fu(is)32 b(exp)s(onen)m(tially)g (small)e(as)i Fo(j)p Fq(x)p Fo(j)c(!)f(1)p Fu(,)32 b(it)0 2766 y(follo)m(ws)f(that,)i(for)f Fq(\025)g Fu(small)e(enough,)1090 3015 y Fj(E)1151 3030 y Fl(n)1220 2934 y Fp(\000)1266 3015 y Fq(z)1315 2974 y Fm(\()p Fl(n;)p Fn(?)p Fm(\))1492 3015 y Fu(\()p Fq(t;)17 b(x)p Fu(\))1702 2974 y Fm(2)1742 2934 y Fp(\001)1815 3015 y Fo(\024)1930 2947 y Fq(r)1977 2911 y Fm(2)p 1930 2992 87 4 v 1949 3083 a Fu(4)2156 3015 y(for)32 b(\()p Fq(t;)17 b(x)p Fu(\))28 b Fo(2)g Fq(A)2710 3030 y Fm(2)3570 3015 y Fu(\(B.20\))0 3248 y(Then)34 b Fq(N)10 b Fu(\()p Fq(r)s Fu(\))27 b Fo(\024)i Fu(1)22 b(+)g(\()p Fq(\025)863 3212 y Fm(7)p Fl(\014)945 3248 y Fq(r)992 3212 y Fm(9)1031 3248 y Fu(\))1069 3212 y Fn(\000)p Fm(1)1163 3248 y Fu(,)33 b(so)g(pro)s(ceeding)f(as)h(in)f (\(B.12\))o(,)h(but)g(no)m(w)g(with)f Fq(T)41 b Fu(=)28 b(1,)k(w)m(e)i(obtain)1589 3439 y Fq(\026)1648 3454 y Fl(n;)p Fn(?)1797 3439 y Fo(\024)28 b Fq(C)7 b Fo(j)17 b Fu(log)f Fq(\025)p Fo(j)1319 b Fu(\(B.21\))0 3630 y(and)33 b(the)g(pro)s(of)e(\014nishes)j(as)f(that)f(of)39 b(\(4.6\).)0 3820 y Ft(Pr)-5 b(o)g(of)39 b(of)60 b Fu(\(6.4\))p Ft(.)e Fu(Note)38 b(that,)h(conditionally)34 b(on)k(the)g Fq(x)2137 3835 y Fl(k)r Fn(\000)p Fm(1)2270 3820 y Fu('s,)i(the)e(random)e(v)-5 b(ariables)37 b Fo(f)p Fq(\021)3445 3835 y Fl(k)3487 3820 y Fu(\(2\);)c Fq(k)40 b Fo(\024)0 3937 y Fq(n)58 3952 y Fl(\025)104 3937 y Fu(\()p Fq(\034)11 b Fu(\))p Fo(g)40 b Fu(are)g(indep)s(enden)m(t)h(and)f(Gaussian)g(with)f(v)-5 b(ariance)40 b(b)s(ounded)g(b)m(y)h Fq(C)7 b(T)14 b Fu(.)66 b(Therefore)41 b(\(6.4\))e(is)0 4053 y(easily)32 b(deduced)i(from)d (the)i(follo)m(wing)d(elemen)m(tary)j(lemma.)0 4206 y Fv(Lemma)40 b(B.1.)j Ft(L)-5 b(et)37 b Fo(f)p Fq(!)922 4221 y Fl(h)966 4206 y Fq(;)d(h)e Fu(=)f(1)p Fq(;)17 b(:)g(:)g(:)f(;)h(N)10 b Fo(g)37 b Ft(b)-5 b(e)37 b(me)-5 b(an)36 b(zer)-5 b(o)37 b(i.i.d.)f(r)-5 b(andom)36 b(variables)g(and)h (set)g Fq(Y)3659 4221 y Fl(n)3737 4206 y Fu(:=)0 4251 y Fp(P)105 4277 y Fl(n)105 4355 y(h)p Fm(=1)240 4326 y Fu(\()p Fq(n)24 b Fo(\000)h Fq(h)f Fu(+)g(1\))729 4289 y Fn(\000)p Fm(1)p Fl(=)p Fm(2)894 4326 y Fq(!)955 4341 y Fl(h)1000 4326 y Ft(.)53 b(Assume)37 b(that)h(for)f(e)-5 b(ach)37 b Fq(p)c Fo(2)g Fu([1)p Fq(;)17 b Fo(1)p Fu(\))37 b Ft(we)g(have)g Fv(E)p Fo(j)p Fq(!)3044 4341 y Fl(i)3071 4326 y Fo(j)3099 4289 y Fl(p)3171 4326 y Fq(<)c Fo(1)p Ft(.)52 b(Then)37 b(for)0 4442 y(e)-5 b(ach)34 b Fq(\020)h(>)28 b Fu(0)34 b Ft(and)g Fq(q)e(<)c Fo(1)34 b Ft(ther)-5 b(e)35 b(exists)f(a)h(c)-5 b(onstant)35 b Fq(C)f Fu(=)28 b Fq(C)7 b Fu(\()p Fq(\020)h(;)17 b(q)t Fu(\))33 b Ft(such)i(that)1307 4694 y Fv(P)1401 4553 y Fp(\022)1480 4694 y Fu(sup)1473 4775 y Fl(n)p Fn(\024)p Fl(N)1650 4694 y Fo(j)p Fq(Y)1735 4709 y Fl(n)1782 4694 y Fo(j)27 b Fq(>)h(N)2029 4653 y Fl(\020)2069 4553 y Fp(\023)2170 4694 y Fo(\024)g Fq(C)7 b(N)2440 4653 y Fn(\000)p Fl(q)3570 4694 y Fu(\(B.22\))0 4980 y Ft(Pr)-5 b(o)g(of.)121 b Fu(Let)44 b Fq(Y)642 4929 y Fm(\()p Fl(n)p Fm(\))621 5008 y Fl(k)791 4980 y Fu(:=)940 4905 y Fp(P)1045 4932 y Fl(k)1045 5009 y(h)p Fm(=1)1180 4980 y Fu(\()p Fq(n)30 b Fo(\000)h Fq(h)f Fu(+)f(1\))1692 4944 y Fn(\000)p Fm(1)p Fl(=)p Fm(2)1857 4980 y Fq(!)1918 4995 y Fl(h)1963 4980 y Fu(,)46 b Fq(k)k Fu(=)d(1)p Fq(;)17 b(:)g(:)g(:)e(;)i(n)p Fu(,)47 b(whic)m(h)d(is)g(a)f (martingale)e(with)0 5096 y(quadratic)32 b(v)-5 b(ariation)1359 5338 y([)p Fq(Y)1465 5297 y Fm(\()p Fl(n)p Fm(\))1567 5338 y Fu(])1594 5353 y Fl(k)1664 5338 y Fu(=)1820 5214 y Fl(k)1767 5244 y Fp(X)1774 5456 y Fl(h)p Fm(=1)2116 5271 y Fu(1)p 1938 5315 406 4 v 1938 5407 a Fq(n)22 b Fo(\000)h Fq(h)f Fu(+)g(1)2375 5338 y Fq(!)2440 5297 y Fm(2)2436 5363 y Fl(h)1882 5532 y Fk(50)p eop %%Page: 51 52 51 51 bop 0 266 a Fu(hence,)27 b(b)m(y)d(the)g(BDG)e(inequalit)m(y)g (\(see)i([16,)h Fo(x)p Fu(6,)g(E.)f(4.1]\),)h(for)e(eac)m(h)h Fq(p)k Fo(2)g Fu([1)p Fq(;)17 b Fo(1)p Fu(\))22 b(there)i(exists)g Fq(C)35 b Fu(=)27 b Fq(C)7 b Fu(\()p Fq(p)p Fu(\))0 382 y(suc)m(h)34 b(that)175 634 y Fv(E)17 b Fu(\()o Fo(j)p Fq(Y)388 649 y Fl(n)434 634 y Fo(j)462 593 y Fl(p)501 634 y Fu(\))68 b(=)f Fv(E)841 553 y Fp(\000)885 634 y Fo(j)p Fq(Y)991 593 y Fm(\()p Fl(n)p Fm(\))970 659 y Fl(n)1093 634 y Fo(j)1121 593 y Fl(p)1161 553 y Fp(\001)1234 634 y Fo(\024)28 b Fq(C)c Fv(E)1524 523 y Fp(\020)1582 553 y(\000)1628 634 y Fu([)p Fq(Y)1733 593 y Fm(\()p Fl(n)p Fm(\))1835 634 y Fu(])1862 649 y Fl(n)1909 553 y Fp(\001)1955 569 y Fl(p=)p Fm(2)2065 523 y Fp(\021)2152 634 y Fo(\024)29 b Fq(C)23 b Fv(E)2442 464 y Fp( )2571 509 y Fl(n)2520 539 y Fp(X)2527 751 y Fl(h)p Fm(=1)2869 566 y Fu(1)p 2691 611 406 4 v 2691 702 a Fq(n)f Fo(\000)h Fq(h)f Fu(+)g(1)3128 634 y Fq(!)3193 593 y Fm(2)3189 659 y Fl(h)3233 464 y Fp(!)3312 486 y Fl(p=)p Fm(2)606 1000 y Fo(\024)67 b Fq(C)843 829 y Fp(")952 875 y Fl(n)901 905 y Fp(X)908 1117 y Fl(h)p Fm(=1)1250 932 y Fu(1)p 1072 977 V 1072 1068 a Fq(n)22 b Fo(\000)h Fq(h)f Fu(+)g(1)1526 1000 y(\()o Fv(E)p Fo(j)p Fq(!)1726 1015 y Fl(h)1770 1000 y Fo(j)1798 958 y Fl(p)1838 1000 y Fu(\))1876 951 y Fm(2)p Fl(=p)1986 829 y Fp(#)2044 851 y Fl(p=)p Fm(2)2182 1000 y Fo(\024)28 b Fq(C)2381 829 y Fp( )2465 875 y Fl(n)p Fn(\000)p Fm(1)2459 905 y Fp(X)2467 1117 y Fl(k)r Fm(=0)2717 932 y Fu(1)p 2630 977 224 4 v 2630 1068 a Fq(k)d Fu(+)d(1)2863 829 y Fp(!)2942 849 y Fl(p=)p Fm(2)3080 1000 y Fo(\024)28 b Fq(C)23 b Fu(\(log)17 b Fq(n)p Fu(\))3555 951 y Fl(p=)p Fm(2)0 1256 y Fu(By)33 b(using)h(\(7.17\))e(w)m(e)h(th)m(us)h(get)579 1515 y Fv(P)673 1374 y Fp(\022)753 1515 y Fu(sup)746 1596 y Fl(n)p Fn(\024)p Fl(N)923 1515 y Fo(j)p Fq(Y)1008 1530 y Fl(n)1054 1515 y Fo(j)28 b Fq(>)f(N)1301 1474 y Fn(\000)p Fl(\020)1397 1374 y Fp(\023)1498 1515 y Fo(\024)h Fq(N)1691 1474 y Fn(\000)p Fl(\020)5 b(p)1878 1390 y(N)1838 1420 y Fp(X)1843 1629 y Fl(n)p Fm(=1)1998 1515 y Fv(E)17 b Fu(\()p Fo(j)p Fq(Y)2212 1530 y Fl(n)2258 1515 y Fo(j)2286 1474 y Fl(p)2325 1515 y Fu(\))28 b Fo(\024)g Fq(C)7 b(N)2661 1474 y Fm(1)p Fn(\000)p Fl(\020)e(p)2843 1515 y Fu(\(log)17 b Fq(N)10 b Fu(\))3150 1466 y Fl(p=)p Fm(2)0 1763 y Fu(taking)32 b Fq(p)g Fu(large)g(enough)h(the)g(lemma)d(follo)m(ws.)2008 b Fd(\003)1453 2351 y Fr(A)m(ckno)n(wledgments)100 2526 y Fu(S.)34 b(Brassesco)i(ac)m(kno)m(wledges)g(the)f(v)m(ery)h(kind)e (hospitalit)m(y)f(of)h(the)g(Departmen)m(t)h(of)e(Mathematics)0 2642 y(of)39 b(the)i(Univ)m(ersit)m(y)f(of)g(Rome)f(T)-8 b(or)39 b(V)-8 b(ergata.)65 b(P)m(art)40 b(of)g(this)f(w)m(ork)i(w)m (as)g(done)f(while)f(P)-8 b(.)40 b(Butt\022)-49 b(a)40 b(w)m(as)0 2758 y(visiting)c(that)h(Departmen)m(t)g(with)g(a)g(F)-8 b(ello)m(wship)36 b(in)g(Mathematics)h(of)g(the)h(\\Istituto)f (Nazionale)f(di)0 2874 y(Alta)31 b(Matematica)h(F)-8 b(rancesco)33 b(Sev)m(eri".)1630 3463 y Fr(References)42 3692 y FA([1])41 b(R.J.)g(Adler)g(:)106 b Fc(A)n(n)42 b(Intr)l(o)l(duction)g(to)g(Continuity,)47 b(Extr)l(ema)42 b(and)h(R)l(elate)l(d)f(T)-6 b(opics)44 b(for)f(Gener)l(al)g(Gaussian) 171 3791 y(Pr)l(o)l(c)l(esses.)56 b FA(Lecture)27 b(Notes)g(Monograph)f (series)h Fb(12)p FA(,)g(Institute)h(of)g(Mathematical)f(Statistics)h (1990.)42 3962 y([2])41 b(L.)f(Bertini,)j(S.)d(Brassesco,)g(P)-7 b(.)40 b(Butt\022)-42 b(a,)43 b(E.)d(Presutti)f(:)102 b Fc(Sto)l(chastic)42 b(phase)g(\014eld)g(e)l(quations:)62 b(existenc)l(e)40 b(and)171 4061 y(uniqueness.)27 b FA(Preprin)n(t.)42 4232 y([3])41 b(L.)27 b(Bertini,)g(P)-7 b(.)27 b(Butt\022)-42 b(a,)27 b(E.)g(Presutti,)f(E.)h(Saada)f(:)64 b Fc(Interfac)l(e)29 b(\015uctuations)g(in)g(a)h(c)l(onserve)l(d)g(system:)38 b(derivation)171 4331 y(and)30 b(long)g(time)g(b)l(ehavior.)g FA(Preprin)n(t.)42 4502 y([4])41 b(L.)25 b(Bertini,)g(P)-7 b(.)25 b(Butt\022)-42 b(a,)26 b(B.)f(R)r(\177)-44 b(udiger)25 b(:)61 b Fc(Interfac)l(e)28 b(dynamics)h(and)f(Stefan)f(pr)l(oblem)i (fr)l(om)f(a)f(micr)l(osc)l(opic)j(c)l(onser-)171 4602 y(vative)g(mo)l(del.)f FA(Rend.)f(Mat.)g(Appl.)g(\(7\))g Fb(19)f FA(\(1999\),)g(547{581.)42 4772 y([5])41 b(L.)18 b(Bertini,)j(E.)d(Presutti,)i(B.)f(R)r(\177)-44 b(udiger,)20 b(E.)e(Saada)f(:)52 b Fc(Dynamic)l(al)22 b(\015uctuations)f(at)g(the)h (critic)l(al)h(p)l(oint:)35 b(c)l(onver)l(genc)l(e)171 4872 y(to)28 b(a)h(nonline)l(ar)f(sto)l(chastic)i(PDE.)53 b FA(T)-7 b(eor.)25 b(V)-7 b(ero)n(y)n(atnost.)24 b(i)i(Primenen.)g Fb(38)f FA(\(1993\),)g(no.)h(4,)g(689{741;)d(translation)171 4971 y(in)28 b(Theory)e(Probab.)g(Appl.)i Fb(38)g FA(\(1993\),)e(no.)h (4,)g(586{629.)42 5142 y([6])41 b(P)-7 b(.)27 b(Billingsley)g(:)64 b Fc(Conver)l(genc)l(e)31 b(of)g(Pr)l(ob)l(ability)g(Me)l(asur)l(es.)56 b FA(New)28 b(Y)-7 b(ork:)36 b(Wiley)28 b(1968.)42 5312 y([7])41 b(S.)32 b(Brassesco)d(:)76 b Fc(Stability)34 b(of)h(the)e(Instanton)g(under)g(smal)t(l)h(r)l(andom)g(p)l(erturb)l (ations.)64 b FA(Sto)r(c)n(h.)32 b(Pro)r(c.)e(Appl.)j Fb(54)171 5412 y FA(\(1994\),)26 b(309{330.)1882 5532 y Fk(51)p eop %%Page: 52 53 52 52 bop 42 266 a FA([8])41 b(S.)23 b(Brassesco,)f(P)-7 b(.)23 b(Butt\022)-42 b(a)23 b(:)59 b Fc(Interfac)l(e)26 b(\015uctuations)f(for)i(the)f(D=1)f(Sto)l(chastic)i(Ginzbur)l(g{L)l (andau)f(e)l(quation)g(with)171 365 y(non{symmetric)k(r)l(e)l(action)g (term.)56 b FA(J.)27 b(Statist.)h(Ph)n(ys.)f Fb(93)g FA(\(1998\),)f(1111{1142.)42 536 y([9])41 b(S.)31 b(Brassesco,)f(P)-7 b(.)31 b(Butt\022)-42 b(a,)32 b(A.)f(De)h(Masi,)g(E.)e(Presutti)h(:)75 b Fc(Interfac)l(e)34 b(\015uctuations)e(and)i(c)l(ouplings)f(in)g(the)h Fa(d)29 b FA(=)f(1)171 635 y Fc(Ginzbur)l(g{L)l(andau)i(e)l(quation)g (with)g(noise.)57 b FA(J.)27 b(Theoret.)g(Probab.)f Fb(11)h FA(\(1998\),)g(25{80.)0 806 y([10])41 b(S.)28 b(Brassesco,)d(A.)k(De)f (Masi,)f(E.)h(Presutti)f(:)66 b Fc(Br)l(ownian)31 b(\015uctuations)e (of)i(the)f(interfac)l(e)h(in)f(the)g(d=1)h(Ginzbur)l(g{)171 906 y(L)l(andau)f(e)l(quation)g(with)g(noise.)57 b FA(Annal.)27 b(Inst.)h(H.)g(P)n(oincar)n(\023)-39 b(e)25 b Fb(31)i FA(\(1995\),)f(81{118.)0 1076 y([11])41 b(G.)27 b(Caginalp,)e(X.)i (Chen)g(:)64 b Fc(Conver)l(genc)l(e)29 b(of)h(the)f(phase)h(\014eld)g (mo)l(del)f(to)g(its)g(sharp)h(interfac)l(e)g(limits.)54 b FA(Europ)r(ean)171 1176 y(J.)27 b(Appl.)h(Math.)g Fb(9)g FA(\(1998\),)e(417{445.)0 1346 y([12])41 b(A.)27 b(De)f(Masi,)h(P)-7 b(.)26 b(F)-7 b(errari,)25 b(J.L.)h(Leb)r(o)n(witz)g(:)63 b Fc(R)l(e)l(action{di\013usion)30 b(e)l(quations)f(for)h(inter)l (acting)f(p)l(article)h(systems.)171 1446 y FA(J.)d(Statist.)h(Ph)n (ys.)f Fb(44)g FA(\(1986\),)f(589{644.)0 1616 y([13])41 b(T.)23 b(F)-7 b(unaki)24 b(:)58 b Fc(The)27 b(sc)l(aling)g(limit)f (for)h(a)f(sto)l(chastic)h(PDE)f(and)h(the)f(sep)l(ar)l(ation)h(of)g (phases.)48 b FA(Prob.)23 b(Theory)f(Relat.)171 1716 y(Fields)27 b Fb(102)g FA(\(1995\))g(221{288.)0 1886 y([14])41 b(P)-7 b(.C.)24 b(Hohen)n(b)r(erg,)h(B.I.)g(Halp)r(erin)f(:) 61 b Fc(The)l(ory)29 b(of)f(dynamic)g(critic)l(al)h(phenomena.)51 b FA(Rev.)25 b(Mo)r(d.)g(Ph)n(ys.)f Fb(49)g FA(\(1977\),)171 1986 y(435{479.)0 2157 y([15])41 b(P)-7 b(.L.)34 b(Lions,)i(P)-7 b(.E.)34 b(Souganidis)g(:)86 b Fc(F)-6 b(ul)t(ly)36 b(nonline)l(ar)h (sto)l(chastic)g(p)l(artial)h(di\013er)l(ential)f(e)l(quations:)52 b(non{smo)l(oth)171 2256 y(e)l(quations)30 b(and)g(applic)l(ations.)58 b FA(C.)28 b(R.)g(Acad.)f(Sci.)h(P)n(aris)e(S)n(\023)-39 b(er.)26 b(I)i(Math.)f Fb(327)g FA(\(1998\),)g(735{741.)0 2427 y([16])41 b(M.)27 b(M)n(\023)-39 b(etivier)27 b(:)64 b Fc(Semimartingales:)40 b(A)30 b(Course)g(on)g(Sto)l(chastic)g(Pr)l(o) l(c)l(esses.)56 b FA(Berlin:)37 b(de)28 b(Gruyter)f(1982.)0 2597 y([17])41 b(H.M.)31 b(Soner)f(:)75 b Fc(Conver)l(genc)l(e)34 b(of)g(the)f(phase{\014eld)i(e)l(quations)e(to)g(the)g(Mul)t (lins{Sekerka)i(pr)l(oblem)f(with)f(kinetic)171 2697 y(under)l(c)l(o)l(oling.)56 b FA(Arc)n(h.)27 b(Rational)g(Mec)n(h.)h (Anal.)f Fb(131)g FA(\(1995\),)g(139{197.)0 2867 y([18])41 b(F.G.)28 b(T)-7 b(ricomi)27 b(:)64 b Fc(Inte)l(gr)l(al)30 b(Equations.)56 b FA(New)28 b(Y)-7 b(ork:)36 b(Do)n(v)n(er)26 b(1985.)1882 5532 y Fk(52)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF