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Fx(,)g(the)g(pro)s(cess)g(w)m(aits)50 3778 y(an)h(exp)s(onen)m(tial)g(time)g(of)g(mean)h Fn(\034)1272 3792 y Fv(0)1327 3778 y Fx(exp\()p Fn(\013E)1626 3792 y Fl(i)1657 3778 y Fx(\))g(\(where)f Fn(\013)h Fx(has)f(the)h(ph)m (ysical)e(meaning)h(of)g Fn(\013)e Fx(=)f Fn(\014)5 b(=\014)3611 3792 y Fl(c)3649 3778 y Fx(\),)50 3923 y(and)41 b(then)h(jumps)f(at)i (random)e(to)i(an)m(y)g(of)f(the)h(other)f(states)i Fn(j)k Fx(with)40 b(equal)i(probabilit)m(y)-8 b(.)75 b(Here)42 b Fn(\034)3669 3937 y Fv(0)50 4069 y Fx(denotes)c(a)g(time-scale)g (that)g(will)d(ha)m(v)m(e)k(to)f(b)s(e)g(c)m(hosen)g(appropriately)e (later.)62 b(The)37 b(problem)f(is)h(that)50 4214 y(while)c(w)m(e)j(w)m (ould)e(w)m(an)m(t)i(the)f(pro)s(cess)g(to)h(reac)m(h)g(eac)m(h)g (state)h(with)c(equal)i(probabilit)m(y)-8 b(,)35 b(this)f(mak)m(es)i (no)50 4360 y(sense)h(giv)m(en)h(that)g(there)f(are)h(in\014nitely)d (man)m(y)i(states.)63 b(Th)m(us)37 b(w)m(e)h(ha)m(v)m(e)g(to)h(in)m (tro)s(duce)d(some)i(cut-o\013)50 4505 y(pro)s(cedure.)45 b(Bouc)m(haud)32 b(prop)s(oses)f(to)i(allo)m(w)f(jumps)e(only)h(to)i (the)g Fn(M)42 b Fx(states)33 b(of)f(largest)h(mass,)f(and)g(to)50 4650 y(tak)m(e)k(the)f(limit)e(as)i Fn(M)43 b Fm(")33 b(1)i Fx(in)f(the)h(end.)53 b(W)-8 b(e)36 b(\014nd)e(it)g(more)h (instructiv)m(e)f(to)i(restrict)e(our)h(pro)s(cess)f(to)50 4796 y(states)25 b(whose)g(energy)f(is)g(larger)g(than)g Fn(E)5 b Fx(,)26 b(where)e Fn(E)30 b Fx(is)24 b(a)g(parameter)h(that)g (will)d(b)s(e)i(tak)m(en)i(to)f Fm(\0001)f Fx(later)3644 4763 y Fv(7)3685 4796 y Fx(.)p 50 4920 546 4 v 159 4982 a Fs(5)197 5007 y Fr(W)-6 b(e)28 b(ha)n(v)n(e)f(c)n(hosen)i(this)g (particular)f(dynamics)g(for)f(tec)n(hnical)j(reasons.)42 b(T)-6 b(o)26 b(study)j(e.g.)41 b(the)29 b(Metrop)r(olis)f(algorithm)50 5098 y(w)n(ould)d(require)i(some)f(extra)f(w)n(ork,)h(but)f(w)n(e)h (exp)r(ect)h(essen)n(tially)h(the)e(same)g(results)h(to)f(hold.)159 5163 y Fs(6)197 5189 y Fr(This)j(is)h(completely)f(analogous)f(to)h (the)g(pro)r(cedure)g(of)f(Ruelle)i(to)f(de\014ne)g(a)g(mo)r(del)f (based)h(on)g(P)n(oisson)f(pro)r(cess)j(as)50 5280 y(the)25 b(in\014nite)h(v)n(olume)f(v)n(ersion)h(of)e(Derrida's)j(REM)d(rather)h (then)h(pro)n(ving)e(the)h(con)n(v)n(ergence)h(of)e(Derrida's)j(mo)r (del)e(to)g(this)50 5371 y(limit.)159 5436 y Fs(7)197 5462 y Fr(This)c(has)f(the)g(adv)l(an)n(tage)f(that)h(via)g(the)g (parameter)g Fg(E)k Fr(w)n(e)c(con)n(trol)h(explicitly)h(the)e (time-scale)i(w)n(e)e(consider,)j(whereas)p eop %%Page: 4 4 4 3 bop 50 295 a Fr(4)3309 b Fo(Se)l(ction)27 b Fr(1)50 513 y Fx(This)h(is)i(v)m(ery)g(con)m(v)m(enien)m(t,)i(since)e(it)g (amoun)m(ts)g(to)i(replace)e(the)g(P)m(oisson)g(pro)s(cess)g Fm(P)38 b Fx(\(from)30 b(Section)h(1.2)50 658 y(of)i([BBG1]\))i(b)m(y)d (its)g(restriction)g Fm(P)1278 672 y Fl(E)1372 658 y Fx(to)h(the)g(half)e(line)g([)p Fn(E)5 b(;)15 b Fm(1)p Fx(\).)49 b(Since)31 b(our)i(new)f(P)m(oisson)g(pro)s(cess)g(has)50 804 y(a)k(\014nite)f(in)m(tensit)m(y)h(measure,)i(it)d(has)h(a)h(v)m (ery)f(useful)f(represen)m(tation:)52 b(Consider)34 b(a)j(random)e(v)-5 b(ariable)50 949 y Fn(n)105 963 y Fl(E)203 949 y Fm(2)36 b Fj(N)50 b Fx(that)38 b(is)e(P)m(oisson)h(distributed)e(with)h (parameter)2169 876 y Fi(R)2229 898 y Fk(1)2212 981 y Fl(E)2322 949 y Fn(e)2364 916 y Fk(\000)p Fl(x)2466 949 y Fn(dx)i Fx(=)e Fn(e)2752 916 y Fk(\000)p Fl(E)2871 949 y Fx(.)61 b(Let)38 b Fn(E)3194 963 y Fl(i)3225 949 y Fn(;)15 b(i)37 b Fm(2)g Fj(N)50 b Fx(b)s(e)36 b(a)50 1095 y(family)d(of)j(i.i.d.)53 b(real)34 b(v)-5 b(alued)35 b(r.v.,)h(indep)s(enden)m(t)d(of)i Fn(n)2022 1109 y Fl(E)2118 1095 y Fx(whose)g(common)g(distribution)d(has)i(densit)m(y)50 1240 y Fn(e)92 1207 y Fl(E)154 1240 y Fn(e)196 1207 y Fk(\000)p Fl(x)298 1240 y Fx(1)-23 b(I)354 1254 y Fl(x)p Fk(\025)p Fl(E)544 1240 y Fx(with)29 b(resp)s(ect)h(to)h(Leb)s(esgue)g (measure.)40 b(Then)30 b Fm(P)2257 1254 y Fl(E)2348 1240 y Fx(is)g(equal)g(in)f(distribution)e(to)1760 1417 y Fl(n)1805 1426 y Fp(E)1743 1445 y Fi(X)1750 1639 y Fl(i)p Fv(=1)1889 1531 y Fn(\016)1929 1545 y Fl(E)1982 1554 y Fp(i)3523 1531 y Fx(\(1)p Fn(:)p Fx(3\))50 1826 y(Giv)m(en)38 b(a)h(realization)f(of)h Fm(P)1033 1840 y Fl(E)1094 1826 y Fx(,)i(w)m(e)e(can)g(no)m(w)f(de\014ne)g(a)h(Mark)m(o)m(v)h(pro)s (cess)e(on)g(the)h(random)f(set)h Fm(S)3539 1840 y Fl(E)3639 1826 y Fm(\021)50 1971 y(f)p Fx(1)p Fn(;)15 b(:)g(:)g(:)i(;)e(M)10 b Fx(\()p Fn(E)5 b Fx(\))p Fm(g)p Fx(.)42 b(Let)28 b Fn(Y)907 1985 y Fl(E)968 1971 y Fx(\()p Fn(n)p Fx(\))p Fn(;)15 b(n)26 b Fm(2)f Fj(N)40 b Fx(b)s(e)27 b(an)g(discrete)g(time)h (Mark)m(o)m(v)h(c)m(hain)e(with)f(state)j(space)f Fm(S)3457 1985 y Fl(E)3519 1971 y Fx(.)40 b(W)-8 b(e)50 2117 y(will)24 b(actually)j(only)f(consider)g(the)h(case)h(where)e Fn(Y)1762 2131 y Fl(E)1823 2117 y Fx(\()p Fn(n)p Fx(\))i(are)f(i.i.d.)38 b(random)26 b(v)-5 b(ariables)26 b(with)f(some)j(distri-)50 2262 y(bution)h Fn(q)s Fx(.)42 b(Next)31 b(w)m(e)g(in)m(tro)s(duce,)f (for)h(eac)m(h)h Fn(i)26 b Fm(2)f Fj(N)7 b Fx(,)37 b(a)31 b(family)e Fn(T)2252 2276 y Fl(n)2302 2262 y Fx(\()p Fn(i)p Fx(\))p Fn(;)15 b(n)26 b Fm(2)g Fj(N)43 b Fx(of)31 b(i.i.d.)40 b(random)30 b(v)-5 b(ariables)50 2407 y(taking)30 b(v)-5 b(alues)30 b(in)f Fj(R)767 2421 y Fv(+)863 2407 y Fx(and)h(ha)m(ving)g(an)g(exp)s(onen)m(tial)g(distribution)c(with)j (rates)i Fn(\034)2912 2421 y Fl(i)2968 2407 y Fm(\021)25 b Fn(\034)3104 2421 y Fv(0)3159 2407 y Fx(exp\()p Fn(\013E)3458 2421 y Fl(i)3489 2407 y Fx(\),)31 b(i.e.)1229 2659 y Fn(P)13 b Fx([)p Fn(T)1378 2673 y Fl(n)1428 2659 y Fx(\()p Fn(i)p Fx(\))26 b Fm(\024)f Fn(t)p Fx(])g Fm(\021)g Fn(F)1888 2673 y Fl(i)1919 2659 y Fx(\()p Fn(t)p Fx(\))h(=)f(1)20 b Fm(\000)g Fn(e)2342 2621 y Fk(\000)p Fl(t=\034)2497 2630 y Fp(i)3523 2659 y Fx(\(1)p Fn(:)p Fx(4\))50 2910 y(No)m(w)31 b(w)m(e)g(set)1484 3069 y Fn(R)1553 3083 y Fl(n)1628 3069 y Fm(\021)1768 2955 y Fl(n)1725 2983 y Fi(X)1724 3178 y Fl(k)r Fv(=1)1872 3069 y Fn(T)1925 3083 y Fl(k)1969 3069 y 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Fn(p)34 b Fx(satisfying)540 445 y Fi(P)636 467 y Fl(n)681 476 y Fp(E)636 540 y Fl(i)p Fv(=1)773 513 y Fn(p)p Fx(\()p Fn(\034)894 527 y Fl(i)925 513 y Fx(\))f(=)f(1,)37 b(the)e(la)m(w)g(of)g(the)g(renew)m(al)g(v)-5 b(ariable)34 b Fn(T)2537 527 y Fl(k)2581 513 y Fx(\()p Fn(Y)2669 527 y Fl(E)2731 513 y Fx(\()p Fn(k)s Fx(\)\))i(can)f(b)s(e)g (expressed)f(in)50 658 y(terms)c(of)h(the)f(pro)s(cess)g Fm(N)968 625 y Fk(\003)955 683 y Fl(E)1046 658 y Fx(as)468 919 y Fn(P)13 b Fx([)p Fn(T)617 933 y Fl(k)662 919 y Fx(\()p Fn(Y)750 933 y Fl(E)811 919 y Fx(\()p Fn(k)s Fx(\)\))27 b Fn(>)e(t)p Fx(])g Fm(\021)g Fx(1)c Fm(\000)f Fn(F)1483 933 y Fl(E)1545 919 y Fx(\()p Fn(t)p Fx(\))25 b(=)1786 804 y Fl(n)1831 813 y Fp(E)1769 832 y Fi(X)1776 1026 y Fl(i)p Fv(=1)1916 919 y Fn(q)1957 933 y Fl(i)1987 919 y Fx(\(1)c Fm(\000)f Fn(F)2237 933 y Fl(i)2267 919 y Fx(\()p Fn(t)p Fx(\)\))26 b(=)2527 795 y Fi(Z)2633 919 y Fm(N)2721 881 y Fk(\003)2708 941 y Fl(E)2769 919 y Fx(\()p Fn(ds)p Fx(\))p Fn(p)p Fx(\()p Fn(s)p 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b(w)m(e)g(assume)f(that)h(the)f(initial)e (distribution)e(of)43 b(the)f(c)m(hain)50 1915 y(coincides)29 b(with)g(the)i(jump)e(distribution,)e(i.e.,)j Fn(P)13 b Fx(\()p Fn(X)1906 1929 y Fl(E)1968 1915 y Fx(\(0\))27 b(=)e Fn(i)p Fx(\))h(=)f Fn(p)p Fx(\()p Fn(\034)2515 1929 y Fl(i)2545 1915 y Fx(\).)159 2133 y(The)33 b(follo)m(wing)f (theorem)i(paraphrases)f(the)h(results)e(on)h(the)h(asymptotic)g(b)s (eha)m(viour)e(for)i(this)e(cor-)50 2278 y(relation)d(function)g(as)i (found)e(b)m(y)h(Bouc)m(haud)h(and)f(Dean)h([BD])50 2496 y Fd(Prop)s(osition)38 b(1.1:)p Ft(De\014ne)1043 2729 y Fn(H)1119 2743 y Fv(0)1160 2729 y Fx(\()p Fn(w)r Fx(\))26 b Fm(\021)1657 2667 y Fx(1)p 1430 2708 500 4 v 1430 2791 a Fn(\031)s Ft(c)-5 b(ose)g(c)16 b Fx(\()p Fn(\031)s(=\013)p Fx(\))1956 2605 y Fi(Z)2047 2627 y Fk(1)2006 2812 y Fl(w)2139 2729 y Fn(dx)2455 2667 y Fx(1)p 2249 2708 457 4 v 2249 2793 a(\(1)21 b(+)f Fn(x)p Fx(\))p Fn(x)2580 2766 y Fv(1)p Fl(=\013)3477 2729 y Fx(\(1)p Fn(:)p Fx(10\))50 2963 y Ft(Then,)33 b(for)g Fn(\013)26 b(>)f Fx(0)p Ft(,)1322 3119 y Fx(lim)1274 3179 y Fl(E)s Fk(#\0001)1523 3058 y Fx(\005)1591 3072 y Fl(E)1653 3058 y Fx(\()p Fn(s;)15 b(t)p Fx(\))p 1523 3098 317 4 v 1527 3182 a Fn(H)1603 3196 y Fv(0)1644 3182 y Fx(\()p Fn(s=t)p Fx(\))1875 3119 y(=)25 b(1)p Fn(;)109 b(P)13 b Ft(-a.s.)1084 b Fx(\(1)p Fn(:)p Fx(11\))50 3330 y Ft(Mor)-5 b(e)g(over,)38 b(the)f(asymptotic)h (b)-5 b(ehavior)38 b(of)e Fn(H)1647 3344 y Fv(0)1688 3330 y Fx(\()p Fn(t=s)p Fx(\))h Ft(when)g Fn(s=t)f Ft(tends)h(to)g(zer) -5 b(o)37 b(or)g Fm(1)p Ft(,)g(r)-5 b(esp)g(e)g(ctively,)39 b(is)50 3476 y(r)-5 b(e)g(adily)34 b(evaluate)-5 b(d:)10 3694 y(\(i\))47 b(If)32 b Fx(\()p Fn(s=t)p Fx(\))26 b Fm(#)g Fx(0)p Ft(,)275 3950 y Fn(H)351 3964 y Fv(0)392 3950 y Fx(\()p Fn(s=t)p Fx(\))g(=)f(1)20 b Fm(\000)1099 3888 y Fx(1)p 872 3929 500 4 v 872 4012 a Fn(\031)s Ft(c)-5 b(ose)g(c)16 b Fx(\()p Fn(\031)s(=\013)p Fx(\))1398 3826 y Fi(Z)1489 3849 y Fl(s=t)1448 4033 y Fv(0)1607 3950 y Fn(dx)1923 3888 y Fx(1)p 1717 3929 457 4 v 1717 4014 a(\(1)22 b(+)e Fn(x)p Fx(\))p Fn(x)2049 3988 y Fv(1)p Fl(=\013)2210 3950 y Fm(\030)25 b Fx(1)c Fm(\000)2707 3888 y Fx(\()p Fn(s=t)p Fx(\))2898 3855 y Fv(1)p Fk(\000)p Fv(1)p Fl(=\013)p 2474 3929 877 4 v 2474 4012 a Fx(\(1)g Fm(\000)f Fx(1)p Fn(=\013)p Fx(\))p Fn(\031)s Ft(c)-5 b(ose)g(c)17 b Fx(\()p Fn(\031)s(=\013)p Fx(\))3477 3950 y(\(1)p Fn(:)p Fx(12\))-18 4262 y Ft(\(ii\))47 b(If)32 b Fx(\()p Fn(s=t)p Fx(\))26 b Fm(")g(1)p Ft(,)733 4511 y Fn(H)809 4525 y Fv(0)850 4511 y Fx(\()p Fn(s=t)p Fx(\))f Fm(\030)1400 4449 y Fx(1)p 1173 4490 500 4 v 1173 4573 a Fn(\031)s Ft(c)-5 b(ose)g(c)16 b Fx(\()p Fn(\031)s(=\013)p Fx(\))1699 4387 y Fi(Z)1790 4410 y Fk(1)1750 4594 y Fl(s=t)1882 4511 y Fn(dx)2104 4449 y Fx(1)p 1992 4490 270 4 v 1992 4575 a Fn(x)2044 4549 y Fv(1+1)p Fl(=\013)2298 4511 y Fx(=)2606 4449 y(\()p Fn(t=s)p Fx(\))2797 4416 y Fv(1)p Fl(=\013)p 2405 4490 720 4 v 2405 4573 a Fx(\(1)p Fn(=\013)p Fx(\))p Fn(\031)s Ft(c)-5 b(ose)g(c)18 b Fx(\()p Fn(\031)s(=\013)p Fx(\))3477 4511 y(\(1)p Fn(:)p Fx(13\))159 4950 y(In)30 b(the)g(remainder)f(of)h(this)g(subsection)f(w)m(e)i(outline)e(the)i (pro)s(of)e(of)i(this)e(theorem.)50 5168 y Fd(Lemma)38 b(1.2)68 b Ft(The)33 b(function)g Fx(\005)1282 5182 y Fl(E)1344 5168 y Fx(\()p Fn(s;)15 b(t)p Fx(\))33 b Ft(de\014ne)-5 b(d)33 b(in)g(\(1.9\))h(satis\014es)f(the)g(e)-5 b(quations)899 5415 y Fx(\005)967 5429 y Fl(E)1028 5415 y Fx(\()p Fn(s;)15 b(t)p Fx(\))26 b(=)f(1)c Fm(\000)f Fn(F)1551 5429 y Fl(E)1613 5415 y Fx(\()p Fn(s)g Fx(+)g Fn(t)p Fx(\))g(+)1981 5292 y Fi(Z)2072 5314 y Fl(t)2031 5498 y Fv(0)2134 5415 y Fx(\005)2202 5429 y Fl(E)2264 5415 y Fx(\()p Fn(;)15 b(t)21 b Fm(\000)e Fn(u)p Fx(\))p Fn(dF)2675 5429 y Fl(E)2738 5415 y Fx(\()p Fn(u)p Fx(\))3477 5409 y(\(1)p Fn(:)p Fx(14\))p eop %%Page: 6 6 6 5 bop 50 295 a Fr(6)3309 b Fo(Se)l(ction)27 b Fr(1)50 731 y Fc(Pro)s(of:)65 b Fx(The)37 b(pro)s(of)g(of)h(this)f(lemma)g(is)g (elemen)m(tary)i(since)e(\005)2302 745 y Fl(E)2364 731 y Fx(\()p 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Fk(\000)p Fv(1)1781 1667 y Fi(Z)1872 1690 y Fk(1)1831 1874 y Fv(1)1964 1791 y Fn(dxe)2105 1753 y Fk(\000)p Fl(t=x)2273 1791 y Fn(x)2325 1753 y Fk(\000)p Fv(\(1+)p Fl(\013)p Fv(\))p Fl(=\013)3477 1791 y Fx(\(1)p Fn(:)p Fx(15\))50 2051 y(whic)m(h)k(is)g(no)h(longer)f(random.)38 b(F)-8 b(rom)26 b(no)m(w)f(on)g(w)m(e)g(will)e(only)h(consider)g(the)h (case)h(when)e Fn(q)k Fx(is)c(the)h(uniform)50 2196 y(measure,)30 b Fn(q)470 2210 y Fl(i)525 2196 y Fx(=)663 2160 y Fv(1)p 632 2175 98 4 v 632 2227 a Fl(n)677 2236 y Fp(E)741 2196 y Fx(.)40 b(Let)31 b(\005)1037 2210 y Fk(1)1115 2196 y Fx(\()p Fn(s;)15 b(t)p Fx(\))30 b(denote)h(the)g(unique)d(solution)h (of)i(the)f(equation)854 2491 y(\005)922 2505 y Fk(1)999 2491 y Fx(\()p Fn(s;)15 b(t)p Fx(\))26 b(=)f(1)20 b Fm(\000)g Fn(F)1521 2505 y Fk(1)1599 2491 y Fx(\()p Fn(s)g Fx(+)g Fn(t)p Fx(\))g(+)1967 2367 y Fi(Z)2058 2389 y Fl(t)2018 2573 y Fv(0)2105 2491 y Fx(\005)2173 2505 y Fk(1)2251 2491 y Fx(\()p Fn(s;)15 b(t)20 b Fm(\000)g Fn(u)p Fx(\))p 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b Fx(\(1)p Fn(:)p Fx(20\))p 50 5102 546 4 v 159 5163 a Fs(8)197 5189 y Fr(Other)26 b(c)n(hoices)h(ma)n(y)e(lead)h(to)f(completely)i (di\013eren)n(t)g(b)r(eha)n(viors.)159 5254 y Fs(9)197 5280 y Fr(In)k(this)h(in)n(tro)r(duction)g(w)n(e)f(will)h(not)f (justify)g(the)h(v)l(arious)f(passages)h(to)e(limits)j(\(whic)n(h)f(is) g(also)f(nev)n(er)h(done)f(in)g(the)50 5371 y(ph)n(ysics)26 b(literature\).)37 b(Note)25 b(ho)n(w)n(ev)n(er)g(that)g(these)i (issues)g(are)f(treated)g(in)f(Section)h(5,)g(and)f(the)h(results)h (pro)n(v)n(en)e(there)h(can)50 5462 y(easily)h(b)r(e)f(used)g(to)g (justify)g(ev)n(erything)f(that)h(w)n(e)g(will)h(do)e(in)h(the)g (presen)n(t)h(section.)p eop %%Page: 7 7 7 6 bop 50 295 a Fo(A)l(ging)27 b(in)h(the)g(REM.)f(Part)h(2.)2699 b Fr(7)50 513 y Fx(Denote)32 b(b)m(y)e Fn(g)534 480 y Fk(\003)607 513 y Fx(the)g(Laplace)h(transform)f(of)g(a)h(function)e Fn(g)s Fx(,)i(i.e.)41 b Fn(g)2356 480 y Fk(\003)2398 513 y Fx(\()p Fn(u)p Fx(\))26 b(=)2642 439 y Fi(R)2703 462 y Fk(1)2685 545 y Fv(0)2795 513 y Fn(e)2837 480 y Fk(\000)p Fl(ut)2969 513 y Fn(g)s Fx(\()p Fn(t)p Fx(\).)42 b(Then)953 804 y Fn(F)1024 766 y Fk(\003)1011 826 y(1)1088 804 y Fx(\()p Fn(u)p Fx(\))26 b(=)f Fn(u)1384 766 y Fk(\000)p Fv(1)1502 804 y Fm(\000)20 b Fn(\013)1651 766 y Fk(\000)p Fv(1)1764 680 y Fi(Z)1855 703 y Fk(1)1815 887 y Fv(1)2163 742 y Fn(dx)p 1958 783 509 4 v 1958 868 a Fx(\()p Fn(ux)h Fx(+)f(1\))p Fn(x)2341 842 y Fv(1)p Fl(=\013)1236 1056 y Fx(=)25 b Fn(u)1384 1018 y Fk(\000)p Fv(1)1502 1056 y Fm(\000)20 b Fn(\013)1651 1018 y Fk(\000)p Fv(1)1749 1056 y Fn(u)1801 1018 y Fv(\(1)p Fk(\000)p Fl(\013)p Fv(\))p Fl(=\013)2102 932 y Fi(Z)2193 955 y Fl(u)p Fk(1)2153 1139 y Fl(u)2518 994 y Fn(dx)p 2339 1035 457 4 v 2339 1120 a Fx(\(1)h(+)f Fn(x)p Fx(\))p Fn(x)2670 1094 y Fv(1)p Fl(=\013)3477 932 y Fx(\(1)p Fn(:)p Fx(21\))50 1336 y(In)34 b(the)g(last)g(expression,)h(the)g(in)m(tegration)f(is)g(understo)s(o)s (d)e(to)j(b)s(e)f(along)h(a)g(transformed)e(path)h(in)g(the)50 1481 y(complex)c(plane)f(if)h Fn(u)g Fx(is)f(complex.)41 b(Note)32 b(that)1683 1448 y Fv(10)573 1634 y Fi(Z)663 1657 y Fk(1)623 1841 y Fv(0)945 1697 y Fn(dx)p 767 1737 V 767 1822 a Fx(\(1)21 b(+)e Fn(x)p Fx(\))p Fn(x)1097 1796 y Fv(1)p Fl(=\013)1259 1758 y Fx(=)25 b(\000\()p Fn(\013)1505 1720 y Fk(\000)p Fv(1)1603 1758 y Fx(\)\000\(1)d Fm(\000)e Fn(\013)1946 1720 y Fk(\000)p Fv(1)2044 1758 y Fx(\))25 b(=)2354 1697 y Fn(\031)p 2211 1737 342 4 v 2211 1820 a Fx(sin)o(\()p Fn(\031)s(=\013)p Fx(\))2589 1758 y(=)g Fn(\031)s Fx(cosec)16 b(\()p Fn(\031)s(=\013)p Fx(\))292 b(\(1)p Fn(:)p Fx(22\))50 2043 y(Th)m(us,)32 b(when)f Fn(u)d Fm(!)g Fx(0,)34 b(the)e(in)m(tegral)g(in)f(\(1.21\))j (con)m(v)m(erges)g(to)f(the)g(constan)m(t)g Fn(\031)s Fx(cosec)17 b(\()p Fn(\031)s(=\013)p Fx(\).)48 b(Similarly)-8 b(,)50 2189 y(w)m(e)31 b(ha)m(v)m(e)g(that)1132 2346 y Fn(f)1187 2309 y Fk(\003)1177 2369 y(1)1254 2346 y Fx(\()p Fn(u)p Fx(\))26 b(=)f Fn(\013)1556 2309 y Fk(\000)p Fv(1)1669 2223 y Fi(Z)1760 2245 y Fk(1)1719 2429 y Fv(1)1970 2285 y Fx(1)p 1863 2325 261 4 v 1863 2409 a(1)20 b(+)g Fn(ux)2134 2346 y(x)2186 2309 y Fk(\000)p Fv(\(1+)p Fl(\013)p Fv(\))p Fl(=\013)2528 2346 y Fn(dx)850 b Fx(\(1)p Fn(:)p Fx(23\))50 2582 y(In)29 b(particular,)g Fn(f)663 2549 y Fk(\003)653 2605 y(1)730 2582 y Fx(\(0\))d(=)f(1,)31 b(and)198 2867 y(1)20 b Fm(\000)g Fn(f)409 2829 y Fk(\003)399 2889 y(1)476 2867 y Fx(\()p Fn(u)p Fx(\))26 b(=)f Fn(\013)778 2829 y Fk(\000)p Fv(1)891 2743 y Fi(Z)982 2766 y Fk(1)942 2950 y Fv(1)1074 2739 y Fi(\022)1141 2867 y Fx(1)c Fm(\000)1416 2805 y Fx(1)p 1309 2846 V 1309 2929 a(1)f(+)g Fn(ux)1580 2739 y Fi(\023)1662 2867 y Fn(x)1714 2829 y Fk(\000)p Fv(\(1+)p Fl(\013)p Fv(\))p Fl(=\013)2056 2867 y Fn(dx)26 b Fx(=)f Fn(\013)2335 2829 y Fk(\000)p Fv(1)2433 2867 y Fn(u)2485 2829 y Fv(1)p Fl(=\013)2625 2743 y Fi(Z)2716 2766 y Fl(u)p Fk(1)2676 2950 y Fl(u)3041 2805 y Fn(dx)p 2862 2846 457 4 v 2862 2931 a Fx(\()p Fn(x)c Fx(+)f(1\))p Fn(x)3193 2905 y Fv(1)p Fl(=\013)3477 2867 y Fx(\(1)p Fn(:)p Fx(24\))50 3152 y(T)-8 b(aking)30 b(the)g(Laplace)h(transform)f (of)g(\(1.18\))j(this)c(implies)f(that)741 3435 y Fn(M)839 3397 y Fk(\003)881 3435 y Fx(\()p Fn(u)p Fx(\))e(=)1207 3373 y Fn(F)1278 3340 y Fk(\003)1265 3396 y(1)1342 3373 y Fx(\()p Fn(u)p Fx(\))p 1136 3414 401 4 v 1136 3497 a(1)20 b Fm(\000)g Fn(f)1347 3471 y Fk(\003)1337 3520 y(1)1414 3497 y Fx(\()p Fn(u)p Fx(\))1573 3435 y(=)2191 3373 y(1)p 1679 3414 1068 4 v 1679 3505 a Fn(\013)1737 3479 y Fk(\000)p Fv(1)1835 3505 y Fn(u)1887 3479 y Fv(\(1+)p Fl(\013)p Fv(\))p Fl(=\013)2188 3432 y Fi(R)2248 3454 y Fl(u)p Fk(1)2231 3537 y Fl(u)2526 3469 y(dx)p 2395 3484 342 4 v 2395 3540 a Fv(\(1+)p Fl(x)p Fv(\))p Fl(x)2625 3522 y Fq(1)p Fp(=\013)2778 3435 y Fm(\000)g Fn(u)2921 3397 y Fk(\000)p Fv(1)3477 3435 y Fx(\(1)p Fn(:)p Fx(25\))50 3754 y(and,)29 b(b)m(y)h(classical)f(results)f(on)i(the)f(asymptotics)h (of)g(the)g(in)m(v)m(erse)f(Laplace)h(transform)f(\(see)i([Do)s(e],)g (V)-8 b(ol.)50 3900 y(2,)31 b(Section)f(7\),)h(this)f(in)f(turn)g (implies)f(that)j(for)f Fn(t)25 b Fm(")g Fx(+)p Fm(1)p Fx(,)1158 4195 y Fn(M)10 b Fx(\()p Fn(t)p Fx(\))26 b Fm(\030)1884 4133 y Fn(t)1917 4100 y Fv(1)p Fl(=\013)p 1492 4174 943 4 v 1492 4257 a Fn(\031)s(\013)1605 4231 y Fk(\000)p Fv(1)1703 4257 y Fx(\000\()p Fn(\013)1853 4231 y Fk(\000)p Fv(1)1951 4257 y Fx(\)cosec)17 b(\()p Fn(\031)s(=\013)p Fx(\))2465 4195 y Fm(\000)j Fx(1)876 b(\(1)p Fn(:)p Fx(26\))159 4551 y(Finally)-8 b(,)31 b(w)m(e)i(can)f (compute)g(the)h(asymptotics)f(of)g(the)g(solution)f(of)h(equation)g (\(1.16\).)48 b(Here)33 b(w)m(e)f(will)50 4697 y(directly)d(mak)m(e)i (use)f(of)h(the)f(fact)i(that)f(the)f(Laplace)h(transform)f(of)g(\005) 2492 4711 y Fk(1)2569 4697 y Fx(\()p Fn(s;)15 b(t)p Fx(\))31 b(is)f(giv)m(en)g(explicitly)e(as)1208 5017 y(\005)1276 4979 y Fk(\003)1276 5039 y(1)1353 5017 y Fx(\()p Fn(u;)15 b(s)p Fx(\))26 b(=)1691 4930 y Fn(\013)1765 4857 y Fi(R)1825 4880 y Fk(1)1808 4963 y Fv(1)1918 4930 y Fn(e)1960 4898 y Fk(\000)p Fl(s=x)2298 4895 y(dx)p 2145 4910 385 4 v 2145 4966 a Fv(\()p Fl(ux)p Fv(+1\))p Fl(x)2418 4947 y Fq(1)p Fp(=\013)p 1691 4996 850 4 v 1915 5079 a Fx(1)21 b Fm(\000)f Fn(f)2127 5053 y Fk(\003)2117 5102 y(1)2194 5079 y Fx(\()p Fn(u)p Fx(\))3477 5017 y(\(1)p Fn(:)p Fx(27\))p 50 5204 546 4 v 159 5283 a Fs(10)230 5309 y Fr(P)n(erforming)26 b(the)h(c)n(hange)g(of)f(v)l(ariable)h Fg(x)c Fr(=)g Fg(y)1594 5283 y Fb(\000)p Fs(1)1703 5309 y Fa(\000)18 b Fr(1,)1868 5240 y Fi(R)1929 5262 y Fb(1)1911 5345 y Fs(0)2149 5279 y Ff(dx)p 2024 5292 324 4 v 2024 5347 a Fs(\(1+)p Ff(x)p Fs(\))p Ff(x)2236 5330 y Fq(1)p Fp(=\013)2381 5309 y Fr(=)2465 5240 y Fi(R)2525 5262 y Fs(1)2508 5345 y(0)2809 5274 y Ff(dy)p 2587 5292 516 4 v 2587 5347 a Fs(\(1)p Fb(\000)p Ff(x)p Fs(\))2763 5330 y Fq(1)p Fp(=\013)2874 5347 y Ff(x)2912 5330 y Fq(1)p Fe(\000)p Fq(1)p Fp(=\013)3141 5309 y Fr(where)27 b(one)g(recog-)50 5451 y(nizes)g(the)f(Beta)g(in)n(tegral)837 5383 y Fi(R)898 5405 y Fs(1)880 5488 y(0)1151 5417 y Ff(dy)p 959 5435 456 4 v 959 5485 a Fs(\(1)p Fb(\000)p Ff(y)r Fs(\))1134 5469 y Fp(\026)p Fe(\000)p Fq(1)1257 5485 y Ff(y)1294 5469 y Fp(\027)s Fe(\000)p Fq(1)1447 5451 y Fr(=)1539 5414 y Fs(\000\()p Ff(\026)p Fs(\)\000\()p Ff(\027)t Fs(\))p 1539 5435 266 4 v 1560 5480 a(\000\()p Ff(\026)p Fs(+)p Ff(\027)t Fs(\))p eop %%Page: 8 8 8 7 bop 50 295 a Fr(8)3309 b Fo(Se)l(ction)27 b Fr(1)50 513 y Fx(w)m(e)k(ha)m(v)m(e)g(already)f(established)f(the)h (asymptotics)h(of)f(1)21 b Fm(\000)f Fn(f)2155 480 y Fk(\003)2145 535 y(1)2221 513 y Fx(\()p Fn(u)p Fx(\))31 b(near)f Fn(u)c Fx(=)f(0.)41 b(W)-8 b(e)31 b(still)d(need)j(to)g(treat) 50 658 y(the)f(n)m(umerator.)41 b(It)31 b(will)c(b)s(e)j(con)m(v)m (enien)m(t)i(to)f(write)559 859 y Fn(\013)617 822 y Fk(\000)p Fv(1)730 736 y Fi(Z)821 758 y Fk(1)781 942 y Fv(1)913 859 y Fn(e)955 822 y Fk(\000)p Fl(s=x)1345 798 y Fn(dx)p 1141 839 509 4 v 1141 923 a Fx(\()p Fn(ux)20 b Fx(+)g(1\))p Fn(x)1523 897 y Fv(1)p Fl(=\013)1685 859 y Fx(=)25 b Fn(\013)1839 822 y Fk(\000)p Fv(1)1953 736 y Fi(Z)2044 758 y Fk(1)2003 942 y Fv(1)2136 859 y Fn(dx)2250 736 y Fi(Z)2341 758 y Fk(1)2301 942 y Fl(s=x)2433 859 y Fn(dv)s(e)2569 822 y Fk(\000)p Fl(v)2913 798 y Fx(1)p 2681 839 V 2681 923 a(\()p Fn(ux)c Fx(+)f(1\))p Fn(x)3064 897 y Fv(1)p Fl(=\013)1589 1123 y Fx(=)p Fn(\013)1718 1086 y Fk(\000)p Fv(1)1831 999 y Fi(Z)1922 1022 y Fk(1)1882 1206 y Fv(0)2015 1123 y Fn(dv)s(e)2151 1086 y Fk(\000)p Fl(v)2267 999 y Fi(Z)2358 1022 y Fk(1)2317 1206 y Fl(s=v)r Fk(^)p Fv(1)2532 1123 y Fn(dx)2874 1062 y Fx(1)p 2642 1102 V 2642 1187 a(\()p Fn(ux)h Fx(+)f(1\))p Fn(x)3025 1161 y Fv(1)p Fl(=\013)1589 1387 y Fx(=)p Fn(\013)1718 1350 y Fk(\000)p Fv(1)1831 1263 y Fi(Z)1922 1286 y Fk(1)1882 1470 y Fv(0)2015 1387 y Fn(dv)s(e)2151 1350 y Fk(\000)p Fl(v)2267 1263 y Fi(Z)2358 1286 y Fk(1)2317 1470 y Fl(s=v)2450 1387 y Fn(dx)2791 1326 y Fx(1)p 2560 1366 V 2560 1451 a(\()p Fn(ux)g Fx(+)g(1\))p Fn(x)2942 1425 y Fv(1)p Fl(=\013)1680 1664 y Fm(\000)g Fn(\013)1829 1627 y Fk(\000)p Fv(1)1943 1541 y Fi(Z)2033 1563 y Fk(1)1993 1747 y Fl(s)2126 1664 y Fn(dv)s(e)2262 1627 y Fk(\000)p Fl(v)2378 1541 y Fi(Z)2469 1563 y Fv(1)2428 1747 y Fl(s=v)2558 1664 y Fn(dx)2900 1603 y Fx(1)p 2668 1644 V 2668 1728 a(\()p Fn(ux)h Fx(+)f(1\))p Fn(x)3051 1702 y Fv(1)p Fl(=\013)3477 1270 y Fx(\(1)p Fn(:)p Fx(28\))50 1899 y(No)m(w)25 b(the)f(\014rst)f(term)h(can)h(b)s(e)e(con)m(v)m (enien)m(tly)i(represen)m(ted)f(as)g Fn(u)2222 1866 y Fl(\013)2298 1899 y Fx(times)f(an)h(explicit)f(Laplace)h(transform:)138 2138 y Fn(\013)196 2101 y Fk(\000)p Fv(1)309 2014 y Fi(Z)400 2037 y Fk(1)360 2221 y Fv(0)492 2138 y Fn(dv)s(e)628 2101 y Fk(\000)p Fl(v)745 2014 y Fi(Z)836 2037 y Fk(1)795 2221 y Fl(s=v)928 2138 y Fn(dx)1269 2077 y Fx(1)p 1038 2117 V 1038 2202 a(\()p Fn(ux)c Fx(+)g(1\))p Fn(x)1420 2176 y Fv(1)p Fl(=\013)1583 2138 y Fx(=)25 b Fn(\013)1737 2101 y Fk(\000)p Fv(1)1835 2138 y Fn(u)1887 2101 y Fv(1)p Fl(=\013)2027 2014 y Fi(Z)2118 2037 y Fk(1)p Fl(=u)2078 2221 y Fv(0)2290 2138 y Fn(dv)s(e)2426 2101 y Fk(\000)p Fl(uv)2586 2014 y Fi(Z)2676 2037 y Fl(u)p Fk(1)2636 2221 y Fl(s=v)2812 2138 y Fn(dx)3127 2077 y Fx(1)p 2922 2117 457 4 v 2922 2202 a(\()p Fn(x)20 b Fx(+)g(1\))p Fn(x)3252 2176 y Fv(1)p Fl(=\013)3477 2138 y Fx(\(1)p Fn(:)p Fx(29\))50 2378 y(Note)34 b(that)f(since)e(all)h(in)m(tegrands)g(v)-5 b(anish)30 b(at)k(in\014nit)m(y)c(in)h(the)i(righ)m(t-half)e(plane,)h (0)p Fn(=u)h Fx(and)f Fn(u)p Fm(1)h Fx(can)f(b)s(e)50 2523 y(replaced)g(with)g(0)h(and)g Fm(1)p Fx(,)g(resp.,)h(i.e.)48 b(the)34 b(in)m(tegration)f(con)m(tours)g(can)g(b)s(e)g(deformed)f(to)i (in)m(tegrations)50 2669 y(along)c(the)h(real)f(line.)39 b(W)-8 b(e)31 b(will)d(sho)m(w)i(that)h(this)f(term)g(is)g(the)g (dominan)m(t)g(one.)159 2887 y(In)g(fact,)h(com)m(bining)e(\(1.24\))k (with)c(\(1.28\))j(w)m(e)f(get)h(from)e(\(1.27\))i(that)296 3167 y(\005)364 3130 y Fk(\003)364 3190 y(1)442 3167 y Fx(\()p Fn(u;)15 b(s)p Fx(\))26 b(=)779 3007 y Fi(R)840 3030 y Fk(1)p Fl(=u)822 3113 y Fv(0)1012 3081 y Fn(dv)s(e)1148 3048 y Fk(\000)p Fl(uv)1308 3007 y Fi(R)1368 3030 y Fl(u)p Fk(1)1351 3113 y Fl(s=v)1503 3081 y Fn(dx)1766 3045 y Fv(1)p 1614 3060 342 4 v 1614 3116 a(\(1+)p Fl(x)p Fv(\))p Fl(x)1844 3097 y Fq(1)p Fp(=\013)p 779 3146 1187 4 v 1115 3164 a Fi(R)1175 3186 y Fk(1)1158 3269 y Fl(u)1410 3201 y(dx)p 1278 3216 342 4 v 1278 3272 a Fv(\(1+)p Fl(x)p Fv(\))p Fl(x)1508 3254 y Fq(1)p Fp(=\013)1997 3167 y Fm(\000)2099 3007 y Fi(R)2160 3030 y Fk(1)2142 3113 y Fl(s)2252 3081 y Fn(dv)s(e)2388 3048 y Fk(\000)p Fl(v)2504 3007 y Fi(R)2565 3030 y Fv(1)2547 3113 y Fl(s=v)2677 3081 y Fn(dx)2980 3045 y Fv(1)p 2787 3060 422 4 v 2787 3116 a(\()p Fl(u)p Fv(+1)p Fl(=x)p Fv(\))p Fl(x)3097 3097 y Fq(1)p Fp(=\013)p 2099 3146 1121 4 v 2305 3237 a Fn(u)2357 3211 y Fv(1)p Fl(=\013)2498 3164 y Fi(R)2558 3186 y Fk(1)2541 3269 y Fl(u)2793 3201 y(dx)p 2661 3216 342 4 v 2661 3272 a Fv(\(1+)p Fl(x)p Fv(\))p Fl(x)2891 3254 y Fq(1)p Fp(=\013)3477 3158 y Fx(\(1)p Fn(:)p Fx(30\))50 3424 y(No)m(w)31 b(the)f(in)m(tegral)h(in)e(the)h(denominator)g(equals) 606 3519 y Fi(Z)696 3541 y Fk(1)656 3726 y Fl(u)978 3581 y Fn(dx)p 800 3622 457 4 v 800 3707 a Fx(\(1)21 b(+)e Fn(x)p Fx(\))p Fn(x)1130 3680 y Fv(1)p Fl(=\013)1292 3643 y Fx(=)1388 3519 y Fi(Z)1479 3541 y Fk(1)1439 3726 y Fv(0)1761 3581 y Fn(dx)p 1582 3622 V 1582 3707 a Fx(\(1)i(+)f Fn(x)p Fx(\))p Fn(x)1913 3680 y Fv(1)p Fl(=\013)2070 3643 y Fm(\000)2161 3519 y Fi(Z)2252 3541 y Fl(u)2211 3726 y Fv(0)2504 3581 y Fn(dx)p 2325 3622 V 2325 3707 a Fx(\(1)h(+)f Fn(x)p Fx(\))p Fn(x)2656 3680 y Fv(1)p Fl(=\013)1292 3916 y Fx(=)25 b Fn(\031)s Fx(cosec)17 b(\()p Fn(\031)s(=\013)p Fx(\))22 b Fm(\000)e Fn(u)2054 3878 y Fv(1)p Fk(\000)p Fv(1)p Fl(=\013)2320 3802 y Fk(1)2290 3830 y Fi(X)2287 4022 y Fl(n)p Fv(=0)2424 3916 y Fx(\()p Fm(\000)p Fx(1\))2610 3878 y Fl(n)2857 3854 y Fn(u)2909 3821 y Fl(n)p 2672 3895 472 4 v 2672 3978 a Fn(n)f Fx(+)h(1)h Fm(\000)f Fx(1)p Fn(=\013)3477 3795 y Fx(\(1)p Fn(:)p Fx(31\))50 4172 y(where)28 b(the)h(last)g(sum)f(is)g(con)m(v)m(ergen)m (t)j(for)e Fm(j)p Fn(u)p Fm(j)d Fn(<)f Fx(1.)40 b(Th)m(us)28 b(the)h(leading)f(singular)e(\(at)k Fn(u)c Fx(=)f(0\))k(term)g(from)50 4317 y(the)h(\014rst)g(term)g(in)f(\(1.30\))k(is)d(giv)m(en)g(b)m(y) 1338 4418 y Fi(R)1399 4440 y Fk(1)1381 4523 y Fv(0)1491 4491 y Fn(dv)s(e)1627 4458 y Fk(\000)p Fl(uv)1786 4418 y Fi(R)1847 4440 y Fk(1)1829 4523 y Fl(s=v)1959 4491 y Fn(dx)2222 4455 y Fv(1)p 2069 4470 342 4 v 2069 4526 a(\(1+)p Fl(x)p Fv(\))p Fl(x)2299 4508 y Fq(1)p Fp(=\013)p 1338 4556 1084 4 v 1629 4640 a Fn(\031)s Fx(cosec)16 b(\()p Fn(\031)s(=\013)p Fx(\))3477 4577 y(\(1)p Fn(:)p Fx(32\))50 4803 y(whic)m(h)29 b(ob)m(viously)g(is)g(the)i(Laplace)g (transform)e(of)i(the)f(function)f Fn(H)2404 4817 y Fv(0)2445 4803 y Fx(\()p Fn(s=t)p Fx(\).)159 5021 y(It)43 b(remains)e(to)j (consider)e(the)h(second)f(term)h(in)f(\(1.30\).)80 b(Here)43 b(the)g(n)m(umerator)g(con)m(v)m(erges)i(to)e(a)50 5167 y(constan)m(t)31 b(as)g Fn(u)f Fx(tends)g(to)h(zero,)h(in)d(fact,)j(at) f Fn(u)25 b Fx(=)g(0)31 b(it)f(equals)353 5276 y Fi(Z)444 5298 y Fk(1)404 5482 y Fl(s)536 5399 y Fn(dv)s(e)672 5362 y Fk(\000)p Fl(v)788 5276 y Fi(Z)879 5298 y Fv(1)839 5482 y Fl(s=v)969 5399 y Fn(dx)1145 5338 y Fx(1)p 1079 5378 178 4 v 1079 5463 a Fn(x)1131 5437 y Fv(1)p Fl(=\013)1292 5399 y Fx(=)1530 5338 y(1)p 1399 5378 306 4 v 1399 5462 a(1)21 b Fm(\000)f Fx(1)p Fn(=\013)1731 5276 y Fi(Z)1822 5298 y Fk(1)1782 5482 y Fl(s)1914 5399 y Fn(dy)s(e)2051 5362 y Fk(\000)p Fl(y)2168 5298 y Fi(h)2211 5399 y Fx(1)g Fm(\000)g Fn(y)2415 5362 y Fv(1)p Fk(\000)p Fv(1)p Fl(=\013)2633 5298 y Fi(i)2701 5399 y Fm(\024)25 b Fn(const:e)3078 5362 y Fk(\000)p Fl(s)3477 5399 y Fx(\(1)p Fn(:)p Fx(33\))p eop %%Page: 9 9 9 8 bop 50 295 a Fo(A)l(ging)27 b(in)h(the)g(REM.)f(Part)h(2.)2699 b Fr(9)50 513 y Fx(Therefore)30 b(the)h(leading)e(asymptotic)h(of)h (the)f(second)h(term)f(is)g(giv)m(en)g(b)m(y)1551 761 y Fn(C)7 b(onst:u)1875 723 y Fk(\000)p Fv(1)p Fl(=\013)2071 761 y Fn(e)2113 723 y Fk(\000)p Fl(s)3477 761 y Fx(\(1)p Fn(:)p Fx(34\))50 1009 y(The)33 b(in)m(v)m(erse)h(Laplace)g(transform)f (of)h(the)f(second)h(term)g(has)f(therefore)i(the)f(leading)e (asymptotic)i(b)s(e-)50 1154 y(ha)m(vior)1330 1300 y Fn(H)1406 1314 y Fv(1)1447 1300 y Fx(\()p Fn(s;)15 b(t)p Fx(\))25 b Fm(\030)g Fn(C)7 b(onst:t)2059 1262 y Fv(1)p Fl(=\013)p Fk(\000)p Fv(1)2291 1300 y Fn(e)2333 1262 y Fk(\000)p Fl(s)3477 1300 y Fx(\(1)p Fn(:)p Fx(35\))50 1506 y(Note)29 b(that)g(while)e(the)h(asymptotics)h(in)d Fn(t)j Fx(lo)s(oks)e(the)i(same)f(as)h(that)g(of)f(the)h(second)f(term) g(of)h Fn(H)3374 1520 y Fv(0)3414 1506 y Fx(\()p Fn(s=t)p Fx(\))g(in)50 1652 y(the)k(case)h Fn(s=t)c Fm(#)g Fx(0,)35 b(due)d(to)i(the)g(exp)s(onen)m(tial)e(deca)m(y)i(in)e Fn(s)p Fx(,)h(this)g(term)g(can)g(b)s(e)g(neglected)g(if)g Fn(s)f Fx(is)g(large.)50 1797 y(Th)m(us)d(w)m(e)i(ha)m(v)m(e)h(no)m(w)e (established)f(the)h(\\aging")i(asymptotics)e(found)f(in)g(Bouc)m (haud.)50 1996 y Fu(1.3.)47 b(The)34 b(renew)m(al)h(equations.)47 b(Statemen)m(t)32 b(of)j(the)g(main)f(results.)159 2267 y Fx(Guided)k(b)m(y)h(Bouc)m(haud's)g(trap)g(mo)s(del,)i(w)m(e)f(can)f (no)m(w)g(construct)h(the)g(setup)e(for)h(the)h(analysis)e(of)50 2412 y(aging)33 b(in)f(the)h(full)e(REM)i(dynamics.)47 b(First)33 b(of)g(all)f(the)h(natural)f(subset)h(of)g(states)h(in)e Fm(S)3168 2426 y Fl(N)3270 2412 y Fx(to)i(pla)m(y)f(the)50 2558 y(r^)-45 b(ole)30 b(of)g(the)h(state)h(space)f(in)e(the)h(trap)g (mo)s(del)g(is)f(the)i(set)1223 2806 y Fn(T)1276 2820 y Fl(N)1345 2806 y Fx(\()p Fn(E)5 b Fx(\))26 b Fm(\021)1609 2732 y Fi(\010)1662 2806 y Fn(\033)i Fm(2)d(S)1883 2820 y Fl(N)1952 2729 y Fi(\014)1952 2783 y(\014)1983 2806 y Fn(E)2050 2820 y Fl(\033)2124 2806 y Fm(\025)g Fn(u)2272 2820 y Fl(N)2341 2806 y Fx(\()p Fn(E)5 b Fx(\))2483 2732 y Fi(\011)3477 2806 y Fx(\(1)p Fn(:)p Fx(36\))50 3054 y(where)30 b(\(recall)g(Section)g(1.1)h(of)g([BBG1]\))833 3318 y Fn(u)885 3332 y Fl(N)954 3318 y Fx(\()p Fn(x)p Fx(\))26 b Fm(\021)1198 3236 y(p)p 1274 3236 280 4 v 82 x Fx(2)p Fn(N)g Fx(ln)13 b(2)21 b(+)1828 3257 y Fn(x)p 1676 3297 356 4 v 1676 3316 a Fm(p)p 1752 3316 280 4 v 77 x Fx(2)p Fn(N)k Fx(ln)14 b(2)2063 3318 y Fm(\000)2165 3257 y Fx(1)p 2165 3297 46 4 v 2165 3381 a(2)2232 3257 y(ln)o(\()p Fn(N)25 b Fx(ln)14 b(2\))21 b(+)f(ln)14 b(4)p Fn(\031)p 2232 3297 684 4 v 2396 3316 a Fm(p)p 2472 3316 280 4 v 77 x Fx(2)p Fn(N)25 b Fx(ln)14 b(2)3477 3318 y(\(1)p Fn(:)p Fx(37\))50 3583 y(W)-8 b(e)36 b(will)31 b(call)j(the)h(set)g Fn(T)917 3597 y Fl(N)986 3583 y Fx(\()p Fn(E)5 b Fx(\))36 b('the)e(top',)j(and)d(frequen)m(tly)f (suppress)g(indices,)h(writing)e Fn(T)45 b Fx(=)32 b Fn(T)13 b Fx(\()p Fn(E)5 b Fx(\))33 b(=)50 3728 y Fn(T)103 3742 y Fl(N)172 3728 y Fx(\()p Fn(E)5 b Fx(\))41 b(whenev)m(er)f(no)f (confusion)g(seems)h(p)s(ossible.)67 b(Moreo)m(v)m(er,)45 b(w)m(e)40 b(will)d(use)j(the)g(con)m(v)m(en)m(tion)h(that)50 3874 y Fn(M)35 b Fm(\021)25 b(j)p Fn(T)347 3888 y Fl(N)416 3874 y Fx(\()p Fn(E)5 b Fx(\))p Fm(j)p Fx(,)32 b(and)e Fn(d)c Fm(\021)f Fx(2)1031 3841 y Fl(M)1112 3874 y Fx(.)159 4092 y(The)f(idea)g(is)f(clearly)h(to)h(observ)m(e)h(the)e(pro)s(cess)g (only)g(at)h(its)f(visits)f(to)i Fn(T)13 b Fx(.)39 b(The)24 b(natural)f(generalization)50 4237 y(of)28 b(Bouc)m(haud's)g (correlation)g(function)f(\005)1510 4251 y Fl(E)1571 4237 y Fx(\()p Fn(s;)15 b(t)p Fx(\))29 b(is)e(therefore)i(the)f (probabilit)m(y)e(that)i(the)g(pro)s(cess)g(do)s(es)50 4383 y(not)d(jump)e(from)i(a)g(state)h(in)e(the)h(top)g(to)h(another)f (state)h(in)d(the)i(top)h(during)c(a)j(time)g(in)m(terv)-5 b(al)24 b(of)h(the)g(form)50 4528 y([)p Fn(n;)15 b(n)21 b Fx(+)h Fn(m)p Fx(].)47 b(There)32 b(is)g(some)h(am)m(biguit)m(y)e(ho) m(w)i(this)e(should)g(b)s(e)h(de\014ned)f(precisely)-8 b(,)33 b(but)f(the)g(follo)m(wing)50 4674 y(de\014nition)g(app)s(ears)h (most)i(con)m(v)m(enien)m(t.)54 b(T)-8 b(o)35 b(form)m(ulate)f(it,)h (let)f(us)g(in)m(tro)s(duce)f(the)i(follo)m(wing)e(random)50 4819 y(times.)46 b(F)-8 b(or)33 b(an)m(y)g Fn(k)e Fm(2)e Fj(N)6 b Fx(,)39 b(let)33 b Fn(k)1145 4833 y Fk(\000)1239 4819 y Fx(denote)f(the)h(last)f(time)g(b)s(efore)g Fn(k)k Fx(at)d(whic)m(h)e(the)h(pro)s(cess)g(has)g(visited)50 4965 y(the)e(top,)h(i.e.)1240 5110 y Fn(k)1287 5124 y Fk(\000)1374 5110 y Fm(\021)25 b Fx(sup)14 b Fm(f)p Fn(l)27 b(<)e(k)k Fm(j)c Fn(\033)s Fx(\()p Fn(l)r Fx(\))h Fm(2)f Fn(T)2262 5124 y Fl(N)2331 5110 y Fx(\()p Fn(E)5 b Fx(\))p Fm(g)959 b Fx(\(1)p Fn(:)p Fx(38\))50 5316 y(No)m(w)31 b(set)795 5462 y(\005\()p Fn(n;)15 b(m;)g(N)5 b(;)15 b(E)5 b Fx(\))27 b Fm(\021)e Fj(P)1531 5388 y Fi(\002)1570 5462 y Fm(8)1621 5478 y Fl(k)r Fk(2)p Fv([)p Fl(n)p Fv(+1)p Fl(;n)p Fv(+)p Fl(m)p Fv(])2094 5462 y Fn(\033)s Fx(\()p Fn(k)s Fx(\))h Fm(62)f Fn(T)2434 5476 y Fl(N)2503 5462 y Fx(\()p Fn(E)5 b Fx(\))p Fm(n)p Fn(\033)s Fx(\()p Fn(k)2827 5476 y Fk(\000)2891 5462 y Fx(\))2926 5388 y Fi(\003)3477 5462 y Fx(\(1)p Fn(:)p Fx(39\))p eop %%Page: 10 10 10 9 bop 50 295 a Fr(10)3270 b Fo(Se)l(ction)27 b Fr(1)50 513 y Fx(Of)43 b(course)i(w)m(e)f(still)e(ha)m(v)m(e)k(to)f(sp)s(ecify) e(the)h(initial)d(distribution.)79 b(T)-8 b(o)44 b(b)s(e)g(as)g(close)h (as)f(p)s(ossible)e(to)50 658 y(Bouc)m(haud,)36 b(the)g(natural)e(c)m (hoice)h(is)g(the)g(uniform)e(distribution)e(on)k Fn(T)2553 672 y Fl(N)2622 658 y Fx(\()p Fn(E)5 b Fx(\))36 b(that)f(w)m(e)h(will)c (denote)k(b)m(y)50 804 y Fn(\031)102 818 y Fl(E)163 804 y Fx(.)49 b(Ho)m(w)m(ev)m(er,)36 b(w)m(e)d(will)e(also)i(need)g(to)g (in)m(tro)s(duce)f(the)i(resp)s(ectiv)m(e)f(functions)e(with)h (starting)h(p)s(oin)m(t)f(in)50 949 y(an)e(arbitrary)f(state)j Fn(\033)s Fx(.)41 b(Th)m(us)29 b(w)m(e)i(set)546 1277 y(\005)614 1291 y Fl(\033)663 1277 y Fx(\()p Fn(m;)15 b(m;)g(N)5 b(;)15 b(E)5 b Fx(\))28 b Fm(\021)c Fj(P)1356 1203 y Fi(\002)1395 1277 y Fm(8)1446 1293 y Fl(k)r Fk(2)p Fv([)p Fl(n)p Fv(+1)p Fl(;n)p Fv(+)p Fl(m)p Fv(])1919 1277 y Fn(\033)s Fx(\()p Fn(k)s Fx(\))j Fm(62)e Fn(T)2260 1291 y Fl(N)2329 1277 y Fx(\()p Fn(E)5 b Fx(\))p Fm(n)p Fn(\033)s Fx(\()p 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b(W)-8 b(e)39 b(are)f(no)m(w)f(ready)g(to)h(state)g(the)g(main)e(theorem)i(of) 50 2466 y(this)29 b(pap)s(er.)50 2684 y Fd(Theorem)42 b(1)i Ft(L)-5 b(et)37 b Fn(\014)g(>)994 2606 y Fm(p)p 1070 2606 197 4 v 78 x Fx(2)15 b(ln)g(2)p Ft(.)53 b(Then)37 b(ther)-5 b(e)38 b(is)e(a)h(se)-5 b(quenc)g(e)36 b Fn(c)2416 2698 y Fl(N)2518 2684 y Fm(\030)c Fx(exp\()p Fn(\014)2851 2607 y Fm(p)p 2927 2607 83 4 v 77 x Fn(N)10 b(u)3062 2698 y Fl(N)3131 2684 y Fx(\()p Fn(E)5 b Fx(\)\))38 b Ft(such)e(that)50 2829 y(for)d(any)g Fn(\017)25 b(>)g Fx(0)821 3061 y(lim)787 3120 y Fl(t;s)p Fk("1)1044 3061 y Fx(lim)995 3120 y Fl(E)s Fk(#\0001)1258 3061 y Fx(lim)1234 3120 y Fl(N)7 b Fk("1)1423 3061 y Fn(P)1509 2933 y Fi(\024)1557 2929 y(\014)1557 2984 y(\014)1557 3038 y(\014)1557 3093 y(\014)1599 2999 y Fx(\005\([)p Fn(c)1766 3013 y Fl(N)1836 2999 y Fn(s)p Fx(])p Fn(;)15 b Fx([)p Fn(c)2008 3013 y Fl(N)2077 2999 y Fn(t)p Fx(])p Fn(;)g(N)5 b(;)15 b(E)5 b Fx(\))p 1599 3040 804 4 v 1834 3123 a(\005)1902 3137 y Fk(1)1980 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Fn(n)25 b(>)g Fx(0)h Fm(j)f Fn(\033)s Fx(\()p Fn(n)p Fx(\))h Fm(2)f Fn(I)7 b(;)15 b(\033)s Fx(\(0\))27 b(=)e Fn(\033)s Fm(g)875 b Fx(\(1)p Fn(:)p Fx(43\))50 4371 y(for)32 b(the)g(\014rst)f (p)s(ositiv)m(e)g(time)h(the)g(pro)s(cess)g(starting)f(in)g Fn(\033)k Fx(reac)m(hes)e(the)g(set)f Fn(I)7 b Fx(,)33 b(Note)g(that)g(it)e(is)g(easy)i(to)50 4516 y(deriv)m(e)g(a)h(renew)m (al)f(equation)g(for)g(the)h(quan)m(tities)f(\(1.40\).)52 b(Namely)-8 b(,)35 b(the)e(ev)m(en)m(t)i(in)d(the)i(probabilit)m(y)d (in)50 4662 y(\(1.40\))h(o)s(ccurs)f(either)17 4880 y(\(i\))46 b(if)29 b Fn(\033)s Fx(\()p Fn(k)s Fx(\))e Fm(62)d Fn(T)13 b Fx(\()p Fn(E)5 b Fx(\))p Fm(n)p Fn(\033)s Fx(,)32 b(for)f(all)e Fn(k)f Fm(2)d Fx([0)p Fn(;)15 b(n)21 b Fx(+)f Fn(m)p Fx(],)31 b(or)-8 5098 y(\(ii\))45 b(if)38 b(there)i(is)f(0)i Fn(<)f(l)i Fm(\024)f Fn(n)p Fx(,)g(s.t.)69 b Fn(l)42 b Fx(=)f(inf)5 b Fm(f)p Fn(k)44 b Fm(\024)d Fn(n)f Fm(j)h Fn(\033)s Fx(\()p Fn(k)s Fx(\))g Fm(2)f Fn(T)13 b Fx(\()p Fn(E)5 b Fx(\))p Fm(n)p 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y(T)10 b Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)1556 2043 y Fx(=)25 b Fn(k)s Fx(])h(=)1925 1982 y Fn(\031)1977 1996 y Fl(E)2039 1982 y Fx(\()p Fn(\033)2129 1949 y Fk(0)2155 1982 y Fx(\))p 1860 2022 396 4 v 1860 2106 a(1)20 b Fm(\000)g Fn(\031)2068 2120 y Fl(E)2130 2106 y Fx(\()p Fn(\033)s Fx(\))2266 2043 y Fj(P)p Fx([)p Fn(\034)2396 2006 y Fl(\033)2386 2068 y(T)10 b Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)2665 2043 y Fx(=)25 b Fn(k)s Fx(])641 b(\(1)p Fn(:)p Fx(45\))50 2277 y(a)m(v)m(eraging)32 b(\(1.44\))g(o)m(v)m(er)g Fn(\033)h Fx(w)m(ould)d(yield)215 2475 y(\005\()p Fn(m;)15 b(n;)g(E)5 b Fx(\))26 b(=)826 2388 y Fi(X)762 2587 y Fl(\033)r Fk(2)p Fl(T)10 b Fv(\()p Fl(E)s Fv(\))1037 2475 y Fn(\031)1089 2489 y Fl(E)1150 2475 y Fx(\()p Fn(\033)s Fx(\))p Fj(P)p Fx([)p Fn(\034)1405 2437 y Fl(\033)1395 2499 y(T)g Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)1675 2475 y Fn(>)25 b(m)20 b Fx(+)g Fn(n)p Fx(])326 2799 y(+)460 2685 y Fl(n)417 2713 y Fi(X)417 2908 y Fl(k)r Fv(=1)628 2713 y Fi(X)564 2912 y Fl(\033)r Fk(2)p Fl(T)10 b Fv(\()p Fl(E)s Fv(\))839 2799 y Fn(\031)891 2813 y Fl(E)952 2799 y Fx(\()p Fn(\033)s Fx(\))p Fj(P)p Fx([)p Fn(\034)1207 2761 y Fl(\033)1197 2824 y(T)g Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)1476 2799 y Fx(=)25 b Fn(k)s Fx(])1780 2713 y Fi(X)1662 2912 y Fl(\033)1706 2894 y Fe(0)1732 2912 y Fk(2)p Fl(T)10 b Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)2121 2738 y Fn(\031)2173 2752 y Fl(E)2234 2738 y Fx(\()p Fn(\033)2324 2705 y Fk(0)2350 2738 y Fx(\))p 2055 2778 V 2055 2861 a(1)21 b Fm(\000)f Fn(\031)2264 2875 y Fl(E)2325 2861 y Fx(\()p Fn(\033)s Fx(\))2462 2799 y(\005)2530 2813 y Fl(\033)2574 2794 y Fe(0)2603 2799 y Fx(\()p Fn(m;)15 b(n)21 b Fm(\000)f Fn(k)s(;)15 b(E)5 b Fx(\))240 3269 y(=)400 3183 y Fi(X)336 3382 y Fl(\033)r Fk(2)p Fl(T)10 b Fv(\()p Fl(E)s Fv(\))611 3269 y Fn(\031)663 3283 y Fl(E)724 3269 y Fx(\()p Fn(\033)s Fx(\))p Fj(P)p Fx([)p Fn(\034)979 3231 y Fl(\033)969 3294 y(T)g Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)1248 3269 y Fn(>)25 b(m)20 b Fx(+)g Fn(n)p 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4879 y Fl(N)s(;E)1306 4865 y Fx(\()p Fn(n)p Fx(\))26 b Fm(\021)1617 4779 y Fi(X)1553 4978 y Fl(\033)r Fk(2)p Fl(T)10 b Fv(\()p Fl(E)s Fv(\))1828 4865 y Fn(\031)1880 4879 y Fl(E)1941 4865 y Fx(\()p Fn(\033)s Fx(\))p Fj(P)p Fx([)p Fn(\034)2196 4828 y Fl(\033)2186 4890 y(T)g Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)2465 4865 y Fx(=)25 b Fn(n)p Fx(])836 b(\(1)p Fn(:)p Fx(48\))50 5127 y(Eq.)30 b(\(1.46\))j(then)d(b)s(ecomes) 769 5380 y(\005\()p Fn(m;)15 b(n;)g(E)5 b Fx(\))27 b(=)e Fn(F)1375 5394 y Fl(N)s(;E)1519 5380 y Fx(\()p Fn(m)20 b Fx(+)g Fn(n)p Fx(\))g(+)1990 5266 y Fl(n)1947 5293 y Fi(X)1946 5489 y Fl(k)r Fv(=1)2094 5380 y Fn(f)2139 5394 y Fl(N)s(;E)2282 5380 y Fx(\()p Fn(k)s Fx(\)\005\()p Fn(m;)15 b(n)21 b Fm(\000)f Fn(k)s(;)15 b(E)5 b Fx(\))488 b(\(1)p Fn(:)p Fx(49\))p eop %%Page: 12 12 12 11 bop 50 295 a Fr(12)3270 b Fo(Se)l(ction)27 b Fr(2)50 513 y Fx(whic)m(h)33 b(has)h(the)h(form)f(of)h(the)g(equation)f(in)g (the)g(trap)h(mo)s(del.)52 b(Unfortunately)-8 b(,)36 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Fi(\001)2704 2894 y Fv(1)p Fl(=)p Fv(2)2833 2951 y Fx(log)17 b Fn(N)10 b Ft(.)62 b(Ther)-5 b(e)40 b(exists)g(a)50 3097 y(subset)c Fm(E)j(\032)31 b Fx(\012)k Ft(with)i Fn(P)13 b Fx(\()p Fm(E)8 b Fx(\))32 b(=)e(1)p Ft(,)37 b(such)f(that)h(for)g(al)5 b(l)36 b Fn(!)e Fm(2)d(E)8 b Ft(,)36 b(for)h(al)5 b(l)36 b Fn(N)46 b Ft(lar)-5 b(ge)37 b(enough,)f(the)h(fol)5 b(lowing)50 3242 y(holds:)159 3460 y(F)-7 b(or)34 b Fn(")25 b(>)g Fx(0)33 b Ft(a)g(c)-5 b(onstant,)34 b(de\014ne)f(the)g(sets)857 3716 y Fm(B)917 3691 y Fk(p)p 977 3691 99 4 v 55 x Fl("N)1080 3716 y Fx(\()p Fn(\033)s Fx(\))26 b(=)f Fm(f)p Fn(\033)1427 3679 y Fk(0)1479 3716 y Fm(2)g(S)1620 3730 y Fl(N)1714 3716 y Fm(j)g(k)p Fn(\033)1864 3679 y Fk(0)1911 3716 y Fm(\000)20 b Fn(\033)s Fm(k)2102 3730 y Fv(2)2168 3716 y Fm(\024)2264 3635 y(p)p 2340 3635 126 4 v 81 x Fn("N)10 b Fm(g)p Fn(;)77 b(\033)28 b Fm(2)d(S)2833 3730 y Fl(N)3523 3716 y Fx(\(2)p Fn(:)p Fx(1\))50 3973 y Ft(and)1254 4120 y Fn(W)1340 4134 y Fl(")1379 4120 y Fx(\()p Fn(I)7 b Fx(\))26 b Fm(\021)1633 4033 y Fi(\\)1618 4228 y Fl(\033)r Fk(2)p Fl(I)1764 4120 y Fm(B)1827 4082 y Fl(c)1824 4103 y Fk(p)p 1883 4103 99 4 v 1883 4157 a Fl("N)1986 4120 y Fx(\()p Fn(\033)s Fx(\))p Fn(;)77 b(I)33 b Fm(\022)25 b(S)2437 4134 y Fl(N)3523 4120 y Fx(\(2)p Fn(:)p Fx(2\))50 4380 y Ft(Then,)47 4598 y(i\))47 b(F)-7 b(or)34 b(al)5 b(l)33 b Fn(")25 b(>)g Fx(0)33 b Ft(ther)-5 b(e)34 b(exists)f(a)g(c)-5 b(onstant)34 b Fn(c)26 b(>)f Fx(0)33 b Ft(such)f(that,)i(for)f(al)5 b(l)33 b Fn(\021)c Fm(2)c Fn(T)45 b Ft(and)34 b(al)5 b(l)33 b Fn(\033)28 b Fm(2)d Fn(W)3368 4612 y Fl(")3407 4598 y Fx(\()p Fn(T)13 b Fx(\))p Ft(,)1139 4749 y Fi(\014)1139 4804 y(\014)1139 4859 y(\014)1169 4854 y Fj(P)1239 4753 y Fi(\020)1294 4854 y Fn(\034)1344 4816 y Fl(\033)1334 4876 y(\021)1418 4854 y Fn(<)25 b(\034)1564 4816 y Fl(\033)1554 4879 y(T)10 b Fk(n)p Fl(\021)1688 4753 y Fi(\021)1763 4854 y Fm(\000)1885 4818 y Fv(1)p 1865 4833 77 4 v 1865 4885 a Fl(M)1952 4749 y Fi(\014)1952 4804 y(\014)1952 4859 y(\014)2007 4854 y Fm(\024)2166 4818 y Fl(d)p 2114 4833 141 4 v 2114 4885 a(N)d(M)2266 4854 y Fx(\(1)21 b(+)f Fn(c\016)s Fx(\()p Fn(N)10 b Fx(\)\))795 b(\(2)p Fn(:)p Fx(3\))20 5183 y Ft(ii\))46 b(Ther)-5 b(e)33 b(exists)g(a)g(c)-5 b(onstant)35 b Fn(c)25 b(>)g Fx(0)33 b Ft(such)g(that,)h(for)f(al)5 b(l)33 b Fn(\021)c Fm(2)24 b Fn(T)46 b Ft(and)40 b Fx(\026)-52 b Fn(\021)29 b Fm(2)c Fn(T)45 b Ft(with)34 b Fn(\021)28 b Fm(6)p Fx(=)k(\026)-52 b Fn(\021)t Ft(,)987 5335 y Fi(\014)987 5389 y(\014)987 5444 y(\014)1017 5439 y Fn(e)1059 5402 y Fl(\014)1103 5347 y Fk(p)p 1164 5347 65 4 v 1164 5402 a Fl(N)6 b(E)1286 5411 y Fq(\026)-36 b Fp(\021)1326 5439 y Fj(P)1396 5338 y Fi(\020)1451 5439 y Fn(\034)1507 5402 y Fv(\026)-42 b Fl(\021)1491 5462 y(\021)1571 5439 y Fn(<)25 b(\034)1722 5395 y Fv(\026)-41 b Fl(\021)1707 5470 y(T)10 b Fk(n)p Fl(\021)1841 5338 y Fi(\021)1915 5439 y Fm(\000)2037 5403 y Fv(1)p 2017 5418 77 4 v 2017 5470 a Fl(M)2104 5335 y Fi(\014)2104 5389 y(\014)2104 5444 y(\014)2159 5439 y Fm(\024)2318 5403 y Fl(d)p 2266 5418 141 4 v 2266 5470 a(N)d(M)2418 5439 y Fx(\(1)21 b(+)f Fn(c\016)s Fx(\()p Fn(N)10 b Fx(\)\))643 b(\(2)p Fn(:)p Fx(4\))p eop %%Page: 13 13 13 12 bop 50 295 a Fo(A)l(ging)27 b(in)h(the)g(REM.)f(Part)h(2.)2661 b Fr(13)-8 513 y Ft(iii\))46 b(Ther)-5 b(e)33 b(exists)g(a)g(c)-5 b(onstant)35 b Fn(c)25 b(>)g Fx(0)33 b Ft(such)g(that,)h(for)f(al)5 b(l)33 b Fn(\021)c Fm(2)24 b Fn(T)46 b Ft(and)40 b Fx(\026)-52 b Fn(\021)29 b Fm(2)c Fn(T)45 b Ft(with)34 b Fn(\021)28 b Fm(6)p Fx(=)k(\026)-52 b Fn(\021)t Ft(,)952 656 y Fi(\014)952 711 y(\014)952 765 y(\014)982 761 y Fj(P)1052 660 y Fi(\020)1107 761 y Fn(\034)1163 723 y Fv(\026)-42 b Fl(\021)1147 783 y(\021)1227 761 y Fn(<)25 b(\034)1378 717 y Fv(\026)-41 b Fl(\021)1363 792 y(T)10 b Fk(nf)p Fl(\021)r(;)c Fv(\026)-42 b Fl(\021)s Fk(g)1633 660 y Fi(\021)1707 761 y Fm(\000)1875 725 y Fv(1)p 1809 740 170 4 v 1809 792 a Fl(M)7 b Fk(\000)p Fv(1)1989 656 y Fi(\014)1989 711 y(\014)1989 765 y(\014)2045 761 y Fm(\024)2278 725 y Fl(d)p 2151 740 291 4 v 2151 792 a(N)g Fv(\()p Fl(M)g Fk(\000)p Fv(1\))2453 761 y Fx(\(1)21 b(+)f Fn(c\016)s Fx(\()p Fn(N)10 b Fx(\)\))608 b(\(2)p Fn(:)p Fx(5\))6 1082 y Ft(iv\))46 b(Ther)-5 b(e)33 b(exists)g(a)g(c)-5 b(onstant)35 b Fn(c)25 b(>)g Fx(0)33 b Ft(such)g(that,)h(for)f(al)5 b(l)33 b Fn(\021)c Fm(2)24 b Fn(T)13 b Ft(,)728 1225 y Fi(\014)728 1280 y(\014)728 1335 y(\014)759 1330 y Fn(e)801 1292 y Fl(\014)845 1238 y Fk(p)p 905 1238 65 4 v 54 x Fl(N)7 b(E)1023 1301 y Fp(\021)1068 1330 y Fj(P)1138 1229 y Fi(\020)1193 1330 y Fn(\034)1243 1286 y Fl(\021)1233 1361 y(T)j Fk(n)p Fl(\021)1392 1330 y Fn(<)25 b(\034)1538 1292 y Fl(\021)1528 1352 y(\021)1582 1229 y Fi(\021)1657 1330 y Fm(\000)1748 1256 y Fi(\000)1789 1330 y Fx(1)c Fm(\000)1977 1294 y Fv(1)p 1957 1309 77 4 v 1957 1361 a Fl(M)2044 1256 y Fi(\001)2085 1225 y(\014)2085 1280 y(\014)2085 1335 y(\014)2141 1330 y Fm(\024)2237 1256 y Fi(\000)2279 1330 y Fx(1)f Fm(\000)2466 1294 y Fv(1)p 2446 1309 V 2446 1361 a Fl(M)2533 1256 y Fi(\001)2614 1294 y Fl(d)p 2601 1309 65 4 v 2601 1361 a(N)2676 1330 y Fx(\(1)h(+)f Fn(c\016)s Fx(\()p Fn(N)10 b Fx(\)\))385 b(\(2)p Fn(:)p Fx(6\))33 1651 y Ft(v\))47 b(Ther)-5 b(e)33 b(exists)g(a)g(c)-5 b(onstant)35 b Fn(c)25 b(>)g Fx(0)33 b Ft(such)g(that,)h(for)f(al)5 b(l)33 b Fn(\033)39 b(=)-56 b Fm(2)25 b Fn(T)13 b Ft(,)832 1825 y Fi(\000)874 1899 y Fx(1)21 b Fm(\000)1062 1863 y Fv(1)p 1042 1878 77 4 v 1042 1930 a Fl(M)1129 1825 y Fi(\001)14 b(\000)1227 1899 y Fx(1)21 b Fm(\000)1408 1863 y Fl(d)p 1395 1878 65 4 v 1395 1930 a(N)1470 1899 y Fx(\(1)g(+)f Fn(c\016)s Fx(\()p Fn(N)10 b Fx(\)\))1932 1825 y Fi(\001)2001 1899 y Fm(\024)25 b Fn(e)2139 1862 y Fl(\014)2183 1807 y Fk(p)p 2243 1807 V 55 x Fl(N)7 b(E)2361 1871 y Fp(\033)2409 1899 y Fj(P)15 b Fx(\()q Fn(\034)2565 1862 y Fl(\033)2555 1922 y(T)2639 1899 y Fn(<)25 b(\034)2785 1862 y Fl(\033)2775 1922 y(\033)2834 1899 y Fx(\))h Fm(\024)f Fx(1)487 b(\(2)p Fn(:)p Fx(7\))6 2220 y Ft(vi\))46 b(F)-7 b(or)32 b(al)5 b(l)32 b Fn(")26 b(>)f Fx(0)31 b Ft(ther)-5 b(e)33 b(exists)e(a)h(c)-5 b(onstant)33 b Fn(c)26 b(>)f Fx(0)31 b Ft(such)h(that,)h(for)f(al)5 b(l)31 b Fn(\033)39 b(=)-55 b Fm(2)25 b Fn(T)44 b Ft(and)32 b(al)5 b(l)37 b Fx(\026)-50 b Fn(\033)28 b Fm(2)d Fn(W)3356 2234 y Fl(")3395 2220 y Fx(\()p Fn(T)30 b Fm([)17 b Fn(\033)s Fx(\))p Ft(,)1219 2468 y Fj(P)1289 2395 y Fi(\000)1332 2468 y Fn(\034)1386 2431 y Fv(\026)-40 b Fl(\033)1372 2491 y(\033)1456 2468 y Fm(\024)25 b Fn(\034)1606 2431 y Fv(\026)-40 b Fl(\033)1592 2491 y(T)1651 2395 y Fi(\001)1718 2468 y Fm(\024)1845 2432 y Fv(1)p 1825 2447 77 4 v 1825 2500 a Fl(M)1932 2468 y Fx(+)2085 2432 y Fl(d)p 2034 2447 141 4 v 2034 2500 a(N)7 b(M)2185 2468 y Fx(\(1)21 b(+)f Fn(c\016)s Fx(\()p Fn(N)10 b Fx(\)\))876 b(\(2)p Fn(:)p Fx(8\))159 2935 y(The)37 b(next)g(statemen)m(t)i(\(Theorem)e (1.4)i(of)e([BBG1]\))i(giv)m(es)f(sharp)e(estimates)i(on)f(mean)g (transition)50 3080 y(times.)50 3298 y Fd(Theorem)k(2.2:)107 b Ft(Assume)35 b(that)i Fn(\013)32 b Fm(\021)e Fn(\014)5 b(=)1648 3221 y Fm(p)p 1725 3221 197 4 v 1725 3298 a Fx(2)15 b(ln)f(2)31 b Fn(>)g Fx(1)p Ft(.)52 b(Then)36 b(ther)-5 b(e)36 b(exists)g(a)g(subset)3268 3275 y Fi(e)3255 3298 y Fm(E)j(\032)31 b Fx(\012)k Ft(with)50 3444 y Fn(P)13 b Fx(\()169 3421 y Fi(e)156 3444 y Fm(E)8 b Fx(\))26 b(=)f(1)p Ft(,)33 b(such)f(that)i(for)g(al)5 b(l)33 b Fn(!)28 b Fm(2)1326 3421 y Fi(e)1314 3444 y Fm(E)8 b Ft(,)32 b(for)h(al)5 b(l)33 b Fn(N)43 b Ft(lar)-5 b(ge)33 b(enough,)g(the)g(fol)5 b(lowing)34 b(holds:)47 3662 y(i\))47 b(F)-7 b(or)34 b(al)5 b(l)33 b Fn(\021)28 b Fm(2)d Fn(T)13 b Fx(\()p Fn(E)5 b Fx(\))p Ft(,)777 3916 y Fj(E)10 b Fx(\()q Fn(\034)917 3873 y Fl(\021)907 3947 y(T)g Fv(\()p Fl(E)t Fv(\))p Fk(n)p Fl(\021)1160 3916 y Fx(\))26 b(=)1432 3855 y(1)p 1328 3896 255 4 v 1328 3983 a(1)21 b Fm(\000)1515 3947 y Fv(1)p 1495 3962 77 4 v 1495 4015 a Fl(M)1609 3816 y Fi(h)1651 3916 y Fn(e)1693 3879 y Fl(\014)1737 3825 y Fk(p)p 1798 3825 65 4 v 1798 3879 a Fl(N)6 b(E)1918 3851 y Fq(+)1915 3897 y Fp(\021)1995 3916 y Fx(+)20 b Fm(W)2176 3933 y Fl(\014)s(;N)s(;T)10 b Fv(\()p Fl(E)s Fv(\))2495 3816 y Fi(i)2553 3916 y Fx(\(1)21 b(+)f Fn(O)s Fx(\(1)p Fn(=)-5 b(N)10 b Fx(\)\))433 b(\(2)p Fn(:)p Fx(9\))20 4267 y Ft(ii\))46 b(F)-7 b(or)34 b(al)5 b(l)33 b Fn(\033)38 b(=)-55 b Fm(2)25 b Fn(T)13 b Fx(\()p Fn(E)5 b Fx(\))p Ft(,)410 4522 y Fj(E)11 b Fx(\()p Fn(\034)550 4484 y Fl(\033)540 4547 y(T)f Fv(\()p Fl(E)t Fv(\))717 4522 y Fx(\))25 b Fm(\024)989 4460 y Fx(1)p 884 4501 255 4 v 884 4589 a(1)c Fm(\000)1072 4553 y Fv(1)p 1052 4568 77 4 v 1052 4620 a Fl(M)1165 4421 y Fi(h)1208 4522 y Fn(e)1250 4484 y Fl(\014)1294 4430 y Fk(p)p 1354 4430 65 4 v 54 x Fl(N)7 b(E)1475 4457 y Fq(+)1472 4503 y Fp(\033)1552 4522 y Fx(+)20 b Fm(W)1733 4538 y Fl(\014)s(;N)s(;T)10 b Fv(\()p Fl(E)s Fv(\))2051 4421 y Fi(i)2110 4522 y Fx(\(1)21 b(+)e Fn(O)s Fx(\(1)p Fn(=)-5 b(N)10 b Fx(\)\))410 4794 y Fj(E)h Fx(\()p Fn(\034)550 4757 y Fl(\033)540 4819 y(T)f Fv(\()p Fl(E)t Fv(\))717 4794 y Fx(\))25 b Fm(\025)989 4733 y Fx(1)p 884 4774 255 4 v 884 4861 a(1)c Fm(\000)1072 4825 y Fv(1)p 1052 4840 77 4 v 1052 4893 a Fl(M)1165 4666 y Fi(\024)1213 4794 y Fn(e)1255 4757 y Fl(\014)1299 4703 y Fk(p)p 1359 4703 65 4 v 54 x Fl(N)7 b(E)1480 4730 y Fq(+)1477 4775 y Fp(\033)1557 4794 y Fx(+)1659 4733 y(1)20 b Fm(\000)g Fn(e)1857 4700 y Fl(E)1919 4733 y Fx(\()p Fn(\013)h Fm(\000)f Fx(1\))p 1659 4774 547 4 v 1761 4857 a(1)h(+)f(1)p Fn(=)-5 b(M)2216 4794 y Fm(W)2306 4811 y Fl(\014)s(;N)s(;T)10 b Fv(\()p Fl(E)s Fv(\))2624 4666 y Fi(\025)2688 4794 y Fx(\(1)21 b(+)e Fn(O)s Fx(\(1)p Fn(=)-5 b(N)10 b Fx(\)\))3477 4670 y(\(2)p Fn(:)p Fx(10\))-8 5145 y Ft(iii\))46 b(F)-7 b(or)34 b(al)5 b(l)33 b Fn(\021)s(;)22 b Fx(\026)-52 b Fn(\021)29 b Fm(2)c Fn(T)13 b Fx(\()p Fn(E)5 b Fx(\))p Ft(,)33 b Fn(\021)c Fm(6)p Fx(=)i(\026)-51 b Fn(\021)s Ft(,)678 5295 y Fi(\014)678 5350 y(\014)678 5405 y(\014)708 5400 y Fj(E)11 b Fx(\()p Fn(\034)848 5356 y Fl(\021)844 5420 y Fv(\026)-42 b Fl(\021)924 5400 y Fm(j)25 b Fn(\034)1024 5356 y Fl(\021)1020 5420 y Fv(\026)-42 b Fl(\021)1094 5400 y Fm(\024)25 b Fn(\034)1240 5356 y Fl(\021)1230 5431 y(T)10 b Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\021)1477 5400 y Fx(\))21 b Fm(\000)f Fj(E)10 b Fx(\()q Fn(\034)1764 5356 y Fl(\021)1754 5431 y(T)g Fv(\()p Fl(E)s Fv(\))q Fk(n)p Fl(\021)2007 5400 y Fx(\))2042 5295 y Fi(\014)2042 5350 y(\014)2042 5405 y(\014)2098 5400 y Fm(\024)2310 5338 y Fx(1)p 2205 5379 255 4 v 2205 5467 a(1)21 b Fm(\000)2393 5431 y Fv(1)p 2373 5446 77 4 v 2373 5498 a Fl(M)2471 5400 y Fm(W)2561 5416 y Fl(\014)s(;N)s(;T)10 b Fv(\()p Fl(E)s Fv(\))2880 5400 y Fn(O)s Fx(\(1)p Fn(=)-5 b(N)10 b Fx(\))287 b(\(2)p Fn(:)p Fx(11\))p eop %%Page: 14 14 14 13 bop 50 295 a Fr(14)3270 b Fo(Se)l(ction)27 b Fr(3)159 513 y Ft(wher)-5 b(e)778 692 y Fm(W)868 708 y Fl(\014)s(;N)s(;T)10 b Fv(\()p Fl(E)s Fv(\))1213 692 y Fm(\021)1320 630 y Fn(e)1362 597 y Fv(\()p Fl(\013)p Fk(\000)p Fv(1\))p Fl(E)s Fv(+)p Fl(\014)1714 543 y Fk(p)p 1776 543 65 4 v 1776 597 a Fl(N)c(u)1883 606 y Fp(N)1942 597 y Fv(\(0\))p 1320 671 720 4 v 1487 754 a Fn(M)k Fx(\()p Fn(\013)22 b Fm(\000)e Fx(1\))2065 564 y Fi(\022)2132 692 y Fx(1)h(+)f Fm(V)2345 706 y Fl(N)s(;E)2488 692 y Fn(e)2530 654 y Fl(E)s(=)p Fv(2)2737 630 y Fn(\013)g Fm(\000)g Fx(1)p 2676 671 337 4 v 2676 689 a Fm(p)p 2752 689 261 4 v 71 x Fx(2)p Fn(\013)h Fm(\000)f Fx(1)3023 564 y Fi(\023)3477 692 y Fx(\(2)p Fn(:)p Fx(12\))50 929 y Ft(and)33 b Fm(V)282 943 y Fl(N)s(;E)458 929 y Ft(is)g(a)g(r)-5 b(andom)35 b(variable)e(of)g(me)-5 b(an)34 b(zer)-5 b(o)33 b(and)h(varianc)-5 b(e)33 b(one.)159 1147 y Fx(W)-8 b(e)31 b(will)d(also)j(mak)m(e)g(use)f (of)g(the)h(follo)m(wing)e(simple)f(corollary)i(to)h(this)e(prop)s (osition:)50 1365 y Fd(Corollary)38 b(2.3:)91 b Ft(Under)33 b(the)g(assumptions)i(and)f(with)f(the)g(notation)h(of)f(The)-5 b(or)g(em)35 b(2.2)e(we)g(have:)47 1583 y(i\))47 b(F)-7 b(or)34 b(al)5 b(l)33 b Fn(\021)s(;)22 b Fx(\026)-52 b Fn(\021)29 b Fm(2)c Fn(T)13 b Ft(,)32 b Fn(\021)d Fm(6)p Fx(=)i(\026)-51 b Fn(\021)s Ft(:)159 1739 y Fi(\014)159 1794 y(\014)159 1848 y(\014)159 1903 y(\014)159 1958 y(\014)374 1837 y Fx(1)p 200 1878 394 4 v 200 1961 a Fm(j)p Fn(T)13 b Fx(\()p Fn(E)5 b Fx(\))21 b Fm(n)g Fn(\021)s Fm(j)720 1812 y Fi(X)625 2011 y Fv(\026)-41 b Fl(\021)r Fk(2)p Fl(T)10 b Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\021)967 1898 y Fj(E)h Fx(\()p Fn(\034)1107 1855 y Fl(\021)1103 1918 y Fv(\026)-42 b Fl(\021)1183 1898 y Fm(j)25 b Fn(\034)1283 1855 y Fl(\021)1279 1918 y Fv(\026)-42 b Fl(\021)1353 1898 y Fm(\024)25 b Fn(\034)1499 1855 y Fl(\021)1489 1929 y(T)10 b Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\021)1736 1898 y Fx(\))p Fm(\000)p Fj(E)h Fx(\()p Fn(\034)1982 1855 y Fl(\021)1978 1918 y Fv(\026)-42 b Fl(\021)2058 1898 y Fm(j)25 b Fn(\034)2158 1855 y Fl(\021)2154 1918 y Fv(\026)-42 b Fl(\021)2228 1898 y Fm(\024)25 b Fn(\034)2374 1855 y Fl(\021)2364 1929 y(T)10 b Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\021)2612 1898 y Fx(\))2647 1739 y Fi(\014)2647 1794 y(\014)2647 1848 y(\014)2647 1903 y(\014)2647 1958 y(\014)2702 1898 y Fm(\024)2914 1837 y Fx(1)p 2809 1878 255 4 v 2809 1965 a(1)21 b Fm(\000)2997 1929 y Fv(1)p 2977 1944 77 4 v 2977 1997 a Fl(M)3075 1898 y Fm(W)3165 1915 y Fl(\014)s(;N)s(;T)10 b Fv(\()p Fl(E)s Fv(\))3484 1898 y Fn(O)s Fx(\(1)p Fn(=)-5 b(N)10 b Fx(\))3477 2114 y(\(2)p Fn(:)p Fx(13\))20 2332 y Ft(ii\))46 b(F)-7 b(or)34 b(al)5 b(l)33 b Fn(\021)28 b Fm(2)d Fn(T)13 b Fx(\()p Fn(E)5 b Fx(\))p Ft(,)491 2608 y Fx(0)25 b Fn(<)g Fj(E)11 b Fx(\()p Fn(\034)797 2564 y Fl(\021)787 2639 y(T)f Fv(\()p Fl(E)t Fv(\))p Fk(n)q Fl(\021)1041 2608 y Fx(\))20 b Fm(\000)g Fj(P)1242 2570 y Fk(\000)p Fv(1)1340 2608 y Fx(\()p Fn(\034)1425 2564 y Fl(\021)1415 2639 y(T)10 b Fk(n)p Fl(\021)1575 2608 y Fn(<)25 b(\034)1721 2570 y Fl(\021)1711 2630 y(\021)1765 2608 y Fx(\))h Fm(\024)2037 2546 y Fx(1)p 1933 2587 255 4 v 1933 2675 a(1)20 b Fm(\000)2120 2639 y Fv(1)p 2100 2654 77 4 v 2100 2706 a Fl(M)2198 2608 y Fm(W)2288 2624 y Fl(\014)s(;N)s(;T)10 b Fv(\()p Fl(E)s Fv(\))2607 2608 y Fx(\(1)21 b(+)f Fn(O)s Fx(\(1)p Fn(=)-5 b(N)10 b Fx(\)\))333 b(\(2)p Fn(:)p Fx(14\))50 3095 y Fc(Pro)s(of)31 b(of)f(Corollary)h(2.3:)83 b Fx(The)26 b(\014rst)f(assertion)h(is)g(an)g(ob)m(vious)g(consequence)h(of)f(the)h (last)f(assertion)50 3241 y(of)i(Theorem)g(2.2.)41 b(The)28 b(second)g(assertion)g(simply)d(follo)m(ws)j(from)f(Eq.)40 b(\(3.8\))30 b(of)e([BBG1])j(and)c(is)g(pro)m(v)m(en)50 3386 y(just)j(as)g(the)h(\014rst)e(assertion)h(of)h(Theorem)f(2.2.)42 b Fm(})159 3605 y Fx(Equipp)s(ed)37 b(with)i(this)h(information)e(w)m (e)j(pro)s(ceed)f(in)f(the)i(next)f(section)h(to)g(analyse)f(the)h (Laplace)50 3750 y(transforms)29 b(of)i(the)f(distribution)d(functions) i(of)i(suc)m(h)f(transition)f(times.)50 4024 y Fw(3.)55 b(Estimates)40 b(on)h(Laplace)f(transforms.)159 4243 y Fx(W)-8 b(e)40 b(will)c(use)j(the)g(metho)s(d)f(of)h(Laplace)g (tranforms)f(to)i(solv)m(e)f(the)g(system)g(of)g(renew)m(al)g (equations)50 4388 y(\(1.44\).)h(Doing)22 b(so)h(this)e(will)f(require) h(precise)g(con)m(trol)i(on)f(the)h(Laplace)f(transforms)g(of)g(the)h (distributions)50 4534 y(functions)37 b(of)h(the)g(probabilit)m(y)e (distributions)f(app)s(earing)i(in)f(these)j(equations.)64 b(In)38 b(this)f(section)h(w)m(e)50 4679 y(deriv)m(e)30 b(the)g(basic)g(estimates)h(on)f(these)h(Laplace)g(transforms.)159 4897 y(As)g(in)e([BEGK1],)k(Section)e(3,)g(the)h(\014rst)e(crucial)g (step)h(is)e(an)i(estimate)h(of)f(the)g(maximal)f(mean)h(time)50 5043 y(to)g(reac)m(h)g(the)g(set)g Fn(T)13 b Fx(\()p Fn(E)5 b Fx(\).)50 5261 y Fd(Lemma)38 b(3.1:)91 b Ft(De\014ne)1471 5421 y Fx(\002\()p Fn(E)5 b Fx(\))26 b Fm(\021)38 b Fx(max)1806 5480 y Fl(\033)r Fk(2S)1944 5489 y Fp(N)2017 5421 y Fj(E)11 b Fn(\034)2122 5384 y Fl(\033)2112 5446 y(T)f Fv(\()p Fl(E)t Fv(\))3523 5421 y Fx(\(3)p Fn(:)p Fx(1\))p eop %%Page: 15 15 15 14 bop 50 295 a Fo(A)l(ging)27 b(in)h(the)g(REM.)f(Part)h(2.)2661 b Fr(15)50 513 y Ft(and)60 742 y Fi(b)50 765 y Fx(\002\()p Fn(E)5 b Fx(\))26 b Fm(\021)f Fx(\(1)p Fm(\000)634 729 y Fv(1)p 547 744 210 4 v 547 796 a Fk(j)p Fl(T)10 b Fv(\()p Fl(E)s Fv(\))p Fk(j)768 765 y Fx(\))803 727 y Fk(\000)p Fv(1)901 765 y Fn(e)943 727 y Fl(\014)987 673 y Fk(p)p 1047 673 65 4 v 54 x Fl(N)d(u)1155 736 y Fp(N)1213 727 y Fv(\(0\)+)p Fl(\013E)1486 637 y Fi(\024)1534 765 y Fx(1)20 b(+)1893 703 y Fn(e)1935 670 y Fk(\000)p Fl(E)p 1701 744 546 4 v 1701 827 a Fm(j)p Fn(T)13 b Fx(\()p Fn(E)5 b Fx(\))p Fm(j)p Fx(\()p Fn(\013)23 b Fm(\000)c Fx(1\))2272 637 y Fi(\022)2339 765 y Fx(1)i(+)f Fm(V)7 b Fn(e)2601 727 y Fl(E)s(=)p Fv(2)2808 703 y Fn(\013)20 b Fm(\000)g Fx(1)p 2747 744 337 4 v 2747 762 a Fm(p)p 2823 762 261 4 v 72 x Fx(2)p Fn(\013)h Fm(\000)f Fx(1)3094 637 y Fi(\023\025)3224 765 y Fx(\()q(1)g(+)g Fn(O)s Fx(\(1)p Fn(=)-5 b(N)10 b Fx(\)\))3523 929 y(\(3)p Fn(:)p Fx(2\))50 1074 y Ft(wher)-5 b(e)33 b Fm(V)40 b Ft(is)33 b(a)g(r)-5 b(andom)35 b(variable)e(of)g(me)-5 b(an)33 b(zer)-5 b(o)34 b(and)f(varianc)-5 b(e)34 b Fx(1)p Ft(.)42 b(Then,)33 b(under)g(the)g(assumptions)h(of)50 1219 y(The)-5 b(or)g(em)34 b(2.2,)1605 1365 y Fx(\002\()p Fn(E)5 b Fx(\))26 b Fm(\024)1951 1342 y Fi(b)1940 1365 y Fx(\002\()p Fn(E)5 b Fx(\))1370 b(\(3)p Fn(:)p Fx(3\))50 1790 y Fc(Pro)s(of:)65 b Fx(F)-8 b(or)39 b Fn(\033)i Fm(62)d Fn(T)13 b Fx(\()p Fn(E)5 b Fx(\),)41 b(the)e(b)s(ound)d Fj(E)10 b Fn(\034)1592 1757 y Fl(\033)1581 1820 y(T)g Fv(\()p Fl(E)t Fv(\))1796 1790 y Fm(\024)1915 1767 y Fi(b)1905 1790 y Fx(\002\()p Fn(E)5 b Fx(\))39 b(follo)m(ws)e(immediately)g(from)h(the)g(estimate)50 1936 y(from)30 b(Theorem)g(2.2,)i(i\).)40 b(If)30 b Fn(\033)e Fm(2)d Fn(T)13 b Fx(\()p Fn(E)5 b Fx(\),)31 b(the)g(forw)m(ard)f (Kolmogoro)m(v)h(equation)f(sho)m(ws)g(that)722 2186 y Fj(E)11 b Fn(\034)827 2149 y Fl(\033)817 2211 y(T)f Fv(\()p Fl(E)t Fv(\))1019 2186 y Fx(=)1192 2100 y Fi(X)1115 2299 y Fl(\033)1159 2281 y Fe(0)1184 2299 y Fk(2)p Fl(T)g Fv(\()p Fl(E)s Fv(\))1415 2186 y Fn(p)1461 2200 y Fl(N)1529 2186 y Fx(\()p Fn(\033)n(;)15 b(\033)1709 2149 y Fk(0)1736 2186 y Fx(\))20 b(+)1959 2100 y Fi(X)1882 2299 y Fl(\033)1926 2281 y Fe(0)1951 2299 y Fk(62)p Fl(T)10 b Fv(\()p Fl(E)s Fv(\))2182 2186 y Fn(p)2228 2200 y Fl(N)2297 2186 y Fx(\()p Fn(\033)n(;)15 b(\033)2477 2149 y Fk(0)2503 2186 y Fx(\)\(1)22 b(+)d Fj(E)11 b Fn(\034)2835 2149 y Fl(\033)2880 2122 y Fe(0)2825 2211 y Fl(T)f Fv(\()p Fl(E)t Fv(\))3002 2186 y Fx(\))486 b(\(3)p Fn(:)p Fx(4\))50 2507 y(Using)29 b(the)i(previous)e(result)g(in)g(\(3.4\))j(one)f(sees)g(that)g(the)f (same)h(estimate)g(holds)e(in)g(this)g(case.)42 b Fm(})159 2725 y Fx(W)-8 b(e)31 b(de\014ne,)f(for)g Fn(\033)f Fm(2)c(S)964 2739 y Fl(N)1033 2725 y Fx(,)30 b Fn(I)7 b(;)15 b(J)35 b Fm(\032)25 b(S)1411 2739 y Fl(N)1480 2725 y Fx(,)31 b(and)f Fn(u)25 b Fm(2)g Fn(D)j Fm(\032)d Fj(C)17 b Fx(,)933 3007 y Fn(G)1005 2969 y Fl(\033)1005 3029 y(I)5 b(;J)1114 3007 y Fx(\()p Fn(u)p Fx(\))26 b Fm(\021)f Fj(E)11 b Fn(e)1455 2969 y Fl(u\034)1539 2942 y Fp(\033)1532 2988 y(I)1593 3007 y Fx(1)-23 b(I)1648 3023 y Fk(f)p Fl(\034)1726 3001 y Fp(\033)1719 3047 y(I)1769 3023 y Fk(\024)p Fl(\034)1867 3001 y Fp(\033)1860 3047 y(J)1910 3023 y Fk(g)1977 3007 y Fm(\021)2106 2893 y Fk(1)2076 2921 y Fi(X)2073 3113 y Fl(n)p Fv(=1)2226 3007 y Fj(P)p Fx([)p Fn(\034)2356 2969 y Fl(\033)2346 3029 y(I)2431 3007 y Fx(=)25 b Fn(n)f Fm(\024)h Fn(\034)2752 2969 y Fl(\033)2742 3029 y(J)2801 3007 y Fx(])697 b(\(3)p Fn(:)p Fx(5\))50 3305 y(where)30 b Fn(D)k Fx(is)c(c)m(hosen)h(suc)m(h)g(that)g(the)g(righ)m(t-hand)f (side)f(of)i(\(3.5\))i(exists.)42 b(Note)32 b(that)g(this)d(is)h(alw)m (a)m(ys)i(the)50 3450 y(case)d(for)e Fn(u)h Fx(s.t.)40 b Fm(<)p Fx(\()p Fn(u)p Fx(\))25 b Fm(\024)g Fx(0,)k(but)e(in)f(fact,)k (for)d Fn(x;)15 b(I)7 b(;)15 b(J)38 b Fx(giv)m(en,)28 b(there)g(will)d(b)s(e)i(some)h Fn(u)2955 3464 y Fv(0)3021 3450 y Fm(\021)d Fn(u)3169 3464 y Fv(0)3210 3450 y Fx(\()p Fn(\033)n(;)15 b(I)7 b(;)15 b(J)9 b Fx(\))27 b Fn(>)e Fx(0,)50 3596 y(s.t.)41 b Fn(G)284 3563 y Fl(\033)284 3621 y(I)5 b(;J)393 3596 y Fx(\()p Fn(u)p Fx(\))31 b(exists)f(for)g (all)f Fn(u)i Fx(with)e Fm(<)p Fx(\()p Fn(u)p Fx(\))c Fm(\024)g Fn(u)1713 3610 y Fv(0)1754 3596 y Fx(.)50 3814 y Fd(Theorem)37 b(3.2:)51 b Ft(F)-7 b(or)34 b(any)f Fn(\033)28 b Fm(2)d Fn(T)13 b Fx(\()p Fn(E)5 b Fx(\))p Ft(,)33 b(the)g(L)-5 b(aplac)g(e)35 b(tr)-5 b(ansform)35 b Fn(G)2512 3781 y Fl(\033)2512 3843 y(T)10 b Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)2764 3814 y Fx(\()p Fn(u)p Fx(\))33 b Ft(c)-5 b(an)34 b(b)-5 b(e)32 b(written)i(as)923 4074 y Fn(G)995 4036 y Fl(\033)995 4099 y(T)10 b Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)1247 4074 y Fx(\()p Fn(u)p Fx(\))26 b(=)1939 4012 y Fn(a)1987 4026 y Fl(\033)p 1502 4053 972 4 v 1502 4136 a Fx(1)21 b Fm(\000)f Fx(\(1)h Fm(\000)f Fn(e)1893 4110 y Fk(\000)p Fl(u)1997 4136 y Fx(\))p Fj(E)11 b Fn(\034)2137 4105 y Fl(\033)2127 4167 y(T)f Fv(\()p Fl(E)t Fv(\))p Fk(n)q Fl(\033)2386 4136 y Fn(b)2425 4150 y Fl(\033)2505 4074 y Fx(+)20 b Fn(R)2665 4088 y Fl(\033)2714 4074 y Fx(\()p Fn(u)p Fx(\))687 b(\(3)p Fn(:)p Fx(6\))50 4360 y Ft(wher)-5 b(e)1315 4506 y Fn(a)1363 4520 y Fl(\033)1437 4506 y Fx(=)25 b(1)c(+)f Fn(O)s Fx(\()1807 4483 y Fi(b)1797 4506 y Fx(\002\()p Fn(E)5 b Fx(\))p Fn(=)p Fj(E)12 b Fn(\034)2161 4468 y Fl(\033)2151 4531 y(T)e Fv(\()p Fl(E)t Fv(\))p Fk(n)p Fl(\033)2409 4506 y Fx(\))1079 b(\(3)p Fn(:)p Fx(7\))1320 4713 y Fn(b)1359 4727 y Fl(\033)1433 4713 y Fx(=)25 b(1)20 b(+)g Fn(O)s Fx(\()1802 4690 y Fi(b)1792 4713 y Fx(\002\()p Fn(E)5 b Fx(\))p Fn(=)p Fj(E)12 b Fn(\034)2156 4676 y Fl(\033)2146 4738 y(T)e Fv(\()p Fl(E)t Fv(\))q Fk(n)p Fl(\033)2405 4713 y Fx(\))1083 b(\(3)p Fn(:)p Fx(8\))50 4921 y Ft(and)32 b Fn(R)294 4935 y Fl(\033)343 4921 y Fx(\()p Fn(u)p Fx(\))h Ft(is)e(analytic)i(in) f(the)g(half-plane)h Fm(<)p Fx(\()p Fn(u)p Fx(\))25 b Fn(<)g Fx(1)p Fn(=)2024 4898 y Fi(b)2014 4921 y Fx(\002)q(\()p Fn(E)5 b Fx(\))p Ft(,)33 b(p)-5 b(erio)g(dic)33 b(with)g(p)-5 b(erio)g(d)33 b Fx(2)p Fn(\031)i Ft(in)d(the)g(imag-)50 5066 y(inary)h(dir)-5 b(e)g(ction,)34 b(and)g(satis\014es)10 5284 y(\(i\))47 b(for)33 b(al)5 b(l)33 b Fm(j)p Fn(u)p Fm(j)26 b(\024)f Fn(a=)761 5261 y Fi(b)751 5284 y Fx(\002\()p Fn(E)5 b Fx(\))p Ft(,)1267 5439 y Fm(j)p Fn(R)1361 5453 y Fl(\033)1410 5439 y Fx(\()p Fn(u)p Fx(\))p Fm(j)26 b(\024)f Fn(C)7 b Fx(\()p Fn(a)p Fx(\))1869 5338 y Fi(\020)1924 5439 y Fn(e)1966 5402 y Fk(\000)p Fl(\014)2067 5347 y Fk(p)p 2127 5347 65 4 v 55 x Fl(N)g(E)2245 5411 y Fp(\033)2303 5416 y Fi(b)2293 5439 y Fx(\002\()p Fn(E)e Fx(\))2506 5338 y Fi(\021)2561 5357 y Fv(2)3523 5439 y Fx(\(3)p Fn(:)p Fx(9\))p eop %%Page: 16 16 16 15 bop 50 295 a Fr(16)3270 b Fo(Se)l(ction)27 b Fr(3)159 513 y Ft(and)-18 731 y(\(ii\))47 b(for)33 b(al)5 b(l)33 b Fn(u)g Ft(with)g Fm(<)p Fx(\()p Fn(u)p Fx(\))26 b Fn(<)f Fx(\(1)c Fm(\000)f Fn(\017)p Fx(\))1300 708 y Fi(b)1290 731 y Fx(\002\()p Fn(E)5 b Fx(\))33 b Ft(and)h Fm(j)p Fx(1)21 b Fm(\000)f Fn(e)1937 698 y Fk(\000)p Fl(u)2041 731 y Fm(j)26 b(\025)f Fx(2)p Fn(\017)2270 698 y Fk(\000)p Fv(1)2368 731 y Fn(e)2410 698 y Fk(\000)p Fl(\014)2511 644 y Fk(p)p 2571 644 65 4 v 54 x Fl(N)7 b(E)2689 707 y Fp(\033)1203 1027 y Fm(j)p Fn(R)1297 1041 y Fl(\033)1346 1027 y Fx(\()p Fn(u)p Fx(\))p Fm(j)26 b(\024)f Fx(2)1979 965 y Fn(e)2021 932 y Fk(\000)p Fl(\014)2122 878 y Fk(p)p 2182 878 V 54 x Fl(N)7 b(E)2300 941 y Fp(\033)p 1672 1006 984 4 v 1672 1105 a Fm(j)p Fx(1)21 b Fm(\000)f Fn(e)1896 1079 y Fk(\000)p Fl(u)2000 1105 y Fm(j)p Fx(\(1)h Fm(\000)f(<)p Fx(\()p Fn(u)p Fx(\))2415 1082 y Fi(b)2405 1105 y Fx(\002\()p Fn(E)5 b Fx(\)\))3477 1027 y(\(3)p Fn(:)p Fx(10\))159 1308 y Ft(Mor)-5 b(e)g(over,)1630 1454 y Fn(a)1678 1468 y Fl(\033)1747 1454 y Fx(+)20 b Fn(R)1907 1468 y Fl(\033)1956 1454 y Fx(\(0\))26 b(=)f(1)1239 b(\(3)p Fn(:)p Fx(11\))159 1881 y(This)31 b(prop)s(osition)f(allo)m(ws)h(in)g(fact)j(to)f(pro)m(v) m(e)g(v)m(ery)g(go)s(o)s(d)f(estimates)h(on)g(the)f(distribution)d (function)50 2027 y(of)h Fn(\034)203 1994 y Fl(\033)193 2056 y(T)10 b Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)446 2027 y Fx(.)40 b(Note)32 b(\014rst)e(that)h(if)1296 2340 y Fm(L)p Fx(\()p Fn(u)p Fx(\))26 b Fm(\021)1635 2226 y Fk(1)1605 2254 y Fi(X)1603 2446 y Fl(n)p Fv(=0)1755 2340 y Fn(e)1797 2303 y Fl(un)1890 2340 y Fj(P)p Fx([)p Fn(\034)2020 2303 y Fl(\033)2010 2365 y(T)10 b Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)2288 2340 y Fn(>)25 b(n)p Fx(])1013 b(\(3)p Fn(:)p Fx(12\))50 2637 y(then)1414 2795 y Fm(L)p Fx(\()p Fn(u)p Fx(\))26 b(=)1731 2714 y Fn(G)1803 2681 y Fl(\033)1803 2743 y(T)10 b Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)2056 2714 y Fx(\()p Fn(u)p Fx(\))20 b Fm(\000)g Fx(1)p 1731 2774 604 4 v 1910 2857 a Fn(e)1952 2831 y Fl(u)2020 2857 y Fm(\000)g Fx(1)3477 2795 y(\(3)p Fn(:)p Fx(13\))50 3079 y Fd(Corollary)50 b(3.3:)74 b Ft(With)41 b(the)g(notation)i(of)e(The)-5 b(or)g(em)43 b(3.2,)h(for)e(any)f Fn(\017)g(>)f Fx(0)h Ft(and)h(for)g(any)g(p)-5 b(ositive)50 3225 y(inte)g(ger)33 b Fn(n)24 b Fm(2)h Fj(N)7 b Ft(,)639 3461 y Fj(P)p Fx([)p Fn(\034)769 3424 y Fl(\033)759 3486 y(T)j Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)1038 3461 y Fx(=)25 b Fn(n)p Fx(])g(=)1492 3400 y Fn(a)1540 3414 y Fl(\033)p 1320 3440 441 4 v 1320 3523 a Fj(E)11 b Fn(\034)1425 3492 y Fl(\033)1415 3554 y(T)f Fv(\()p Fl(E)t Fv(\))p Fk(n)p Fl(\033)1673 3523 y Fn(b)1712 3537 y Fl(\033)1772 3461 y Fn(e)1814 3419 y Fk(\000)p Fl(n=)p Fh(E)p Fl(\034)2031 3391 y Fp(\033)2024 3440 y(T)j Fq(\()p Fp(E)r Fq(\))p Fe(n)p Fp(\033)2249 3419 y Fl(b)2281 3428 y Fp(\033)1330 3714 y Fx(+)20 b Fn(O)1508 3613 y Fi(\020)1562 3714 y Fn(e)1604 3677 y Fk(\000)p Fl(n)p Fv(\(1)p Fk(\000)p Fl(\017)p Fv(\))p Fl(=)1925 3673 y Fi(b)1922 3677 y Fv(\002)q(\()p Fl(E)s Fv(\))2097 3714 y Fn(e)2139 3677 y Fk(\000)p Fl(\014)2240 3622 y Fk(p)p 2301 3622 65 4 v 2301 3677 a Fl(N)6 b(E)2418 3686 y Fp(\033)2466 3714 y Fn(\017)2503 3677 y Fk(\000)p Fv(1)2616 3714 y Fx(ln)2707 3613 y Fi(\020)2771 3691 y(b)2761 3714 y Fx(\002\()p Fn(E)f Fx(\))p Fn(\017)3011 3613 y Fi(\021)q(\021)3477 3590 y Fx(\(3)p Fn(:)p Fx(14\))50 3960 y Ft(and)33 b(\(for)h Fn(n)25 b(>)g Fx(0)p Ft(\))309 4215 y Fj(P)p Fx([)p Fn(\034)439 4178 y Fl(\033)429 4240 y(T)10 b Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)708 4215 y Fn(>)25 b(n)p Fx(])g(=)g Fn(a)1053 4229 y Fl(\033)1102 4215 y Fn(e)1144 4173 y Fk(\000)p Fl(n=)p Fh(E)p Fl(\034)1361 4146 y Fp(\033)1354 4195 y(T)13 b Fq(\()p Fp(E)r Fq(\))p Fe(n)p Fp(\033)1579 4173 y Fl(b)1611 4182 y Fp(\033)1679 4215 y Fx(+)20 b Fn(O)1857 4115 y Fi(\020)1911 4215 y Fn(e)1953 4178 y Fk(\000)p Fl(n)p Fv(\(1)p Fk(\000)p Fl(\017)p Fv(\))p Fl(=)2274 4174 y Fi(b)2271 4178 y Fv(\002)q(\()p Fl(E)s Fv(\))2447 4215 y Fn(e)2489 4178 y Fk(\000)p Fl(\014)2590 4124 y Fk(p)p 2650 4124 V 54 x Fl(N)6 b(E)2767 4187 y Fp(\033)2826 4193 y Fi(b)2815 4216 y Fx(\002)2886 4215 y(\()p Fn(E)f Fx(\))p Fn(\017)3065 4178 y Fk(\000)p Fv(1)3164 4115 y Fi(\021)3477 4214 y Fx(\(3)p Fn(:)p Fx(15\))50 4687 y Fc(Pro)s(of)44 b(of)g(Theorem)f(3.2:)121 b Fx(Our)37 b(analysis)g(of)h(the)g(Laplace)h(transforms)e(will)e(follo)m(w)j (closely)g(the)50 4832 y(strategy)43 b(emplo)m(y)m(ed)g(in)d([BEGK1],) 47 b(but)41 b(some)i(simpli\014cations)38 b(will)i(o)s(ccur)i(due)f(to) i(the)f(particular)50 4978 y(prop)s(erties)29 b(of)h(the)h(mo)s(del)e (at)i(hand.)p eop %%Page: 17 17 17 16 bop 50 295 a Fo(A)l(ging)27 b(in)h(the)g(REM.)f(Part)h(2.)2661 b Fr(17)50 513 y Fu(3.1)35 b(A)g(priori)h(estimates)d(on)i(Laplace)g (transforms.)159 756 y Fx(As)26 b(in)f([BEGK1],)k(Lemma)d(3.1)h (implies)d(immediate)h(con)m(trol)i(on)f(the)g(Laplace)h(transforms)e Fn(g)3417 723 y Fl(\033)3461 696 y Fe(0)3414 779 y Fl(\033)3491 756 y Fx(\()p Fn(u)p Fx(\))h Fm(\021)50 902 y Fn(G)122 869 y Fl(\033)166 841 y Fe(0)122 931 y Fl(\033)n(;T)10 b Fv(\()p Fl(E)s Fv(\))354 902 y Fx(\()p Fn(u)p Fx(\):)50 1120 y Fd(Lemma)38 b(3.4:)91 b Ft(F)-7 b(or)34 b(al)5 b(l)33 b Fn(\017)25 b(>)g Fx(0)p Ft(,)33 b(and)g(for)h(al)5 b(l)33 b(r)-5 b(e)g(al)34 b Fn(u)25 b Fm(\024)g Fx(\(1)c Fm(\000)f Fn(\017)p Fx(\))p Fn(=)2406 1097 y Fi(b)2396 1120 y Fx(\002\()p Fn(E)5 b Fx(\))p Ft(,)34 b(for)f(al)5 b(l)33 b Fn(\033)n(;)15 b(\033)3091 1087 y Fk(0)3143 1120 y Fm(2)25 b(S)3284 1134 y Fl(N)3353 1120 y Ft(,)1348 1362 y Fn(g)1394 1325 y Fl(\033)1438 1297 y Fe(0)1391 1385 y Fl(\033)1468 1362 y Fx(\()p Fn(u)p Fx(\))h Fm(\024)1911 1301 y Fx(1)p 1723 1341 423 4 v 1723 1441 a(1)20 b Fm(\000)g Fn(u)1941 1418 y Fi(b)1931 1441 y Fx(\002\()p Fn(E)5 b Fx(\))2181 1362 y Fm(\024)25 b Fn(\017)2314 1325 y Fk(\000)p Fv(1)3477 1362 y Fx(\(3)p Fn(:)p Fx(16\))50 1817 y Fc(Pro)s(of:)53 b Fx(The)32 b(pro)s(of)g(is)g(iden)m(tical)g(to) i(the)f(pro)s(of)f(of)h(Lemma)g(3.2)h(of)f([BEGK1].)49 b(Just)32 b(note)i(that)f(if)f(w)m(e)50 1962 y(set)1086 2185 y Fn(v)1130 2199 y Fl(u)1178 2185 y Fx(\()p Fn(\033)1268 2147 y Fk(0)1294 2185 y Fx(\))26 b Fm(\021)1451 1971 y Fi(8)1451 2053 y(>)1451 2080 y(<)1451 2244 y(>)1451 2271 y(:)1562 2052 y Fn(g)1608 2020 y Fl(\033)1652 1992 y Fe(0)1605 2075 y Fl(\033)1682 2052 y Fx(\()p Fn(u)p Fx(\))p Fn(;)107 b Fx(for)45 b Fn(\033)2145 2020 y Fk(0)2196 2052 y Fm(62)25 b Fn(T)13 b Fx(\()p Fn(E)5 b Fx(\))21 b Fm([)f Fn(\033)1562 2189 y Fx(1)p Fn(;)304 b Fx(for)45 b Fn(\033)2145 2156 y Fk(0)2196 2189 y Fx(=)25 b Fn(\033)1562 2325 y Fx(0)p Fn(;)304 b Fx(for)45 b Fn(\033)2145 2292 y Fk(0)2196 2325 y Fm(2)25 b Fn(T)13 b Fx(\()p Fn(E)5 b Fx(\))p Fm(n)p Fn(\033)3477 2185 y Fx(\(3)p Fn(:)p Fx(17\))50 2481 y(then)30 b Fn(v)301 2495 y Fl(u)379 2481 y Fx(is)f(the)i(unique)d(solution)h(of)i(the)f(Diric)m(hlet)g (problem)1037 2675 y(\(1)21 b Fm(\000)f Fn(e)1271 2638 y Fl(u)1318 2675 y Fn(P)1376 2689 y Fl(N)1446 2675 y Fx(\))p Fn(v)1525 2689 y Fl(u)1573 2675 y Fx(\()p Fn(\033)1663 2638 y Fk(0)1689 2675 y Fx(\))25 b(=)g(0)p Fn(;)107 b Fx(if)90 b Fn(\033)2221 2638 y Fk(0)2272 2675 y Fm(62)25 b Fn(T)13 b Fx(\()p Fn(E)5 b Fx(\))21 b Fm([)f Fn(\033)1507 2848 y(v)1551 2862 y Fl(u)1598 2848 y Fx(\()p Fn(\033)s Fx(\))26 b(=)f(1)1481 3021 y Fn(v)1525 3035 y Fl(u)1573 3021 y Fx(\()p Fn(\033)1663 2983 y Fk(0)1689 3021 y Fx(\))g(=)g(0)92 b(if)e Fn(\033)2181 2983 y Fk(0)2232 3021 y Fm(2)24 b Fn(T)13 b Fx(\()p Fn(E)5 b Fx(\))p Fm(n)p Fn(\033)3477 2848 y Fx(\(3)p Fn(:)p Fx(18\))50 3221 y(Setting)30 b Fn(w)428 3235 y Fl(u)476 3221 y Fx(\()p Fn(\033)566 3188 y Fk(0)592 3221 y Fx(\))25 b Fm(\021)g Fn(v)792 3235 y Fl(u)840 3221 y Fx(\()p Fn(\033)930 3188 y Fk(0)956 3221 y Fx(\))20 b Fm(\000)g Fn(v)1146 3235 y Fv(0)1187 3221 y Fx(\()p Fn(\033)1277 3188 y Fk(0)1303 3221 y Fx(\),)31 b(w)m(e)g(see)g(that)g Fn(w)1937 3235 y Fl(u)2015 3221 y Fx(solv)m(es)785 3421 y(\(1)21 b Fm(\000)f Fn(P)1035 3435 y Fl(N)1105 3421 y Fx(\))p Fn(w)1205 3435 y Fl(u)1253 3421 y Fx(\()p Fn(\033)1343 3383 y Fk(0)1369 3421 y Fx(\))25 b(=)g(\(1)c Fm(\000)f Fn(e)1759 3383 y Fk(\000)p Fl(u)1864 3421 y Fx(\))p Fn(v)1943 3435 y Fl(u)1991 3421 y Fx(\()p Fn(\033)2081 3383 y Fk(0)2107 3421 y Fx(\))p Fn(;)106 b Fx(if)90 b Fn(\033)2472 3383 y Fk(0)2523 3421 y Fm(62)25 b Fn(T)13 b Fx(\()p Fn(E)5 b Fx(\))21 b Fm([)f Fn(\033)1140 3593 y(w)1205 3607 y Fl(u)1253 3593 y Fx(\()p Fn(\033)1343 3556 y Fk(0)1369 3593 y Fx(\))25 b(=)g(0)92 b(if)d Fn(\033)1860 3556 y Fk(0)1911 3593 y Fm(2)25 b Fn(T)13 b Fx(\()p Fn(E)5 b Fx(\))21 b Fm([)f Fn(\033)3477 3507 y Fx(\(3)p Fn(:)p Fx(19\))50 3793 y(The)30 b(solution)f(of)h(\(3.19\))j(can)d(b)s(e)g (represen)m(ted)h(as)1150 4134 y Fn(w)1215 4148 y Fl(u)1262 4134 y Fx(\()p Fn(\033)1352 4096 y Fk(0)1378 4134 y Fx(\))26 b(=)f Fj(E)1605 3996 y Fl(\034)1646 3968 y Fp(\033)1685 3950 y Fe(0)1639 4017 y Fp(T)13 b Fq(\()p Fp(E)r Fq(\))p Fe([)p Fp(\033)1873 3996 y Fk(\000)p Fv(1)1723 4048 y Fi(X)1729 4240 y Fl(t)p Fv(=1)1966 4134 y Fx(\(1)21 b Fm(\000)f Fn(e)2200 4096 y Fk(\000)p Fl(u)2305 4134 y Fx(\))p Fn(v)2384 4148 y Fl(u)2432 4134 y Fx(\()p Fn(X)2542 4148 y Fl(t)2575 4134 y Fx(\))867 b(\(3)p Fn(:)p Fx(20\))50 4412 y(implying)27 b(that)770 4748 y Fn(v)814 4762 y Fl(u)861 4748 y Fx(\()p Fn(\033)951 4710 y Fk(0)977 4748 y Fx(\))f(=)f Fj(P)p Fx([)p Fn(\034)1264 4710 y Fl(\033)1308 4683 y Fe(0)1254 4770 y Fl(\033)1363 4748 y Fx(=)g Fn(\034)1509 4710 y Fl(\033)1553 4683 y Fe(0)1499 4772 y Fl(T)10 b Fv(\()p Fl(E)s Fv(\))p Fk([)p Fl(\033)1763 4748 y Fx(])21 b(+)f(\(1)h Fm(\000)f Fn(e)2134 4710 y Fk(\000)p Fl(u)2238 4748 y Fx(\))p Fj(E)2344 4610 y Fl(\034)2385 4582 y Fp(\033)2424 4564 y Fe(0)2378 4631 y Fp(T)13 b Fq(\()p Fp(E)r Fq(\))p Fe([)p Fp(\033)2612 4610 y Fk(\000)p Fv(1)2462 4661 y Fi(X)2468 4854 y Fl(t)p Fv(=1)2720 4748 y Fn(v)2764 4762 y Fl(u)2812 4748 y Fx(\()p Fn(X)2922 4762 y Fl(t)2955 4748 y Fx(\))487 b(\(3)p Fn(:)p Fx(21\))50 5031 y(Setting)30 b Fn(S)5 b Fx(\()p Fn(u)p Fx(\))26 b Fm(\021)f Fx(max)837 5048 y Fl(\033)881 5029 y Fe(0)906 5048 y Fk(62)p Fl(T)10 b Fv(\()p Fl(E)s Fv(\))p Fk([)p Fl(\033)1235 5031 y Fn(v)1279 5045 y Fl(u)1326 5031 y Fx(\()p Fn(\033)1416 4998 y Fk(0)1442 5031 y Fx(\),)31 b(\(3.21\))i(implies)984 5253 y Fn(S)5 b Fx(\()p Fn(u)p Fx(\))26 b Fm(\024)f Fx(1)20 b(+)g(\(1)h Fm(\000)f Fn(e)1679 5216 y Fk(\000)p Fl(u)1784 5253 y Fx(\))120 b(max)1834 5316 y Fl(\033)1878 5298 y Fe(0)1903 5316 y Fk(62)p Fl(T)10 b Fv(\()p Fl(E)s Fv(\))p Fk([)p Fl(\033)2227 5253 y Fj(E)h Fn(\034)2332 5216 y Fl(\033)2376 5188 y Fe(0)2322 5278 y Fl(T)f Fv(\()p Fl(E)t Fv(\))p Fk([)p Fl(\033)2592 5253 y Fn(S)5 b Fx(\()p Fn(u)p Fx(\))1193 5466 y Fm(\024)25 b Fx(1)20 b(+)g Fn(u)1507 5443 y Fi(b)1497 5466 y Fx(\002\()p Fn(E)5 b Fx(\))p Fn(S)g Fx(\()p Fn(u)p Fx(\))3477 5353 y(\(3)p Fn(:)p Fx(22\))p eop %%Page: 18 18 18 17 bop 50 295 a Fr(18)3270 b Fo(Se)l(ction)27 b Fr(3)50 513 y Fx(and)j(hence)1505 675 y Fn(S)5 b Fx(\()p Fn(u)p Fx(\))26 b Fm(\024)2010 614 y Fx(1)p 1821 655 423 4 v 1821 754 a(1)21 b Fm(\000)f Fn(u)2040 731 y Fi(b)2030 754 y Fx(\002)o(\()p Fn(E)5 b Fx(\))3477 675 y(\(3)p Fn(:)p Fx(23\))50 930 y(whic)m(h)29 b(pro)m(v)m(es)i(the)g(lemma.)p Fm(})159 1148 y Fx(This)d(basic)i(estimate)h(can)g(b)s(e)f(impro)m(v)m (ed)g(in)f(certain)h(cases:)50 1366 y Fd(Lemma)38 b(3.5:)51 b Ft(L)-5 b(et)33 b Fn(\033)28 b Fm(2)d Fn(T)13 b Fx(\()p Fn(E)5 b Fx(\))p Ft(.)42 b(Then,)33 b(for)g Fn(u)g Ft(as)g(in)g(L)-5 b(emma)34 b(3.4,)10 1584 y(\(i\))794 1755 y Fn(G)866 1718 y Fl(\033)866 1780 y(T)10 b Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)n(;\033)1180 1755 y Fx(\()p Fn(u)p Fx(\))26 b Fm(\024)f Fn(e)1466 1718 y Fk(\000)p Fl(\014)1567 1664 y Fk(p)p 1627 1664 65 4 v 54 x Fl(N)7 b(E)1745 1727 y Fp(\033)1970 1694 y Fn(e)2012 1661 y Fl(u)p 1804 1735 423 4 v 1804 1834 a Fx(1)20 b Fm(\000)g Fn(u)2022 1811 y Fi(b)2012 1834 y Fx(\002\()p Fn(E)5 b Fx(\))2262 1755 y Fm(\024)25 b Fx(2)p Fn(\017)2440 1718 y Fk(\000)p Fv(1)2538 1755 y Fj(P)p Fx([)p Fn(\034)2668 1718 y Fl(\033)2658 1780 y(T)10 b Fv(\()p Fl(E)s Fv(\))2855 1755 y Fn(<)25 b(\034)3001 1718 y Fl(\033)2991 1778 y(\033)3050 1755 y Fx(])402 b(\(3)p Fn(:)p Fx(24\))-18 2088 y Ft(\(ii\))437 2296 y Fn(G)509 2258 y Fl(\033)509 2321 y(\033)n(;T)10 b Fv(\()p Fl(E)s Fv(\))742 2296 y Fx(\()p Fn(u)p Fx(\))26 b Fm(\024)f Fn(e)1028 2258 y Fl(u)1091 2140 y Fi( )1163 2296 y Fx(1)20 b(+)g Fn(e)1361 2258 y Fk(\000)p Fl(\014)1462 2204 y Fk(p)p 1523 2204 65 4 v 1523 2258 a Fl(N)6 b(E)1640 2267 y Fp(\033)1888 2234 y Fx(1)p 1699 2275 423 4 v 1699 2374 a(1)21 b Fm(\000)f Fn(u)1918 2351 y Fi(b)1908 2374 y Fx(\002)o(\()p Fn(E)5 b Fx(\))2132 2140 y Fi(!)2230 2296 y Fm(\024)25 b Fx(1)20 b(+)g(2)p Fn(\017)2564 2258 y Fk(\000)p Fv(1)2662 2296 y Fj(P)p Fx([)p Fn(\034)2792 2258 y Fl(\033)2782 2321 y(T)10 b Fv(\()p Fl(E)s Fv(\))2979 2296 y Fn(<)25 b(\034)3125 2258 y Fl(\033)3115 2318 y(\033)3174 2296 y Fx(])278 b(\(3)p Fn(:)p Fx(25\))50 2740 y Fc(Pro)s(of:)71 b Fx(Let)41 b(us)f(\014rst)g(pro)m(v)m(e)i(\(i\).)72 b(This)39 b(go)s(es)i(essen)m(tially)f(along)h(the)g(same)g(lines)e(as) i(the)g(pro)s(of)f(of)50 2886 y(Lemma)30 b(3.4.)42 b(De\014ne)946 3237 y Fn( )1005 3251 y Fl(u)1053 3237 y Fx(\()p Fn(\033)1143 3199 y Fk(0)1169 3237 y Fx(\))25 b Fm(\021)1325 3023 y Fi(8)1325 3105 y(>)1325 3132 y(<)1325 3296 y(>)1325 3323 y(:)1436 3097 y Fn(G)1508 3065 y Fl(\033)1552 3037 y Fe(0)1508 3127 y Fl(T)10 b Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)n(;\033)1823 3097 y Fx(\()p Fn(u)p Fx(\))p Fn(;)107 b Fx(for)45 b Fn(\033)2286 3065 y Fk(0)2337 3097 y Fm(62)25 b Fn(T)13 b Fx(\()p Fn(E)5 b 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Fx(,)31 b(\(i\))f(is)g(pro)m(v)m(en.)p eop %%Page: 19 19 19 18 bop 50 295 a Fo(A)l(ging)27 b(in)h(the)g(REM.)f(Part)h(2.)2661 b Fr(19)159 513 y Fx(In)30 b(the)g(same)h(w)m(a)m(y)-8 b(,)803 743 y Fn(G)875 706 y Fl(\033)875 768 y(\033)n(;T)10 b Fv(\()p Fl(E)s Fv(\))1107 743 y Fx(\()p Fn(u)p Fx(\))26 b(=)f Fn(e)1393 706 y Fl(u)1441 743 y Fn(p)1487 757 y Fl(N)1556 743 y Fx(\()p Fn(\033)n(;)15 b(\033)s Fx(\))22 b(+)d Fn(e)1925 706 y Fl(u)2007 657 y Fi(X)1988 852 y Fl(\033)2032 834 y Fe(0)2057 852 y Fk(6)p Fv(=)p Fl(\033)2173 743 y Fn(p)2219 757 y Fl(N)2288 743 y Fx(\()p Fn(\033)n(;)c(\033)2468 706 y Fk(0)2494 743 y Fx(\))p Fn(G)2601 706 y Fl(\033)2645 678 y Fe(0)2601 768 y Fl(\033)n(;T)10 b Fv(\()p Fl(E)s Fv(\))2834 743 y Fx(\()p Fn(u)p Fx(\))1255 1069 y Fm(\024)1351 914 y Fi( )1423 1069 y Fx(1)21 b(+)f Fn(e)1622 1032 y Fk(\000)p Fl(\014)1723 978 y Fk(p)p 1783 978 65 4 v 54 x Fl(N)7 b(E)1901 1041 y Fp(\033)2148 1008 y Fx(1)p 1960 1048 423 4 v 1960 1148 a(1)20 b Fm(\000)g Fn(u)2178 1125 y 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5434 y(T)f Fk(n)p Fl(\033)2060 5409 y Fx(])20 b Fm(\000)2238 5334 y Fj(E)11 b Fn(\034)2343 5301 y Fl(\033)2333 5356 y(\033)2398 5334 y Fx(1)-23 b(I)2453 5350 y Fk(f)p Fl(\034)2531 5332 y Fp(\033)2524 5368 y(\033)2575 5350 y Fk(\024)p Fl(\034)2673 5328 y Fp(\033)2666 5374 y(T)2716 5350 y Fk(g)p 2207 5388 582 4 v 2207 5471 a Fx(1)21 b Fm(\000)e Fj(P)p Fx([)p Fn(\034)2493 5445 y Fl(\033)2483 5494 y(\033)2569 5471 y Fm(\024)24 b Fn(\034)2714 5440 y Fl(\033)2704 5498 y(T)2763 5471 y Fx(])3477 5409 y(\(3)p Fn(:)p Fx(35\))p eop %%Page: 20 20 20 19 bop 50 295 a Fr(20)3270 b Fo(Se)l(ction)27 b Fr(3)50 513 y Fx(whic)m(h)i(implies)f(immediately)g(that)1488 677 y Fj(E)11 b Fn(\034)1593 644 y Fl(\033)1583 699 y(\033)1648 677 y Fx(1)-23 b(I)1703 693 y Fk(f)p Fl(\034)1781 675 y Fp(\033)1774 711 y(\033)1825 693 y Fk(\024)p Fl(\034)1923 671 y Fp(\033)1916 720 y(T)7 b Fq(\()p Fp(E)r Fq(\))2063 693 y Fk(g)p 1311 746 972 4 v 1311 829 a Fj(P)p Fx([)p Fn(\034)1441 798 y Fl(\033)1431 860 y(T)j Fv(\()p Fl(E)s 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Fx(\))1203 4611 y Fv(2)1255 4416 y Fi(\014)1255 4471 y(\014)1255 4525 y(\014)1255 4580 y(\014)1255 4634 y(\014)1311 4575 y Fm(\024)1432 4514 y Fn(e)1474 4481 y Fk(\000)p Fl(\014)1575 4426 y Fk(p)p 1635 4426 65 4 v 55 x Fl(N)d(E)1753 4490 y Fp(\033)1801 4514 y Fn(C)g(\015)1925 4481 y Fk(\000)p Fv(2)p 1417 4554 619 4 v 1417 4637 a Fj(P)p Fx([)p Fn(\034)1547 4606 y Fl(\033)1537 4668 y(T)j Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)1816 4637 y Fn(<)25 b(\034)1962 4611 y Fl(\033)1952 4660 y(\033)2011 4637 y Fx(])2173 4491 y Fi(b)2163 4514 y Fx(\002\()p Fn(E)5 b Fx(\))2376 4481 y Fv(2)p 2058 4554 465 4 v 2058 4637 a Fx(\()p Fj(E)11 b Fn(\034)2198 4606 y Fl(\033)2188 4668 y(T)f Fv(\()p Fl(E)t Fv(\))p Fk(n)p Fl(\033)2446 4637 y Fx(\))2481 4611 y Fv(2)2559 4575 y Fm(\024)24 b Fx(2)p Fn(C)7 b(\015)2823 4538 y Fk(\000)p Fv(3)3047 4491 y Fi(b)3037 4514 y Fx(\002\()p Fn(E)e Fx(\))3250 4481 y Fv(2)p 2932 4554 V 2932 4637 a Fx(\()p Fj(E)11 b Fn(\034)3072 4606 y Fl(\033)3062 4668 y(T)f Fv(\()p Fl(E)t 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295 a Fo(A)l(ging)27 b(in)h(the)g(REM.)f(Part)h(2.)2661 b Fr(21)50 513 y Fx(This)27 b(can)i(b)s(e)f(b)s(ounded)e(in)i(the)h (same)g(w)m(a)m(y)h(as)f(b)s(efore,)g(using)e(the)i(Cauc)m(h)m(y)g (estimates)g(under)e(the)i(same)50 658 y(assumptions)f(on)j Fn(u)f Fx(\(with)g(a)g(di\013eren)m(t)g(constan)m(t)i Fn(C)7 b Fx(\),)30 b(b)m(y)1072 806 y Fi(\014)1072 861 y(\014)1072 915 y(\014)1072 970 y(\014)1139 876 y Fn(d)p 1113 917 100 4 v 1113 1000 a(du)1223 938 y(G)1295 900 y Fl(\033)1295 963 y(T)10 b Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)n(;\033)1610 938 y Fx(\()c(~)-51 b Fn(u)p Fx(\))1732 806 y Fi(\014)1732 861 y(\014)1732 915 y(\014)1732 970 y(\014)1788 938 y Fm(\024)25 b Fn(e)1926 900 y Fk(\000)p Fl(\014)2027 846 y Fk(p)p 2087 846 65 4 v 54 x Fl(N)7 b(E)2205 909 y Fp(\033)2253 938 y Fn(C)g(\015)2377 900 y Fk(\000)p Fv(2)2484 915 y Fi(b)2474 938 y Fx(\002\()p Fn(E)e Fx(\))790 b(\(3)p Fn(:)p Fx(46\))50 1217 y(This)29 b(sho)m(ws)i(that)h(\(I)s(I\))e(can)i(b)s(e)e(estimated)h(as)h(a)f 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3906 y Fx(\))2794 3873 y Fv(2)2835 3832 y Fi(\001)2877 3906 y Fx(.)159 4124 y(This)23 b(implies)f(that)k(the)f(function)f Fn(G)1430 4091 y Fl(\033)1430 4153 y(T)10 b Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)1682 4124 y Fx(\()p Fn(u)p Fx(\))25 b(has)g(simple)e(p)s(oles)h(at)i Fn(u)2647 4138 y Fl(\033)2696 4124 y Fx(\()p Fn(mod)9 b Fx(+)g Fn(i)p Fx(2)p Fn(\031)s Fx(\),)28 b(and)d(all)e(other)50 4269 y(p)s(oles)29 b(satisfy)i Fm(<)p Fx(\()p Fn(u)p Fx(\))26 b Fm(\025)885 4246 y Fi(b)875 4269 y Fx(\002\()p Fn(E)5 b Fx(\))1088 4236 y Fk(\000)p Fv(1)1186 4269 y Fx(,)31 b Fm(=)p Fx(\()p Fn(u)p Fx(\))26 b(=)g(0)31 b(or)g Fm(=)p Fx(\()p Fn(u)p Fx(\))26 b(=)f Fn(\031)s Fx(.)42 b(Moreo)m(v)m(er,)33 b(Prop)s(osition)c(3.6)i (implies)d(that)50 4415 y(the)i(residue)f(at)i Fn(u)678 4429 y Fl(\033)758 4415 y Fx(equals)582 4706 y(res)709 4720 y Fl(u)752 4729 y Fp(\033)825 4706 y Fx(=)974 4626 y Fn(G)1046 4593 y Fl(\033)1046 4655 y(T)10 b Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)1298 4626 y Fx(\()p Fn(u)1385 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b(will)g(pro)m(v)m(e)i(v)m(ery)h(helpful)c(in)h(obtaining)h(go)s(o)s(d) h(b)s(ounds.)50 5308 y(As)41 b(a)h(\014rst)f(observ)-5 b(ation,)44 b(w)m(e)e(note)g(that)g(the)f(domain)f(of)i(v)-5 b(alidit)m(y)40 b(of)h(\(3.31\))j(can)e(b)s(e)e(immediately)50 5453 y(extended)30 b(to)h(the)g(set)g Fn(\032)25 b(<)g Fj(E)11 b Fn(\034)1119 5420 y Fl(\033)1109 5482 y(T)f Fv(\()p Fl(E)s Fv(\))q Fk(n)p Fl(\033)1367 5453 y Fn(=)1422 5430 y Fi(b)1412 5453 y Fx(\002\()p Fn(E)5 b Fx(\).)p eop %%Page: 22 22 22 21 bop 50 295 a Fr(22)3270 b Fo(Se)l(ction)27 b Fr(3)159 513 y Fx(W)-8 b(e)39 b(will)34 b(no)m(w)k(estimate)g(the)g (di\013erence)f(b)s(et)m(w)m(een)h Fn(G)2077 480 y Fl(\033)2077 542 y(T)10 b Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)2329 513 y Fx(\()p Fn(u)p Fx(\))38 b(and)f(the)h(con)m(tribution)e(from)h (the)50 658 y(p)s(ole)29 b(at)i Fn(u)407 672 y Fl(\033)456 658 y Fx(.)41 b(W)-8 b(e)32 b(set)952 951 y Fn(R)1021 965 y Fl(\033)1070 951 y Fx(\()p Fn(u)p Fx(\))26 b(=)f Fn(G)1386 914 y Fl(\033)1386 976 y(T)10 b Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)1638 951 y Fx(\()p Fn(u)p Fx(\))21 b(+)2000 871 y Fn(G)2072 838 y Fl(\033)2072 900 y(T)10 b Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)n(;T)g Fv(\()p Fl(E)s Fv(\))2508 871 y Fx(\()p Fn(u)2595 885 y Fl(\033)2644 871 y Fx(\))p 1883 930 914 4 v 1883 1021 a(\()p Fn(u)20 b Fm(\000)g Fn(u)2133 1035 y Fl(\033)2182 1021 y Fx(\))2250 986 y Fl(d)p 2229 1001 81 4 v 2229 1053 a(du)2320 1021 y Fn(G)2392 990 y Fl(\033)2392 1052 y(\033)n(;T)10 b Fv(\()p Fl(E)s Fv(\))2625 1021 y Fx(\()p Fn(u)2712 1035 y Fl(\033)2761 1021 y Fx(\))3477 951 y(\(3)p Fn(:)p Fx(50\))50 1284 y(W)-8 b(e)27 b(\014rst)f(giv)m(e)h(a)g(uniform)d(estimate)j(of)g (the)f(mo)s(dulus)e(of)j Fn(R)2122 1298 y Fl(\033)2197 1284 y Fx(on)f(the)h(disk)e Fm(j)p Fn(u)p Fm(j)h(\024)f Fn(a=)2985 1261 y Fi(b)2975 1284 y Fx(\002\()p Fn(E)5 b Fx(\),)28 b Fn(a)e(<)f Fx(1.)39 b(Note)50 1429 y(that)27 b(a)g(straigh)m(tforw)m(ard)g(computation)g(and)f(the)h(use)g(of)g(T)-8 b(a)m(ylor)27 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1859 3421 4 v 1156 1950 a(\(1)21 b Fm(\000)f Fn(G)1420 1918 y Fl(\033)1420 1981 y(\033)n(;T)10 b Fv(\()p Fl(E)s Fv(\))1653 1950 y Fx(\()p Fn(u)p Fx(\)\)\()p Fn(u)21 b Fm(\000)f Fn(u)2061 1964 y Fl(\033)2110 1950 y Fx(\))2178 1914 y Fl(d)p 2157 1929 81 4 v 2157 1981 a(du)2248 1950 y Fn(G)2320 1918 y Fl(\033)2320 1981 y(\033)n(;T)10 b Fv(\()p Fl(E)s Fv(\))2553 1950 y Fx(\()p Fn(u)2640 1964 y Fl(\033)2689 1950 y Fx(\))123 2206 y(=)262 2090 y Fl(d)p 241 2105 V 241 2157 a(du)333 2125 y Fn(G)405 2092 y Fl(\033)405 2155 y(\033)n(;T)g Fv(\()p Fl(E)s Fv(\))637 2125 y Fx(\()p Fn(u)724 2139 y Fl(\033)773 2125 y Fx(\))841 2090 y Fl(d)p 820 2105 V 820 2157 a(du)911 2125 y Fn(G)983 2092 y Fl(\033)983 2155 y(T)g Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)n(;T)g Fv(\()p Fl(E)s Fv(\))1420 2125 y Fx(\()c(~)-51 b Fn(u)p Fx(\))21 b Fm(\000)1664 2090 y Fv(1)p 1664 2105 37 4 v 1664 2157 a(2)1712 2125 y Fn(G)1784 2092 y Fl(\033)1784 2155 y(T)10 b Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)n(;T)g Fv(\()p Fl(E)s 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Fn(=)2300 5435 y Fi(b)2290 5458 y Fx(\002)q(\()p Fn(E)5 b Fx(\))p eop %%Page: 23 23 23 22 bop 50 295 a Fo(A)l(ging)27 b(in)h(the)g(REM.)f(Part)h(2.)2661 b Fr(23)50 513 y Fx(T)-8 b(o)24 b(do)g(so,)i(w)m(e)e(rely)f(on)h (\(3.32\).)41 b(W)-8 b(e)25 b(will)c(use)j(\(3.24\))i(to)f(b)s(ound)d (the)i(n)m(umerator)g Ft(uniformly)h Fx(in)d(the)j(imag-)50 658 y(inary)32 b(part)i(of)f Fn(u)p Fx(,)i(while)c(the)j(denominator)f (will)e(pro)m(vide)i(extra)h(deca)m(y)h(in)d(the)i(imaginary)e (direction.)50 804 y(Note)g(that)e(b)m(y)h(\(3.30\),)715 1002 y Fi(\014)715 1057 y(\014)715 1111 y(\014)746 1107 y Fn(G)818 1069 y Fl(\033)818 1131 y(\033)n(;T)10 b Fv(\()p Fl(E)s Fv(\))1050 1107 y Fx(\()p Fn(u)p Fx(\))21 b Fm(\000)f Fn(p)1330 1121 y Fl(N)1399 1107 y Fx(\()p Fn(\033)n(;)15 b(\033)s Fx(\))p Fn(e)1656 1069 y Fl(u)1705 1002 y Fi(\014)1705 1057 y(\014)1705 1111 y(\014)1761 1107 y Fm(\024)25 b Fn(e)1899 1069 y Fk(\000)p Fl(\014)2000 1015 y Fk(p)p 2060 1015 65 4 v 54 x Fl(N)6 b(E)2177 1078 y 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4175 y Fn(;)15 b Fj(C)2290 4142 y Fl(M)2376 4175 y Fx(\))p Ft(,)33 b Fn(K)2556 4142 y Fk(\003)2549 4200 y Fl(E)2611 4175 y Fx(\()p Fn(u)p Fx(\))g Ft(is)10 4393 y(\(i\))47 b(Perio)-5 b(dic)33 b(with)h(p)-5 b(erio)g(d)34 b Fx(2)p Fn(\031)i Ft(in)d(the)g(imaginary)h(dir)-5 b(e)g(ction.)-18 4611 y(\(ii\))47 b(Mer)-5 b(omorphic)35 b(in)d Fj(C)56 b Ft(with)34 b(p)-5 b(oles)34 b(only)f(on)g(the)g(p)-5 b(ositive)34 b(r)-5 b(e)g(al)34 b(axis)f(and)h(its)f Fx(2)p Fn(\031)j Ft(tr)-5 b(anslates.)-46 4829 y(\(iii\))47 b(F)-7 b(or)34 b Fn(\033)28 b Fm(6)p Fx(=)d Fn(\033)560 4796 y Fk(0)611 4829 y Fm(2)g Fn(T)13 b Fx(\()p Fn(E)5 b Fx(\))p Ft(,)1403 5022 y Fn(K)1487 4985 y Fk(\003)1480 5045 y Fl(\033)n(;\033)1586 5026 y Fe(0)1616 5022 y Fx(\()p Fn(u)p Fx(\))26 b(=)1935 4942 y Fn(G)2007 4909 y Fl(\033)2007 4971 y(\033)2051 4953 y Fe(0)2075 4971 y Fl(;T)10 b Fv(\()p Fl(E)s Fv(\))2268 4942 y Fx(\()p Fn(u)p Fx(\))p 1871 5001 584 4 v 1871 5085 a(1)20 b Fm(\000)g Fn(G)2099 5053 y Fl(\033)2099 5115 y(\033)n(;T)10 b Fv(\()p Fl(E)s Fv(\))2332 5085 y Fx(\()p Fn(u)p Fx(\))3523 5022 y(\(4)p Fn(:)p Fx(8\))159 5462 y(The)30 b(follo)m(wing)f(observ)-5 b(ation)30 b(will)d(b)s(e)j(extremely)h(useful:)p eop %%Page: 26 26 26 25 bop 50 295 a Fr(26)3270 b Fo(Se)l(ction)27 b Fr(4)50 513 y Fd(Lemma)38 b(4.2:)51 b Ft(F)-7 b(or)33 b(any)h Fn(u)25 b Fm(2)g Fj(C)56 b Ft(for)33 b(which)h Fn(G)1763 480 y Fl(\033)1763 542 y(T)10 b Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)n(;T)g Fv(\()p Fl(E)s Fv(\))2199 513 y Fx(\()p Fn(u)p Fx(\))33 b Ft(is)g(\014nite,)1203 688 y Fi(X)1085 887 y Fl(\033)1129 869 y Fe(0)1154 887 y Fk(2)p Fl(T)10 b Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)1466 774 y Fn(G)1538 737 y Fl(\033)1538 799 y(\033)1582 781 y Fe(0)1607 799 y Fl(;T)g Fv(\()p Fl(E)s Fv(\))1799 774 y Fx(\()p Fn(u)p Fx(\))26 b(=)f Fn(G)2115 737 y Fl(\033)2115 799 y(T)10 b Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)n(;T)g Fv(\()p Fl(E)s Fv(\))2552 774 y Fx(\()p Fn(u)p Fx(\))849 b(\(4)p Fn(:)p Fx(9\))50 1252 y Fc(Pro)s(of:)47 b Fx(it)30 b(is)g(enough)g(to)h(pro)m(v)m(e)g(\(4.9\))h(for)e Fn(u)h Fx(in)e(the)h(negativ)m(e)i(imaginary)d(half)g(plane.)40 b(No)m(w)1153 1509 y(1)-23 b(I)1209 1523 y Fl(\034)1250 1501 y Fp(\033)1243 1550 y(T)7 b Fq(\()p Fp(E)r Fq(\))p Fe(n)p Fp(\033)1462 1523 y Fk(\024)p Fl(\034)1560 1501 y Fp(\033)1553 1550 y(T)g Fq(\()p Fp(E)r Fq(\))1730 1509 y Fx(=)1943 1423 y Fi(X)1826 1622 y Fl(\033)1870 1604 y Fe(0)1895 1622 y Fk(2)p Fl(T)j Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)2207 1509 y Fx(1)-23 b(I)2263 1523 y Fl(\034)2304 1501 y Fp(\033)2297 1554 y(\033)2336 1541 y Fe(0)2364 1523 y Fk(\024)p Fl(\034)2462 1501 y Fp(\033)2455 1550 y(T)7 b Fq(\()p Fp(E)r Fq(\))3477 1509 y Fx(\(4)p Fn(:)p Fx(10\))50 1831 y(Th)m(us)265 2048 y Fn(G)337 2010 y Fl(\033)337 2073 y(T)j Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)n(;T)g Fv(\()p Fl(E)s Fv(\))773 2048 y Fx(\()p Fn(u)p Fx(\))26 b(=)p Fj(E)11 b Fn(e)1089 2006 y Fl(u\034)1173 1978 y Fp(\033)1166 2027 y(T)i Fq(\()p Fp(E)r Fq(\))p Fe(n)p Fp(\033)1396 2048 y Fx(1)-23 b(I)1452 2062 y Fl(t)1479 2040 y Fp(\033)1479 2088 y(T)8 b Fq(\()p Fp(E)r Fq(\))p Fe(n)p Fp(\033)1699 2062 y Fk(\024)p Fl(\034)1797 2040 y Fp(\033)1790 2088 y(T)f Fq(\()p Fp(E)r Fq(\))1966 2048 y Fx(=)25 b Fj(E)2256 1962 y Fi(X)2132 2161 y Fl(\033)2177 2143 y Fe(0)2207 2161 y Fk(2)p Fl(T)10 b Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)2519 2048 y Fn(e)2561 2006 y Fl(u\034)2645 1978 y Fp(\033)2638 2027 y(T)e Fq(\()p Fp(E)r Fq(\))p Fe(n)p Fp(\033)2863 2048 y Fx(1)-23 b(I)2918 2062 y Fl(\034)2959 2040 y Fp(\033)2952 2093 y(\033)2991 2080 y Fe(0)3019 2062 y Fk(\024)p Fl(\034)3117 2040 y Fp(\033)3110 2088 y(T)8 b Fq(\()p Fp(E)r Fq(\))1017 2318 y Fx(=)1231 2231 y Fi(X)1113 2431 y Fl(\033)1157 2412 y Fe(0)1182 2431 y Fk(2)p Fl(T)i Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)1495 2318 y Fj(E)g Fn(e)1592 2280 y Fl(u\034)1676 2253 y Fp(\033)1669 2306 y(\033)1708 2293 y Fe(0)1746 2318 y Fx(1)-23 b(I)1802 2332 y Fl(\034)1843 2309 y Fp(\033)1836 2363 y(\033)1875 2350 y Fe(0)1903 2332 y Fk(\024)p Fl(\034)2001 2309 y Fp(\033)1994 2358 y(T)7 b Fq(\()p Fp(E)r Fq(\))2171 2318 y Fx(=)2384 2231 y Fi(X)2267 2431 y Fl(\033)2311 2412 y Fe(0)2336 2431 y Fk(2)p Fl(T)j Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)2648 2318 y Fn(G)2720 2280 y Fl(\033)2720 2342 y(\033)2764 2324 y Fe(0)2789 2342 y Fl(;T)g Fv(\()p Fl(E)s Fv(\))2981 2318 y Fx(\()p Fn(u)p Fx(\))3477 2227 y(\(4)p Fn(:)p Fx(11\))50 2640 y Fm(})159 2858 y Fx(An)30 b(immediate,)g(but)f(imp)s(ortan)m(t)h(consequence)h(of)f(Lemma)h(4.2)g (is)f(that)1624 3116 y Fm(k)p Fn(K)1753 3078 y Fk(\003)1746 3138 y Fl(E)1808 3116 y Fx(\(0\))p Fm(k)d Fx(=)e(1)1341 b(\(4)p Fn(:)p Fx(12\))50 3373 y(The)30 b(\014rst)h(step)g(to)m(w)m (ards)h(con)m(trol)f(in)f(the)h(complex)g(plane)f(will)f(b)s(e)h(to)i (sho)m(w)f(that)h Fm(k)p Fn(K)3091 3340 y Fk(\003)3084 3398 y Fl(E)3146 3373 y Fx(\()p Fn(u)p Fx(\))p Fm(k)g Fx(decreases)50 3519 y(do)m(wn)e(from)g(zero)h(along)f(the)h(imaginary) e(axis)h(in)f(the)h(strip)f Fm(=)p Fx(\()p Fn(u)p Fx(\))d Fm(2)f Fx([)p Fm(\000)p Fn(\031)s(;)15 b(\031)s Fx(].)50 3737 y Fd(Lemma)38 b(4.3:)51 b Ft(L)-5 b(et)33 b Fn(v)28 b Fm(2)d Fx([)p Fm(\000)p Fn(\031)s(;)15 b(\031)s Fx(])33 b Ft(and)h(set)1512 3971 y Fx(\026)1499 3994 y(\002)25 b Fm(\021)g Fn(e)1733 3957 y Fl(\014)1777 3902 y Fk(p)p 1837 3902 65 4 v 55 x Fl(N)7 b(u)1945 3966 y Fp(N)2003 3957 y Fv(\(0\)+)p Fl(\013E)3477 3994 y Fx(\(4)p Fn(:)p Fx(13\))50 4252 y Ft(R)-5 b(e)g(c)g(al)5 b(l)34 b Fn(M)h Fx(=)25 b Fm(j)p Fn(T)13 b Fx(\()p Fn(E)5 b Fx(\))p Fm(j)34 b Ft(and)g Fn(d)25 b Fx(=)g(2)1217 4219 y Fl(M)1298 4252 y Ft(.)42 b(Then)33 b(\(for)g Fn(N)43 b Ft(lar)-5 b(ge)33 b(enough\),)383 4516 y Fm(k)p Fn(K)512 4478 y Fk(\003)505 4538 y Fl(E)567 4516 y Fx(\()p Fn(iv)s Fx(\))p Fm(k)27 b(\024)1991 4454 y Fx(1)p 894 4495 2240 4 v 894 4513 a Fi(q)p 985 4513 1291 4 v 105 x Fx(2\(1)21 b Fm(\000)f Fx(cos)c Fn(v)s Fx(\))1454 4595 y(\026)1441 4618 y(\002)1512 4592 y Fv(2)1568 4544 y Fi(\000)1610 4618 y Fx(1)21 b Fm(\000)f Fn(O)s Fx(\()1887 4595 y(\026)1874 4618 y(\002)1945 4592 y Fk(\000)p Fv(1)2042 4618 y Fx(\))2077 4544 y Fi(\001)2139 4618 y Fx(+)g(1)h Fm(\000)2464 4582 y Fv(4)p 2398 4597 170 4 v 2398 4649 a Fl(M)7 b Fk(\000)p Fv(1)2593 4618 y Fx(\(1)21 b(+)f Fn(O)s Fx(\()p Fn(d=)-5 b(N)10 b Fx(\)\))3477 4516 y(\(4)p Fn(:)p Fx(14\))159 5012 y(Before)31 b(pro)m(ving)f(the)g (lemma,)h(w)m(e)f(will)e(note)j(the)g(ob)m(vious)e(consequence)j(that) 50 5231 y Fd(Corollary)38 b(4.4:)91 b Ft(Under)33 b(the)g(assumptions)i (and)f(notations)g(of)f(L)-5 b(emma)34 b(4.3,)10 5449 y(\(i\))47 b(If)267 5426 y Fx(\026)254 5449 y(\002)p Fm(j)p Fn(v)s Fm(j)26 b Fn(>)651 5413 y Fv(3)p 555 5428 229 4 v 555 5438 a Fk(p)p 615 5438 170 4 v 50 x Fl(M)7 b Fk(\000)p Fv(1)795 5449 y Ft(,)32 b(then)h Fm(k)p Fn(K)1186 5416 y Fk(\003)1179 5474 y Fl(E)1241 5449 y Fx(\()p Fn(iv)s Fx(\))p Fm(k)28 b Fn(<)c Fx(1)p Ft(.)p eop %%Page: 27 27 27 26 bop 50 295 a Fo(A)l(ging)27 b(in)h(the)g(REM.)f(Part)h(2.)2661 b Fr(27)-18 513 y Ft(\(ii\))47 b(F)-7 b(or)34 b(any)f Fx(0)26 b Fn(<)f(\017)g(<)g Fx(1)p Ft(,)32 b(if)660 790 y Fx(2\(1)22 b Fm(\000)e Fx(cos)c Fn(v)s Fx(\))26 b Fm(\025)1251 767 y Fx(\026)1238 790 y(\002)1309 752 y Fk(\000)p Fv(2)1422 662 y Fi(\024)1559 728 y Fn(\017)p 1481 769 194 4 v 1481 852 a Fx(1)21 b Fm(\000)e Fn(\017)1696 728 y Fx(2)i Fm(\000)f Fn(\017)p 1696 769 V 1696 852 a Fx(1)h Fm(\000)f Fn(\017)1921 790 y Fx(+)2285 728 y(9)p 2023 769 572 4 v 2023 852 a(\()p Fn(m)g Fm(\000)g Fx(1\)\(1)i Fm(\000)e Fn(\017)p Fx(\))2605 662 y Fi(\025)2668 790 y Fx(\(1)h(+)f Fn(O)s Fx(\()p Fn(d=)-5 b(N)10 b Fx(\)\))270 b(\(4)p Fn(:)p Fx(15\))159 1067 y Ft(then)33 b Fm(k)p Fn(K)490 1034 y Fk(\003)483 1092 y Fl(E)545 1067 y Fx(\()p Fn(iv)s Fx(\))p Fm(k)27 b(\024)e Fx(1)c Fm(\000)f Fn(\017)p Ft(.)-46 1285 y(\(iii\))47 b(Under)33 b(the)g(same)g(assumptions)i(as)e(in)f(\(i\),)375 1551 y Fm(k)p Fn(M)518 1513 y Fk(\003)508 1573 y Fl(E)570 1551 y Fx(\()p Fn(iv)s Fx(\))p Fm(k)27 b(\024)2051 1489 y Fx(1)p 897 1530 2354 4 v 897 1548 a Fi(p)p 988 1548 1264 4 v 81 x Fx(1)21 b(+)f(2)1203 1606 y(\026)1190 1629 y(\002)1261 1603 y Fv(2)1302 1629 y Fx(\(1)h Fm(\000)e Fx(cos)d Fn(v)s Fx(\)\(1)22 b Fm(\000)e Fn(O)s Fx(\()2025 1606 y(\026)2012 1629 y(\002)2083 1603 y Fk(\000)p Fv(1)2180 1629 y Fx(\)\))h Fm(\000)f Fx(1)h Fm(\000)2596 1593 y Fv(4)p 2530 1608 170 4 v 2530 1660 a Fl(M)7 b Fk(\000)p Fv(1)2710 1629 y Fx(\(1)21 b(+)f Fn(O)s Fx(\()p Fn(d=)-5 b(N)10 b Fx(\)\))3477 1551 y(\(4)p Fn(:)p Fx(16\))50 2028 y Fc(Pro)s(of:)47 b Fx(T)-8 b(o)31 b(b)s(ound)d(the)j(norm)f(of)g Fn(K)1375 1995 y Fk(\003)1368 2053 y Fl(E)1430 2028 y Fx(,)g(w)m(e)h(use)f(simply)e(that)1135 2236 y Fi(X)1075 2435 y Fl(\033)1119 2417 y Fe(0)1144 2435 y Fk(2)p Fl(T)10 b Fk(n)p Fl(\033)1342 2323 y Fm(j)p Fn(K)1451 2285 y Fk(\003)1444 2345 y Fl(\033)n(;\033)1550 2327 y Fe(0)1580 2323 y Fx(\()p Fn(iv)s Fx(\))p Fm(j)27 b Fx(=)1887 2176 y Fi(P)1983 2271 y Fl(\033)2027 2253 y Fe(0)2052 2271 y Fk(2)p Fl(T)10 b Fk(n)p Fl(\033)2255 2244 y Fm(j)p Fn(G)2352 2211 y Fl(\033)2352 2269 y(\033)2396 2251 y Fe(0)2421 2269 y Fl(;T)2500 2244 y Fx(\()p Fn(iv)s Fx(\))p Fm(j)p 1887 2302 788 4 v 2007 2385 a(j)p Fx(1)21 b Fm(\000)f Fn(G)2261 2354 y Fl(\033)2261 2412 y(\033)n(;T)2380 2385 y Fx(\()p Fn(iv)s Fx(\))p Fm(j)1780 2646 y(\024)1908 2565 y Fj(P)p Fx([)p Fn(\034)2038 2532 y Fl(\033)2028 2594 y(T)10 b Fk(n)p Fl(\033)2193 2565 y Fm(\024)25 b Fn(\034)2339 2532 y Fl(\033)2329 2590 y(T)2388 2565 y Fx(])p 1887 2625 548 4 v 1887 2708 a Fm(j)p Fx(1)c Fm(\000)f Fn(G)2141 2677 y Fl(\033)2141 2735 y(\033)n(;T)2260 2708 y Fx(\()p Fn(iv)s Fx(\))p Fm(j)3477 2484 y Fx(\(4)p Fn(:)p Fx(17\))50 2933 y(Th)m(us)29 b(the)i(k)m(ey)g(p)s(oin)m(t)e(is)h(to)h (b)s(ound)d(the)j(denominator)e(from)h(b)s(elo)m(w.)40 b(No)m(w)187 3223 y Fm(=)p Fn(G)325 3186 y Fl(\033)325 3248 y(\033)n(;T)10 b Fv(\()p Fl(E)s Fv(\))557 3223 y Fx(\()p Fn(iv)s Fx(\))27 b(=)860 3110 y Fk(1)831 3137 y Fi(X)828 3330 y Fl(n)p Fv(=1)980 3223 y Fx(sin)o(\()p Fn(v)s(n)p Fx(\))p Fj(P)p Fx([)p Fn(\034)1394 3186 y Fl(\033)1384 3246 y(\033)1469 3223 y Fx(=)e Fn(\034)1615 3186 y Fl(\033)1605 3248 y(T)10 b Fv(\()p Fl(E)s Fv(\))1801 3223 y Fx(=)25 b Fn(n)p Fx(])732 3525 y(=)g(sin)o(\()p Fn(v)s Fx(\))p Fn(p)1103 3539 y Fl(N)1172 3525 y Fx(\()p Fn(\033)n(;)15 b(\033)s Fx(\))22 b(+)1577 3439 y Fi(X)1500 3638 y Fl(\033)1544 3620 y Fe(0)1569 3638 y Fk(62)p Fl(T)10 b Fv(\()p Fl(E)s Fv(\))1800 3525 y Fn(p)1846 3539 y Fl(N)1914 3525 y Fx(\()p Fn(\033)n(;)15 b(\033)2094 3488 y Fk(0)2121 3525 y Fx(\))2204 3411 y Fk(1)2174 3439 y Fi(X)2171 3631 y Fl(n)p Fv(=1)2323 3525 y Fx(sin)o(\()p Fn(v)s Fx(\()p Fn(n)21 b Fx(+)f(1\)\))p Fj(P)p Fx([)p Fn(\034)2964 3488 y Fl(\033)3008 3460 y Fe(0)2954 3548 y Fl(\033)3065 3525 y Fx(=)25 b Fn(\034)3211 3488 y Fl(\033)3255 3460 y Fe(0)3201 3550 y Fl(T)10 b Fv(\()p Fl(E)s Fv(\))3397 3525 y Fx(=)24 b Fn(n)p Fx(])732 3777 y Fm(\021)h Fn(p)874 3791 y Fl(N)943 3777 y Fx(\()p Fn(\033)n(;)15 b(\033)s Fx(\))g(sin)g Fn(v)24 b Fx(+)c Fn(d)1507 3791 y Fl(\033)1556 3777 y Fx(\()p Fn(v)s Fx(\))3477 3886 y(\(4)p Fn(:)p Fx(18\))50 4031 y(where)298 4309 y Fm(j)p Fn(d)370 4323 y Fl(\033)419 4309 y Fx(\()p Fn(v)s Fx(\))p Fm(j)27 b(\024)e Fn(e)726 4271 y Fk(\000)p Fl(\014)827 4217 y Fk(p)p 887 4217 65 4 v 54 x Fl(N)7 b(E)1005 4280 y Fp(\033)1087 4222 y Fi(X)1068 4417 y Fl(\033)1112 4399 y Fe(0)1137 4417 y Fk(\030)p Fl(\033)1283 4247 y Fx(1)p 1264 4288 83 4 v 1264 4371 a Fn(N)1358 4309 y Fj(P)p Fx([)p Fn(\034)1488 4271 y Fl(\033)1532 4244 y Fe(0)1478 4331 y Fl(\033)1588 4309 y Fx(=)25 b Fn(\034)1734 4271 y Fl(\033)1778 4244 y Fe(0)1724 4334 y Fl(T)10 b Fv(\()p Fl(E)s Fv(\))1894 4309 y Fx(])26 b Fm(\024)f Fx(2)2097 4247 y Fn(e)2139 4214 y Fk(\000)p Fl(\014)2240 4160 y Fk(p)p 2301 4160 65 4 v 2301 4214 a Fl(N)6 b(E)2418 4223 y Fp(\033)p 2097 4288 369 4 v 2152 4371 a Fm(j)p Fn(T)13 b Fx(\()p Fn(E)5 b Fx(\))p Fm(j)2477 4309 y Fx(\(1)21 b(+)f Fn(O)s Fx(\()p Fm(j)p Fn(T)13 b Fx(\()p Fn(E)5 b Fx(\))p Fm(j)p Fn(=)-5 b(N)10 b Fx(\)\))250 b(\(4)p Fn(:)p Fx(19\))50 4617 y(where)30 b(w)m(e)h(used)e(the)i(b)s(ound)d(\(2.3\))k(from)e(Prop)s(osition)e (2.1.)889 4876 y Fm(<)970 4775 y Fi(\020)1024 4876 y Fx(1)21 b Fm(\000)f Fn(G)1253 4839 y Fl(\033)1253 4901 y(\033)n(;T)10 b Fv(\()p Fl(E)s Fv(\))1485 4876 y Fx(\()p Fn(iv)s Fx(\))1633 4775 y Fi(\021)1714 4876 y Fx(=)25 b Fn(p)1856 4890 y Fl(N)1925 4876 y Fx(\()p Fn(\033)n(;)15 b(\033)s Fx(\)\(1)22 b Fm(\000)e Fx(cos)c Fn(v)s Fx(\))21 b(+)f Fn(c)2703 4890 y Fl(\033)2752 4876 y Fx(\()p Fn(v)s Fx(\))608 b(\(4)p Fn(:)p Fx(20\))50 5135 y(where)159 5413 y Fj(P)p Fx([)p Fn(\034)289 5376 y Fl(\033)279 5438 y(T)10 b Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)557 5413 y Fx(=)25 b Fn(\034)703 5376 y Fl(\033)693 5438 y(T)10 b Fv(\()p Fl(E)s Fv(\))864 5413 y Fx(])26 b Fm(\024)f Fn(c)1050 5427 y Fl(\033)1099 5413 y Fx(\()p Fn(v)s Fx(\))h Fm(\024)f Fj(P)p Fx([)p Fn(\034)1468 5376 y Fl(\033)1458 5438 y(T)10 b Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)1737 5413 y Fx(=)24 b Fn(\034)1882 5376 y Fl(\033)1872 5438 y(T)10 b Fv(\()p Fl(E)s Fv(\))2043 5413 y Fx(])21 b(+)f(2)2236 5352 y Fn(e)2278 5319 y Fk(\000)p Fl(\014)2379 5264 y Fk(p)p 2439 5264 65 4 v 55 x Fl(N)7 b(E)2557 5328 y Fp(\033)p 2236 5392 369 4 v 2291 5475 a Fm(j)p Fn(T)13 b Fx(\()p Fn(E)5 b Fx(\))p Fm(j)2616 5413 y Fx(\(1)21 b(+)f Fn(O)s Fx(\()p Fm(j)p Fn(T)13 b Fx(\()p Fn(E)5 b Fx(\))p Fm(j)p Fn(=)-5 b(N)10 b Fx(\)\))111 b(\(4)p Fn(:)p Fx(21\))p eop %%Page: 28 28 28 27 bop 50 295 a Fr(28)3270 b Fo(Se)l(ction)27 b Fr(4)50 513 y Fx(Th)m(us)i(w)m(e)i(ha)m(v)m(e)h(that)258 772 y Fm(j)p Fx(1)21 b Fm(\000)f Fn(G)512 734 y Fl(\033)512 794 y(\033)n(;T)631 772 y Fx(\()p Fn(iv)s Fx(\))p Fm(j)27 b(\025)927 626 y Fi(r)p 1018 626 2252 4 v 146 x Fx(\()p Fn(p)1099 786 y Fl(N)1168 772 y Fx(\()p Fn(\033)n(;)15 b(\033)s Fx(\))g(sin)g Fn(v)s Fx(\))1608 745 y Fv(2)1670 772 y Fx(+)1761 671 y Fi(\020)1815 772 y Fn(p)1861 786 y Fl(N)1930 772 y Fx(\()p Fn(\033)n(;)g(\033)s Fx(\)\(1)22 b Fm(\000)e Fx(cos)c Fn(v)s Fx(\))21 b(+)f Fj(P)p Fx([)p Fn(\034)2799 740 y Fl(\033)2789 803 y(T)10 b Fk(n)p Fl(\033)2954 772 y Fx(=)25 b Fn(\034)3100 740 y Fl(\033)3090 798 y(T)3149 772 y Fx(])3174 671 y Fi(\021)3228 690 y Fv(2)826 954 y Fm(\000)20 b(j)p Fn(d)989 968 y Fl(\033)1038 954 y Fx(\()p Fn(v)s Fx(\))p Fm(j)i(\000)e(j)p Fn(c)1357 968 y Fl(\033)1406 954 y Fx(\()p Fn(v)s Fx(\))i Fm(\000)d Fj(P)p Fx([)p Fn(\034)1765 916 y Fl(\033)1755 979 y(T)10 b Fk(n)p Fl(\033)1920 954 y Fx(=)25 b Fn(\034)2066 916 y Fl(\033)2056 976 y(T)2115 954 y Fx(])p Fm(j)3477 829 y Fx(\(4)p Fn(:)p Fx(22\))50 1169 y(T)-8 b(o)31 b(simplify)c(the)j (notation,)h(set)g Fn(p)1261 1183 y Fl(N)1355 1169 y Fm(\021)25 b Fn(p)1497 1183 y Fl(N)1566 1169 y Fx(\()p Fn(\033)n(;)15 b(\033)s Fx(\),)32 b Fj(P)1893 1183 y Fl(\033)1967 1169 y Fm(\021)25 b Fj(P)p Fx([)p Fn(\034)2193 1136 y Fl(\033)2183 1198 y(T)10 b Fk(n)p Fl(\033)2348 1169 y Fx(=)25 b Fn(\034)2494 1136 y Fl(\033)2484 1194 y(T)2543 1169 y Fx(].)41 b(Let)322 1411 y Fn(Y)k Fm(\021)25 b Fx(\()p Fn(p)597 1425 y Fl(N)681 1411 y Fx(sin)14 b Fn(v)s Fx(\))890 1373 y Fv(2)952 1411 y Fx(+)20 b(\()p Fn(p)1124 1425 y Fl(N)1193 1411 y Fx(\(1)h Fm(\000)f Fx(cos)15 b Fn(v)s Fx(\))21 b(+)f Fj(P)1770 1425 y Fl(\033)1820 1411 y Fx(\))1855 1365 y Fv(2)1921 1411 y Fx(=)25 b(2)p Fn(p)2108 1425 y Fl(N)2177 1411 y Fx(\(1)c Fm(\000)f Fx(cos)c Fn(v)s Fx(\)\()p Fn(p)2669 1425 y Fl(N)2759 1411 y Fx(+)k Fj(P)2905 1425 y Fl(\033)2954 1411 y Fx(\))h(+)f Fj(P)3156 1373 y Fv(2)3156 1433 y Fl(\033)3477 1411 y Fx(\(4)p Fn(:)p Fx(23\))50 1653 y(Th)m(us)29 b(w)m(e)i(ha)m(v)m(e)h(in) d(fact)i(that)197 1895 y Fm(j)p Fx(1)21 b Fm(\000)f Fn(G)451 1857 y Fl(\033)451 1917 y(\033)n(;T)569 1895 y Fx(\()p Fn(iv)s Fx(\))p Fm(j)27 b(\025)865 1813 y Fi(p)p 956 1813 1189 4 v 82 x Fx(2)p Fn(p)1047 1909 y Fl(N)1116 1895 y Fx(\(1)21 b Fm(\000)f Fx(cos)c Fn(v)s Fx(\)\()p Fn(p)1608 1909 y Fl(N)1698 1895 y Fx(+)k Fj(P)1844 1909 y Fl(\033)1893 1895 y Fx(\))h(+)f Fj(P)2095 1869 y Fv(2)2095 1917 y Fl(\033)2165 1895 y Fm(\000)2293 1833 y Fx(4)p 2266 1874 99 4 v 2266 1957 a Fn(M)2375 1895 y(e)2417 1857 y Fk(\000)p Fl(\014)2518 1803 y Fk(p)p 2579 1803 65 4 v 2579 1857 a Fl(N)6 b(E)2696 1866 y Fp(\033)2744 1895 y Fx(\(1)21 b(+)f Fn(O)s Fx(\()p Fn(M)5 b(=)-5 b(N)10 b Fx(\)\))148 b(\(4)p Fn(:)p Fx(24\))50 2137 y(whic)m(h)29 b(together)j(with)d(\(4.17\))j(giv)m(es)f(that)183 2303 y Fi(X)123 2503 y Fl(\033)167 2484 y Fe(0)192 2503 y Fk(2)p Fl(T)10 b Fk(n)p Fl(\033)390 2390 y Fm(j)p Fn(K)499 2352 y Fk(\003)492 2412 y Fl(\033)n(;\033)598 2394 y Fe(0)628 2390 y Fx(\()p Fn(iv)s Fx(\))p Fm(j)27 b(\024)2105 2328 y Fj(P)2160 2342 y Fl(\033)p 935 2369 2444 4 v 935 2389 a Fi(p)p 1026 2389 1189 4 v 77 x Fx(2)p Fn(p)1117 2480 y Fl(N)1186 2466 y Fx(\(1)21 b Fm(\000)f Fx(cos)c Fn(v)s Fx(\)\()p Fn(p)1678 2480 y Fl(N)1768 2466 y Fx(+)k Fj(P)1914 2480 y Fl(\033)1963 2466 y Fx(\))h(+)f Fj(P)2165 2440 y Fv(2)2165 2489 y Fl(\033)2235 2466 y Fm(\000)2356 2431 y Fv(4)p 2336 2446 77 4 v 2336 2498 a Fl(M)2423 2466 y Fn(e)2465 2440 y Fk(\000)p Fl(\014)2566 2386 y Fk(p)p 2627 2386 65 4 v 2627 2440 a Fl(N)6 b(E)2744 2449 y Fp(\033)2792 2466 y Fx(\(1)21 b(+)f Fn(O)s Fx(\()p Fn(M)5 b(=)-5 b(N)10 b Fx(\)\))828 2684 y(=)2258 2623 y(1)p 935 2663 2691 4 v 935 2681 a Fi(q)p 1026 2681 1283 4 v 111 x Fx(2)p Fn(p)1117 2806 y Fl(N)1186 2792 y Fj(P)1241 2753 y Fk(\000)p Fv(2)1241 2803 y Fl(\033)1339 2792 y Fx(\(1)21 b Fm(\000)f Fx(cos)c Fn(v)s Fx(\)\()p Fn(p)1831 2806 y Fl(N)1921 2792 y Fx(+)k Fj(P)2067 2806 y Fl(\033)2117 2792 y Fx(\))g(+)g(1)h Fm(\000)2450 2756 y Fv(4)p 2430 2771 77 4 v 2430 2824 a Fl(M)2518 2792 y Fj(P)2573 2753 y Fk(\000)p Fv(1)2573 2803 y Fl(\033)2671 2792 y Fn(e)2713 2766 y Fk(\000)p Fl(\014)2814 2712 y Fk(p)p 2874 2712 65 4 v 54 x Fl(N)6 b(E)2991 2775 y Fp(\033)3040 2792 y Fx(\(1)21 b(+)f Fn(O)s Fx(\()p Fn(M)5 b(=)-5 b(N)10 b Fx(\)\))3477 2919 y(\(4)p Fn(:)p Fx(25\))50 3064 y(No)m(w)31 b(recall)f(from)f(Prop)s(osition)g(2.1,)j(\(iii\),)d(that)796 3251 y(1)p 692 3292 255 4 v 692 3380 a(1)20 b Fm(\000)879 3344 y Fv(1)p 859 3359 77 4 v 859 3411 a Fl(M)957 3313 y Fx(\(1)h Fm(\000)f Fn(O)s Fx(\()p Fn(d=)-5 b(N)10 b Fx(\)\))27 b Fm(\024)e Fj(P)1674 3275 y Fk(\000)p Fv(1)1674 3335 y Fl(\033)1772 3313 y Fn(e)1814 3275 y Fk(\000)p Fl(\014)1915 3221 y Fk(p)p 1975 3221 65 4 v 54 x Fl(N)7 b(E)2093 3284 y Fp(\033)2166 3313 y Fm(\024)2378 3251 y Fx(1)p 2273 3292 255 4 v 2273 3380 a(1)21 b Fm(\000)2460 3344 y Fv(1)p 2440 3359 77 4 v 2440 3411 a Fl(M)2538 3313 y Fx(\(1)g(+)f Fn(O)s Fx(\()p Fn(d=)-5 b(N)10 b Fx(\)\))400 b(\(4)p Fn(:)p Fx(26\))50 3580 y(It)30 b(follo)m(ws)g (readily)f(that)679 3858 y Fn(p)725 3872 y Fl(N)814 3858 y Fx(+)20 b Fj(P)960 3872 y Fl(\033)1035 3858 y Fx(=)25 b(1)20 b Fm(\000)g Fn(e)1329 3821 y Fk(\000)p Fl(\014)1430 3766 y Fk(p)p 1490 3766 65 4 v 55 x Fl(N)7 b(E)1608 3830 y Fp(\033)1676 3858 y Fx(+)20 b Fj(P)1822 3872 y Fl(\033)1897 3858 y Fm(\025)25 b Fx(1)20 b Fm(\000)2160 3797 y Fn(e)2202 3764 y Fk(\000)p Fl(\014)2303 3709 y Fk(p)p 2364 3709 V 2364 3764 a Fl(N)6 b(E)2481 3773 y Fp(\033)p 2160 3837 369 4 v 2296 3921 a Fn(M)2540 3858 y Fx(\(1)21 b Fm(\000)f Fn(O)s Fx(\()p Fn(d=)-5 b(N)10 b Fx(\)\))398 b(\(4)p Fn(:)p Fx(27\))50 4100 y(and)30 b(hence)739 4246 y(1)c Fn(>)f(p)952 4260 y Fl(N)1021 4246 y Fx(\()p Fn(p)1102 4260 y Fl(N)1191 4246 y Fx(+)20 b Fj(P)1337 4260 y Fl(\033)1386 4246 y Fx(\))26 b Fm(\025)f Fx(1)c Fm(\000)f Fn(e)1742 4208 y Fk(\000)p Fl(\014)1843 4154 y Fk(p)p 1903 4154 65 4 v 54 x Fl(N)6 b(E)2020 4217 y Fp(\033)2068 4246 y Fx(\(1)21 b(+)f(1)p Fn(=)-5 b(M)10 b Fx(\)\(1)23 b(+)d Fn(O)s Fx(\()p Fn(d=)-5 b(N)10 b Fx(\)\))458 b(\(4)p Fn(:)p Fx(28\))50 4449 y(Since)29 b(b)m(y)h(de\014nition)e(of)j Fn(T)13 b Fx(,)30 b(min)1194 4466 y Fl(\033)r Fk(2)p Fl(T)10 b Fv(\()p Fl(E)s Fv(\))1473 4372 y Fm(p)p 1549 4372 83 4 v 77 x Fn(N)g(E)1699 4463 y Fl(\033)1773 4449 y Fm(\025)25 b Fn(u)1921 4463 y Fl(N)1990 4449 y Fx(\()p Fn(E)5 b Fx(\),)32 b(this)d(implies)840 4691 y(min)842 4750 y Fl(\033)r Fk(2)p Fl(T)1006 4691 y Fn(p)1052 4705 y Fl(N)1121 4691 y Fx(\()p Fn(p)1202 4705 y Fl(N)1291 4691 y Fx(+)20 b Fj(P)1437 4705 y Fl(\033)1487 4691 y Fx(\))25 b Fm(\025)g Fx(1)c Fm(\000)1812 4668 y Fx(\026)1800 4691 y(\002)1871 4654 y Fk(\000)p Fv(1)1968 4691 y Fx(\(1)g(+)f(1)p Fn(=)-5 b(M)10 b Fx(\)\(1)22 b(+)e Fn(O)s Fx(\()p Fn(d=)-5 b(N)10 b Fx(\)\))559 b(\(4)p Fn(:)p Fx(29\))50 4934 y(and)288 5176 y Fm(k)p Fn(K)417 5138 y Fk(\003)410 5198 y Fl(E)472 5176 y Fx(\()p Fn(iv)s Fx(\))p Fm(k)27 b(\024)1991 5114 y Fx(1)p 799 5155 2431 4 v 799 5173 a Fi(p)p 890 5173 1497 4 v 903 5231 a Fx(\026)890 5254 y(\002)961 5228 y Fv(2)1001 5254 y Fx(2\(1)22 b Fm(\000)d Fx(cos)d Fn(v)s Fx(\)\(1)22 b Fm(\000)1663 5231 y Fx(\026)1650 5254 y(\002)1721 5228 y Fk(\000)p Fv(1)1818 5254 y Fx(\(1)f(+)f(1)p Fn(=)-5 b(M)10 b Fx(\))22 b(+)e(1)g Fm(\000)2575 5218 y Fv(4)p 2508 5233 170 4 v 2508 5285 a Fl(M)7 b Fk(\000)p Fv(1)2688 5254 y Fx(\(1)21 b(+)f Fn(O)s Fx(\()p Fn(d=)-5 b(N)10 b Fx(\)\))3477 5176 y(\(4)p Fn(:)p Fx(30\))50 5462 y(whic)m(h)29 b(pro)m(v)m(es)i(the)g(Lemma.)p Fm(})p eop %%Page: 29 29 29 28 bop 50 295 a Fo(A)l(ging)27 b(in)h(the)g(REM.)f(Part)h(2.)2661 b Fr(29)159 513 y Fx(The)30 b(pro)s(of)f(of)i(Corollary)e(4.4)i(is)f (an)g(exercise)h(in)e(simple)f(algebra)j(and)e(is)h(left)g(to)h(the)f (reader.)159 731 y(Next)h(w)m(e)f(use)g(these)h(results)d(to)j(extend)f (similar)e(b)s(ounds)f(somewhat)k(in)m(to)f(the)g(p)s(ositiv)m(e)f (imaginary)50 876 y(half)e(plane.)39 b(The)28 b(imp)s(ortan)m(t)f(p)s (oin)m(t)g(p)s(ermitting)f(this)h(is)h(that)g(w)m(e)h(will)d(need)i(to) h(T)-8 b(a)m(ylor-expand)28 b(in)f(the)50 1022 y(real)g(part)h(of)g Fn(u)f Fx(only)g(Diric)m(hlet)g(Green's)h(functions)e(with)h(exclusion) f(set)i Fn(T)41 b Fx(and)27 b(these)h(are)g(analytic)g(up)50 1167 y(to)j Fm(<)p Fx(\()p Fn(u)p Fx(\))26 b Fm(\031)f Fx(1)p Fn(=)571 1144 y Fi(b)561 1167 y Fx(\002.)41 b(Let)31 b(us)e(\014rst)h(\014x)g(some)h(notation.)50 1385 y Fu(Notation:)58 b Fx(As)35 b(b)s(efore)g(the)h(letter)g Fn(u)e Fm(2)f Fj(C)59 b Fx(denotes)36 b(a)g(complex)g(n)m(um)m(b)s(er.)54 b(Its)36 b(real)f(and)g(imaginary)50 1531 y(parts)30 b(will)e(alw)m(a)m(ys)j(b)s(e)e(called)h Fn(w)j Fx(and)c Fn(v)s Fx(:)1665 1683 y Fn(u)c Fx(=)g Fn(w)e Fx(+)c Fn(iv)1386 b Fx(\(4)p Fn(:)p Fx(31\))50 1899 y(F)-8 b(or)31 b(giv)m(en)f Fn(u)c Fm(2)e Fj(C)18 b Fx(,)37 b(w)m(e)31 b(will)c(denote)k(b)m(y)f Fn(z)g Fm(2)25 b Fj(C)54 b Fx(the)30 b(n)m(um)m(b)s(er)1663 2160 y Fn(z)g Fx(=)1841 2137 y Fi(b)1831 2160 y Fx(\002)o(\()p Fn(E)5 b Fx(\))p Fn(u)1382 b Fx(\(4)p Fn(:)p Fx(32\))50 2421 y(The)30 b(real)g(and)f(imaginary)g(parts)i(of)f Fn(z)35 b Fx(will)27 b(alw)m(a)m(ys)k(b)s(e)f(called)g Fn(r)i Fx(and)e Fn(s)p Fx(:)1682 2682 y Fn(z)f Fx(=)c Fn(r)e Fx(+)d Fn(is)1399 b Fx(\(4)p Fn(:)p Fx(33\))50 2943 y(Th)m(us)1657 3048 y Fn(r)28 b Fx(=)1832 3025 y Fi(b)1822 3048 y Fx(\002)o(\()p Fn(E)5 b Fx(\))p Fn(w)1658 3220 y(s)25 b Fx(=)1832 3197 y Fi(b)1822 3220 y Fx(\002)o(\()p Fn(E)5 b Fx(\))p Fn(v)3477 3128 y Fx(\(4)p Fn(:)p Fx(34\))50 3400 y(T)-8 b(o)31 b(simplify)c(the)j(notation)h(the)f(dep)s(endence)g (on)g Fn(u)g Fx(of)h Fn(z)j Fx(\(or)d(on)f Fn(w)r Fx(,)h(resp.)f Fn(v)s Fx(,)h(of)g Fn(r)s Fx(,)f(resp.)f Fn(s)p Fx(\))i(will)c(nev)m (er)50 3545 y(b)s(e)32 b(made)h(explicit.)47 b(No)33 b(confusion)f(should)e(arise)j(from)f(this)g(as,)i(up)e(un)m(til)f (Section)i(4.7)3166 3512 y Fv(13)3244 3545 y Fx(,)h(the)f(letters)50 3691 y Fn(u;)15 b(w)r(;)g(v)35 b Fx(and)30 b Fn(z)t(;)15 b(r)m(;)g(s)31 b Fx(will)d(b)s(e)i(used)f(exclusiv)m(ely)g(according)i (to)g(the)f(relations)g(sp)s(eci\014ed)e(ab)s(o)m(v)m(e.)159 3909 y(F)-8 b(or)31 b(ready)f(reference)h(w)m(e)g(mak)m(e)g(the)g (follo)m(wing)e(de\014nitions.)50 4127 y Fd(De\014nition)40 b(4.5:)106 b Ft(L)-5 b(et)36 b Fx(0)c Fn(<)e(C)1276 4141 y Fv(1)1317 4127 y Fn(;)15 b(C)1422 4141 y Fv(2)1494 4127 y Fn(<)31 b Fm(1)p Ft(,)36 b(and)h Fx(0)31 b Fn(<)g(\015)36 b(<)31 b Fx(1)36 b Ft(b)-5 b(e)35 b(numeric)-5 b(al)37 b(c)-5 b(onstants.)53 b(With)36 b(the)50 4272 y(ab)-5 b(ove)33 b(notation)h(we)f(de\014ne)g(the)g(sets:)440 4520 y Fn(D)515 4534 y Fv(1)556 4520 y Fx(\()p Fn(C)656 4534 y Fv(1)697 4520 y Fx(\))25 b Fm(\021)853 4419 y Fi(n)914 4520 y Fn(u)g Fm(2)g Fj(C)64 b Fx(:)1249 4430 y Fi(p)p 1340 4430 279 4 v 90 x Fn(r)1384 4493 y Fv(2)1444 4520 y Fx(+)20 b Fn(s)1578 4493 y Fv(2)1644 4520 y Fm(\025)25 b Fn(C)1805 4534 y Fv(1)1846 4520 y Fn(=)1891 4438 y Fm(p)p 1967 4438 99 4 v 82 x Fn(M)2065 4419 y Fi(o)347 4751 y Fn(D)422 4765 y Fv(2)463 4751 y Fx(\()p Fn(C)563 4765 y Fv(2)604 4751 y Fn(;)15 b(\015)5 b Fx(\))26 b Fm(\021)853 4622 y Fi(\032)922 4751 y Fn(u)f Fm(2)g Fj(C)64 b Fx(:)40 b(0)26 b Fm(\024)f Fn(r)j(<)d Fx(min)1755 4622 y Fi(\022)1975 4689 y Fn(\015)5 b(s)2070 4656 y Fv(2)p 1832 4730 422 4 v 1832 4824 a Fn(C)1897 4838 y Fv(2)1938 4748 y Fm(p)p 2014 4748 240 4 v 76 x Fx(1)21 b(+)e Fn(s)2213 4797 y Fv(2)2265 4751 y Fn(;)c Fx(1)21 b Fm(\000)f Fn(\015)2514 4622 y Fi(\023)2596 4751 y Fn(;)61 b(v)28 b Fm(2)d Fx([)p Fm(\000)p Fn(\031)s(;)15 b(\031)s Fx(])3111 4622 y Fi(\033)616 4951 y Fn(D)691 4965 y Fv(3)757 4951 y Fm(\021)25 b(f)q Fn(u)g Fm(2)g Fj(C)64 b Fx(:)41 b Fm(\000)p Fx(1)25 b Fm(\024)g Fn(r)j(<)d Fx(0)p Fn(;)46 b Fm(j)p Fn(s)p Fm(j)25 b Fn(<)g Fx(1)p Fm(g)616 5124 y Fn(D)691 5138 y Fv(4)757 5124 y Fm(\021)g(f)q Fn(u)g Fm(2)g Fj(C)64 b Fx(:)41 b Fm(j)p Fn(r)s Fm(j)25 b Fn(<)g Fx(1)p Fn(;)46 b Fm(j)p Fn(s)p Fm(j)26 b Fn(<)e Fx(1)p Fm(g)3477 4808 y Fx(\(4)p Fn(:)p Fx(35\))p 50 5375 546 4 v 159 5436 a Fs(13)230 5462 y Fr(There,)j(the)f(letter)h Fg(s)f Fr(will)h(retriev)n(e)g(the)f (initial)h(meaning)e(it)h(w)n(as)g(giv)n(en)g(in)g(Theorem)f(1.)p eop %%Page: 30 30 30 29 bop 50 295 a Fr(30)3270 b Fo(Se)l(ction)27 b Fr(4)50 513 y Fd(Lemma)45 b(4.6:)65 b Ft(Ther)-5 b(e)38 b(exist)g(c)-5 b(onstants)40 b Fx(0)35 b Fn(<)g(C)q(;)15 b(C)1958 480 y Fk(0)2018 513 y Fn(<)34 b Fm(1)k Ft(such)g(that,)i(for)f(al)5 b(l)38 b Fx(0)d Fn(<)f(\015)40 b(<)34 b Fx(1)k Ft(and)h(al)5 b(l)50 658 y Fn(u)25 b Fm(2)g Fn(D)288 672 y Fv(2)329 658 y Fx(\()p Fn(C)436 625 y Fk(0)461 658 y Fn(;)15 b(\015)5 b Fx(\))p Ft(,)217 894 y Fm(k)p Fn(K)346 857 y Fk(\003)339 917 y Fl(E)401 894 y Fx(\()p Fn(u)p Fx(\))p Fm(k)27 b(\024)1790 833 y Fx(1)20 b(+)g Fn(C)7 b(\015)2070 800 y Fk(\000)p Fv(1)2168 833 y Fn(r)p 701 873 2599 4 v 701 891 a Fi(q)p 792 891 1264 4 v 116 x Fx(1)21 b(+)959 984 y Fi(b)949 1007 y Fx(\002)1020 980 y Fv(2)1060 1007 y Fx(2\(1)h Fm(\000)e Fx(cos)c Fn(v)s Fx(\)\(1)21 b Fm(\000)f Fn(O)s Fx(\()1829 984 y(\026)1816 1007 y(\002)1887 980 y Fk(\000)p Fv(1)1985 1007 y Fx(\)\))h Fm(\000)2244 971 y Fv(4)p 2177 986 170 4 v 2177 1038 a Fl(M)7 b Fk(\000)p Fv(1)2358 1007 y Fx(\(1)21 b(+)e Fn(O)s Fx(\()p Fn(d=)-5 b(N)10 b Fx(\)\))22 b Fm(\000)e Fn(C)3081 980 y Fk(0)3106 1007 y Fn(\015)3158 980 y Fk(\000)p Fv(1)3256 1007 y Fn(r)3477 894 y Fx(\(4)p Fn(:)p Fx(36\))50 1347 y Fc(Pro)s(of:)47 b Fx(As)29 b(in)g(the)g(pro)s(of)g(of)h(Lemma)g(4.3,)h(w)m(e)f(b)s (egin)e(b)m(y)i(writing)d(the)j(analogue)g(of)g(\(4.17\))i(and)d(again) 50 1493 y(w)m(e)39 b(b)s(ound)f(the)h(n)m(umerator)g(b)m(y)g(the)h(v)-5 b(alue)38 b(obtained)h(when)f(putting)g(its)g(imaginary)g(part)h(equal) g(to)50 1638 y(zero.)i(This)29 b(yields)1244 1812 y Fm(k)p Fn(K)1373 1774 y Fk(\003)1366 1834 y Fl(E)1428 1812 y Fx(\()p Fn(u)p Fx(\))p Fm(k)e(\024)1729 1665 y Fi(P)1824 1761 y Fl(\033)1868 1742 y Fe(0)1893 1761 y Fk(2)p Fl(T)10 b Fk(n)p Fl(\033)2097 1733 y Fm(j)p Fn(G)2194 1700 y Fl(\033)2194 1758 y(\033)2238 1740 y Fe(0)2262 1758 y Fl(;T)2341 1733 y Fx(\()p Fn(w)r Fx(\))p Fm(j)p 1729 1791 777 4 v 1754 1874 a(j)p Fx(1)21 b Fm(\000)e Fn(G)2007 1843 y Fl(\033)2007 1901 y(\033)n(;T)2126 1874 y Fx(\()p Fn(w)k Fx(+)d Fn(iv)s Fx(\))p Fm(j)1526 2099 y Fx(=)1761 2018 y Fn(G)1833 1985 y Fl(\033)1833 2047 y(T)10 b Fk(n)p Fl(\033)n(;T)2042 2018 y Fx(\()p Fn(w)r Fx(\))p 1607 2078 726 4 v 1607 2161 a Fm(j)p Fx(1)21 b Fm(\000)f Fn(G)1861 2130 y Fl(\033)1861 2188 y(\033)n(;T)1980 2161 y Fx(\()p Fn(w)j Fx(+)d Fn(iv)s Fx(\))p Fm(j)3477 1955 y Fx(\(4)p Fn(:)p Fx(37\))50 2318 y(W)-8 b(e)30 b(no)m(w)g(T)-8 b(a)m(ylor)29 b(expand)g(b)s(oth)f(the)i(n)m(umerator)f(and)g(the)g (denominator.)40 b(Note)31 b(that)f(w)m(e)f(will)e(only)h(b)s(e)50 2463 y(in)m(terested)k(in)e Fn(w)g Fm(\024)d Fx(\(1)22 b Fm(\000)f Fn(\015)5 b Fx(\))p Fn(=)1105 2440 y Fi(b)1095 2463 y Fx(\002)q(.)44 b(F)-8 b(or)33 b(the)f(n)m(umerator)g(w)m(e)g (will)d(use)i(\(3.45\))j(together)f(with)e(the)h(b)s(ound)50 2609 y(\(3.46\))g(to)g(write,)d(for)i(0)25 b Fm(\024)g Fn(w)j Fm(\024)d Fx(\(1)c Fm(\000)f Fn(\015)5 b Fx(\))p Fn(=)1510 2586 y Fi(b)1500 2609 y Fx(\002,)937 2823 y Fn(G)1009 2785 y Fl(\033)1009 2848 y(T)10 b Fk(n)p Fl(\033)n(;T)1218 2823 y Fx(\()p Fn(w)r Fx(\))27 b Fm(\024)e Fj(P)p Fx([)p Fn(\034)1608 2785 y Fl(\033)1598 2848 y(T)10 b Fk(n)p Fl(\033)1762 2823 y Fm(\024)25 b Fn(\034)1908 2785 y Fl(\033)1898 2845 y(T)1957 2823 y Fx(])c(+)f Fn(C)7 b(w)r(\015)2285 2785 y Fk(\000)p Fv(1)2393 2800 y Fi(b)2383 2823 y Fx(\002)o Fn(e)2495 2785 y Fk(\000)p Fl(\014)2596 2731 y Fk(p)p 2657 2731 65 4 v 2657 2785 a Fl(N)f(E)2774 2794 y Fp(\033)3477 2823 y Fx(\(4)p Fn(:)p Fx(38\))50 3037 y(On)40 b(the)i(other)g(hand,)h (from)e(\(3.30\))i(and)e(the)h(Cauc)m(h)m(y)f(b)s(ound)f(w)m(e)i(get)g (that,)j(again)d(for)f(0)j Fm(\024)62 b Fx(~)-64 b Fn(w)46 b Fm(\024)50 3183 y Fx(\(1)21 b Fm(\000)f Fn(\015)5 b Fx(\))p Fn(=)384 3160 y Fi(b)374 3183 y Fx(\002,)687 3214 y Fi(\014)687 3269 y(\014)687 3323 y(\014)687 3378 y(\014)762 3285 y Fn(d)p 728 3325 115 4 v 728 3409 a(dw)854 3346 y(G)926 3309 y Fl(\033)926 3371 y(\033)n(;T)10 b Fv(\()p Fl(E)s Fv(\))1159 3346 y Fx(\()p Fn(iv)24 b Fx(+)38 b(~)-63 b Fn(w)r Fx(\))1486 3214 y Fi(\014)1486 3269 y(\014)1486 3323 y(\014)1486 3378 y(\014)1542 3346 y Fm(\024)25 b(j)p Fn(e)1720 3309 y Fv(~)-51 b Fl(w)1764 3346 y Fm(j)21 b Fx(+)f Fn(C)7 b(\015)2025 3309 y Fk(\000)p Fv(1)2122 3346 y Fn(e)2164 3309 y Fk(\000)p Fl(\014)2265 3254 y Fk(p)p 2325 3254 65 4 v 55 x Fl(N)g(E)2443 3318 y Fp(\033)2501 3323 y Fi(b)2491 3346 y Fx(\002\()p Fn(E)e Fx(\))26 b Fm(\024)f Fn(C)2898 3309 y Fk(0)2923 3346 y Fn(\015)2975 3309 y Fk(\000)p Fv(1)3477 3346 y Fx(\(4)p Fn(:)p Fx(39\))50 3550 y(This)j(implies)g(again)970 3696 y Fm(j)p Fx(1)21 b Fm(\000)f Fn(G)1224 3658 y Fl(\033)1224 3718 y(\033)n(;T)1343 3696 y Fx(\()p Fn(iv)k Fx(+)c Fn(w)r Fx(\))p Fm(j)26 b(\025)f(j)p Fx(1)c Fm(\000)f Fn(G)2071 3658 y Fl(\033)2071 3718 y(\033)n(;T)2190 3696 y Fx(\()p Fn(iv)s Fx(\))p Fm(j)i(\000)d Fn(w)r(\015)2594 3658 y Fk(\000)p Fv(1)2693 3696 y Fn(C)2765 3658 y Fk(0)3477 3696 y Fx(\(4)p Fn(:)p Fx(40\))50 3887 y(As)26 b(w)m(e)g(already)g(ha)m (v)m(e)h(b)s(ounded)d(the)i(\014rst)f(term)h(on)g(the)g(righ)m(t)g(in)e (the)i(pro)s(of)g(of)g(Lemma)g(4.3,)i(w)m(e)e(readily)50 4032 y(arriv)m(e)k(at)255 4119 y Fi(X)194 4319 y Fl(\033)238 4300 y Fe(0)263 4319 y Fk(2)p Fl(T)10 b Fk(n)p Fl(\033)462 4206 y Fm(j)p Fn(K)571 4168 y Fk(\003)564 4228 y Fl(\033)n(;\033)670 4210 y Fe(0)700 4206 y Fx(\()p Fn(u)p Fx(\))p Fm(j)204 4535 y(\024)1305 4473 y Fx(1)21 b(+)f Fn(C)7 b(w)1612 4450 y Fi(b)1601 4473 y Fx(\002)o Fn(\015)1723 4440 y Fk(\000)p Fv(1)1821 4473 y Fn(e)1863 4440 y Fk(\000)p Fl(\014)1964 4386 y Fk(p)p 2025 4386 V 2025 4440 a Fl(N)f(E)2142 4449 y Fp(\033)2190 4473 y Fj(P)2245 4440 y Fk(\000)p Fv(1)2245 4496 y Fl(\033)p 311 4514 3027 4 v 311 4532 a Fi(q)p 402 4532 1283 4 v 111 x Fx(1)21 b(+)e(2)p Fj(P)658 4604 y Fk(\000)p Fv(2)658 4654 y Fl(\033)757 4643 y Fn(p)803 4657 y Fl(N)872 4643 y Fx(\(1)i Fm(\000)f Fx(cos)c Fn(v)s Fx(\)\()p Fn(p)1364 4657 y Fl(N)1454 4643 y Fx(+)j Fj(P)1599 4657 y Fl(\033)1649 4643 y Fx(\))h Fm(\000)1806 4607 y Fv(4)p Fl(e)1877 4589 y Fe(\000)p Fp(\014)1964 4550 y Fe(p)p 2014 4550 54 3 v 39 x Fp(N)5 b(E)2114 4598 y(\033)p 1806 4622 356 4 v 1905 4674 a Fl(M)i Fh(P)2026 4683 y Fp(\033)2173 4643 y Fx(\(1)21 b(+)f Fn(O)s Fx(\()p Fn(M)5 b(=)-5 b(N)10 b Fx(\)\))21 b Fm(\000)f Fn(C)2942 4617 y Fk(0)2967 4643 y Fn(\015)3019 4617 y Fk(\000)p Fv(1)3117 4643 y Fj(P)3172 4604 y Fk(\000)p Fv(1)3172 4654 y Fl(\033)3270 4643 y Fn(w)3477 4424 y Fx(\(4)p Fn(:)p Fx(41\))50 4833 y(Pro)s(ceeding)33 b(from)h(there)g(on)g(exactly)h(as)f(in)f(the)h(pro) s(of)f(of)h(Lemma)g(4.3)h(w)m(e)g(then)f(get,)i(using)d(relation)50 4979 y(\(4.34\),)235 5128 y Fi(X)174 5327 y Fl(\033)218 5309 y Fe(0)243 5327 y Fk(2)p Fl(T)10 b Fk(n)p Fl(\033)441 5215 y Fm(j)p Fn(K)550 5177 y Fk(\003)543 5237 y Fl(\033)n(;\033)649 5219 y Fe(0)680 5215 y Fx(\()p Fn(u)p Fx(\))p Fm(j)26 b(\024)2057 5153 y Fx(1)20 b(+)g Fn(C)7 b(\015)2337 5120 y Fk(\000)p Fv(1)2434 5153 y Fn(r)p 960 5194 2616 4 v 960 5212 a Fi(q)p 1051 5212 1305 4 v 111 x Fx(1)20 b(+)g Fj(P)1262 5284 y Fk(\000)p Fv(2)1262 5333 y Fl(\033)1360 5323 y Fx(2\(1)i Fm(\000)e Fx(cos)c Fn(v)s Fx(\)\(1)21 b Fm(\000)f Fn(O)s Fx(\()2129 5300 y(\026)2116 5323 y(\002)2187 5296 y Fk(\000)p Fv(1)2285 5323 y Fx(\)\))h Fm(\000)2544 5287 y Fv(4)p 2477 5302 170 4 v 2477 5354 a Fl(M)7 b Fk(\000)p Fv(1)2658 5323 y Fx(\(1)21 b(+)e Fn(O)s Fx(\()p Fn(d=)-5 b(N)10 b Fx(\)\))22 b Fm(\000)3320 5282 y Fl(C)3376 5264 y Fe(0)3401 5282 y Fl(\015)3443 5264 y Fe(\000)p Fq(1)3528 5282 y Fl(r)p 3320 5302 244 4 v 3373 5375 a Fh(P)3418 5384 y Fp(\033)3458 5371 y Fi(b)3455 5375 y Fv(\002)3477 5462 y Fx(\(4)p Fn(:)p Fx(42\))p eop %%Page: 31 31 31 30 bop 50 295 a Fo(A)l(ging)27 b(in)h(the)g(REM.)f(Part)h(2.)2661 b Fr(31)50 513 y Fx(Since)27 b(w)m(e)h(need)g(to)g(tak)m(e)i(the)e (maxim)m(um)e(o)m(v)m(er)k(all)c Fn(\033)j Fm(2)c Fn(T)13 b Fx(,)28 b(it)f(is)g(imp)s(ortan)m(t)g(to)i(restrict)e Fn(r)k Fx(as)d(a)g(function)50 658 y(of)34 b Fn(v)j Fx(in)c(suc)m(h)g (a)i(w)m(a)m(y)g(that)f(the)g(maxim)m(um)f(will)e(b)s(e)j(tak)m(en)h (on)f(b)m(y)g(the)g Fn(\033)j Fx(that)d(maximises)f Fj(P)3372 672 y Fl(\033)3421 658 y Fx(.)52 b(Some)50 804 y(elemen)m(tary)31 b(algebra)f(sho)m(ws)g(that)h(this)f(will)d(b)s(e)j(the)h(case)g(pro)m (vided)e(that)984 962 y Fi(\020)1038 1063 y Fx(2\(1)22 b Fm(\000)e Fx(cos)c Fn(v)s Fx(\))1505 1040 y Fi(b)1495 1063 y Fx(\002)1566 1030 y Fv(2)1607 1063 y Fx(\(1)21 b Fm(\000)f Fn(O)s Fx(\()1919 1040 y(\026)1906 1063 y(\002)1977 1030 y Fk(\000)p Fv(1)2074 1063 y Fx(\)\))2144 962 y Fi(\021)2199 980 y Fv(2)p 980 1136 1264 4 v 980 1236 a Fx(1)h(+)f(2\(1)h Fm(\000)f Fx(cos)c Fn(v)s Fx(\))1603 1213 y Fi(b)1593 1236 y Fx(\002)1664 1209 y Fv(2)1705 1236 y Fx(\(1)21 b Fm(\000)f Fn(O)s Fx(\()2017 1213 y(\026)2004 1236 y(\002)2075 1209 y Fk(\000)p Fv(1)2173 1236 y Fx(\)\))2280 1157 y Fm(\025)2376 1083 y Fi(\000)2417 1157 y Fn(C)2489 1120 y Fk(0)2514 1157 y Fn(\015)2566 1120 y Fk(\000)p Fv(1)2664 1157 y Fn(r)2708 1083 y Fi(\001)2749 1101 y Fv(2)3477 1157 y Fx(\(4)p Fn(:)p Fx(43\))50 1416 y(or)986 1575 y Fn(r)28 b Fm(\024)1409 1514 y Fx(2\(1)21 b Fm(\000)f Fx(cos)c Fn(v)s Fx(\))1875 1491 y Fi(b)1865 1514 y Fx(\002)1936 1481 y Fv(2)1977 1514 y Fx(\(1)21 b Fm(\000)f Fn(O)s Fx(\()2289 1491 y(\026)2276 1514 y(\002)2347 1481 y Fk(\000)p Fv(1)2445 1514 y Fx(\)\))p 1162 1554 1601 4 v 1162 1688 a Fn(\015)1214 1661 y Fk(\000)p Fv(1)1312 1688 y Fn(C)1384 1661 y Fk(0)1409 1572 y Fi(q)p 1499 1572 1264 4 v 1499 1688 a Fx(1)h(+)f(2\(1)h Fm(\000)f Fx(cos)c Fn(v)s Fx(\))2122 1665 y Fi(b)2112 1688 y Fx(\002)2183 1661 y Fv(2)2224 1688 y Fx(\(1)21 b Fm(\000)f Fn(O)s Fx(\()2536 1665 y(\026)2523 1688 y(\002)2594 1661 y Fk(\000)p Fv(1)2692 1688 y Fx(\)\))3477 1575 y(\(4)p Fn(:)p Fx(44\))50 1873 y(Since)28 b(this)g(is)h(a)g (serious)g(condition)e(only)i(if)f Fn(v)33 b Fx(is)28 b(v)m(ery)i(small)d(w)m(e)j(see,)h(using)c(relation)i(\(4.34\),)j(that) e(this)50 2019 y(condition)f(reduces)h(to)1305 2164 y Fn(r)e(<)d Fx(min)1637 2036 y Fi(\022)1853 2103 y Fn(\015)5 b(s)1948 2070 y Fv(2)p 1715 2143 413 4 v 1715 2237 a Fn(C)1787 2211 y Fk(0)1812 2161 y Fm(p)p 1887 2161 240 4 v 1887 2237 a Fx(1)21 b(+)f Fn(s)2087 2211 y Fv(2)2138 2164 y Fn(;)15 b Fx(1)21 b Fm(\000)f Fn(\015)2387 2036 y Fi(\023)3477 2164 y Fx(\(4)p Fn(:)p Fx(45\))50 2386 y(On)29 b(this)h(domain)f(w)m(e)i(can)f(th)m(us)g(estimate)h(the)g (norm)f(of)g Fn(K)2148 2353 y Fk(\003)2141 2411 y Fl(E)2233 2386 y Fx(b)m(y)217 2665 y Fm(k)p Fn(K)346 2628 y Fk(\003)339 2688 y Fl(E)401 2665 y Fx(\()p Fn(u)p Fx(\))p Fm(k)d(\024)1790 2604 y Fx(1)20 b(+)g Fn(C)7 b(\015)2070 2571 y Fk(\000)p Fv(1)2168 2604 y Fn(r)p 701 2644 2599 4 v 701 2662 a Fi(q)p 792 2662 1264 4 v 116 x Fx(1)21 b(+)959 2755 y Fi(b)949 2778 y Fx(\002)1020 2751 y Fv(2)1060 2778 y Fx(2\(1)h Fm(\000)e Fx(cos)c Fn(v)s Fx(\)\(1)21 b Fm(\000)f Fn(O)s Fx(\()1829 2755 y(\026)1816 2778 y(\002)1887 2751 y Fk(\000)p Fv(1)1985 2778 y Fx(\)\))h Fm(\000)2244 2742 y Fv(4)p 2177 2757 170 4 v 2177 2809 a Fl(M)7 b Fk(\000)p Fv(1)2358 2778 y Fx(\(1)21 b(+)e Fn(O)s Fx(\()p Fn(d=)-5 b(N)10 b Fx(\)\))22 b Fm(\000)e Fn(C)3081 2751 y Fk(0)3106 2778 y Fn(\015)3158 2751 y Fk(\000)p Fv(1)3256 2778 y Fn(r)3477 2665 y Fx(\(4)p Fn(:)p Fx(46\))50 3004 y(This)28 b(pro)m(v)m(es)j(the)g(lemma.)p Fm(})159 3222 y Fx(As)36 b(in)f(the)h(case)h(of)f(Lemma)g(4.3,)j(w)m(e)e(get)g(as)f(an)g (immediate)f(corollary)h(an)g(upp)s(er)e(b)s(ound)g(on)i(the)50 3367 y(norm)29 b(of)i(the)g(resolv)m(en)m(t.)50 3585 y Fd(Corollary)36 b(4.7:)89 b Ft(F)-7 b(or)32 b(al)5 b(l)32 b Fx(0)26 b Fn(<)e(\015)31 b(<)25 b Fx(1)31 b Ft(ther)-5 b(e)32 b(exists)f(a)h(c)-5 b(onstant)33 b Fx(0)25 b Fn(<)g(L)g(<)g Fm(1)31 b Ft(\(dep)-5 b(ending)33 b(on)e Fn(C)q(;)15 b(C)3684 3552 y Fk(0)50 3731 y Ft(and)33 b Fn(\015)5 b Ft(\))33 b(such)g(that,)h(for)f(al)5 b(l)33 b Fn(u)25 b Fm(2)g Fn(D)1282 3745 y Fv(1)1323 3731 y Fx(\(4\))d Fm(\\)d Fn(D)1615 3745 y Fv(2)1656 3731 y Fx(\()p Fn(L;)c(\015)5 b Fx(\))p Ft(,)1620 3985 y Fm(k)p Fn(K)1749 3948 y Fk(\003)1742 4008 y Fl(E)1804 3985 y Fx(\()p Fn(u)p Fx(\))p Fm(k)27 b Fn(<)e Fx(1)1338 b(\(4)p Fn(:)p Fx(47\))50 4240 y Ft(and)123 4495 y Fm(k)p Fn(M)266 4457 y Fk(\003)256 4517 y Fl(E)318 4495 y Fx(\()p Fn(u)p Fx(\))p Fm(k)26 b(\024)1911 4433 y Fx(1)21 b(+)f Fn(C)7 b(\015)2192 4400 y Fk(\000)p Fv(1)2289 4433 y Fn(r)p 618 4474 3008 4 v 618 4492 a Fi(q)p 709 4492 1264 4 v 115 x Fx(1)21 b(+)876 4584 y Fi(b)866 4607 y Fx(\002)937 4581 y Fv(2)977 4607 y Fx(2\(1)h Fm(\000)d Fx(cos)d Fn(v)s Fx(\)\(1)22 b Fm(\000)e Fn(O)s Fx(\()1746 4584 y(\026)1733 4607 y(\002)1804 4581 y Fk(\000)p Fv(1)1901 4607 y Fx(\)\))h Fm(\000)f Fx(1)h Fm(\000)2317 4571 y Fv(4)p 2251 4586 170 4 v 2251 4638 a Fl(M)7 b Fk(\000)p Fv(1)2431 4607 y Fx(\(1)21 b(+)f Fn(O)s Fx(\()p Fn(d=)-5 b(N)10 b Fx(\)\))22 b Fm(\000)d Fx(\()p Fn(C)28 b Fx(+)19 b Fn(C)3372 4581 y Fk(0)3397 4607 y Fx(\))p Fn(\015)3484 4581 y Fk(\000)p Fv(1)3582 4607 y Fn(r)3477 4729 y Fx(\(4)p Fn(:)p Fx(48\))159 5093 y(Finally)25 b(w)m(e)j(will)d(need)i(an)g(estimate)h(on)f Fm(k)p Fn(M)1724 5060 y Fk(\003)1714 5118 y Fl(E)1776 5093 y Fx(\()p Fn(u)p Fx(\))p Fm(k)i Fx(in)d(the)h(case)i(when)d Fm(j)p Fn(u)p Fm(j)i Fx(is)e(v)m(ery)i(small)e(and)h Fn(w)g Fm(\024)e Fx(0)50 5238 y(that)31 b(sho)m(ws)f(that)h(there,)g (the)f(negativ)m(e)i(real)e(part)g(helps)f(to)i(depress)e Fm(k)p Fn(K)2644 5205 y Fk(\003)2637 5263 y Fl(E)2700 5238 y Fx(\()p Fn(u)p Fx(\))p Fm(k)d Fn(<)f Fx(1)31 b(do)m(wn)f(from)g (one.)p eop %%Page: 32 32 32 31 bop 50 295 a Fr(32)3270 b Fo(Se)l(ction)27 b Fr(4)50 513 y Fd(Lemma)38 b(4.8:)51 b Ft(F)-7 b(or)33 b Fn(M)43 b Ft(lar)-5 b(ge)34 b(enough,)10 731 y(\(i\))47 b(for)33 b(al)5 b(l)33 b Fn(u)25 b Fm(2)g Fn(D)672 745 y Fv(3)713 731 y Ft(,)1327 883 y Fm(k)p Fn(K)1456 845 y Fk(\003)1449 905 y Fl(E)1511 883 y Fx(\()p Fn(u)p Fx(\))p Fm(k)h(\024)2148 821 y Fx(1)p 1811 862 721 4 v 1811 880 a Fm(p)p 1887 880 436 4 v 76 x Fx(1)20 b(+)g Fn(r)2087 930 y Fv(2)2148 956 y Fx(+)g Fn(s)2282 930 y Fv(2)2342 956 y Fm(\000)2464 920 y Fv(5)p 2444 935 77 4 v 2444 987 a Fl(M)3477 883 y Fx(\(4)p Fn(:)p Fx(49\))10 1193 y Ft(\(i\))47 b(for)33 b(al)5 b(l)33 b Fn(u)25 b Fm(2)g Fn(D)672 1207 y Fv(1)713 1193 y Fx(\(4\))d Fm(\\)e Fn(D)1006 1207 y Fv(3)1047 1193 y Ft(,)32 b Fm(k)p Fn(K)1236 1160 y Fk(\003)1229 1218 y Fl(E)1291 1193 y Fx(\()p Fn(u)p Fx(\))p Fm(k)27 b Fn(<)e Fx(1)33 b Ft(and)1243 1437 y Fm(k)p Fn(M)1386 1400 y Fk(\003)1376 1460 y Fl(E)1438 1437 y Fx(\()p Fn(u)p Fx(\))p Fm(k)26 b(\024)2154 1376 y Fx(1)p 1738 1416 877 4 v 1738 1434 a Fm(p)p 1814 1434 436 4 v 76 x Fx(1)21 b(+)f Fn(r)2015 1484 y Fv(2)2075 1510 y Fx(+)g Fn(s)2209 1484 y Fv(2)2270 1510 y Fm(\000)f Fx(1)i Fm(\000)2548 1474 y Fv(5)p 2528 1489 77 4 v 2528 1542 a Fl(M)3477 1437 y Fx(\(4)p Fn(:)p Fx(50\))50 1891 y Fc(Pro)s(of:)65 b Fx(The)38 b(pro)s(of)f(of)i(this)e(estimate)i(go)s(es)g(quite)e (along)h(the)h(lines)d(of)j(the)f(pro)s(of)f(of)i(the)f(previous)50 2037 y(lemmas.)74 b(Ho)m(w)m(ev)m(er,)47 b(to)c(simplify)38 b(things,)44 b(w)m(e)e(b)s(ound)e(the)i(Green)g(function)e(in)h(the)h (n)m(umerator)g(of)50 2182 y(\(4.37\))30 b(b)m(y)f(its)e(v)-5 b(alue)28 b(at)h(zero)g(and,)g(instead)e(of)i(using)d(\(4.40\))31 b(in)c(the)h(denominator,)h(w)m(e)f(go)h(bac)m(k)h(to)f(the)50 2328 y(estimates)i(\(4.18\))h(and)e(\(4.20\))j(whic)m(h)c(w)m(e)i(mo)s (dify)d(sligh)m(tly)h(to)i(yield,)e(for)h Fn(w)e Fm(\024)d Fx(0,)65 2600 y Fm(=)p Fn(G)203 2563 y Fl(\033)203 2625 y(\033)n(;T)10 b Fv(\()p Fl(E)s Fv(\))435 2600 y Fx(\()p Fn(iv)24 b Fx(+)c Fn(w)r Fx(\))26 b(=)917 2486 y Fk(1)887 2514 y Fi(X)884 2706 y Fl(n)p Fv(=1)1036 2600 y Fn(e)1078 2563 y Fl(nw)1198 2600 y Fx(sin)n(\()p Fn(v)s(n)p Fx(\))p Fj(P)p Fx([)p Fn(\034)1611 2563 y Fl(\033)1601 2623 y(\033)1687 2600 y Fx(=)f Fn(\034)1833 2563 y Fl(\033)1823 2625 y(T)10 b Fv(\()p Fl(E)s Fv(\))2019 2600 y Fx(=)25 b Fn(n)p Fx(])788 2902 y(=)g Fn(e)926 2864 y Fl(w)1001 2902 y Fx(sin)n(\()p Fn(v)s Fx(\))p Fn(p)1275 2916 y Fl(N)1345 2902 y Fx(\()p Fn(\033)n(;)15 b(\033)s Fx(\))22 b(+)1749 2816 y Fi(X)1673 3015 y Fl(\033)1717 2997 y Fe(0)1742 3015 y Fk(62)p Fl(T)10 b Fv(\()p Fl(E)s Fv(\))1972 2902 y Fn(p)2018 2916 y Fl(N)2087 2902 y Fx(\()p Fn(\033)n(;)15 b(\033)2267 2864 y Fk(0)2293 2902 y Fx(\))2376 2788 y Fk(1)2347 2816 y Fi(X)2343 3008 y Fl(n)p Fv(=1)2496 2902 y Fn(e)2538 2864 y Fl(w)r(n)2657 2902 y Fx(sin)o(\()p Fn(v)s Fx(\()p Fn(n)21 b Fx(+)f(1\)\))p Fj(P)p Fx([)p Fn(\034)3298 2864 y Fl(\033)3342 2837 y Fe(0)3288 2924 y Fl(\033)3399 2902 y Fx(=)k Fn(\034)3544 2864 y Fl(\033)3588 2837 y Fe(0)3534 2927 y Fl(T)10 b Fv(\()p Fl(E)s Fv(\))3730 2902 y Fx(=)25 b Fn(n)p Fx(])788 3153 y Fm(\024)g Fn(p)930 3167 y Fl(N)999 3153 y Fx(\()p Fn(\033)n(;)15 b(\033)s Fx(\))p Fn(e)1256 3116 y Fl(w)1332 3153 y Fx(sin)f Fn(v)23 b Fx(+)d Fn(d)1664 3167 y Fl(\033)1713 3153 y Fx(\()p Fn(v)s Fx(\))3477 3263 y(\(4)p Fn(:)p Fx(51\))50 3408 y(with)29 b Fn(d)304 3422 y Fl(\033)353 3408 y Fx(\()p Fn(v)s Fx(\))i(from)f(\(4.18\).)43 b(Similarly)-8 b(,)742 3644 y Fm(<)823 3543 y Fi(\020)877 3644 y Fx(1)21 b Fm(\000)e Fn(G)1105 3606 y Fl(\033)1105 3669 y(\033)n(;T)10 b Fv(\()p Fl(E)s Fv(\))1338 3644 y Fx(\()p Fn(iv)24 b Fx(+)c Fn(w)r Fx(\))1665 3543 y Fi(\021)1745 3644 y Fx(=)25 b Fn(p)1887 3658 y Fl(N)1956 3644 y Fx(\()p Fn(\033)n(;)15 b(\033)s Fx(\)\(1)23 b Fm(\000)d Fn(e)2407 3606 y Fl(w)2481 3644 y Fx(cos)c Fn(v)s Fx(\))21 b(+)f Fn(c)2851 3658 y Fl(\033)2900 3644 y Fx(\()p Fn(v)s Fx(\))460 b(\(4)p Fn(:)p Fx(52\))50 3880 y(with)29 b Fn(c)296 3894 y Fl(\033)345 3880 y Fx(\()p Fn(v)s Fx(\))j(from)d(\(4.20\).)43 b(On)30 b(the)g(other)h(hand)50 4116 y Fm(j)p Fj(P)130 4130 y Fl(\033)191 4116 y Fx(+)11 b Fn(p)319 4130 y Fl(N)388 4116 y Fx(\(1)g Fm(\000)g Fn(e)603 4078 y Fl(u)651 4116 y Fx(\))p Fm(j)711 4078 y Fv(2)778 4116 y Fx(=)25 b Fj(P)929 4078 y Fv(2)929 4138 y Fl(\033)990 4116 y Fx(+)11 b(2)p Fn(p)1163 4130 y Fl(N)1232 4116 y Fx(\(1)g Fm(\000)g Fx(cos)17 b Fn(v)s Fx(\)\()p Fn(p)1706 4130 y Fl(N)1787 4116 y Fx(+)11 b Fj(P)1924 4130 y Fl(\033)1973 4116 y Fx(\))g Fm(\000)g Fx(2)k(cos)i Fn(v)s(p)2392 4130 y Fl(N)2461 4116 y Fx(\()p Fn(e)2538 4078 y Fl(w)2609 4116 y Fm(\000)11 b Fx(1\)\()p Fn(p)2852 4130 y Fl(N)2933 4116 y Fx(+)g Fj(P)3070 4130 y Fl(\033)3120 4116 y Fx(\))g(+)g Fn(p)3294 4078 y Fv(2)3294 4138 y Fl(N)3363 4116 y Fx(\()p Fn(e)3440 4078 y Fv(2)p Fl(w)3547 4116 y Fm(\000)g Fx(1\))3477 4261 y(\(4)p Fn(:)p Fx(53\))50 4407 y(F)-8 b(or)31 b Fn(w)i Fx(small,)c(w)m(e)i(can)f(expand)g Fn(e)1232 4374 y Fl(w)1322 4407 y Fx(to)h(second)f(order)g(and,)g(using)f(that)i Fn(w)d Fm(\024)d Fx(0,)31 b(w)m(e)f(get)625 4602 y Fm(j)p Fj(P)705 4616 y Fl(\033)775 4602 y Fx(+)20 b Fn(p)912 4616 y Fl(N)980 4602 y Fx(\(1)h Fm(\000)f Fn(e)1214 4565 y Fl(u)1262 4602 y Fx(\))p Fm(j)1322 4565 y Fv(2)650 4775 y Fx(=)25 b Fj(P)801 4738 y Fv(2)801 4798 y Fl(\033)871 4775 y Fx(+)20 b(2)p Fn(p)1053 4789 y Fl(N)1122 4775 y Fx(\(1)h Fm(\000)f Fx(cos)c Fn(v)s Fx(\)\()p Fn(p)1614 4789 y Fl(N)1704 4775 y Fx(+)k Fj(P)1850 4789 y Fl(\033)1899 4775 y Fx(\))h Fm(\000)f Fx(2)p Fn(w)r(p)2204 4789 y Fl(N)2273 4775 y Fx([)p Fn(p)2344 4789 y Fl(N)2433 4775 y Fm(\000)g Fx(cos)c Fn(v)s Fx(\()p Fn(p)2789 4789 y Fl(N)2879 4775 y Fx(+)k Fj(P)3025 4789 y Fl(\033)3074 4775 y Fx(\)])736 4948 y(+)g(+)p Fn(w)965 4910 y Fv(2)1006 4948 y Fn(p)1052 4962 y Fl(N)1121 4948 y Fx([2)p Fn(p)1237 4962 y Fl(N)1326 4948 y Fm(\000)g Fx(cos)c Fn(v)s Fx(\()p Fn(p)1682 4962 y Fl(N)1772 4948 y Fx(+)k Fj(P)1918 4962 y Fl(\033)1967 4948 y Fx(\)])h(+)f Fn(O)s Fx(\()p Fn(w)2313 4910 y Fv(3)2354 4948 y Fx(\))650 5121 y(=)25 b Fj(P)801 5083 y Fv(2)801 5143 y Fl(\033)871 5121 y Fx(+)20 b(2)p Fn(p)1053 5135 y Fl(N)1122 5121 y Fx(\(1)h Fm(\000)f Fx(cos)c Fn(v)s Fx(\)\()p Fn(p)1614 5135 y Fl(N)1704 5121 y Fx(+)k Fj(P)1850 5135 y Fl(\033)1899 5121 y Fx(\)\(1)h Fm(\000)f Fn(w)r Fx(\))h Fm(\000)f Fx(2)p Fn(w)r(p)2498 5135 y Fl(N)2568 5121 y Fx(\(1)h Fm(\000)f Fn(p)2806 5135 y Fl(N)2875 5121 y Fx(\))736 5294 y(+)g Fn(w)894 5256 y Fv(2)935 5294 y Fn(p)981 5308 y Fl(N)1050 5294 y Fx([2)p Fn(p)1166 5308 y Fl(N)1256 5294 y Fm(\000)f Fx(cos)d Fn(v)s Fx(\()p Fn(p)1611 5308 y Fl(N)1701 5294 y Fx(+)k Fj(P)1847 5308 y Fl(\033)1896 5294 y Fx(\)])h(+)f Fn(O)s Fx(\()p Fn(w)2242 5256 y Fv(3)2284 5294 y Fx(\))650 5466 y Fm(\025)25 b Fj(P)801 5429 y Fv(2)801 5489 y Fl(\033)871 5466 y Fx(+)20 b Fn(v)1009 5429 y Fv(2)1070 5466 y Fx(+)g Fn(w)1228 5429 y Fv(2)1289 5466 y Fx(+)g Fn(O)s Fx(\()p Fn(w)1554 5429 y Fv(3)1596 5466 y Fx(\))3477 5034 y(\(4)p Fn(:)p Fx(54\))p eop %%Page: 33 33 33 32 bop 50 295 a Fo(A)l(ging)27 b(in)h(the)g(REM.)f(Part)h(2.)2661 b Fr(33)50 513 y Fx(Th)m(us)1007 586 y Fi(X)946 786 y Fl(\033)990 767 y Fe(0)1015 786 y Fk(2)p Fl(T)10 b Fk(n)p Fl(\033)1214 673 y Fm(j)p Fn(K)1323 635 y Fk(\003)1316 695 y Fl(\033)n(;\033)1422 677 y Fe(0)1452 673 y Fx(\()p Fn(u)p Fx(\))p Fm(j)26 b(\024)2244 611 y Fx(1)p 1732 652 1071 4 v 1732 670 a Fi(q)p 1823 670 771 4 v 115 x Fx(1)20 b(+)g Fj(P)2034 746 y Fk(\000)p Fv(2)2034 796 y Fl(\033)2143 762 y Fi(b)2132 785 y Fx(\002)2203 759 y Fv(2)2244 785 y Fx(\()p Fn(s)2322 759 y Fv(2)2383 785 y Fx(+)g Fn(r)2518 759 y Fv(2)2558 785 y Fx(\))h Fm(\000)2735 749 y Fv(5)p 2716 764 77 4 v 2716 817 a Fl(M)3477 673 y Fx(\(4)p Fn(:)p Fx(55\))50 980 y(and)30 b(since)f(this)h(is)f (clearly)h(monotone)h(in)e Fj(P)1589 994 y Fl(\033)1638 980 y Fx(,)i(it)f(follo)m(ws)f(that)1260 1251 y Fm(k)p Fn(K)1389 1213 y Fk(\003)1382 1273 y Fl(E)1444 1251 y Fx(\()p Fn(u)p Fx(\))p Fm(k)e(\024)2094 1189 y Fx(1)p 1744 1230 745 4 v 1744 1248 a Fm(p)p 1820 1248 460 4 v 76 x Fx(1)21 b(+)f Fn(s)2020 1298 y Fv(2)2080 1324 y Fx(+)g Fn(w)2238 1298 y Fv(2)2300 1324 y Fm(\000)2422 1288 y Fv(5)p 2402 1303 77 4 v 2402 1355 a Fl(M)3477 1251 y Fx(\(4)p Fn(:)p Fx(56\))50 1556 y(and)30 b(hence,)g(for)h Fn(u)25 b Fm(2)g Fn(D)882 1570 y Fv(1)923 1556 y Fx(\(4\),)31 b Fm(k)p Fn(K)1223 1523 y Fk(\003)1216 1581 y Fl(E)1278 1556 y Fx(\()p Fn(u)p Fx(\))p Fm(k)c Fn(<)e Fx(1)31 b(and)1176 1834 y Fm(k)p Fn(M)1319 1797 y Fk(\003)1309 1857 y Fl(E)1371 1834 y Fx(\()p Fn(u)p Fx(\))p Fm(k)c(\024)2099 1773 y Fx(1)p 1672 1813 901 4 v 1672 1831 a Fm(p)p 1747 1831 460 4 v 1747 1907 a Fx(1)21 b(+)f Fn(s)1947 1881 y Fv(2)2008 1907 y Fx(+)f Fn(w)2165 1881 y Fv(2)2227 1907 y Fm(\000)h Fx(1)h Fm(\000)2505 1871 y Fv(5)p 2485 1886 77 4 v 2485 1939 a Fl(M)3477 1834 y Fx(\(4)p Fn(:)p Fx(57\))50 2134 y Fm(})50 2344 y Fu(4.3.)47 b(P)m(erturbativ)m(e)35 b(estimates)e(for)i (small)g Fn(u)p Fu(.)159 2627 y Fx(The)g(a)h(priori)d(b)s(ounds)h (obtained)h(in)f(the)i(last)g(subsection)f(will)e(su\016ce)i(to)i(sho)m (w)e(that)i(the)e(con)m(tri-)50 2772 y(butions)f(from)i Fn(u)g Fx(a)m(w)m(a)m(y)i(from)d(zero)i(in)e(the)h(Laplace)h(in)m(v)m (ersion)d(form)m(ula)i(are)g(sub-dominan)m(t.)56 b(In)36 b(the)50 2918 y(neigh)m(b)s(orho)s(o)s(d)27 b(of)j(zero)h(w)m(e)f(ha)m (v)m(e)i(to)e(pro)s(ceed)g(more)g(carefully)f(and)g(extract)i(the)f (dominan)m(t)f(con)m(tribu-)50 3063 y(tion)36 b(to)i(the)g(resolv)m(en) m(t,)i(while)35 b(estimating)h(the)i(remainders.)59 b(This)36 b(will)e(b)s(e)j(done)g(b)m(y)g(decomp)s(osing)50 3208 y Fn(K)134 3175 y Fk(\003)127 3233 y Fl(E)188 3208 y Fx(\()p Fn(u)p Fx(\))d(in)d(a)h(suitable)f(w)m(a)m(y)-8 b(,)35 b(the)e(idea)f(b)s(eing)f(that)i(the)f(leading)f(term)i(should)d (allo)m(w)i(exact)i(computa-)50 3354 y(tions;)28 b(in)e(fact,)k(w)m(e)e (will)d(w)m(an)m(t)k(this)d(term)i(to)g(b)s(e)f(a)h(matrix)f(with)g (constan)m(t)i(columns.)38 b(T)-8 b(o)28 b(this)f(end)g(note)50 3499 y(that)k(for)f Fn(\033)e Fm(6)p Fx(=)d Fn(\033)617 3466 y Fk(0)643 3499 y Fx(,)31 b(b)m(y)f(T)-8 b(a)m(ylor's)31 b(form)m(ula,)225 3785 y Fn(K)309 3748 y Fk(\003)302 3808 y Fl(\033)n(;\033)408 3790 y Fe(0)438 3785 y Fx(\()p Fn(u)p Fx(\))26 b(=)880 3724 y(1)p 668 3764 471 4 v 668 3848 a(1)20 b Fm(\000)g Fn(G)896 3816 y Fl(\033)896 3874 y(\033)n(;T)1015 3848 y Fx(\()p Fn(u)p Fx(\))1164 3657 y Fi(\022)1231 3785 y Fn(G)1303 3748 y Fl(\033)1303 3808 y(\033)1347 3790 y Fe(0)1371 3808 y Fl(;T)1450 3785 y Fx(\(0\))h(+)f Fn(u)1766 3724 y(d)p 1740 3764 100 4 v 1740 3848 a(du)1851 3785 y(G)1923 3748 y Fl(\033)1923 3808 y(\033)1967 3790 y Fe(0)1991 3808 y Fl(;T)2070 3785 y Fx(\(0\))h(+)2308 3724 y Fn(u)2360 3691 y Fv(2)p 2308 3764 93 4 v 2332 3848 a Fx(2)2449 3724 y Fn(d)2496 3691 y Fv(2)p 2423 3764 141 4 v 2423 3848 a Fn(du)2522 3821 y Fv(2)2574 3785 y Fn(G)2646 3748 y Fl(\033)2646 3808 y(\033)2690 3790 y Fe(0)2714 3808 y Fl(;T)2793 3785 y Fx(\()6 b(~)-51 b Fn(u)q Fx(\))2916 3657 y Fi(\023)682 4059 y Fx(=)1001 3997 y(1)p 789 4038 471 4 v 789 4121 a(1)20 b Fm(\000)g Fn(G)1017 4090 y Fl(\033)1017 4148 y(\033)n(;T)1136 4121 y Fx(\()p Fn(u)p Fx(\))1285 3930 y Fi(\022)1352 4059 y Fj(P)p Fx([)p Fn(\034)1482 4021 y Fl(\033)1472 4081 y(\033)1516 4063 y Fe(0)1571 4059 y Fm(\024)25 b Fn(\034)1717 4021 y Fl(\033)1707 4081 y(T)1766 4059 y Fx(])c(+)f Fn(u)p Fj(E)10 b Fn(\034)2059 4021 y Fl(\033)2049 4081 y(\033)2093 4063 y Fe(0)2129 4059 y Fx(1)-23 b(I)2184 4075 y Fk(f)p Fl(\034)2262 4053 y Fp(\033)2255 4106 y(\033)2294 4093 y Fe(0)2323 4075 y Fk(\024)p Fl(\034)2421 4053 y Fp(\033)2414 4099 y(T)2464 4075 y Fk(g)2526 4059 y Fx(+)2627 3997 y Fn(u)2679 3964 y Fv(2)p 2627 4038 93 4 v 2651 4121 a Fx(2)2768 3997 y Fn(d)2815 3964 y Fv(2)p 2742 4038 141 4 v 2742 4121 a Fn(du)2841 4095 y Fv(2)2893 4059 y Fn(G)2965 4021 y Fl(\033)2965 4081 y(\033)3009 4063 y Fe(0)3034 4081 y Fl(;T)3112 4059 y Fx(\()6 b(~)-51 b Fn(u)q Fx(\))3235 3930 y Fi(\023)3477 3928 y Fx(\(4)p Fn(:)p Fx(58\))50 4356 y(where)46 b(~)-51 b Fn(u)40 b Fx(is)g(on)h(the)f(ra)m(y)h(b)s(et) m(w)m(een)h(0)f(and)f Fn(u)p Fx(.)71 b(The)40 b(idea)g(is)g(of)h (course)f(that)i(since)d Fn(u)i Fx(is)f(small,)i(the)50 4501 y(quadratic)31 b(term)h(is)f(a)h(small)f(p)s(erturbation)1592 4468 y Fv(14)1699 4501 y Fx(while)f(the)i(constan)m(t)h(and)e(linear)g (terms)g(are)i(essen)m(tially)50 4647 y(indep)s(enden)m(t)28 b(of)j Fn(\033)724 4614 y Fk(0)749 4647 y Fx(,)g(the)f(deviations)g(b)s (eing)f(treatable)i(as)f(p)s(erturbations)f(as)h(w)m(ell.)159 4865 y(Let)38 b(us)e(\014rst)h(establish)e(a)j(b)s(ound)d(on)i(the)g (second)h(order)e(con)m(tribution.)60 b(The)37 b(notation)g(and)g(def-) 50 5011 y(initions)f(of)j(the)g(presen)m(t)f(are)h(the)g(same)g(as)g (in)e(the)i(previous)f(one)g(\(recall)h(in)e(particular)g(De\014nition) 50 5156 y(4.5\).)p 50 5284 546 4 v 159 5345 a Fs(14)230 5371 y Fr(It)26 b(will)i(b)r(ecome)e(clear)i(only)e(later)h(wh)n(y)f(w) n(e)g(expand)g(to)g(second)h(order)g(and)f(are)g(not)g(con)n(ten)n(t)g (with)h(the)g(\014rst)g(order)50 5462 y(as)f(b)r(efore.)p eop %%Page: 34 34 34 33 bop 50 295 a Fr(34)3270 b Fo(Se)l(ction)27 b Fr(4)50 517 y Fd(Lemma)38 b(4.9:)51 b Ft(Denote)32 b(by)h Fn(K)1200 470 y Fk(\003)p Fv(\(2\))1193 543 y Fl(E)1367 517 y Ft(the)g(matrix)h (with)f(entries)1029 821 y Fn(K)1113 774 y Fk(\003)p Fl(;)p Fv(\(2\))1106 848 y Fl(\033)n(;\033)1212 830 y Fe(0)1269 821 y Fx(\()p Fn(u)p Fx(\))26 b(=)1513 635 y Fi(8)1513 717 y(<)1513 880 y(:)1646 679 y Fq(1)p 1646 689 31 4 v 1646 725 a(2)1688 703 y Fl(u)1731 676 y Fq(2)1797 679 y Fp(d)1831 660 y Fq(2)p 1777 689 108 4 v 1777 735 a Fp(du)1849 722 y Fq(2)1896 703 y Fl(G)1953 676 y Fp(\033)1953 729 y(\033)1992 716 y Fe(0)2016 729 y Fp(;T)2087 703 y Fv(\()5 b(~)-41 b Fl(u)p Fv(\))p 1635 745 552 4 v 1733 797 a(1)p Fk(\000)p Fl(G)1883 775 y Fp(\033)1883 821 y(\033)n(;T)1989 797 y Fv(\()p Fl(u)p Fv(\))2198 766 y Fn(;)108 b Ft(if)60 b Fn(\033)29 b Fm(6)p Fx(=)c Fn(\033)2679 733 y Fk(0)1624 934 y Fx(0)p Fn(;)637 b Ft(if)60 b Fn(\033)29 b Fx(=)c Fn(\033)2679 901 y Fk(0)3477 821 y Fx(\(4)p Fn(:)p Fx(59\))50 1117 y Ft(F)-7 b(or)34 b Fx(0)25 b Fn(<)g(\015)31 b(<)25 b Fx(1)p Ft(,)32 b(let)h(the)g(c)-5 b(onstant)35 b Fn(L)d Ft(b)-5 b(e)32 b(chosen)i(such)f(that)1165 1271 y Fi(\010)1218 1344 y Fn(u)26 b Fm(2)e Fj(C)49 b Fm(j)26 b Fn(r)i Fm(\024)d Fn(s)1731 1307 y Fv(2)1771 1344 y Fn(=)p Fx(4)1861 1271 y Fi(\011)1940 1344 y Fm(\022)g Fn(D)2111 1358 y Fv(2)2152 1344 y Fx(\()p Fn(L;)15 b(\015)5 b Fx(\))22 b Fm(\\)d Fn(D)2553 1358 y Fv(4)3477 1344 y Fx(\(4)p Fn(:)p Fx(60\))50 1572 y Ft(Then,)41 b(ther)-5 b(e)41 b(exists)f(a)g(c)-5 b(onstant)41 b Fn(C)k(>)38 b Fx(0)i Ft(such)g(that)g(for)h(al)5 b(l)40 b(for)g Fn(u)e Fm(2)g Fn(D)2684 1586 y Fv(2)2725 1572 y Fx(\()p Fn(L;)15 b(\015)5 b Fx(\))26 b Fm(\\)f Fn(D)3136 1586 y Fv(4)3217 1572 y Ft(and)40 b Fn(N)50 b Ft(lar)-5 b(ge)50 1718 y(enough,)1101 1880 y Fm(k)p Fn(K)1230 1833 y Fk(\003)p Fv(\(2\))1223 1907 y Fl(E)1365 1880 y Fx(\()p Fn(u)p Fx(\))p Fm(k)26 b(\024)1871 1819 y Fn(\015)1923 1786 y Fk(\000)p Fv(2)2021 1819 y Fn(C)7 b Fx(\()p Fn(s)2171 1786 y Fv(2)2231 1819 y Fx(+)20 b Fn(r)2366 1786 y Fv(2)2407 1819 y Fx(\))p 1665 1859 983 4 v 1665 1877 a Fi(p)p 1756 1877 598 4 v 78 x Fx(1)h(+)f(\()p Fn(s)1991 1929 y Fv(2)2051 1955 y Fx(+)g Fn(r)2186 1929 y Fv(2)2227 1955 y Fx(\))p Fn(=)p Fx(2)h Fm(\000)f Fx(5)p Fn(=)-5 b(M)3477 1880 y Fx(\(4)p Fn(:)p Fx(61\))50 2295 y Fu(Remark:)56 b Fx(The)34 b(assumption)f (\(4.60\))k(is)d(made)h(for)g(con)m(v)m(enience)g(only)f(as)h(it)g (allo)m(ws)f(to)h(simplify)d(the)50 2441 y(expressions)d(of)h(our)g (estimates.)50 2659 y Fu(Remark:)44 b Fx(Note)29 b(also)e(that)h(the)f (b)s(ound)f(\(4.61\))j(simply)c(b)s(eha)m(v)m(es,)k(for)e(small)2797 2636 y Fi(b)2787 2659 y Fx(\002)o Fn(u)p Fx(,)h(lik)m(e)f Fn(\015)3178 2626 y Fk(\000)p Fv(2)3276 2659 y Fn(C)7 b Fx(\()p Fn(s)3426 2626 y Fv(2)3480 2659 y Fx(+)14 b Fn(r)3609 2626 y Fv(2)3649 2659 y Fx(\).)50 2877 y Fc(Pro)s(of:)52 b Fx(T)-8 b(o)33 b(b)s(ound)e(the)h(denominator)g(w)m(e)h(pro)s(ceed)f (as)h(in)e(the)i(pro)s(ofs)f(of)g(Lemma)h(4.6)h(and)d(4.8)j(with)50 3023 y(the)c(di\013erence)g(that,)h(for)g Fn(r)c(>)e Fx(0,)31 b(the)g(b)s(ound)d(\(4.54\))33 b(b)s(ecomes,)e(using)e(that)i Fn(r)c Fm(\024)e Fn(s)2969 2990 y Fv(2)3010 3023 y Fn(=)p Fx(4,)926 3250 y Fm(j)p Fj(P)1006 3264 y Fl(\033)1076 3250 y Fx(+)20 b Fn(p)1213 3264 y Fl(N)1282 3250 y Fx(\(1)h Fm(\000)f Fn(e)1516 3213 y Fl(u)1564 3250 y Fx(\))p Fm(j)1624 3213 y Fv(2)1690 3250 y Fm(\025)25 b Fj(P)1841 3213 y Fv(2)1841 3273 y Fl(\033)1911 3250 y Fx(+)20 b(\()p Fn(v)2084 3213 y Fv(2)2146 3250 y Fx(+)g Fn(w)2304 3213 y Fv(2)2345 3250 y Fx(\))p Fn(=)p Fx(2)h(+)f Fn(O)s Fx(\()p Fn(w)2756 3213 y Fv(3)2798 3250 y Fx(\))644 b(\(4)p Fn(:)p Fx(62\))50 3478 y(F)-8 b(or)31 b(the)g(n)m(umerator)f(w)m(e)h(use)f(that)669 3620 y Fi(X)608 3819 y Fl(\033)652 3801 y Fe(0)677 3819 y Fk(2)p Fl(T)10 b Fk(n)p Fl(\033)876 3574 y Fi(\014)876 3629 y(\014)876 3683 y(\014)876 3738 y(\014)943 3645 y Fn(d)990 3612 y Fv(2)p 917 3685 141 4 v 917 3768 a Fn(du)1016 3742 y Fv(2)1068 3706 y Fn(G)1140 3668 y Fl(\033)1140 3728 y(\033)1184 3710 y Fe(0)1209 3728 y Fl(;T)1287 3706 y Fx(\()c(~)-51 b Fn(u)q Fx(\))1410 3574 y Fi(\014)1410 3629 y(\014)1410 3683 y(\014)1410 3738 y(\014)1466 3706 y Fm(\024)1622 3620 y Fi(X)1562 3819 y Fl(\033)1606 3801 y Fe(0)1631 3819 y Fk(2)p Fl(T)10 b Fk(n)p Fl(\033)1866 3645 y Fn(d)1913 3612 y Fv(2)p 1840 3685 V 1840 3768 a Fn(du)1939 3742 y Fv(2)1991 3706 y Fn(G)2063 3668 y Fl(\033)2063 3728 y(\033)2107 3710 y Fe(0)2132 3728 y Fl(;T)2211 3706 y Fx(\()p Fm(<)c Fx(~)-51 b Fn(u)p Fx(\))25 b(=)2557 3645 y Fn(d)2604 3612 y Fv(2)p 2531 3685 V 2531 3768 a Fn(du)2630 3742 y Fv(2)2682 3706 y Fn(G)2754 3668 y Fl(\033)2754 3731 y(T)10 b Fk(n)p Fl(\033)n(;T)2963 3706 y Fx(\()p Fm(<)c Fx(~)-51 b Fn(u)p Fx(\))326 b(\(4)p Fn(:)p Fx(63\))50 4024 y(and,)28 b(since)g Fm(<)6 b Fx(~)-51 b Fn(u)25 b Fm(\024)g Fx(\(1)17 b Fm(\000)f Fn(\015)5 b Fx(\))p Fn(=)1036 4001 y Fi(b)1026 4024 y Fx(\002)q(,)29 b(b)s(ound)d(the)j(last)f(quan)m(tit)m(y)h(in)e(the)i(r.h.s)f(pro)s (ceeding)f(as)i(in)e(the)i(pro)s(of)f(of)50 4169 y(Prop)s(osition)g (3.6)k(\(see)f(the)g(treatmen)m(t)h(of)e(the)h(term)f(\(I)s(I\))g (therein\).)40 b Fm(})159 4388 y Fx(What)f(remains)f(of)h Fn(K)966 4355 y Fk(\003)959 4413 y Fl(E)1059 4388 y Fx(after)h (subtraction)e(of)h Fn(K)1968 4340 y Fk(\003)p Fv(\(2\))1961 4414 y Fl(E)2141 4388 y Fx(is)f(almost)h(of)g(the)g(desired)e(form)i (\(i.e.)66 b(has)50 4533 y(almost)44 b(constan)m(t)h(columns\);)50 b(ho)m(w)m(ev)m(er,)e(a)d(few)e(cosmetic)i(c)m(hanges)g(need)e(to)i(b)s (e)e(made:)68 b(\014rst,)47 b(the)50 4679 y(matrix)30 b(elemen)m(ts)550 4906 y Fn(K)634 4859 y Fk(\003)p Fv(\(1\))627 4933 y Fl(\033)n(;\033)733 4915 y Fe(0)768 4906 y Fx(\()p Fn(u)p Fx(\))c Fm(\021)1236 4845 y Fx(1)p 1023 4885 471 4 v 1023 4969 a(1)21 b Fm(\000)f Fn(G)1252 4937 y Fl(\033)1252 4995 y(\033)n(;T)1371 4969 y Fx(\()p Fn(u)p Fx(\))1504 4906 y Fj(P)p Fx([)p Fn(\034)1634 4869 y Fl(\033)1624 4929 y(\033)1668 4911 y Fe(0)1724 4906 y Fm(\024)25 b Fn(\034)1870 4869 y Fl(\033)1860 4929 y(T)1919 4906 y Fx(])15 b(\(1)21 b(+)f Fn(u)p Fj(E)26 b Fx([)6 b Fn(\034)2354 4869 y Fl(\033)2344 4929 y(\033)2388 4911 y Fe(0)2417 4906 y Fm(j)p Fn(\034)2492 4869 y Fl(\033)2482 4929 y(\033)2526 4911 y Fe(0)2581 4906 y Fm(\024)25 b Fn(\034)2727 4869 y Fl(\033)2717 4929 y(T)2776 4906 y Fx(]\))31 b Fn(;)61 b(\033)28 b Fm(6)p Fx(=)d Fn(\033)3184 4869 y Fk(0)3477 4906 y Fx(\(4)p Fn(:)p Fx(64\))50 5168 y(ha)m(v)m(e)31 b(to)h(b)s(e)d(replaced)h(b)m(y)g(their)g(leading,)f Fn(\033)1590 5135 y Fk(0)1616 5168 y Fx(-indep)s(enden)m(t)f(part)213 5397 y Fn(K)297 5349 y Fk(\003)p Fv(\(0\))290 5423 y Fl(\033)n(;\033)396 5405 y Fe(0)432 5397 y Fx(\()p Fn(u)p Fx(\))e Fm(\021)899 5335 y Fx(1)p 687 5376 V 687 5459 a(1)20 b Fm(\000)g Fn(G)915 5428 y Fl(\033)915 5486 y(\033)n(;T)1034 5459 y Fx(\()p Fn(u)p Fx(\))1183 5296 y Fi(\020)1268 5361 y Fv(1)p 1248 5376 77 4 v 1248 5428 a Fl(M)1335 5397 y Fj(P)p Fx([)p Fn(\034)1465 5359 y Fl(\033)1455 5421 y(T)10 b Fk(n)p Fl(\033)1620 5397 y Fn(<)25 b(\034)1766 5359 y Fl(\033)1756 5419 y(T)1815 5397 y Fx(])1855 5296 y Fi(\020)1910 5397 y Fx(1)20 b(+)g Fn(u)p Fj(E)11 b Fx([)p Fn(\034)2248 5359 y Fl(\033)2238 5421 y(T)f Fk(n)p Fl(\033)2383 5397 y Fm(j)p Fn(\034)2458 5359 y Fl(\033)2448 5421 y(T)g Fk(n)p Fl(\033)2612 5397 y Fx(=)25 b Fn(\034)2758 5359 y Fl(\033)2748 5419 y(T)2807 5397 y Fx(])2832 5296 y Fi(\021\021)2971 5397 y Fn(;)61 b(\033)28 b Fm(6)p Fx(=)d Fn(\033)3288 5359 y Fk(0)3477 5397 y Fx(\(4)p Fn(:)p Fx(65\))p eop %%Page: 35 35 35 34 bop 50 295 a Fo(A)l(ging)27 b(in)h(the)g(REM.)f(Part)h(2.)2661 b Fr(35)50 513 y Fx(As)26 b(sho)m(wn)g(in)e(the)j(next)f(lemma,)h(this) e(replacemen)m(t)i(can)f(b)s(e)g(done)g(at)h(the)f(cost)i(of)e(error)g (terms)g(of)g(order)50 658 y(at)31 b(most)g Fn(O)s Fx(\(1)p Fn(=)-5 b(N)10 b Fx(\).)50 876 y Fd(Lemma)45 b(4.10:)64 b Ft(Denote)38 b(by)f Fn(K)1287 829 y Fk(\003)p Fv(\(0\))1280 903 y Fl(E)1459 876 y Ft(and)i Fn(K)1725 829 y Fk(\003)p Fv(\(1\))1718 903 y Fl(E)1897 876 y Ft(the)f(matrix)h(with)f (o\013-diagonal)h(entries)f(given)f(r)-5 b(e-)50 1022 y(sp)g(e)g(ctively)39 b(by)e(\(4.65\))j(and)f(\(4.64\))g(and)g(zer)-5 b(o)39 b(diagonals.)59 b(Then,)40 b(under)e(the)g(assumptions)i(and)f (with)50 1167 y(the)33 b(notation)h(of)f(L)-5 b(emma)34 b(4.9)f(and)h(Pr)-5 b(op)g(osition)35 b(2.2)f(we)e(have,)h(for)g Fn(N)43 b Ft(lar)-5 b(ge)33 b(enough,)719 1392 y Fm(k)p Fn(K)848 1355 y Fk(\003)p Fv(\(0\))983 1392 y Fx(\()p Fn(u)p Fx(\))21 b Fm(\000)f Fn(K)1301 1355 y Fk(\003)p Fv(\(1\))1435 1392 y Fx(\()p Fn(u)p Fx(\))p Fm(k)27 b(\024)1949 1331 y Fx(1)20 b(+)g(3)2150 1255 y Fm(p)p 2227 1255 279 4 v 2227 1331 a Fn(s)2270 1304 y Fv(2)2330 1331 y Fx(+)g Fn(r)2465 1304 y Fv(2)p 1736 1371 983 4 v 1736 1389 a Fi(p)p 1826 1389 598 4 v 1826 1467 a Fx(1)h(+)f(\()p Fn(s)2061 1441 y Fv(2)2122 1467 y Fx(+)g Fn(r)2257 1441 y Fv(2)2297 1467 y Fx(\))p Fn(=)p Fx(2)i Fm(\000)e Fx(5)p Fn(=)-5 b(M)2729 1392 y(O)s Fx(\(1)p Fn(=)g(N)10 b Fx(\))438 b(\(4)p Fn(:)p Fx(66\))159 1809 y(Second,)39 b(since)d(the)h(matrix)g Fn(K)1280 1776 y Fk(\003)p Fv(\(0\))1414 1809 y Fx(\()p Fn(u)p Fx(\))h(has)f(zero)h(diagonal,)g(w)m(e)f(still)e(ha)m(v)m(e)k (to)f(compare)f(it)g(to)h(the)50 1955 y(matrix)30 b Fm(K)416 1922 y Fk(\003)p Fv(\(0\))581 1955 y Fx(with)f(en)m(tries)194 2154 y Fm(K)264 2106 y Fk(\003)p Fv(\(0\))263 2180 y Fl(\033)n(;\033)369 2162 y Fe(0)399 2154 y Fx(\()p Fn(u)p Fx(\))d Fm(\021)866 2092 y Fx(1)p 654 2133 471 4 v 654 2216 a(1)21 b Fm(\000)f Fn(G)883 2185 y Fl(\033)883 2243 y(\033)n(;T)1001 2216 y Fx(\()p Fn(u)p Fx(\))1150 2053 y Fi(\020)1235 2118 y Fv(1)p 1215 2133 77 4 v 1215 2185 a Fl(M)1303 2154 y Fj(P)p Fx([)p Fn(\034)1433 2116 y Fl(\033)1423 2178 y(T)10 b Fk(n)p Fl(\033)1587 2154 y Fn(<)25 b(\034)1733 2116 y Fl(\033)1723 2176 y(T)1782 2154 y Fx(])1822 2053 y Fi(\020)1877 2154 y Fx(1)c(+)e Fn(u)p Fj(E)11 b Fx([)p Fn(\034)2215 2116 y Fl(\033)2205 2178 y(T)f Fk(n)p Fl(\033)2350 2154 y Fm(j)p Fn(\034)2425 2116 y Fl(\033)2415 2178 y(T)g Fk(n)p Fl(\033)2579 2154 y Fx(=)25 b Fn(\034)2725 2116 y Fl(\033)2715 2176 y(T)2774 2154 y Fx(])2799 2053 y Fi(\021)q(\021)2938 2154 y Fn(;)61 b Fm(8)p Fn(\033)n(;)15 b(\033)3220 2116 y Fk(0)3271 2154 y Fm(2)25 b Fn(T)13 b Fx(\()p Fn(E)5 b Fx(\))3477 2332 y(\(4)p Fn(:)p Fx(67\))50 2478 y(This)28 b(in)m(v)m(olv)m(es)j (con)m(trolling)e(the)i(norm)e(of)i(the)f(diagonal)g(matrix)g Fm(K)2435 2445 y Fk(\003)p Fv(\(0\))2570 2478 y Fx(\()p Fn(u)p Fx(\))21 b Fm(\000)f Fn(K)2888 2445 y Fk(\003)p Fv(\(0\))3023 2478 y Fx(\()p Fn(u)p Fx(\):)50 2696 y Fd(Lemma)38 b(4.11:)52 b Ft(L)-5 b(et)33 b Fm(K)970 2663 y Fk(\003)p Fv(\(0\))1138 2696 y Ft(b)-5 b(e)32 b(the)i(matrix)g (de\014ne)-5 b(d)34 b(in)e(\(4.67\).)44 b(Under)33 b(the)g(assumptions) i(and)f(with)50 2841 y(the)f(notation)h(of)f(L)-5 b(emma)34 b(4.10)g(we)f(have,)f(for)i Fn(N)42 b Ft(lar)-5 b(ge)34 b(enough,)602 3066 y Fm(kK)717 3029 y Fk(\003)p Fv(\(0\))852 3066 y Fx(\()p Fn(u)p Fx(\))21 b Fm(\000)f Fn(K)1170 3029 y Fk(\003)p Fv(\(0\))1305 3066 y Fx(\()p Fn(u)p Fx(\))p Fm(k)26 b(\024)1841 3005 y Fx(1)21 b(+)1998 2929 y Fm(p)p 2073 2929 279 4 v 2073 3005 a Fn(s)2116 2978 y Fv(2)2177 3005 y Fx(+)f Fn(r)2312 2978 y Fv(2)p 1605 3045 983 4 v 1605 3063 a Fi(p)p 1696 3063 598 4 v 78 x Fx(1)h(+)f(\()p Fn(s)1931 3115 y Fv(2)1992 3141 y Fx(+)g Fn(r)2127 3115 y Fv(2)2167 3141 y Fx(\))p Fn(=)p Fx(2)h Fm(\000)f Fx(5)p Fn(=)-5 b(M)2599 3066 y(O)s Fx(\(1)p Fn(=)p Fx(\()p Fn(M)32 b Fm(\000)20 b Fx(1\)\))320 b(\(4)p Fn(:)p Fx(68\))50 3483 y Fc(Pro)s(of)36 b(of)f(Lemma)d(4.10:)88 b Fx(F)-8 b(or)31 b Fn(\033)n(;)15 b(\033)1438 3450 y Fk(0)1490 3483 y Fm(2)25 b Fn(T)13 b Fx(,)30 b Fn(\033)e Fm(6)p Fx(=)d Fn(\033)1928 3450 y Fk(0)1954 3483 y Fx(,)31 b(let)f Fn(\024)2193 3497 y Fl(\033)n(;\033)2299 3479 y Fe(0)2329 3483 y Fx(\()p Fn(u)p Fx(\))h(b)s(e)f(de\014ned)f(through) 1176 3702 y Fn(K)1260 3654 y Fk(\003)p Fv(\(0\))1253 3728 y Fl(\033)n(;\033)1359 3710 y Fe(0)1395 3702 y Fx(\()p Fn(u)p Fx(\))21 b Fm(\000)f Fn(K)1713 3654 y Fk(\003)p Fv(\(1\))1706 3728 y Fl(\033)n(;\033)1812 3710 y Fe(0)1847 3702 y Fx(\()p Fn(u)p Fx(\))26 b(=)2182 3640 y Fn(\024)2234 3654 y Fl(\033)n(;\033)2340 3636 y Fe(0)2370 3640 y Fx(\()p Fn(u)p Fx(\))p 2102 3681 471 4 v 2102 3764 a(1)21 b Fm(\000)f Fn(G)2331 3733 y Fl(\033)2331 3791 y(\033)n(;T)2450 3764 y Fx(\()p Fn(u)p Fx(\))3477 3702 y(\(4)p Fn(:)p Fx(69\))50 3934 y(Since)29 b(the)i(denominator)e(in)g(\(4.69\))j(has)e(already)g (b)s(een)g(dealt)g(with)f(in)g(Lemma)h(4.9,)i(what)e(w)m(e)h(need)f(is) 50 4079 y(an)i(upp)s(er)d(b)s(ound)h(on)i Fm(j)p Fn(\024)926 4093 y Fl(\033)n(;\033)1032 4075 y Fe(0)1062 4079 y Fx(\()p Fn(u)p Fx(\))p Fm(j)p Fx(.)46 b(Appropriately)30 b(sorting)h(out)h(the) h(di\013eren)m(t)e(terms)h(con)m(tributing)e(to)50 4225 y Fn(\024)102 4239 y Fl(\033)n(;\033)208 4220 y Fe(0)238 4225 y Fx(\()p Fn(u)p Fx(\))h(w)m(e)g(ma)m(y)g(write,)545 4445 y Fm(j)p Fn(\024)622 4459 y Fl(\033)n(;\033)728 4441 y Fe(0)758 4445 y Fx(\()p Fn(u)p Fx(\))p Fm(j)c(\024)1018 4313 y Fi(\014)1018 4368 y(\014)1018 4422 y(\014)1018 4477 y(\014)1048 4445 y Fj(P)p Fx([)p Fn(\034)1178 4408 y Fl(\033)1168 4468 y(\033)1212 4449 y Fe(0)1267 4445 y Fm(\024)e Fn(\034)1413 4408 y Fl(\033)1403 4468 y(T)1462 4445 y Fx(])c Fm(\000)1636 4384 y Fx(1)p 1609 4424 99 4 v 1609 4508 a Fn(M)1718 4445 y(e)1760 4408 y Fk(\000)p Fl(\014)1861 4353 y Fk(p)p 1922 4353 65 4 v 1922 4408 a Fl(N)6 b(E)2039 4417 y Fp(\033)2087 4313 y Fi(\014)2087 4368 y(\014)2087 4422 y(\014)2087 4477 y(\014)2133 4344 y(\020)2187 4445 y Fx(1)21 b(+)f Fm(j)p Fn(u)p Fm(j)p Fj(E)11 b Fx([)p Fn(\034)2576 4408 y Fl(\033)2566 4470 y(T)f Fk(n)p Fl(\033)2711 4445 y Fm(j)p Fn(\034)2786 4408 y Fl(\033)2776 4470 y(T)g Fk(n)p Fl(\033)2940 4445 y Fx(=)25 b Fn(\034)3086 4408 y Fl(\033)3076 4468 y(T)3135 4445 y Fx(])3160 4344 y Fi(\021)932 4673 y Fx(+)o Fm(j)p Fn(u)p Fm(j)p Fj(P)p Fx([)p Fn(\034)1234 4635 y Fl(\033)1224 4695 y(\033)1268 4677 y Fe(0)1324 4673 y Fm(\024)g Fn(\034)1470 4635 y Fl(\033)1460 4695 y(T)1519 4673 y Fx(])1559 4568 y Fi(\014)1559 4623 y(\014)1559 4677 y(\014)1590 4673 y Fj(E)11 b Fx([)p Fn(\034)1720 4635 y Fl(\033)1710 4695 y(\033)1754 4677 y Fe(0)1814 4673 y Fm(j)26 b Fn(\034)1915 4635 y Fl(\033)1905 4695 y(\033)1949 4677 y Fe(0)2003 4673 y Fm(\024)f Fn(\034)2149 4635 y Fl(\033)2139 4695 y(T)2198 4673 y Fx(])c Fm(\000)f Fj(E)10 b Fx([)q Fn(\034)2465 4635 y Fl(\033)2455 4697 y(T)g Fk(n)p Fl(\033)2624 4673 y Fm(j)26 b Fn(\034)2725 4635 y Fl(\033)2715 4697 y(T)10 b Fk(n)p Fl(\033)2879 4673 y Fx(=)25 b Fn(\034)3025 4635 y Fl(\033)3015 4695 y(T)3074 4673 y Fx(])3099 4568 y Fi(\014)3099 4623 y(\014)3099 4677 y(\014)3477 4545 y Fx(\(4)p Fn(:)p Fx(70\))50 4862 y(Plugging)k(in)g(the)h(estimates)h(of) g(Prop)s(osition)d(2.1,)k(ii\),)388 5103 y Fm(j)p Fn(\024)465 5117 y Fl(\033)n(;\033)571 5098 y Fe(0)601 5103 y Fx(\()p Fn(u)p Fx(\))p Fm(j)26 b(\024)856 5041 y Fn(e)898 5008 y Fk(\000)p Fl(\014)999 4954 y Fk(p)p 1059 4954 V 54 x Fl(N)7 b(E)1177 5017 y Fp(\033)p 856 5082 369 4 v 991 5165 a Fn(M)1236 4974 y Fi(\024)1284 5002 y(\020)1338 5103 y Fx(1)21 b(+)f Fm(j)p Fn(u)p Fm(j)p Fj(E)11 b Fx([)p Fn(\034)1727 5065 y Fl(\033)1717 5127 y(T)f Fk(n)p Fl(\033)1862 5103 y Fm(j)p Fn(\034)1937 5065 y Fl(\033)1927 5127 y(T)g Fk(n)p Fl(\033)2091 5103 y Fx(=)25 b Fn(\034)2237 5065 y Fl(\033)2227 5125 y(T)2286 5103 y Fx(])2311 5002 y Fi(\021)2381 5103 y Fn(O)s Fx(\(1)p Fn(=)-5 b(N)10 b Fx(\))774 5385 y(+)p Fm(j)p Fn(u)p Fm(j)962 5280 y Fi(\014)962 5335 y(\014)962 5389 y(\014)993 5385 y Fj(E)h Fx([)p Fn(\034)1123 5347 y Fl(\033)1113 5407 y(\033)1157 5389 y Fe(0)1218 5385 y Fm(j)25 b Fn(\034)1318 5347 y Fl(\033)1308 5407 y(\033)1352 5389 y Fe(0)1407 5385 y Fm(\024)g Fn(\034)1553 5347 y Fl(\033)1543 5407 y(T)1601 5385 y Fx(])c Fm(\000)f Fj(E)10 b Fx([)q Fn(\034)1868 5347 y Fl(\033)1858 5409 y(T)g Fk(n)p Fl(\033)2028 5385 y Fm(j)25 b Fn(\034)2128 5347 y Fl(\033)2118 5409 y(T)10 b Fk(n)p Fl(\033)2282 5385 y Fx(=)25 b Fn(\034)2428 5347 y Fl(\033)2418 5407 y(T)2477 5385 y Fx(])2502 5280 y Fi(\014)2502 5335 y(\014)2502 5389 y(\014)2548 5385 y Fx(\(1)c(+)f Fn(O)s Fx(\(1)p Fn(=)-5 b(N)10 b Fx(\)\))3085 5229 y Fi(#)3477 5246 y Fx(\(4)p Fn(:)p Fx(71\))p eop %%Page: 36 36 36 35 bop 50 295 a Fr(36)3270 b Fo(Se)l(ction)27 b Fr(4)50 513 y Fx(and)k(w)m(e)i(are)f(left)g(to)h(b)s(ound)d(the)i(exp)s(ected)h (transition)d(time)i Fj(E)11 b Fx([)p Fn(\034)2367 480 y Fl(\033)2357 542 y(T)f Fk(n)p Fl(\033)2502 513 y Fm(j)p Fn(\034)2577 480 y Fl(\033)2567 542 y(T)g Fk(n)p Fl(\033)2734 513 y Fx(=)27 b Fn(\034)2882 480 y Fl(\033)2872 538 y(T)2931 513 y Fx(],)33 b(together)h(with)d(the)50 672 y(di\013erence)454 567 y Fi(\014)454 622 y(\014)454 676 y(\014)484 672 y Fj(E)11 b Fx([)p Fn(\034)614 639 y Fl(\033)604 697 y(\033)648 678 y Fe(0)709 672 y Fm(j)25 b Fn(\034)809 639 y Fl(\033)799 697 y(\033)843 678 y Fe(0)898 672 y Fm(\024)g Fn(\034)1044 639 y Fl(\033)1034 697 y(T)1093 672 y Fx(])20 b Fm(\000)g Fj(E)10 b Fx([)q Fn(\034)1359 639 y Fl(\033)1349 701 y(T)g Fk(n)p Fl(\033)1519 672 y Fm(j)25 b Fn(\034)1619 639 y Fl(\033)1609 701 y(T)10 b Fk(n)p Fl(\033)1773 672 y Fx(=)25 b Fn(\034)1919 639 y Fl(\033)1909 697 y(T)1968 672 y Fx(])1993 567 y Fi(\014)1993 622 y(\014)1993 676 y(\014)2024 672 y Fx(.)39 b(T)-8 b(o)28 b(deal)f(with)g(the)g(latter,)i (\014rst)e(observ)m(e)h(that)50 850 y(di\013eren)m(tiating)h(the)h (renew)m(al)h(equation)f Fn(G)1567 817 y Fl(\033)1567 879 y(\033)1611 861 y Fe(0)1635 879 y Fl(;T)10 b Fk(n)p Fl(\033)1796 850 y Fx(\()p Fn(u)p Fx(\))26 b(=)2083 787 y Fl(G)2140 760 y Fp(\033)2140 813 y(\033)2179 800 y Fe(0)2203 813 y Fp(;T)2274 787 y Fv(\()p Fl(u)p Fv(\))p 2051 829 357 4 v 2051 882 a(1)p Fk(\000)p Fl(G)2201 860 y Fp(\033)2201 905 y(\033)n(;T)2307 882 y Fv(\()p Fl(u)p Fv(\))2418 850 y Fx(,)k(w)m(e)h(ha)m(v)m(e)299 1103 y Fn(d)p 273 1144 100 4 v 273 1227 a(du)384 1165 y(G)456 1127 y Fl(\033)456 1187 y(\033)500 1169 y Fe(0)524 1187 y Fl(;T)603 1165 y Fx(\(0\))26 b(=)f(\(1)c Fm(\000)f Fj(P)p Fx([)p Fn(\034)1162 1127 y Fl(\033)1152 1187 y(\033)1237 1165 y Fm(\024)25 b Fn(\034)1383 1127 y Fl(\033)1373 1187 y(T)1432 1165 y Fx(]\))p Fj(E)11 b Fn(\034)1597 1127 y Fl(\033)1587 1187 y(\033)1632 1169 y Fe(0)1667 1165 y Fx(1)-23 b(I)1722 1181 y Fk(f)p Fl(\034)1800 1159 y Fp(\033)1793 1212 y(\033)1832 1199 y Fe(0)1861 1181 y Fk(\024)p Fl(\034)1959 1159 y Fp(\033)1952 1208 y(T)7 b Fe(n)p Fp(\033)2074 1181 y Fk(g)2136 1165 y Fm(\000)19 b Fj(P)p Fx([)p Fn(\034)2356 1127 y Fl(\033)2346 1187 y(\033)2390 1169 y Fe(0)2446 1165 y Fm(\024)25 b Fn(\034)2592 1127 y Fl(\033)2582 1190 y(T)10 b Fk(n)p Fl(\033)2720 1165 y Fx(])p Fj(E)h Fn(\034)2850 1127 y Fl(\033)2840 1187 y(\033)2905 1165 y Fx(1)-23 b(I)2961 1181 y Fk(f)p Fl(\034)3039 1163 y Fp(\033)3032 1199 y(\033)3082 1181 y Fk(\024)p Fl(\034)3180 1159 y Fp(\033)3173 1205 y(T)3223 1181 y Fk(g)744 1402 y Fx(=)25 b Fj(P)p Fx([)p Fn(\034)970 1364 y Fl(\033)960 1424 y(\033)1004 1406 y Fe(0)1060 1402 y Fm(\024)g Fn(\034)1206 1364 y Fl(\033)1196 1424 y(T)1255 1402 y Fx(])1295 1274 y Fi(\022)1362 1402 y Fj(E)10 b Fx([)q Fn(\034)1492 1364 y Fl(\033)1482 1424 y(\033)1526 1406 y Fe(0)1586 1402 y Fm(j)26 b Fn(\034)1687 1364 y Fl(\033)1677 1424 y(\033)1721 1406 y Fe(0)1775 1402 y Fm(\024)f Fn(\034)1921 1364 y Fl(\033)1911 1426 y(T)10 b Fk(n)p Fl(\033)2050 1402 y Fx(])20 b Fm(\000)2229 1326 y Fj(E)10 b Fn(\034)2333 1293 y Fl(\033)2323 1349 y(\033)2388 1326 y Fx(1)-23 b(I)2444 1343 y Fk(f)p Fl(\034)2522 1324 y Fp(\033)2515 1361 y(\033)2565 1343 y Fk(\024)p Fl(\034)2663 1321 y Fp(\033)2656 1366 y(T)2706 1343 y Fk(g)p 2197 1381 582 4 v 2197 1464 a Fx(1)21 b Fm(\000)f Fj(P)p Fx([)p Fn(\034)2484 1438 y Fl(\033)2474 1487 y(\033)2559 1464 y Fm(\024)25 b Fn(\034)2705 1433 y Fl(\033)2695 1491 y(T)2754 1464 y Fx(])2790 1274 y Fi(\023)3477 1288 y Fx(\(4)p Fn(:)p Fx(72\))50 1680 y(implying)i(that)781 1853 y Fj(E)f Fx([)6 b Fn(\034)932 1815 y Fl(\033)922 1875 y(\033)966 1857 y Fe(0)1020 1853 y Fm(j)26 b Fn(\034)1121 1815 y Fl(\033)1111 1875 y(\033)1155 1857 y Fe(0)1209 1853 y Fm(\024)f Fn(\034)1355 1815 y Fl(\033)1345 1875 y(T)1404 1853 y Fx(])h(=)f Fj(E)10 b Fx([)q Fn(\034)1681 1815 y Fl(\033)1671 1875 y(\033)1715 1857 y Fe(0)1775 1853 y Fm(j)26 b Fn(\034)1876 1815 y Fl(\033)1866 1875 y(\033)1910 1857 y Fe(0)1964 1853 y Fm(\024)f Fn(\034)2110 1815 y Fl(\033)2100 1878 y(T)10 b Fk(n)p Fl(\033)2239 1853 y Fx(])20 b Fm(\000)2417 1777 y Fj(E)11 b Fn(\034)2522 1744 y Fl(\033)2512 1800 y(\033)2577 1777 y Fx(1)-23 b(I)2633 1794 y Fk(f)p Fl(\034)2711 1775 y Fp(\033)2704 1812 y(\033)2754 1794 y Fk(\024)p Fl(\034)2852 1772 y Fp(\033)2845 1817 y(T)2895 1794 y Fk(g)p 2386 1832 V 2386 1915 a Fx(1)21 b Fm(\000)f Fj(P)p Fx([)p Fn(\034)2673 1889 y Fl(\033)2663 1938 y(\033)2748 1915 y Fm(\024)25 b Fn(\034)2894 1884 y Fl(\033)2884 1942 y(T)2943 1915 y Fx(])3477 1853 y(\(4)p Fn(:)p Fx(73\))50 2091 y(and,)47 b(since)c(the)i(last)f(term)g(in)f(the)h(r.h.s.)81 b(is)44 b Fn(\033)1823 2058 y Fk(0)1848 2091 y Fx(-indep)s(enden)m(t,)i(w)m(e)f (can)f(express)g(our)f(conditional)50 2236 y(exp)s(ectation)31 b(in)e(the)h(follo)m(wing,)f(remark)-5 b(ably)30 b(useful)e(form:)268 2500 y Fj(E)e Fx([)6 b Fn(\034)419 2463 y Fl(\033)409 2523 y(\033)453 2504 y Fe(0)508 2500 y Fm(j)25 b Fn(\034)608 2463 y Fl(\033)598 2523 y(\033)642 2504 y Fe(0)697 2500 y Fm(\024)g Fn(\034)843 2463 y Fl(\033)833 2523 y(T)892 2500 y Fx(])g(=)1130 2439 y(1)p 1024 2479 258 4 v 1024 2562 a Fm(j)p Fn(T)33 b Fm(n)21 b Fn(\033)s Fm(j)1368 2414 y Fi(X)1307 2613 y Fl(\033)1351 2595 y Fe(0)1376 2613 y Fk(2)p Fl(T)10 b Fk(n)p Fl(\033)1575 2500 y Fj(E)26 b Fx([)6 b Fn(\034)1726 2463 y Fl(\033)1716 2523 y(\033)1760 2504 y Fe(0)1814 2500 y Fm(j)26 b Fn(\034)1915 2463 y Fl(\033)1905 2523 y(\033)1949 2504 y Fe(0)2003 2500 y Fm(\024)f Fn(\034)2149 2463 y Fl(\033)2139 2523 y(T)2198 2500 y Fx(])942 2833 y(+)1013 2678 y Fi(\()1086 2833 y Fj(E)11 b Fx([)p Fn(\034)1216 2796 y Fl(\033)1206 2856 y(\033)1250 2838 y Fe(0)1311 2833 y Fm(j)25 b Fn(\034)1411 2796 y Fl(\033)1401 2856 y(\033)1445 2838 y Fe(0)1500 2833 y Fm(\024)g Fn(\034)1646 2796 y Fl(\033)1636 2858 y(T)10 b Fk(n)p Fl(\033)1774 2833 y Fx(])21 b Fm(\000)2028 2772 y Fx(1)p 1922 2813 V 1922 2896 a Fm(j)p Fn(T)33 b Fm(n)20 b Fn(\033)s Fm(j)2265 2747 y Fi(X)2205 2946 y Fl(\033)2249 2928 y Fe(0)2274 2946 y Fk(2)p Fl(T)10 b Fk(n)p Fl(\033)2472 2833 y Fj(E)h Fx([)p Fn(\034)2602 2796 y Fl(\033)2592 2856 y(\033)2636 2838 y Fe(0)2697 2833 y Fm(j)25 b Fn(\034)2797 2796 y Fl(\033)2787 2856 y(\033)2831 2838 y Fe(0)2886 2833 y Fm(\024)g Fn(\034)3032 2796 y Fl(\033)3022 2858 y(T)10 b Fk(n)p Fl(\033)3161 2833 y Fx(])3186 2678 y Fi(\))3477 2698 y Fx(\(4)p Fn(:)p Fx(74\))50 3166 y(Next)31 b(observ)m(e)g(that)g(b)m(y)f(\(4.9\),)1162 3097 y Fi(P)1258 3193 y Fl(\033)1302 3175 y Fe(0)1327 3193 y Fk(2)p Fl(T)10 b Fk(n)p Fl(\033)1530 3166 y Fj(P)p Fx([)p Fn(\034)1660 3133 y Fl(\033)1650 3191 y(\033)1694 3172 y Fe(0)1749 3166 y Fm(\024)25 b Fn(\034)1895 3133 y Fl(\033)1885 3191 y(T)1944 3166 y Fx(])h(=)e Fj(P)p Fx([)p Fn(\034)2220 3133 y Fl(\033)2210 3195 y(T)10 b Fk(n)p Fl(\033)2375 3166 y Fm(\024)25 b Fn(\034)2521 3133 y Fl(\033)2511 3191 y(T)2570 3166 y Fx(],)31 b(as)g(w)m(ell)e(as) 1135 3346 y Fi(X)1075 3545 y Fl(\033)1119 3527 y Fe(0)1144 3545 y Fk(2)p Fl(T)10 b Fk(n)p Fl(\033)1342 3432 y Fj(E)h Fn(\034)1447 3395 y Fl(\033)1437 3455 y(\033)1481 3437 y Fe(0)1516 3432 y Fx(1)-23 b(I)1572 3449 y Fk(f)p Fl(\034)1650 3427 y Fp(\033)1643 3480 y(\033)1682 3467 y Fe(0)1710 3449 y Fk(\024)p Fl(\034)1808 3427 y Fp(\033)1801 3472 y(T)1851 3449 y Fk(g)1918 3432 y Fx(=)25 b Fj(E)11 b Fn(\034)2119 3395 y Fl(\033)2109 3457 y(T)f Fk(n)p Fl(\033)2253 3432 y Fx(1)-23 b(I)2309 3449 y Fk(f)p Fl(\034)2387 3427 y Fp(\033)2380 3475 y(T)8 b Fe(n)p Fp(\033)2502 3449 y Fk(\024)p Fl(\034)2600 3427 y Fp(\033)2593 3472 y(T)2643 3449 y Fk(g)3477 3432 y Fx(\(4)p Fn(:)p Fx(75\))50 3764 y(hold)26 b(\(\(4.75\))k(is)d(obtained)g(b)m(y)h(di\013eren)m(tiating)e (\(4.9\))k(and)d(setting)g Fn(u)f Fx(=)f(0\).)40 b(On)27 b(the)h(other)g(hand,)f(using)50 3910 y(\(2.4\))32 b(from)e(Prop)s (osition)e(2.1,)k(the)f(\014rst)e(term)i(in)e(the)h(r.h.s)g(of)h (\(4.74\))h(ma)m(y)f(th)m(us)f(b)s(e)g(rewritten)f(as)292 4184 y(1)p 186 4224 V 186 4307 a Fm(j)p Fn(T)k Fm(n)21 b Fn(\033)s Fm(j)530 4159 y Fi(X)469 4358 y Fl(\033)513 4340 y Fe(0)538 4358 y Fk(2)p Fl(T)10 b Fk(n)p Fl(\033)737 4245 y Fj(E)26 b Fx([)6 b Fn(\034)888 4208 y Fl(\033)878 4268 y(\033)922 4249 y Fe(0)976 4245 y Fm(j)26 b Fn(\034)1077 4208 y Fl(\033)1067 4268 y(\033)1111 4249 y Fe(0)1166 4245 y Fm(\024)e Fn(\034)1311 4208 y Fl(\033)1301 4268 y(T)1360 4245 y Fx(])i(=)1567 4159 y Fi(X)1507 4358 y Fl(\033)1551 4340 y Fe(0)1576 4358 y Fk(2)p Fl(T)10 b Fk(n)p Fl(\033)1785 4162 y Fj(E)h Fn(\034)1890 4129 y Fl(\033)1880 4187 y(\033)1924 4169 y Fe(0)1959 4162 y Fx(1)-23 b(I)2015 4178 y Fk(f)p Fl(\034)2093 4156 y Fp(\033)2086 4210 y(\033)2125 4197 y Fe(0)2153 4178 y Fk(\024)p Fl(\034)2251 4156 y Fp(\033)2244 4202 y(T)2294 4178 y Fk(g)p 1785 4224 551 4 v 1808 4307 a Fj(P)p Fx([)p Fn(\034)1938 4276 y Fl(\033)1928 4338 y(T)10 b Fk(n)p Fl(\033)2093 4307 y Fm(\024)25 b Fn(\034)2239 4276 y Fl(\033)2229 4334 y(T)2288 4307 y Fx(])2362 4062 y Fi(0)2362 4226 y(@)2559 4184 y Fx(1)p 2453 4224 258 4 v 2453 4307 a Fm(j)p Fn(T)33 b Fm(n)20 b Fn(\033)s Fm(j)2807 4159 y Fi(X)2736 4358 y Fl(\033)2780 4340 y Fe(00)2825 4358 y Fk(2)p Fl(T)10 b Fk(n)p Fl(\033)3034 4184 y Fj(P)p Fx([)p Fn(\034)3164 4151 y Fl(\033)3154 4209 y(\033)3198 4190 y Fe(0)q(0)3274 4184 y Fm(\024)25 b Fn(\034)3420 4151 y Fl(\033)3410 4209 y(T)3469 4184 y Fx(])p 3034 4224 460 4 v 3044 4307 a Fj(P)p Fx([)p Fn(\034)3174 4276 y Fl(\033)3164 4334 y(\033)3208 4316 y Fe(0)3264 4307 y Fm(\024)g Fn(\034)3410 4276 y Fl(\033)3400 4334 y(T)3459 4307 y Fx(])3505 4062 y Fi(1)3505 4226 y(A)1411 4500 y Fx(=)g Fj(E)11 b Fx([)p Fn(\034)1637 4462 y Fl(\033)1627 4524 y(T)f Fk(n)p Fl(\033)1797 4500 y Fm(j)25 b Fn(\034)1897 4462 y Fl(\033)1887 4524 y(T)10 b Fk(n)p Fl(\033)2051 4500 y Fx(=)25 b Fn(\034)2197 4462 y Fl(\033)2187 4522 y(T)2246 4500 y Fx(]\(1)c(+)f Fn(O)s Fx(\(1)p Fn(=)-5 b(N)10 b Fx(\)\))3477 4618 y(\(4)p Fn(:)p Fx(76\))50 4763 y(Since)34 b(the)h(term)h(in)d(braces)j(in)e (the)h(last)g(line)e(of)j(\(4.74\))h(w)m(as)f(estimated)f(in)f (Corollary)g(2.3,)k(inserting)50 4908 y(\(2.13\))c(and)c(\(4.76\))k(in) c(\(4.74\),)k(w)m(e)e(obtain)f(that,)i(under)d(the)h(assumptions)f(and) h(with)f(the)i(notation)g(of)50 5054 y(Prop)s(osition)c(2.2,)50 5212 y Fi(\014)50 5266 y(\014)50 5321 y(\014)80 5316 y Fj(E)e Fx([)6 b Fn(\034)231 5279 y Fl(\033)221 5339 y(\033)265 5321 y Fe(0)320 5316 y Fm(j)25 b Fn(\034)420 5279 y Fl(\033)410 5339 y(\033)454 5321 y Fe(0)509 5316 y Fm(\024)g Fn(\034)655 5279 y Fl(\033)645 5339 y(T)704 5316 y Fx(])20 b Fm(\000)g Fj(E)11 b Fx([)p Fn(\034)970 5279 y Fl(\033)960 5341 y(T)f Fk(n)p Fl(\033)1130 5316 y Fm(j)25 b Fn(\034)1230 5279 y Fl(\033)1220 5341 y(T)10 b Fk(n)p Fl(\033)1384 5316 y Fx(=)25 b Fn(\034)1530 5279 y Fl(\033)1520 5339 y(T)1579 5316 y Fx(])1604 5212 y Fi(\014)1604 5266 y(\014)1604 5321 y(\014)1660 5316 y Fm(\024)g Fn(O)s Fx(\(1)p Fn(=)-5 b(N)10 b Fx(\))2081 5215 y Fi(\020)2137 5316 y Fj(E)g Fx([)q Fn(\034)2267 5279 y Fl(\033)2257 5341 y(T)g Fk(n)p Fl(\033)2427 5316 y Fm(j)25 b Fn(\034)2527 5279 y Fl(\033)2517 5341 y(T)10 b Fk(n)p Fl(\033)2681 5316 y Fx(=)25 b Fn(\034)2827 5279 y Fl(\033)2817 5339 y(T)2876 5316 y Fx(])20 b(+)g(\(1)h Fm(\000)3235 5281 y Fv(1)p 3215 5296 77 4 v 3215 5348 a Fl(M)3302 5316 y Fx(\))3337 5279 y Fk(\000)p Fv(1)3435 5316 y Fm(W)3525 5330 y Fl(\014)s(;N)s(;T)3730 5215 y Fi(\021)3477 5462 y Fx(\(4)p Fn(:)p Fx(77\))p eop %%Page: 37 37 37 36 bop 50 295 a Fo(A)l(ging)27 b(in)h(the)g(REM.)f(Part)h(2.)2661 b Fr(37)50 513 y Fx(Therefore,)30 b(collecting)g(\(4.77\))j(and)d (\(4.71\),)135 801 y Fm(j)p Fn(\024)212 815 y Fl(\033)n(;\033)318 796 y Fe(0)348 801 y Fx(\()p Fn(u)p Fx(\))p Fm(j)d(\024)629 739 y Fn(e)671 706 y Fk(\000)p Fl(\014)772 652 y Fk(p)p 832 652 65 4 v 54 x Fl(N)6 b(E)949 715 y Fp(\033)p 629 780 369 4 v 764 863 a Fn(M)1008 672 y Fi(\024)1056 801 y Fx(1)21 b(+)f Fm(j)p Fn(u)p Fm(j)1330 700 y Fi(\020)1385 801 y Fx(2)p Fj(E)11 b Fx([)p Fn(\034)1560 763 y Fl(\033)1550 825 y(T)f Fk(n)p Fl(\033)1695 801 y Fm(j)p Fn(\034)1770 763 y Fl(\033)1760 825 y(T)g Fk(n)p Fl(\033)1924 801 y Fx(=)25 b Fn(\034)2070 763 y Fl(\033)2060 823 y(T)2119 801 y Fx(])20 b(+)g(\(1)h Fm(\000)2478 765 y Fv(1)p 2458 780 77 4 v 2458 832 a Fl(M)2545 801 y Fx(\))2580 763 y Fk(\000)p Fv(1)2678 801 y Fm(W)2768 815 y Fl(\014)s(;N)s(;T)2973 700 y Fi(\021)3028 645 y(#)3081 801 y Fn(O)s Fx(\(1)p Fn(=)-5 b(N)10 b Fx(\))86 b(\(4)p Fn(:)p Fx(78\))50 1089 y(and)33 b(w)m(e)h(are)g(left)g(to)g(b)s(ound)e(the)i(term)g Fj(E)10 b Fx([)q Fn(\034)1594 1056 y Fl(\033)1584 1118 y(T)g Fk(n)p Fl(\033)1728 1089 y Fm(j)p Fn(\034)1803 1056 y Fl(\033)1793 1118 y(T)g Fk(n)p Fl(\033)1963 1089 y Fx(=)31 b Fn(\034)2115 1056 y Fl(\033)2105 1114 y(T)2164 1089 y Fx(])j(from)f(ab)s(o)m(v)m(e.)52 b(T)-8 b(o)34 b(do)g(so,)h(w)m(e)f(pro)s(ceed)f(as)50 1234 y(in)h(\(4.72\),)40 b(\(4.73\),)f(but)c(this)f(time)i(using)e(\(3.35\))k(and)d(the)g(fact)i (that)f Fj(E)11 b Fn(\034)2663 1201 y Fl(\033)2653 1257 y(\033)2718 1234 y Fx(1)-23 b(I)2773 1250 y Fk(f)p Fl(\034)2851 1232 y Fp(\033)2844 1269 y(\033)2895 1250 y Fk(\024)p Fl(\034)2993 1229 y Fp(\033)2986 1274 y(T)3036 1250 y Fk(g)3111 1234 y Fm(\025)34 b Fj(P)p Fx([)p Fn(\034)3346 1201 y Fl(\033)3336 1257 y(\033)3430 1234 y Fx(=)f(1])i(=)50 1380 y(1)20 b Fm(\000)g Fn(e)248 1347 y Fk(\000)p Fl(\014)349 1292 y Fk(p)p 410 1292 65 4 v 410 1347 a Fl(N)6 b(E)527 1356 y Fp(\033)575 1380 y Fx(,)31 b(w)m(e)g(obtain)f(that)387 1623 y Fj(E)11 b Fx([)p Fn(\034)517 1586 y Fl(\033)507 1648 y(T)f Fk(n)p Fl(\033)652 1623 y Fm(j)p Fn(\034)727 1586 y Fl(\033)717 1648 y(T)g Fk(n)p Fl(\033)881 1623 y Fx(=)25 b Fn(\034)1027 1586 y Fl(\033)1017 1646 y(T)1076 1623 y Fx(])g Fm(\024)g Fj(E)11 b Fx([)p Fn(\034)1352 1586 y Fl(\033)1342 1648 y(T)f Fk(n)p Fl(\033)1487 1623 y Fx(])20 b Fm(\000)1874 1562 y Fx(1)p 1634 1603 526 4 v 1634 1686 a Fj(P)p Fx(\()p Fn(\034)1774 1654 y Fl(\033)1764 1717 y(T)10 b Fk(n)p Fl(\033)1929 1686 y Fn(<)25 b(\034)2075 1660 y Fl(\033)2065 1708 y(\033)2124 1686 y Fx(\))2190 1623 y(+)2688 1562 y(1)p 2292 1603 838 4 v 2292 1700 a Fn(e)2334 1674 y Fl(\014)2378 1620 y Fk(p)p 2439 1620 65 4 v 2439 1674 a Fl(N)6 b(E)2556 1683 y Fp(\033)2604 1700 y Fj(P)p Fx(\()p Fn(\034)2744 1669 y Fl(\033)2734 1731 y(T)k Fk(n)p Fl(\033)2899 1700 y Fn(<)25 b(\034)3045 1674 y Fl(\033)3035 1723 y(\033)3094 1700 y Fx(\))1126 1904 y Fm(\024)1338 1842 y Fx(1)p 1233 1883 255 4 v 1233 1970 a(1)c Fm(\000)1421 1935 y Fv(1)p 1401 1950 77 4 v 1401 2002 a Fl(M)1514 1830 y Fi(\000)1555 1904 y Fx(1)g(+)f Fm(W)1802 1920 y Fl(\014)s(;N)s(;T)10 b Fv(\()p Fl(E)s Fv(\))2121 1830 y Fi(\001)2178 1904 y Fx(\(1)21 b(+)f Fn(O)s Fx(\(1)p Fn(=)-5 b(N)10 b Fx(\)\))3477 1775 y(\(4)p Fn(:)p Fx(79\))50 2176 y(where)20 b(the)h(second)g(line)e(follo)m(ws)g (from)h(the)h(b)s(ound)e(\(2.14\))k(of)e(Corollary)e(2.3)j(together)g (with)d(the)i(estimate)50 2321 y(\(2.6\))32 b(of)f(Prop)s(osition)d (2.1.)42 b(Inserting)29 b(this)g(b)s(ound)f(in)h(\(4.78\))k(yields,)541 2609 y Fm(j)p Fn(\024)618 2623 y Fl(\033)n(;\033)724 2605 y Fe(0)755 2609 y Fx(\()p Fn(u)p Fx(\))p Fm(j)26 b(\024)1035 2548 y Fn(e)1077 2515 y Fk(\000)p Fl(\014)1178 2461 y Fk(p)p 1238 2461 65 4 v 54 x Fl(N)6 b(E)1355 2524 y Fp(\033)p 1035 2589 369 4 v 1170 2672 a Fn(M)1415 2481 y Fi(\024)1463 2609 y Fx(1)20 b(+)g(3)p Fm(j)p Fn(u)p Fm(j)p Fx(\(1)i Fm(\000)1990 2574 y Fv(1)p 1970 2589 77 4 v 1970 2641 a Fl(M)2057 2609 y Fx(\))2092 2572 y Fk(\000)p Fv(1)2205 2536 y Fi(\000)2247 2609 y Fx(1)e(+)g Fm(W)2493 2626 y Fl(\014)s(;N)s(;T)10 b Fv(\()p Fl(E)s Fv(\))2812 2536 y Fi(\001)2854 2454 y(#)2907 2609 y Fn(O)s Fx(\(1)p Fn(=)-5 b(N)10 b Fx(\))260 b(\(4)p Fn(:)p Fx(80\))50 2892 y(Th)m(us)65 3145 y Fm(k)p Fn(K)194 3108 y Fk(\003)p Fv(\(0\))329 3145 y Fx(\()p Fn(u)p Fx(\))21 b Fm(\000)f 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Fv(1)p 2527 3423 77 4 v 2527 3475 a Fl(M)2614 3443 y Fx(\))2649 3406 y Fk(\000)p Fv(1)2762 3370 y Fi(\000)2803 3443 y Fx(1)f(+)f Fm(W)3050 3460 y Fl(\014)s(;N)s(;T)10 b Fv(\()p Fl(E)s Fv(\))3369 3370 y Fi(\001)3410 3288 y(#)3464 3443 y Fn(O)s Fx(\(1)p Fn(=)-5 b(N)10 b Fx(\))3477 3634 y(\(4)p Fn(:)p Fx(81\))50 3780 y(and)30 b(observing)f(that,)i(b)m(y)f(assertion)g (\(v\))h(of)g(Prop)s(osition)d(2.1,)1004 4022 y(\(1)21 b Fm(\000)1227 3986 y Fv(1)p 1207 4001 V 1207 4054 a Fl(M)1294 4022 y Fx(\))p Fn(e)1371 3985 y Fk(\000)p Fl(\014)1472 3930 y Fk(p)p 1533 3930 65 4 v 1533 3985 a Fl(N)6 b(E)1650 3994 y Fp(\033)1724 4022 y Fx(=)25 b Fn(G)1892 3985 y Fl(\033)1892 4047 y(T)10 b Fk(n)p Fl(\033)n(;T)2100 4022 y Fx(\(0\)\(1)23 b(+)d Fn(O)s Fx(\(1)p Fn(=)-5 b(N)10 b Fx(\)\))723 b(\(4)p Fn(:)p Fx(82\))50 4265 y(w)m(e)31 b(\014nally)d(arriv)m(e)i(at)112 4548 y Fm(k)p Fn(K)241 4510 y Fk(\003)p Fv(\(0\))376 4548 y Fx(\()p Fn(u)p Fx(\))21 b Fm(\000)e Fn(K)693 4510 y Fk(\003)p Fv(\(1\))828 4548 y Fx(\()p Fn(u)p Fx(\))p Fm(k)26 b(\024)f Fx(max)1129 4606 y Fl(\033)r Fk(2)p Fl(T)1375 4467 y Fn(G)1447 4434 y Fl(\033)1447 4496 y(T)10 b Fk(n)p Fl(\033)n(;T)1655 4467 y Fx(\(0\))p 1313 4527 521 4 v 1313 4610 a Fm(j)p Fx(1)21 b Fm(\000)f Fn(G)1567 4579 y Fl(\033)1567 4637 y(\033)n(;T)1685 4610 y Fx(\()p Fn(u)p Fx(\))p Fm(j)1844 4419 y Fi(\024)1892 4548 y Fx(1)h(+)f(3)p Fm(j)p Fn(u)p Fm(j)p Fx(\(1)i Fm(\000)2420 4512 y Fv(1)p 2400 4527 77 4 v 2400 4579 a Fl(M)2487 4548 y Fx(\))2522 4510 y Fk(\000)p Fv(1)2635 4474 y Fi(\000)2676 4548 y Fx(1)f(+)f Fm(W)2923 4564 y Fl(\014)s(;N)s(;T)10 b Fv(\()p Fl(E)s Fv(\))3242 4474 y Fi(\001)3284 4392 y(#)3337 4548 y Fn(O)s Fx(\(1)p Fn(=)-5 b(N)10 b Fx(\))3477 4739 y(\(4)p Fn(:)p Fx(83\))50 4884 y(F)-8 b(rom)31 b(there)f(on,)h(the)f(pro)s(of)g(pro)s (ceeds)g(exactly)h(as)g(the)f(pro)s(ofs)g(of)g(Lemma)h(4.6,)h(4.8)f (and)f(4.9,)h(yielding)181 5164 y Fm(k)p Fn(K)310 5127 y Fk(\003)p Fv(\(0\))445 5164 y Fx(\()p Fn(u)p Fx(\))21 b Fm(\000)f Fn(K)763 5127 y Fk(\003)p Fv(\(1\))897 5164 y Fx(\()p Fn(u)p Fx(\))p Fm(k)27 b(\024)1197 5097 y Fx(1)21 b(+)f(3)1399 5021 y Fm(p)p 1475 5021 279 4 v 76 x Fn(s)1518 5070 y Fv(2)1579 5097 y Fx(+)g Fn(r)1714 5070 y Fv(2)1754 5097 y Fx(\(1)h Fm(\000)1977 5061 y Fv(1)p 1957 5076 77 4 v 1957 5128 a Fl(M)2044 5097 y Fx(\))2079 5064 y Fk(\000)p Fv(1)2192 5023 y Fi(\000)2234 5097 y Fx(1)f(+)g Fm(W)2480 5113 y Fl(\014)s(;N)s(;T)10 b Fv(\()p Fl(E)s Fv(\))2799 5023 y Fi(\001)2866 5074 y(b)2856 5097 y Fx(\002)2927 5064 y Fk(\000)p Fv(1)p 1197 5143 1827 4 v 1619 5162 a Fi(p)p 1710 5162 598 4 v 77 x Fx(1)21 b(+)f(\()p Fn(s)1945 5213 y Fv(2)2006 5239 y Fx(+)g Fn(r)2141 5213 y Fv(2)2181 5239 y Fx(\))p Fn(=)p Fx(2)i Fm(\000)d Fx(5)p Fn(=)-5 b(M)3035 5164 y(O)s Fx(\(1)p Fn(=)g(N)10 b Fx(\))132 b(\(4)p Fn(:)p Fx(84\))50 5462 y(whic)m(h,)29 b(since)h(\(1)21 b Fm(\000)781 5426 y Fv(1)p 761 5441 77 4 v 761 5493 a Fl(M)848 5462 y Fx(\))883 5429 y Fk(\000)p Fv(1)996 5388 y Fi(\000)1038 5462 y Fx(1)f(+)g Fm(W)1284 5478 y Fl(\014)s(;N)s(;T)10 b Fv(\()p Fl(E)s Fv(\))1603 5388 y Fi(\001)1670 5439 y(b)1660 5462 y Fx(\002)1731 5429 y Fk(\000)p Fv(1)1853 5462 y Fm(\024)25 b Fx(1,)31 b(giv)m(es)g (\(4.68\),)i(pro)m(ving)d(Lemma)g(4.10.)42 b Fm(})p eop %%Page: 38 38 38 37 bop 50 295 a Fr(38)3270 b Fo(Se)l(ction)27 b Fr(4)50 513 y Fc(Pro)s(of)36 b(of)f(Lemma)d(4.11:)88 b Fx(By)31 b(de\014nition)d(of)i Fm(K)1850 480 y Fk(\003)p Fv(\(0\))1986 513 y Fx(\()p Fn(u)p Fx(\))h(and)f Fn(K)2400 480 y Fk(\003)p Fv(\(0\))2534 513 y Fx(\()p Fn(u)p Fx(\),)190 728 y Fm(kK)305 691 y Fk(\003)p Fv(\(0\))441 728 y Fx(\()p Fn(u)p Fx(\))21 b Fm(\000)f Fn(K)759 691 y Fk(\003)p Fv(\(0\))894 728 y Fx(\()p Fn(u)p Fx(\))p Fm(k)26 b Fx(=)f(max)1195 787 y Fl(\033)r Fk(2)p Fl(T)1367 728 y Fm(jK)1462 691 y Fk(\003)p Fv(\(0\))1461 751 y Fl(\033)n(;\033)1598 728 y Fx(\()p Fn(u)p Fx(\))p Fm(j)1087 998 y(\024)1299 936 y Fx(1)p 1194 977 255 4 v 1194 1060 a Fn(M)30 b Fm(\000)20 b Fx(1)1475 998 y(max)1486 1056 y Fl(\033)r Fk(2)p Fl(T)1670 931 y Fx(\(1)h Fm(\000)1893 895 y Fv(1)p 1873 910 77 4 v 1873 962 a Fl(M)1960 931 y Fx(\))1995 898 y Fk(\000)p Fv(1)2093 931 y Fn(e)2135 898 y Fk(\000)p Fl(\014)2236 844 y Fk(p)p 2296 844 65 4 v 54 x Fl(N)7 b(E)2414 907 y Fp(\033)p 1670 977 792 4 v 1805 1060 a Fm(j)p Fx(1)21 b Fm(\000)f Fn(G)2059 1029 y Fl(\033)2059 1087 y(\033)n(;T)2178 1060 y Fx(\()p Fn(u)p Fx(\))p Fm(j)2488 897 y Fi(\020)2542 998 y Fx(1)h(+)f Fm(j)p Fn(u)p Fm(j)p Fj(E)11 b Fx([)p Fn(\034)2931 960 y Fl(\033)2921 1022 y(T)f Fk(n)p Fl(\033)3066 998 y Fm(j)p Fn(\034)3141 960 y Fl(\033)3131 1022 y(T)g Fk(n)p Fl(\033)3295 998 y Fx(=)25 b Fn(\034)3441 960 y Fl(\033)3431 1020 y(T)3490 998 y Fx(])3515 897 y Fi(\021)3477 1176 y Fx(\(4)p Fn(:)p Fx(85\))50 1322 y(Eq.)40 b(\(4.79\))33 b(then)d(yields)f(the)h(b)s(ound)50 1612 y Fm(kK)165 1575 y Fk(\003)p Fv(\(0\))301 1612 y Fx(\()p Fn(u)p Fx(\))p Fm(\000)p Fn(K)578 1575 y Fk(\003)p Fv(\(0\))713 1612 y Fx(\()p Fn(u)p Fx(\))p Fm(k)c(\024)1118 1551 y Fx(1)p 1013 1592 255 4 v 1013 1675 a Fn(M)k Fm(\000)20 b Fx(1)1294 1612 y(max)1305 1671 y Fl(\033)r Fk(2)p Fl(T)1489 1546 y Fx(\(1)h Fm(\000)1712 1510 y Fv(1)p 1692 1525 77 4 v 1692 1577 a Fl(M)1779 1546 y Fx(\))p Fn(e)1856 1513 y Fk(\000)p Fl(\014)1957 1458 y Fk(p)p 2018 1458 65 4 v 2018 1513 a Fl(N)6 b(E)2135 1522 y Fp(\033)p 1489 1592 695 4 v 1576 1675 a Fm(j)p Fx(1)21 b Fm(\000)f Fn(G)1830 1643 y Fl(\033)1830 1702 y(\033)n(;T)1948 1675 y Fx(\()p Fn(u)p Fx(\))p Fm(j)2194 1484 y Fi(\024)2242 1612 y Fx(1+)p Fm(j)p Fn(u)p Fm(j)p Fx(\(1)p Fm(\000)2643 1577 y Fv(1)p 2623 1592 77 4 v 2623 1644 a Fl(M)2710 1612 y Fx(\))2745 1575 y Fk(\000)p Fv(1)2858 1539 y Fi(\000)2900 1612 y Fx(1)h(+)f Fm(W)3147 1629 y Fl(\014)s(;N)s(;T)10 b Fv(\()p Fl(E)s Fv(\))3466 1539 y Fi(\001)3507 1457 y(#)3560 1612 y Fn(O)s Fx(\(1)p Fn(=)-5 b(N)10 b Fx(\))3477 1803 y(\(4)p Fn(:)p Fx(86\))50 1949 y(whic)m(h,)33 b(up)e(to)j(some)g(constan)m(ts,) h(is)d(iden)m(tical)g(to)i(that)f(of)g(\(4.81\).)51 b(F)-8 b(rom)34 b(there)f(on)g(the)g(pro)s(of)g(follo)m(ws)50 2094 y(that)e(of)f(Lemma)h(4.10.)p Fm(})159 2312 y Fx(Let)g(us)e(in)m (tro)s(duce)h(the)g(decomp)s(osition)1311 2556 y Fn(K)1395 2519 y Fk(\003)1437 2556 y Fx(\()p Fn(u)p Fx(\))c Fm(\021)f(K)1751 2519 y Fk(\003)p Fv(\(0\))1886 2556 y Fx(\()p Fn(u)p Fx(\))c(+)f Fm(K)2190 2519 y Fk(\003)p Fv(\(1\))2326 2556 y Fx(\()p Fn(u)p Fx(\))1029 b(\(4)p Fn(:)p Fx(87\))50 2801 y(and)30 b(note)h(that)g Fm(K)696 2768 y Fk(\003)p Fv(\(1\))831 2801 y Fx(\()p Fn(u)p Fx(\))g(can)g(b)s(e)e(written)h(in)f (the)i(form)515 3045 y Fm(K)585 3007 y Fk(\003)p Fv(\(1\))721 3045 y Fx(\()p Fn(u)p Fx(\))26 b Fm(\021)f Fx(\()p Fn(K)1084 3007 y Fk(\003)p Fv(\(0\))1219 3045 y Fx(\()p Fn(u)p Fx(\))c Fm(\000)f(K)1523 3007 y Fk(\003)p Fv(\(0\))1658 3045 y Fx(\()p Fn(u)p Fx(\)\))h(+)f(\()p Fn(K)2046 3007 y Fk(\003)p Fv(\(1\))2181 3045 y Fx(\()p Fn(u)p Fx(\))h Fm(\000)f Fn(K)2499 3007 y Fk(\003)p Fv(\(0\))2634 3045 y Fx(\()p Fn(u)p Fx(\)\))h(+)f Fn(K)2987 3007 y Fk(\003)p Fv(\(2\))3121 3045 y Fx(\()p Fn(u)p Fx(\))234 b(\(4)p Fn(:)p Fx(88\))50 3289 y(The)30 b(follo)m(wing)f(corollary)g(then)h(is) g(an)g(immediate)f(consequence)j(of)e(the)h(previous)e(three)h (lemmata.)50 3507 y Fd(Corollary)36 b(4.12:)51 b Ft(Under)31 b(the)h(assumptions)i(and)e(with)h(the)e(notation)j(of)d(L)-5 b(emma)33 b(4.9)f(and)g(L)-5 b(emma)50 3652 y(4.10)34 b(we)e(have,)h(for)h Fn(N)42 b Ft(lar)-5 b(ge)34 b(enough,)238 3922 y Fm(kK)353 3885 y Fk(\003)p Fv(\(1\))488 3922 y Fx(\()p Fn(u)p Fx(\))p Fm(k)27 b(\024)789 3861 y Fn(\015)841 3828 y Fk(\000)p Fv(2)939 3861 y Fn(C)7 b Fx(\()p Fn(s)1089 3828 y Fv(2)1149 3861 y Fx(+)20 b Fn(r)1284 3828 y Fv(2)1324 3861 y Fx(\))h(+)f(\(1)h(+)f(3)1708 3785 y Fm(p)p 1784 3785 279 4 v 76 x Fn(s)1827 3834 y Fv(2)1887 3861 y Fx(+)g Fn(r)2022 3834 y Fv(2)2063 3861 y Fx(\))15 b(max)g(\()q Fn(O)s Fx(\(1)p Fn(=)p Fx(\()p Fn(M)32 b Fm(\000)19 b Fx(1\)\))p Fn(;)c(O)s Fx(\(1)p Fn(=)-5 b(N)10 b Fx(\)\))p 789 3901 2491 4 v 1542 3919 a Fi(p)p 1633 3919 598 4 v 78 x Fx(1)21 b(+)f(\()p Fn(s)1868 3971 y Fv(2)1929 3997 y Fx(+)g Fn(r)2064 3971 y Fv(2)2104 3997 y Fx(\))p Fn(=)p Fx(2)h Fm(\000)f Fx(5)p Fn(=)-5 b(M)3477 3922 y Fx(\(4)p Fn(:)p Fx(89\))159 4384 y(The)31 b(leading)f(con)m (tribution)g(to)j Fn(K)1377 4351 y Fk(\003)1418 4384 y Fx(\()p Fn(u)p Fx(\))f(th)m(us)g(comes)g(from)f(the)h(matrix)f Fm(K)2784 4351 y Fk(\003)p Fv(\(0\))2919 4384 y Fx(\()p Fn(u)p Fx(\))i(whose)e(sp)s(ectrum)50 4530 y(is)42 b(easily)h (analysed.)79 b(In)43 b(particular,)i Fm(K)1522 4497 y Fk(\003)p Fv(\(0\))1658 4530 y Fx(\()p Fn(u)p Fx(\))f(has)f(a)h (unique)e(non)h(zero)h(eigen)m(v)-5 b(alue)43 b(of)h(algebraic)50 4675 y(m)m(ultiplicit)m(y)27 b(one,)k(denoted)g(b)m(y)f Fn(\025)p Fx(\()p Fn(u)p Fx(\),)h(and)f(giv)m(en)g(b)m(y:)1487 4919 y Fn(\025)p Fx(\()p Fn(u)p Fx(\))c Fm(\021)1791 4833 y Fi(X)1784 5028 y Fl(\033)r Fk(2)p Fl(T)1945 4919 y Fm(K)2015 4872 y Fk(\003)p Fv(\(0\))2014 4946 y Fl(\033)n(;\033)2120 4928 y Fe(0)2150 4919 y Fx(\()p Fn(u)p Fx(\))1205 b(\(4)p Fn(:)p Fx(90\))50 5218 y(The)30 b(corresp)s(onding)e(left)i(eigen)m(v)m (ector)j(is)c(prop)s(ortional)f(to)j(\(1)p Fn(;)15 b Fx(1)p Fn(;)g(:)g(:)g(:)k(;)c Fx(1\).)42 b(Similarly)-8 b(,)27 b(de\014ning)1156 5462 y Fn(M)1254 5424 y Fk(\003)p Fv(\(0\))1389 5462 y Fx(\()p Fn(u)p Fx(\))f Fm(\021)f Fx([1)-23 b(I)21 b Fm(\000)f(K)1895 5424 y Fk(\003)p Fv(\(0\))2030 5462 y Fx(\()p Fn(u)p Fx(\)])2177 5424 y Fk(\000)p Fv(1)2276 5462 y Fm(K)2346 5424 y Fk(\003)p Fv(\(0\))2481 5462 y Fx(\()p Fn(u)p Fx(\))874 b(\(4)p Fn(:)p Fx(91\))p eop %%Page: 39 39 39 38 bop 50 295 a Fo(A)l(ging)27 b(in)h(the)g(REM.)f(Part)h(2.)2661 b Fr(39)50 513 y Fx(w)m(e)31 b(decomp)s(ose)f(the)h(Laplace)f (transform)g(of)h(the)f(resolv)m(en)m(t)h(\(de\014ned)f(in)f(\(4.6\)\)) j(in)m(to)1277 748 y Fn(M)1375 710 y Fk(\003)1417 748 y Fx(\()p Fn(u)p Fx(\))26 b Fm(\021)e Fn(M)1758 710 y Fk(\003)p Fv(\(0\))1893 748 y Fx(\()p Fn(u)p Fx(\))d(+)f Fn(M)2225 710 y Fk(\003)p Fv(\(1\))2360 748 y Fx(\()p Fn(u)p Fx(\))995 b(\(4)p 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Fm(\000)g(K)1881 1694 y Fk(\003)p Fv(\(0\))2017 1732 y Fx(\()p Fn(u)p Fx(\)])2164 1694 y Fk(\000)p Fv(1)1329 1904 y Fn(\032)p Fx(\()p Fn(u)p Fx(\))26 b Fm(\021)f Fx(max)1804 1831 y Fi(\000)1846 1904 y Fm(j)p Fx(1)c Fm(\000)e Fn(\025)p Fx(\()p Fn(u)p Fx(\))p Fm(j)2227 1867 y Fk(\000)p Fv(1)2326 1904 y Fn(;)c Fx(1)2411 1831 y Fi(\001)3477 1814 y Fx(\(4)p Fn(:)p Fx(93\))50 2063 y Ft(Then,)930 2210 y Fn(M)1028 2173 y Fk(\003)p Fv(\(1\))1163 2210 y Fx(\()p Fn(u)p Fx(\))26 b(=)f Fn(R)q Fx(\()p Fn(u)p Fx(\))p Fm(K)1669 2173 y Fk(\003)p Fv(\(1\))1805 2210 y Fx(\()p Fn(u)p Fx(\))p Fn(R)q Fx(\()p Fn(u)p Fx(\))2452 2149 y(1)p 2130 2189 687 4 v 2130 2274 a(1)-23 b(I)22 b Fm(\000)e Fn(R)q Fx(\()p Fn(u)p Fx(\))p Fm(K)2560 2248 y Fk(\003)p Fv(\(1\))2695 2274 y Fx(\()p Fn(u)p Fx(\))3477 2210 y(\(4)p Fn(:)p Fx(94\))50 2442 y Ft(and,)33 b(if)g Fm(k)p Fn(R)q Fx(\()p Fn(u)p Fx(\))p Fm(K)650 2409 y Fk(\003)p Fv(\(1\))786 2442 y Fx(\()p Fn(u)p Fx(\))p Fm(k)26 b Fn(<)f Fx(1)p Ft(,)1212 2705 y Fm(k)p 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3512 a(1)-23 b(I)21 b Fm(\000)f Fn(K)2614 3485 y Fk(\003)2656 3512 y Fx(\()p Fn(u)p Fx(\))3477 3449 y(\(4)p Fn(:)p Fx(96\))50 3661 y(Th)m(us)536 3791 y Fn(M)634 3753 y Fk(\003)676 3791 y Fx(\()p Fn(u)p Fx(\))26 b(=)f Fn(M)1018 3753 y Fk(\003)p Fv(\(0\))1153 3791 y Fx(\()p Fn(u)p Fx(\))c(+)e Fn(R)q Fx(\()p Fn(u)p Fx(\))p Fm(K)1648 3753 y Fk(\003)p Fv(\(1\))1784 3791 y Fx(\()p Fn(u)p Fx(\))i(+)f Fn(R)q Fx(\()p Fn(u)p Fx(\))p Fm(K)2280 3753 y Fk(\003)p Fv(\(1\))2416 3791 y Fx(\()p Fn(u)p Fx(\))2735 3729 y(1)p 2549 3770 V 2549 3853 a(1)-23 b(I)22 b Fm(\000)d Fn(K)2800 3827 y Fk(\003)2842 3853 y Fx(\()p Fn(u)p Fx(\))2976 3791 y Fn(K)3060 3753 y Fk(\003)3101 3791 y Fx(\()p Fn(u)p Fx(\))824 4032 y(=)25 b Fn(M)1018 3995 y Fk(\003)p Fv(\(0\))1153 4032 y Fx(\()p Fn(u)p Fx(\))c(+)e Fn(R)q Fx(\()p Fn(u)p Fx(\))p Fm(K)1648 3995 y Fk(\003)p Fv(\(1\))1784 4032 y Fx(\()p Fn(u)p Fx(\))2103 3971 y(1)p 1917 4011 V 1917 4095 a(1)-23 b(I)22 b Fm(\000)e Fn(K)2169 4068 y Fk(\003)2210 4095 y Fx(\()p Fn(u)p Fx(\))3477 3917 y(\(4)p Fn(:)p Fx(97\))50 4249 y(Eq.)40 b(\(4.94\))33 b(then)d(results)f(from)h (\(4.97\))j(together)f(with)d(the)h(iden)m(tit)m(y)1346 4429 y(1)p 1161 4470 V 1161 4553 a(1)-23 b(I)21 b Fm(\000)f Fn(K)1412 4527 y Fk(\003)1453 4553 y Fx(\()p Fn(u)p Fx(\))1612 4491 y(=)25 b Fn(R)q Fx(\()p Fn(u)p Fx(\))2232 4429 y(1)p 1911 4470 688 4 v 1911 4555 a(1)-23 b(I)21 b Fm(\000)f Fn(R)q Fx(\()p Fn(u)p Fx(\))p Fm(K)2340 4529 y Fk(\003)p Fv(\(1\))2476 4555 y Fx(\()p Fn(u)p Fx(\))3477 4491 y(\(4)p Fn(:)p Fx(98\))50 4757 y(W)-8 b(e)31 b(no)m(w)g(turn)e(to)i(the)g(pro)s (of)e(of)i(\(4.96\).)43 b(It)30 b(follo)m(ws)g(from)g(the)g(sp)s (ectral)g(prop)s(erties)f(of)h Fm(K)3216 4724 y Fk(\003)p Fv(\(0\))3352 4757 y Fx(\()p Fn(u)p Fx(\))h(that)890 4992 y Fm(k)p Fx([1)-23 b(I)21 b Fm(\000)f(K)1197 4954 y Fk(\003)p Fv(\(0\))1333 4992 y Fx(\()p Fn(u)p Fx(\)])1480 4954 y Fk(\000)p Fv(1)1578 4992 y Fm(k)26 b Fx(=)f(max)1929 4918 y Fi(\000)1971 4992 y Fm(j)p Fx(1)c Fm(\000)f Fn(\025)p Fx(\()p 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Fx([1)-23 b(I)21 b Fm(\000)f Fn(R)q Fx(\()p Fn(u)p Fx(\))p Fm(K)751 761 y Fk(\003)p Fv(\(1\))887 798 y Fx(\()p Fn(u)p Fx(\)])1034 761 y Fk(\000)p Fv(1)1133 798 y Fm(k)25 b(\024)g Fx([1)c Fm(\000)f(k)p Fn(R)q Fx(\()p Fn(u)p Fx(\))p Fm(K)1788 761 y Fk(\003)p Fv(\(1\))1924 798 y Fx(\()p Fn(u)p Fx(\))p Fm(k)p Fx(])2116 761 y Fk(\000)p Fv(1)2241 798 y Fm(\024)25 b Fx([1)c Fm(\000)e Fn(\032)p Fx(\()p Fn(u)p Fx(\))p Fm(kK)2802 761 y Fk(\003)p Fv(\(1\))2939 798 y Fx(\()p Fn(u)p Fx(\))p Fm(k)p Fx(])3131 761 y Fk(\000)p Fv(1)3432 798 y Fx(\(4)p Fn(:)p Fx(101\))50 1084 y(The)30 b(lemma)g(is)f(pro)m(v) m(en.)41 b Fm(})159 1302 y Fx(A)m(t)34 b(this)d(stage)j(w)m(e)f(see)h (that)f(to)g(fully)e(con)m(trol)i(the)g(b)s(eha)m(vior)f(of)g(b)s(oth)g Fn(M)2776 1269 y Fk(\003)p Fv(\(0\))2911 1302 y Fx(\()p Fn(u)p Fx(\))i(and)e Fn(M)3344 1269 y Fk(\003)p Fv(\(1\))3479 1302 y Fx(\()p Fn(u)p Fx(\))h(in)50 1447 y(a)e(small)d(neigh)m(b)s (orho)s(o)s(d)g(of)j(the)g(origin)d(requires)h(a)i(precise)f(con)m (trol)h(of)f(1)21 b Fm(\000)f Fn(\025)p Fx(\()p Fn(u)p Fx(\).)41 b(Observ)m(e)30 b(that)456 1778 y(1)21 b Fm(\000)f Fn(\025)p Fx(\()p Fn(u)p Fx(\))26 b(=)956 1717 y(1)p 921 1757 117 4 v 921 1841 a Fm(j)p Fn(T)13 b Fm(j)1071 1692 y Fi(X)1063 1887 y Fl(\033)r Fk(2)p Fl(T)1225 1623 y Fi(")1278 1778 y Fx(1)20 b Fm(\000)1482 1698 y Fn(G)1554 1665 y Fl(\033)1554 1727 y(T)10 b Fk(n)p Fl(\033)n(;T)1762 1698 y Fx(\(0\))p 1445 1757 471 4 v 1445 1841 a(1)21 b Fm(\000)f Fn(G)1674 1809 y Fl(\033)1674 1867 y(\033)n(;T)1792 1841 y Fx(\()p Fn(u)p Fx(\))1941 1677 y Fi(\020)1996 1778 y Fx(1)g(+)g Fn(u)p Fj(E)11 b Fx([)p Fn(\034)2334 1741 y Fl(\033)2324 1803 y(T)f Fk(n)p Fl(\033)2469 1778 y Fm(j)p Fn(\034)2544 1741 y Fl(\033)2534 1803 y(T)g Fk(n)p Fl(\033)2698 1778 y Fx(=)25 b Fn(\034)2844 1741 y Fl(\033)2834 1801 y(T)2893 1778 y Fx(])2918 1677 y Fi(\021)2972 1623 y(#)3432 1778 y Fx(\(4)p Fn(:)p Fx(102\))50 2118 y(so)36 b(that)h(1)25 b Fm(\000)f Fn(\025)p Fx(\()p Fn(u)p Fx(\))37 b(tak)m(es)g(the)g(form)f(of)g(a)h(sum)e(o)m(v)m(er)j Fn(\033)s Fx(.)59 b(The)35 b(ev)-5 b(aluation)36 b(of)h(suc)m(h)f(sums) f(is)g(a)i(rather)50 2264 y(in)m(v)m(olv)m(ed)42 b(question)g(whose)h (treatmen)m(t)h(is)e(the)h(ob)5 b(ject)44 b(of)f(the)g(next)g (subsection.)77 b(The)42 b(analysis)g(of)50 2409 y Fn(M)148 2376 y Fk(\003)p Fv(\(0\))283 2409 y Fx(\()p Fn(u)p Fx(\))d(and)e Fn(M)726 2376 y Fk(\003)p Fv(\(1\))861 2409 y Fx(\()p Fn(u)p Fx(\))i(will)d(then)h(b)s(e)h(brough)m(t)g(to)h(a)g(close)f(in)f (Section)h(4.5.)65 b(As)38 b(for)g(the)h(presen)m(t)50 2554 y(section,)31 b(w)m(e)f(conclude)g(it)g(with)f(the)i(analysis)e (of)h(the)h(summands)d(of)j(\(4.102\).)50 2773 y Fd(Lemma)38 b(4.14:)51 b Ft(R)-5 b(e)g(c)g(al)5 b(l)34 b(that)g Fn(u)25 b Fx(=)g Fn(z)t(=)1468 2750 y Fi(b)1458 2773 y Fx(\002)q(\()p Fn(E)5 b Fx(\))33 b Ft(and)h(set)1306 3058 y Fn(z)1348 3072 y Fl(\033)1422 3058 y Fm(\021)1518 2984 y Fi(\000)1560 3058 y Fx(1)21 b Fm(\000)1747 3022 y Fv(1)p 1727 3037 77 4 v 1727 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y(\033)n(;T)1155 4303 y Fx(\()p Fn(u)p Fx(\))1303 4140 y Fi(\020)1358 4240 y Fx(1)g(+)g Fn(u)p Fj(E)11 b Fx([)p Fn(\034)1696 4203 y Fl(\033)1686 4265 y(T)f Fk(n)p Fl(\033)1831 4240 y Fm(j)p Fn(\034)1906 4203 y Fl(\033)1896 4265 y(T)g Fk(n)p Fl(\033)2060 4240 y Fx(=)25 b Fn(\034)2206 4203 y Fl(\033)2196 4263 y(T)2255 4240 y Fx(])2280 4140 y Fi(\021)2355 4240 y Fm(\000)2558 4179 y Fn(z)p 2456 4220 249 4 v 2456 4303 a(z)g Fm(\000)20 b Fn(z)2656 4317 y Fl(\033)2716 4081 y Fi(\014)2716 4136 y(\014)2716 4190 y(\014)2716 4245 y(\014)2716 4300 y(\014)2772 4240 y Fm(\024)k Fn(C)7 b Fx(\()p Fn(\016)s Fx(\))p Fm(j)p Fn(z)t Fm(j)284 b Fx(\(4)p Fn(:)p Fx(105\))50 4571 y Ft(for)33 b(some)g(c)-5 b(onstant)35 b Fx(0)25 b Fn(<)g(C)7 b Fx(\()p Fn(\016)s Fx(\))27 b Fn(<)e Fm(1)32 b Ft(that)i(only)f(dep)-5 b(ends)35 b(on)e Fn(\016)s Ft(.)50 4790 y Fc(Pro)s(of:)47 b Fx(Let)31 b(us)f(write)411 5088 y(1)20 b Fm(\000)615 5007 y Fn(G)687 4974 y Fl(\033)687 5036 y(T)10 b Fk(n)p Fl(\033)n(;T)896 5007 y Fx(\(0\))p 578 5067 471 4 v 578 5150 a(1)21 b Fm(\000)f Fn(G)807 5119 y Fl(\033)807 5177 y(\033)n(;T)926 5150 y Fx(\()p Fn(u)p Fx(\))1074 4987 y Fi(\020)1129 5088 y Fx(1)g(+)g Fn(u)p Fj(E)11 b Fx([)p Fn(\034)1467 5050 y Fl(\033)1457 5112 y(T)f Fk(n)p Fl(\033)1602 5088 y Fm(j)p Fn(\034)1677 5050 y Fl(\033)1667 5112 y(T)g Fk(n)p Fl(\033)1831 5088 y Fx(=)25 b Fn(\034)1977 5050 y Fl(\033)1967 5110 y(T)2026 5088 y Fx(])2051 4987 y Fi(\021)340 5385 y Fx(=)426 5229 y Fi( )498 5385 y Fx(1)c Fm(\000)702 5304 y Fn(G)774 5271 y Fl(\033)774 5333 y(T)10 b Fk(n)p Fl(\033)n(;T)983 5304 y Fx(\(0\))p 665 5364 V 665 5447 a(1)21 b Fm(\000)f Fn(G)894 5416 y Fl(\033)894 5474 y(\033)n(;T)1013 5447 y Fx(\()p Fn(u)p Fx(\))1146 5229 y Fi(!)1234 5284 y(\020)1288 5385 y Fx(1)h(+)e Fn(u)p Fj(E)11 b Fx([)p Fn(\034)1626 5347 y Fl(\033)1616 5409 y(T)f Fk(n)p Fl(\033)1761 5385 y Fm(j)p Fn(\034)1836 5347 y Fl(\033)1826 5409 y(T)g Fk(n)p Fl(\033)1990 5385 y Fx(=)25 b Fn(\034)2136 5347 y Fl(\033)2126 5407 y(T)2185 5385 y Fx(])2210 5284 y Fi(\021)2285 5385 y Fm(\000)20 b Fn(u)p Fj(E)10 b Fx([)q Fn(\034)2558 5347 y Fl(\033)2548 5409 y(T)g Fk(n)p Fl(\033)2692 5385 y Fm(j)p Fn(\034)2767 5347 y Fl(\033)2757 5409 y(T)g Fk(n)p Fl(\033)2922 5385 y Fx(=)24 b Fn(\034)3067 5347 y Fl(\033)3057 5407 y(T)3116 5385 y Fx(])3432 5241 y(\(4)p Fn(:)p Fx(106\))p eop %%Page: 41 41 41 40 bop 50 295 a Fo(A)l(ging)27 b(in)h(the)g(REM.)f(Part)h(2.)2661 b Fr(41)159 513 y Fx(Recall)22 b(that)i(w)m(e)f(denote)g(b)m(y)g Fn(u)1197 527 y Fl(\033)1268 513 y Fx(the)g(smallest)f(real)g(n)m(um)m (b)s(er)g(that)h(solv)m(es)g(the)g(equation)f Fn(G)3276 480 y Fl(\033)3276 538 y(\033)n(;T)3395 513 y Fx(\()p Fn(u)p Fx(\))k(=)f(1.)50 658 y(W)-8 b(e)31 b(will)d(\014rst)i(lo)s(ok)g (at)h(the)f(term)h(in)e(round)g(brac)m(k)m(ets.)168 904 y(1)21 b Fm(\000)372 824 y Fn(G)444 791 y Fl(\033)444 853 y(T)10 b Fk(n)p Fl(\033)n(;T)653 824 y Fx(\(0\))p 335 884 471 4 v 335 967 a(1)21 b Fm(\000)f Fn(G)564 936 y Fl(\033)564 994 y(\033)n(;T)683 967 y Fx(\()p Fn(u)p Fx(\))842 904 y(=)949 832 y Fn(G)1021 799 y Fl(\033)1021 857 y(\033)n(;T)1139 832 y Fx(\(0\))i Fm(\000)e Fn(G)1439 799 y Fl(\033)1439 857 y(\033)n(;T)1557 832 y Fx(\()p Fn(u)p Fx(\))p 949 884 732 4 v 1079 967 a(1)h Fm(\000)f Fn(G)1308 936 y Fl(\033)1308 994 y(\033)n(;T)1427 967 y Fx(\()p Fn(u)p Fx(\))842 1183 y(=)25 b Fm(\000)1053 1111 y Fn(G)1125 1078 y Fl(\033)1125 1136 y(\033)n(;T)1244 1111 y Fx(\(0\))d Fm(\000)e Fn(G)1544 1078 y Fl(\033)1544 1136 y(\033)n(;T)1662 1111 y Fx(\()p Fn(u)p Fx(\))p 1020 1162 801 4 v 1020 1253 a(\()p Fn(u)g Fm(\000)g Fn(u)1270 1267 y Fl(\033)1319 1253 y Fx(\))1387 1218 y Fl(d)p 1365 1233 81 4 v 1365 1285 a(du)1457 1253 y Fn(G)1529 1222 y Fl(\033)1529 1280 y(\033)n(;T)1648 1253 y Fx(\()p Fn(u)1735 1267 y Fl(\033)1784 1253 y Fx(\))837 1488 y(+)g(\()p Fn(G)1035 1450 y Fl(\033)1035 1510 y(\033)n(;T)1154 1488 y Fx(\(0\))h Fm(\000)f Fn(G)1453 1450 y Fl(\033)1453 1510 y(\033)n(;T)1572 1488 y Fx(\()p Fn(u)p Fx(\)\))1744 1333 y Fi( )2040 1426 y Fx(1)p 1828 1467 471 4 v 1828 1550 a(1)h Fm(\000)f Fn(G)2057 1519 y Fl(\033)2057 1577 y(\033)n(;T)2175 1550 y Fx(\()p Fn(u)p Fx(\))2329 1488 y(+)2808 1426 y(1)p 2431 1467 801 4 v 2431 1558 a(\()p Fn(u)h Fm(\000)f Fn(u)2682 1572 y Fl(\033)2731 1558 y Fx(\))2798 1522 y Fl(d)p 2777 1537 81 4 v 2777 1589 a(du)2869 1558 y Fn(G)2941 1527 y Fl(\033)2941 1585 y(\033)n(;T)3059 1558 y Fx(\()p Fn(u)3146 1572 y Fl(\033)3196 1558 y Fx(\))3242 1333 y Fi(!)842 1791 y Fx(=)1055 1729 y Fn(u)p 949 1770 265 4 v 949 1853 a(u)g Fm(\000)g Fn(u)1164 1867 y Fl(\033)1244 1791 y Fm(\000)1346 1718 y Fn(G)1418 1685 y Fl(\033)1418 1743 y(\033)n(;T)1536 1718 y Fx(\(0\))i Fm(\000)e Fn(G)1836 1685 y Fl(\033)1836 1743 y(\033)n(;T)1955 1718 y Fx(\()p Fn(u)p Fx(\))h(+)e Fn(u)2273 1682 y Fl(d)p 2251 1697 81 4 v 2251 1749 a(du)2343 1718 y Fn(G)2415 1685 y Fl(\033)2415 1743 y(\033)n(;T)2534 1718 y Fx(\()p Fn(u)2621 1732 y Fl(\033)2670 1718 y Fx(\))p 1346 1770 1361 4 v 1626 1861 a(\()p Fn(u)h Fm(\000)g Fn(u)1876 1875 y Fl(\033)1925 1861 y Fx(\))1993 1825 y Fl(d)p 1971 1840 81 4 v 1971 1892 a(du)2063 1861 y Fn(G)2135 1830 y Fl(\033)2135 1888 y(\033)n(;T)2254 1861 y Fx(\()p Fn(u)2341 1875 y Fl(\033)2390 1861 y Fx(\))837 2089 y(+)938 2016 y(\()p Fn(G)1045 1983 y Fl(\033)1045 2041 y(\033)n(;T)1165 2016 y Fx(\(0\))h Fm(\000)f Fn(G)1464 1983 y Fl(\033)1464 2041 y(\033)n(;T)1583 2016 y Fx(\()p Fn(u)p Fx(\)\))1755 1943 y Fi(\000)1798 2016 y Fx(\()p Fn(u)g Fm(\000)g Fn(u)2048 2030 y Fl(\033)2097 2016 y Fx(\))2165 1981 y Fl(d)p 2143 1996 V 2143 2048 a(du)2235 2016 y Fn(G)2307 1983 y Fl(\033)2307 2041 y(\033)n(;T)2426 2016 y Fx(\()p Fn(u)2513 2030 y Fl(\033)2562 2016 y Fx(\))h(+)f(1)g Fm(\000)g Fn(G)2937 1983 y Fl(\033)2937 2041 y(\033)n(;T)3056 2016 y Fx(\()p Fn(u)p Fx(\))3178 1943 y Fi(\001)p 938 2068 2283 4 v 1409 2159 a Fx(\(1)h Fm(\000)f Fn(G)1673 2128 y Fl(\033)1673 2186 y(\033)n(;T)1792 2159 y Fx(\()p Fn(u)p Fx(\)\)\()p Fn(u)i Fm(\000)d Fn(u)2200 2173 y Fl(\033)2249 2159 y Fx(\))2317 2123 y Fl(d)p 2296 2138 81 4 v 2296 2191 a(du)2387 2159 y Fn(G)2459 2128 y Fl(\033)2459 2186 y(\033)n(;T)2578 2159 y Fx(\()p Fn(u)2665 2173 y Fl(\033)2715 2159 y Fx(\))842 2335 y(=)1055 2274 y Fn(u)p 949 2314 265 4 v 949 2398 a(u)h Fm(\000)g Fn(u)1164 2412 y Fl(\033)1224 2335 y Fx(\(1)h(+)1433 2312 y Fi(e)1416 2335 y Fn(R)1485 2349 y Fl(\033)1534 2335 y Fx(\()p Fn(u)p Fx(\)\))g(+)1820 2312 y Fi(b)1803 2335 y Fn(R)1872 2349 y Fl(\033)1921 2335 y Fx(\()p Fn(u)p Fx(\))3432 1606 y(\(4)p Fn(:)p Fx(107\))67 2574 y Fi(e)50 2597 y Fn(R)119 2611 y Fl(\033)198 2597 y Fx(and)392 2574 y Fi(b)375 2597 y Fn(R)444 2611 y Fl(\033)523 2597 y Fx(b)s(eing)29 b(de\014ned)g(through)1008 2855 y Fi(e)991 2878 y Fn(R)1060 2892 y Fl(\033)1109 2878 y Fx(\()p Fn(u)p Fx(\))d Fm(\021)1364 2805 y Fx(\()6 b(~)-51 b Fn(u)21 b Fm(\000)e Fn(u)1614 2819 y Fl(\033)1663 2805 y Fx(\))1731 2769 y Fl(d)1769 2742 y Fq(2)p 1710 2784 117 4 v 1710 2837 a Fl(du)1791 2818 y Fq(2)1837 2805 y Fn(G)1909 2772 y Fl(\033)1909 2830 y(\033)n(;T)2028 2805 y Fx(\()6 b(^)-51 b Fn(u)p Fx(\))p 1364 2857 787 4 v 1557 2912 a Fl(d)p 1536 2927 81 4 v 1536 2979 a(du)1627 2948 y Fn(G)1699 2917 y Fl(\033)1699 2975 y(\033)n(;T)1818 2948 y Fx(\()p Fn(u)1905 2962 y Fl(\033)1954 2948 y Fx(\))1008 3166 y Fi(b)991 3189 y Fn(R)1060 3203 y Fl(\033)1109 3189 y Fx(\()p Fn(u)p Fx(\))26 b Fm(\021)f Fn(u)1442 3127 y(d)p 1416 3168 100 4 v 1416 3251 a(du)1526 3189 y(G)1598 3151 y Fl(\033)1598 3211 y(\033)n(;T)1717 3189 y Fx(\()6 b(~)-51 b Fn(u)p Fx(\))2060 3080 y Fv(1)p 2060 3095 37 4 v 2060 3147 a(2)2140 3080 y Fl(d)2178 3053 y Fq(2)p 2118 3095 117 4 v 2118 3147 a Fl(du)2199 3129 y Fq(2)2245 3116 y Fn(G)2317 3083 y Fl(\033)2317 3141 y(\033)n(;T)2436 3116 y Fx(\()6 b(\024)-51 b Fn(u)q Fx(\))p 1850 3168 907 4 v 1883 3223 a Fl(d)p 1862 3238 81 4 v 1862 3290 a(du)1953 3259 y Fn(G)2025 3228 y Fl(\033)2025 3286 y(\033)n(;T)2144 3259 y Fx(\()p Fn(u)2231 3233 y Fk(0)2257 3259 y Fx(\))2325 3223 y Fl(d)p 2303 3238 V 2303 3290 a(du)2395 3259 y Fn(G)2467 3228 y Fl(\033)2467 3286 y(\033)n(;T)2586 3259 y Fx(\()p Fn(u)2673 3273 y Fl(\033)2722 3259 y Fx(\))3432 3028 y(\(4)p Fn(:)p Fx(108\))50 3464 y(where)38 b(~)-51 b Fn(u)33 b Fx(is)f(on)h(the)g(ra)m (y)g(b)s(et)m(w)m(een)g(0)h(and)e Fn(u)p Fx(,)39 b(^)-50 b Fn(u)32 b Fx(on)h(the)g(ra)m(y)h(b)s(et)m(w)m(een)39 b(~)-51 b Fn(u)33 b Fx(and)f Fn(u)2864 3478 y Fl(\033)2913 3464 y Fx(,)i(and)e(b)s(oth)38 b(\024)-51 b Fn(u)33 b Fx(and)f Fn(u)3684 3431 y Fk(0)50 3609 y Fx(are)f(on)f(the)g(ra)m(y)h (b)s(et)m(w)m(een)g Fn(u)g Fx(and)e Fn(u)1303 3623 y Fl(\033)1352 3609 y Fx(,)i(and)f Fn(u)1637 3576 y Fk(0)1662 3609 y Fx(.)159 3827 y(The)g(v)-5 b(arious)29 b(\014rst)h(and)g(second) h(deriv)-5 b(ativ)m(es)29 b(en)m(tering)i(the)g(expressions)e(of)2871 3804 y Fi(b)2854 3827 y Fn(R)2923 3841 y Fl(\033)2972 3827 y Fx(\()p Fn(u)p Fx(\))i(and)3319 3804 y Fi(e)3302 3827 y Fn(R)3371 3841 y Fl(\033)3420 3827 y Fx(\()p Fn(u)p Fx(\))g(can)50 3973 y(b)s(e)g(b)s(ounded)e(with)i(the)g(help)g(of)g (\(3.52\))j(and)d(\(3.54\).)47 b(W)-8 b(e)33 b(then)e(get)i(that)f(on)g (the)f(disk)g Fm(j)p Fn(u)p Fm(j)c(\024)g Fn(\016)s(=)3479 3950 y Fi(b)3469 3973 y Fx(\002)r(\()p Fn(E)5 b Fx(\),)50 4118 y(0)26 b Fn(<)e(\016)30 b(<)25 b Fx(1,)1503 4264 y Fm(j)1545 4241 y Fi(b)1528 4264 y Fn(R)1597 4278 y Fl(\033)1646 4264 y Fx(\()p Fn(u)p Fx(\))p Fm(j)h(\024)f Fn(c)p Fx(\()p Fn(\016)s Fx(\))p Fn(z)2109 4278 y Fl(\033)2160 4264 y Fm(j)p Fn(z)t Fm(j)1176 b Fx(\(4)p Fn(:)p Fx(109\))50 4464 y(where)31 b Fn(z)356 4478 y Fl(\033)438 4464 y Fx(is)g(de\014ned)f(in)h(\(4.103\))k(and)c(0)e Fn(<)e(c)p Fx(\()p Fn(\016)s Fx(\))j Fn(<)e Fm(1)k Fx(only)f(dep)s(ends)f(on)i Fn(\016)s Fx(.)46 b(Similarly)-8 b(,)30 b(using)g(that)39 b(~)-51 b Fn(u)50 4609 y Fx(is)29 b(on)h(the)h(ra)m(y)g(b)s(et)m(w)m (een)g(0)g(and)e Fn(u)p Fx(,)1216 4843 y Fm(j)1259 4820 y Fi(e)1241 4843 y Fn(R)1310 4857 y Fl(\033)1360 4843 y Fx(\()p Fn(u)p Fx(\))p Fm(j)d(\024)f Fn(c)1668 4806 y Fk(0)1694 4843 y Fx(\()p Fn(\016)s Fx(\))p Fn(z)1849 4857 y Fl(\033)1899 4843 y Fx(\()p Fm(j)p Fn(z)t Fm(j)d Fx(+)e Fm(j)2178 4820 y Fi(b)2168 4843 y Fx(\002\()p Fn(E)5 b Fx(\))p Fn(u)2433 4857 y Fl(\033)2483 4843 y Fm(j)p Fx(\))889 b(\(4)p Fn(:)p Fx(110\))50 5077 y(for)21 b(some)h(0)k Fn(<)f(c)605 5044 y Fk(0)631 5077 y Fx(\()p Fn(\016)s Fx(\))i Fn(<)e Fm(1)p Fx(.)37 b(Recall)22 b(from)f(Section)g (3.2)i(\(form)m(ula)e(\(3.48\)\))j(that)f Fn(u)2829 5091 y Fl(\033)2903 5077 y Fm(\031)3139 5041 y Fv(1)p 3010 5056 296 4 v 3010 5108 a Fh(E)p Fl(\034)3088 5086 y Fp(\033)3081 5135 y(T)12 b Fq(\()p Fp(E)r Fq(\))p Fe(n)p Fp(\033)3316 5077 y Fx(;)25 b(ho)m(w)m(ev)m(er,)50 5224 y(insp)s(ecting)j(the)j(pro) s(of)e(of)i(Prop)s(osition)d(3.6)k(\(see)f(also)f(\(2.9\)\))j(an)d (alternativ)m(e)h(represen)m(tation)f(is)980 5458 y Fn(u)1032 5472 y Fl(\033)1107 5458 y Fx(=)24 b Fn(G)1274 5420 y Fl(\033)1274 5482 y(T)10 b Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)n(;T)g Fv(\()p Fl(E)s Fv(\))1711 5458 y Fx(\(0\)\(1)22 b(+)e Fn(O)s Fx(\()p Fn(e)2168 5420 y Fk(\000)p Fl(\014)2269 5366 y Fk(p)p 2329 5366 65 4 v 54 x Fl(N)7 b(E)2447 5429 y Fp(\033)2505 5435 y Fi(b)2495 5458 y Fx(\002\()p Fn(E)e Fx(\)\)\))654 b(\(4)p Fn(:)p Fx(111\))p eop %%Page: 42 42 42 41 bop 50 295 a Fr(42)3270 b Fo(Se)l(ction)27 b Fr(4)50 513 y Fx(and)j(this)f(will)f(b)s(e)h(ev)m(en)i(more)g(con)m(v)m(enien)m (t)g(here)g(as,)f(using)f(\(4.82\),)k(w)m(e)e(then)f(ha)m(v)m(e)1396 734 y Fi(b)1386 757 y Fx(\002)o(\()p Fn(E)5 b Fx(\))p Fn(u)1650 771 y Fl(\033)1726 757 y Fx(=)25 b Fn(z)1864 771 y Fl(\033)1913 757 y Fx(\(1)c(+)f Fn(O)s Fx(\()p Fn(z)2254 771 y Fl(\033)2303 757 y Fx(\)\))1059 b(\(4)p Fn(:)p Fx(112\))50 1002 y(The)30 b(b)s(ound)e(\(4.110\))33 b(th)m(us)d(b)s(ecomes)1343 1247 y Fm(j)1385 1224 y Fi(e)1368 1247 y Fn(R)1437 1261 y Fl(\033)1486 1247 y Fx(\()p Fn(u)p Fx(\))p Fm(j)c(\024)f Fn(c)1794 1209 y Fk(0)q(0)1841 1247 y Fx(\()p Fn(\016)s Fx(\))p Fn(z)1996 1261 y Fl(\033)2047 1247 y Fx(\()p Fm(j)p Fn(z)t Fm(j)c Fx(+)f Fn(z)2332 1261 y Fl(\033)2381 1247 y Fx(\))1016 b(\(4)p Fn(:)p Fx(113\))159 1564 y(W)-8 b(e)31 b(no)m(w)g(come)g(to)g(the)g(main)e (con)m(tribution)g(to)i(the)g(r.h.s.)f(of)g(\(4.107\),)k(namely)29 b(to)i(the)g(term)50 1710 y Fn(u=)p Fx(\()p Fn(u)21 b Fm(\000)f Fn(u)398 1724 y Fl(\033)447 1710 y Fx(\).)41 b(Using)29 b(\(4.112\))k(w)m(e)e(can)g(write)1485 1893 y Fn(u)p 1379 1934 265 4 v 1379 2017 a(u)20 b Fm(\000)g Fn(u)1594 2031 y Fl(\033)1679 1954 y Fx(=)1887 1893 y Fn(z)p 1786 1934 249 4 v 1786 2017 a(z)k Fm(\000)c Fn(z)1985 2031 y Fl(\033)2065 1954 y Fx(+)p 2156 1881 70 4 v 20 w Fn(R)2226 1968 y Fl(\033)2275 1954 y Fx(\()p Fn(z)t Fx(\))1041 b(\(4)p Fn(:)p Fx(114\))50 2202 y(where)p 638 2288 V 638 2361 a Fn(R)708 2375 y Fl(\033)757 2361 y Fx(\()p Fn(z)t Fx(\))26 b Fm(\021)1120 2300 y Fn(z)t Fx(\()p Fn(u)1253 2314 y Fl(\033)1313 2277 y Fi(b)1302 2300 y Fx(\002\()p Fn(E)5 b Fx(\))21 b Fm(\000)f Fn(z)1669 2314 y Fl(\033)1719 2300 y Fx(\))p 1006 2340 863 4 v 1006 2440 a(\()p Fn(z)25 b Fm(\000)20 b Fn(z)1241 2454 y Fl(\033)1290 2440 y Fx(\)\()p Fn(z)25 b Fm(\000)20 b Fn(u)1570 2454 y Fl(\033)1629 2417 y Fi(b)1619 2440 y Fx(\002\()p Fn(E)5 b Fx(\)\))1904 2361 y(=)2421 2300 y Fn(z)t(O)s Fx(\()p Fn(z)2620 2267 y Fv(2)2616 2322 y Fl(\033)2666 2300 y Fx(\))p 2011 2340 1100 4 v 2011 2424 a(\()p Fn(z)25 b Fm(\000)20 b Fn(z)2246 2438 y Fl(\033)2295 2424 y Fx(\)\()p Fn(z)25 b Fm(\000)20 b Fn(z)2565 2438 y Fl(\033)2614 2424 y Fx(\(1)h(+)f Fn(O)s Fx(\()p Fn(z)2955 2438 y Fl(\033)3005 2424 y Fx(\)\)\))3432 2361 y(\(4)p Fn(:)p Fx(115\))50 2608 y(T)-8 b(o)31 b(b)s(ound)d(this)h(term)i(w)m(e) f(use)h(that)g(on)f(the)g(set)1771 2535 y Fi(\010)1824 2608 y Fn(z)g Fm(2)25 b Fj(C)49 b Fm(j)25 b Fn(r)j(<)d(s)2331 2575 y Fv(2)2371 2608 y Fn(=)p Fx(4)2461 2535 y Fi(\011)2515 2608 y Fx(:)1624 2853 y Fm(j)p Fn(z)g Fm(\000)20 b Fn(z)1849 2867 y Fl(\033)1898 2853 y Fm(j)25 b(\025)g Fn(z)2086 2867 y Fl(\033)3432 2853 y Fx(\(4)p Fn(:)p Fx(116\))50 3098 y(and)341 3346 y Fm(j)p Fn(z)g Fm(\000)20 b Fn(z)566 3360 y Fl(\033)615 3346 y Fx(\(1)h(+)f Fn(O)s Fx(\()p Fn(z)956 3360 y Fl(\033)1005 3346 y Fx(\)\))p Fm(j)26 b(\025)1222 3218 y Fi(\032)1321 3274 y Fn(z)1363 3288 y Fl(\033)1412 3274 y Fx(\(1)21 b(+)f Fn(O)s Fx(\()p Fn(z)1753 3288 y Fl(\033)1802 3274 y Fx(\)\))p Fn(;)385 b Fx(if)60 b Fn(z)2438 3288 y Fl(\033)2487 3274 y Fx(\(1)21 b(+)f Fn(O)s Fx(\()p Fn(z)2828 3288 y Fl(\033)2877 3274 y Fx(\)\))26 b Fm(\024)f Fx(2)1321 3418 y(2)1366 3341 y Fi(p)p 1457 3341 709 4 v 77 x Fn(z)1499 3432 y Fl(\033)1548 3418 y Fx(\(1)c(+)f Fn(O)s Fx(\()p Fn(z)1889 3432 y Fl(\033)1939 3418 y Fx(\)\))h Fm(\000)f Fx(1)p Fn(;)91 b Fx(otherwise)3432 3346 y(\(4)p Fn(:)p Fx(117\))50 3612 y(Therefore,)30 b(for)g Fn(z)g Fm(2)25 b Fn(D)859 3626 y Fl(\016)899 3612 y Fx(,)999 3874 y Fm(j)p 1024 3801 70 4 v Fn(R)1094 3888 y Fl(\033)1143 3874 y Fx(\()p Fn(z)t Fx(\))p Fm(j)i(\024)1790 3812 y(j)p Fn(z)t Fm(j)p Fn(O)s Fx(\()p Fn(z)2039 3779 y Fv(2)2035 3835 y Fl(\033)2085 3812 y Fx(\))p 1418 3853 1075 4 v 1418 3936 a Fm(j)p Fn(z)d Fm(\000)c Fn(z)1642 3950 y Fl(\033)1691 3936 y Fm(j)15 b(j)p Fn(z)26 b Fm(\000)20 b Fn(z)1957 3950 y Fl(\033)2006 3936 y Fx(\(1)h(+)f Fn(O)s Fx(\()p Fn(z)2347 3950 y Fl(\033)2396 3936 y Fx(\)\))p Fm(j)2528 3874 y(\024)25 b Fn(c)p Fm(j)p Fn(z)t Fm(j)673 b Fx(\(4)p Fn(:)p Fx(118\))50 4131 y(for)30 b(some)h(constan)m(t)g Fn(c)26 b(>)f Fx(0.)159 4349 y(Inserting)i(\(4.114\))k(in)d(\(4.107\),) k(and)c(plugging)f(the)h(resulting)f(expression)g(in)h(\(4.106\),)j(w)m (e)f(ma)m(y)f(no)m(w)50 4494 y(write)705 4652 y(1)20 b Fm(\000)909 4571 y Fn(G)981 4538 y Fl(\033)981 4601 y(T)10 b Fk(n)p Fl(\033)n(;T)1189 4571 y Fx(\(0\))p 872 4631 471 4 v 872 4714 a(1)21 b Fm(\000)f Fn(G)1101 4683 y Fl(\033)1101 4741 y(\033)n(;T)1219 4714 y Fx(\()p Fn(u)p Fx(\))1368 4551 y Fi(\020)1423 4652 y Fx(1)g(+)g Fn(u)p Fj(E)11 b Fx([)p Fn(\034)1761 4615 y Fl(\033)1751 4677 y(T)f Fk(n)p Fl(\033)1896 4652 y Fm(j)p Fn(\034)1971 4615 y Fl(\033)1961 4677 y(T)g Fk(n)p Fl(\033)2125 4652 y Fx(=)25 b Fn(\034)2271 4615 y Fl(\033)2261 4675 y(T)2320 4652 y Fx(])2345 4551 y Fi(\021)2425 4652 y Fx(=)g Fn(I)2568 4615 y Fv(0)2561 4675 y Fl(\033)2609 4652 y Fx(\()p Fn(u)p Fx(\))c(+)f Fn(I)2890 4615 y Fv(1)2883 4675 y Fl(\033)2932 4652 y Fx(\()p Fn(u)p Fx(\))378 b(\(4)p Fn(:)p Fx(119\))50 4885 y(where)257 5125 y Fn(I)304 5088 y Fv(0)297 5148 y Fl(\033)346 5125 y Fx(\()p Fn(u)p Fx(\))26 b Fm(\021)590 4997 y Fi(\022)769 5064 y Fn(z)p 667 5104 249 4 v 667 5188 a(z)f Fm(\000)20 b Fn(z)867 5202 y Fl(\033)947 5125 y Fx(+)p 1038 5052 70 4 v 20 w Fn(R)1108 5139 y Fl(\033)1157 5125 y Fx(\()p Fn(z)t Fx(\))1273 4997 y Fi(\023)1356 5125 y Fx(\(1)h(+)1565 5102 y Fi(e)1548 5125 y Fn(R)1617 5139 y Fl(\033)1666 5125 y Fx(\()p Fn(u)p Fx(\)\))g(+)1952 5102 y Fi(b)1935 5125 y Fn(R)2004 5139 y Fl(\033)2053 5125 y Fx(\()p Fn(u)p Fx(\))257 5397 y Fn(I)304 5359 y Fv(1)297 5419 y Fl(\033)346 5397 y Fx(\()p Fn(u)p Fx(\))26 b Fm(\021)f Fn(z)647 5316 y Fj(E)10 b Fx([)q Fn(\034)777 5283 y Fl(\033)767 5345 y(T)g Fk(n)p Fl(\033)911 5316 y Fm(j)p Fn(\034)986 5283 y Fl(\033)976 5345 y(T)g Fk(n)p Fl(\033)1140 5316 y Fx(=)25 b Fn(\034)1286 5283 y Fl(\033)1276 5341 y(T)1335 5316 y Fx(])p 647 5376 714 4 v 907 5452 a Fi(b)897 5475 y Fx(\002)o(\()p Fn(E)5 b Fx(\))1387 5269 y Fi(\024\022)1614 5335 y Fn(z)p 1512 5376 249 4 v 1512 5459 a(z)25 b Fm(\000)20 b Fn(z)1712 5473 y Fl(\033)1792 5397 y Fx(+)p 1883 5324 70 4 v 20 w Fn(R)1953 5411 y Fl(\033)2002 5397 y Fx(\()p Fn(z)t Fx(\))2118 5269 y Fi(\023)2201 5397 y Fx(\(1)h(+)2410 5374 y Fi(e)2393 5397 y Fn(R)2462 5411 y Fl(\033)2511 5397 y Fx(\()p Fn(u)p Fx(\)\))g(+)2797 5374 y Fi(b)2780 5397 y Fn(R)2849 5411 y Fl(\033)2898 5397 y Fx(\()p Fn(u)p Fx(\))g Fm(\000)f Fx(1)3177 5269 y Fi(\025)3432 5268 y Fx(\(4)p Fn(:)p Fx(120\))p eop %%Page: 43 43 43 42 bop 50 295 a Fo(A)l(ging)27 b(in)h(the)g(REM.)f(Part)h(2.)2661 b Fr(43)50 513 y Fx(Assume)42 b(that)i Fn(z)51 b Fm(2)46 b Fn(D)884 527 y Fl(\016)924 513 y Fx(.)79 b(Since)43 b Fj(E)10 b Fx([)p Fn(\034)1408 480 y Fl(\033)1398 542 y(T)g Fk(n)p Fl(\033)1543 513 y Fm(j)p Fn(\034)1618 480 y Fl(\033)1608 542 y(T)g Fk(n)p Fl(\033)1794 513 y Fx(=)46 b Fn(\034)1961 480 y Fl(\033)1951 538 y(T)2010 513 y Fx(])2045 490 y Fi(b)2035 513 y Fx(\002\()p Fn(E)5 b Fx(\))2248 480 y Fk(\000)p Fv(1)2393 513 y Fm(\024)46 b Fx(1,)h(it)c(readily)f(follo)m(ws)g(from)h(the)50 658 y(estimates)31 b(\(4.109\),)i(\(4.113\),)g(\(4.118\),)h(and)c(the)g(b)s (ound)1327 773 y Fi(\014)1327 827 y(\014)1327 882 y(\014)1327 936 y(\014)1469 843 y Fn(z)p 1368 883 249 4 v 1368 967 a(z)24 b Fm(\000)c Fn(z)1567 981 y Fl(\033)1627 773 y Fi(\014)1627 827 y(\014)1627 882 y(\014)1627 936 y(\014)1683 904 y Fx(=)1779 773 y Fi(\014)1779 827 y(\014)1779 882 y(\014)1779 936 y(\014)1809 904 y Fx(1)h(+)2055 843 y Fn(z)2097 857 y Fl(\033)p 1977 883 V 1977 967 a Fn(z)j Fm(\000)c Fn(z)2176 981 y Fl(\033)2236 773 y Fi(\014)2236 827 y(\014)2236 882 y(\014)2236 936 y(\014)2292 904 y Fm(\024)25 b Fx(2)999 b(\(4)p Fn(:)p Fx(121\))50 1151 y(whic)m(h,)29 b(b)m(y)h(\(4.116\),)k(holds)29 b(for)h(all)f Fn(z)h Fm(2)24 b Fn(D)1530 1165 y Fl(\016)1571 1151 y Fx(,)30 b(that)1534 1379 y Fm(j)p Fn(I)1606 1341 y Fv(1)1599 1401 y Fl(\033)1648 1379 y Fx(\()p Fn(u)p Fx(\))p Fm(j)c(\024)f Fn(C)1989 1341 y Fk(0)2014 1379 y Fx(\()p Fn(\016)s Fx(\))p Fm(j)p Fn(z)t Fm(j)1209 b Fx(\(4)p Fn(:)p Fx(122\))50 1607 y(for)35 b(some)i(constan)m(t)g Fn(C)872 1574 y Fk(0)897 1607 y Fx(\()p Fn(\016)s Fx(\))f Fn(>)e Fx(0.)57 b(T)-8 b(o)36 b(treat)h(the)f(term)g Fn(I)2077 1574 y Fv(0)2070 1629 y Fl(\033)2119 1607 y Fx(\()p Fn(u)p Fx(\))g(note)h (that)f(using)f(in)f(turn)h(\(4.113\))j(and)50 1752 y(\(4.116\),)694 1784 y Fi(\014)694 1839 y(\014)694 1893 y(\014)694 1948 y(\014)836 1854 y Fn(z)p 735 1895 V 735 1978 a(z)24 b Fm(\000)c Fn(z)934 1992 y Fl(\033)1012 1893 y Fi(e)994 1916 y Fn(R)1063 1930 y Fl(\033)1112 1916 y Fx(\()p Fn(u)p Fx(\))1234 1784 y Fi(\014)1234 1839 y(\014)1234 1893 y(\014)1234 1948 y(\014)1291 1916 y Fm(\024)25 b Fn(c)1426 1878 y Fk(00)1472 1916 y Fx(\()p Fn(\016)s Fx(\))1653 1854 y Fn(z)1695 1868 y Fl(\033)1745 1854 y Fm(j)p Fn(z)t Fm(j)p 1596 1895 300 4 v 1596 1978 a(j)p Fn(z)i Fm(\000)20 b Fn(z)1823 1992 y Fl(\033)1872 1978 y Fm(j)1908 1916 y Fx(\()p Fm(j)p Fn(z)t Fm(j)h Fx(+)f Fn(z)2193 1930 y Fl(\033)2242 1916 y Fx(\))26 b Fm(\024)f Fn(c)2438 1878 y Fk(00)2485 1916 y Fx(\()p Fn(\016)s Fx(\))p Fm(j)p Fn(z)t Fm(j)p Fx(\()p Fm(j)p Fn(z)t Fm(j)e Fx(+)d Fn(z)2981 1930 y Fl(\033)3031 1916 y Fx(\))366 b(\(4)p Fn(:)p Fx(123\))50 2126 y(Therefore,)1328 2153 y Fi(\014)1328 2208 y(\014)1328 2262 y(\014)1328 2317 y(\014)1358 2285 y Fn(I)1405 2247 y Fv(0)1398 2307 y Fl(\033)1447 2285 y Fx(\()p Fn(u)p Fx(\))21 b Fm(\000)1793 2223 y Fn(z)p 1692 2264 249 4 v 1692 2347 a(z)j Fm(\000)c Fn(z)1891 2361 y Fl(\033)1951 2153 y Fi(\014)1951 2208 y(\014)1951 2262 y(\014)1951 2317 y(\014)2007 2285 y Fm(\024)25 b Fn(C)2175 2247 y Fk(0)o(0)2221 2285 y Fx(\()p Fn(\016)s Fx(\))p Fm(j)p Fn(z)t Fm(j)1002 b Fx(\(4)p Fn(:)p Fx(124\))50 2500 y(for)37 b(some)g(constan)m(t)h Fn(C)875 2467 y Fk(00)921 2500 y Fx(\()p Fn(\016)s Fx(\))g Fn(>)e Fx(0.)61 b(Com)m(bining)34 b(\(4.119\))40 b(together)e(with)e(\(4.123\))k(and)c(\(4.124\))k (yields)50 2646 y(\(4.105\).)j(This)29 b(concludes)g(the)i(pro)s(of)f (of)g(Lemma)h(4.13.)p Fm(})50 2840 y Fu(4.4)k(P)m(oisson)i(con)m(v)m (ergence.)159 3107 y Fx(Finally)d(w)m(e)j(need)e(to)i(con)m(trol)g(the) f(con)m(v)m(ergence)j(of)d(v)-5 b(arious)35 b(in)m(tegral)h(functions)f (of)h(the)g(v)-5 b(ariables)50 3252 y Fn(z)92 3266 y Fl(\033)141 3252 y Fx(.)47 b(W)-8 b(e)34 b(will)c(do)i(this)g(in)f(a)i (general)g(setting)f(\014rst)g(and)g(then)g(apply)g(this)f(to)i(the)g (v)-5 b(arious)32 b(o)s(ccurrences)50 3398 y(later)e(on.)159 3616 y(Note)i(\014rst)d(that)i(b)m(y)f(\(4.103\))k(and)29 b(\(3.2\),)146 3828 y Fn(z)188 3842 y Fl(\033)262 3828 y Fx(=)c(\(1)c Fm(\000)f Fx(1)p Fn(=)-5 b(M)10 b Fx(\))p Fn(e)810 3791 y Fk(\000)p Fl(\014)911 3737 y Fk(p)p 973 3737 65 4 v 973 3791 a Fl(N)c(E)1090 3800 y Fp(\033)1149 3805 y Fi(b)1138 3828 y Fx(\002\()p Fn(E)f Fx(\))171 4035 y(=)25 b Fn(e)309 3997 y Fk(\000)p Fl(\013)p Fv(\()p Fl(u)484 3968 y Fe(\000)p Fq(1)484 4021 y Fp(N)570 3997 y Fv(\()p Fl(E)651 4006 y Fp(\033)695 3997 y Fv(\))p Fk(\000)p Fl(E)s Fv(\))885 3906 y Fi(\022)952 4035 y Fx(1)20 b(+)1311 3973 y Fn(e)1353 3940 y Fk(\000)p Fl(E)p 1119 4014 546 4 v 1119 4097 a Fm(j)p Fn(T)13 b Fx(\()p Fn(E)5 b Fx(\))p Fm(j)p Fx(\()p Fn(\013)23 b Fm(\000)c Fx(1\))1690 3906 y Fi(\022)1757 4035 y Fx(1)i(+)f Fm(V)1970 4049 y Fl(N)s(;E)2113 4035 y Fn(e)2155 3997 y Fl(E)s(=)p Fv(2)2362 3973 y Fn(\013)g Fm(\000)g Fx(1)p 2301 4014 337 4 v 2301 4032 a Fm(p)p 2377 4032 261 4 v 71 x Fx(2)p Fn(\013)h Fm(\000)f Fx(1)2648 3906 y Fi(\023\023)2797 4035 y Fx(\(1)h(+)f Fn(O)s Fx(\(1)p Fn(=)-5 b(N)10 b Fx(\)\))171 4277 y Fm(\021)622 4216 y Fx(1)p 278 4257 733 4 v 278 4361 a Fn(e)320 4327 y Fl(\013)p Fv(\()p Fl(u)438 4298 y Fe(\000)p Fq(1)438 4351 y Fp(N)524 4327 y Fv(\()p Fl(E)605 4336 y Fp(\033)649 4327 y Fv(\))p Fk(\000)p Fl(E)s Fv(\))824 4361 y Fn(\034)864 4375 y Fl(E)s(;N)3432 4081 y Fx(\(4)p Fn(:)p Fx(125\))50 4558 y(only)29 b(dep)s(ends)g(on)h Fn(\033)j Fx(through)d Fn(u)1208 4519 y Fk(\000)p Fv(1)1208 4585 y Fl(N)1305 4558 y Fx(\()p Fn(E)1407 4572 y Fl(\033)1457 4558 y Fx(\).)41 b(As)30 b(has)g(b)s(een)f(explained)g(in)59 b(Section)30 b(1,)h(the)f(p)s(oin)m(t)f(pro)s(cess)659 4786 y Fm(N)747 4749 y Fk(\003)734 4809 y Fl(N)s(;E)903 4786 y Fm(\021)1122 4700 y Fi(X)999 4901 y Fl(\033)r Fk(2f\000)p Fv(1)p Fl(;)p Fv(1)p Fk(g)1317 4883 y Fp(N)1391 4786 y Fn(\016)1431 4815 y Fv(exp)q Fk(f)p Fl(\013)p Fv(\()p Fk(\000)p Fl(E)s Fv(+)p Fl(u)1866 4785 y Fe(\000)p Fq(1)1866 4839 y Fp(N)1952 4815 y Fv(\()p Fl(E)2033 4824 y Fp(\033)2077 4815 y Fv(\)\))p Fk(g)2201 4786 y Fx(=)2420 4700 y Fi(X)2297 4901 y Fl(\033)r Fk(2f\000)p Fv(1)p Fl(;)p Fv(1)p Fk(g)2615 4883 y Fp(N)2689 4786 y Fn(\016)2729 4803 y Fv(1)p Fl(=)p Fv(\()p Fl(z)2865 4812 y Fp(\033)2910 4803 y Fl(\034)2944 4812 y Fp(N)r(;E)3067 4803 y Fv(\))3432 4786 y Fx(\(4)p Fn(:)p Fx(126\))50 5098 y(con)m(v)m(erges)22 b(w)m(eakly)e(to)h(the)g(P)m(oisson)f(p)s (oin)m(t)f(pro)s(cess)h Fm(N)1925 5065 y Fk(\003)1912 5123 y Fl(E)1993 5098 y Fx(on)g([1)p Fn(;)15 b Fm(1)p Fx(\))22 b(with)d(in)m(tensit)m(y)g(measure)i Fn(\013)3328 5065 y Fk(\000)p Fv(1)3426 5098 y Fn(e)3468 5065 y Fl(E)3529 5098 y Fn(x)3581 5065 y Fk(\000)p Fv(1)p Fk(\000)p Fv(1)p Fl(=\013)3857 5098 y Fn(dx)p Fx(.)159 5316 y(W)-8 b(e)34 b(will)d(no)m(w)i(sho)m(w)g(ho)m(w)g(to)h(mak)m(e)h(use)d(of)i(the)f (con)m(v)m(ergence)j(of)d(our)g(p)s(oin)m(t)f(pro)s(cesses)h(to)h(P)m (oisson)50 5462 y(p)s(oin)m(t)28 b(pro)s(cesses)h(in)f(the)h(analysis)f (of)h(the)g(asymptotic)h(b)s(eha)m(vior)e(of)h(our)g(functions)e(as)j (b)s(oth)e Fn(N)39 b Fx(and)28 b Fn(E)p eop %%Page: 44 44 44 43 bop 50 295 a Fr(44)3270 b Fo(Se)l(ction)27 b Fr(4)50 513 y Fx(tend)e(to)i(in\014nit)m(y)-8 b(.)38 b(As)26 b(a)g(\014rst)f(example)h(w)m(e)g(will)e(explain)g(ho)m(w)i(to)h(con)m (trol)f(the)g(b)s(eha)m(vior)f(of)h(the)h(random)50 658 y(co)s(e\016cien)m(ts)k Fn(\034)550 672 y Fl(N)s(;E)693 658 y Fx(.)50 876 y Fd(Lemma)38 b(4.15:)51 b Ft(Set)33 b Fn(\034)938 890 y Fk(1)1040 876 y Fm(\021)1146 841 y Fl(\013)p Fk(\000)p Fv(1)p 1146 856 141 4 v 1193 908 a Fl(\013)1298 876 y Ft(.)41 b(Then,)1111 1118 y Fx(lim)1062 1178 y Fl(E)s Fk(#\0001)1325 1118 y Fx(lim)1301 1178 y Fl(N)7 b Fk("1)1490 1118 y Fn(\034)1530 1132 y Fl(N)s(;E)1698 1118 y Fx(=)25 b Fn(\034)1834 1132 y Fk(1)1911 1118 y Fn(;)109 b Ft(in)32 b(Pr)-5 b(ob)g(ability.)829 b Fx(\(4)p Fn(:)p Fx(127\))50 1578 y Fc(Pro)s(of:)53 b Fn(\034)425 1592 y Fl(N)s(;E)601 1578 y Fx(dep)s(ends)31 b(on)i(t)m(w)m(o)h(random) e(v)-5 b(ariables,)32 b Fm(V)2055 1592 y Fl(N)s(;E)2231 1578 y Fx(\(de\014ned)g(in)g(Eq.)48 b(\(3.2\))34 b(of)f([BBG1]\))i(and) 50 1724 y Fm(j)p Fn(T)13 b Fx(\()p Fn(E)5 b Fx(\))p Fm(j)p Fx(.)80 b(Let)44 b(us)e(\014rst)h(lo)s(ok)g(at)h Fm(V)1309 1738 y Fl(N)s(;E)1452 1724 y Fx(.)79 b(W)-8 b(e)44 b(w)m(an)m(t)g(to)g (sho)m(w)f(that)h Fm(V)2588 1738 y Fl(N)s(;E)2731 1724 y Fn(e)2773 1691 y Fl(E)s(=)p Fv(2)2952 1724 y Fx(tends)f(to)g(zero.)80 b(By)50 1869 y(Cheb)m(yc)m(hev's)31 b(inequalit)m(y)d(of)j(order)f (four,)g(w)m(e)g(ha)m(v)m(e)i(that)1298 2140 y Fj(P)p Fx([)p Fm(jV)1459 2154 y Fl(N)s(;E)1603 2140 y Fn(e)1645 2102 y Fv(+)p Fl(E)s(=)p Fv(2)1836 2140 y Fm(j)25 b Fn(>)g(\017)p Fx(])h Fm(\024)2184 2067 y Fj(E)11 b Fm(V)2302 2034 y Fv(4)2295 2092 y Fl(N)s(;E)p 2177 2119 275 4 v 2177 2202 a Fn(\017)2214 2176 y Fv(4)2254 2202 y Fn(e)2296 2176 y Fk(\000)p Fv(2)p Fl(E)3432 2140 y Fx(\(4)p Fn(:)p Fx(128\))50 2381 y(But)24 b(\(see)g([BKL],)g(Lemma)g(3.3,)i(where)d(ho)m(w)m(ev)m (er)i(the)f(normalisation)e(of)h Fm(V)2639 2395 y Fl(N)2731 2381 y Fx(is)g(di\013eren)m(t\))g(the)h(momen)m(ts)50 2527 y(of)30 b(the)h(random)f(v)-5 b(ariable)29 b Fm(V)1041 2541 y Fl(N)s(;E)1214 2527 y Fx(con)m(v)m(erge,)k(as)e Fn(N)k Fm(")25 b(1)p Fx(,)31 b(and)f(in)f(particular)1263 2794 y(lim)1239 2853 y Fl(N)7 b Fk("1)1428 2794 y Fj(E)k Fm(V)1546 2757 y Fv(4)1539 2817 y Fl(N)s(;E)1714 2794 y Fx(=)1820 2733 y(\(2)p Fn(\013)22 b Fm(\000)e Fx(1\))2151 2700 y Fv(2)p 1820 2773 373 4 v 1876 2856 a Fx(4)p Fn(\013)h Fm(\000)f Fx(1)2203 2794 y Fn(e)2245 2757 y Fk(\000)p Fl(E)2384 2794 y Fx(+)g(3)912 b(\(4)p Fn(:)p Fx(129\))50 3044 y(Therefore,)30 b(there)h(exists)f Fn(N)1044 3058 y Fv(0)1085 3044 y Fx(,)g(suc)m(h)g(that)h(for)f(all)g Fn(N)35 b(>)25 b(N)2085 3058 y Fv(0)2126 3044 y Fx(,)30 b(and)g(for)g Fm(\000)p Fn(E)35 b Fx(large)c(enough,)1388 3305 y Fj(P)p Fx([)p Fm(jV)1549 3319 y Fl(N)s(;E)1693 3305 y Fn(e)1735 3267 y Fl(E)s(=)p Fv(2)1871 3305 y Fm(j)25 b Fn(>)g(\017)p Fx(])h Fm(\024)2211 3243 y Fx(4)p Fn(e)2298 3210 y Fl(E)p 2211 3284 150 4 v 2247 3367 a Fn(\017)2284 3341 y Fv(4)3432 3305 y Fx(\(4)p Fn(:)p Fx(130\))50 3551 y(Next)31 b(w)m(e)g(note)g(that)f Fm(j)p Fn(T)13 b Fx(\()p Fn(E)5 b Fx(\))p Fm(j)27 b Fx(=)1185 3478 y Fi(R)1246 3500 y Fk(1)1228 3583 y Fl(E)1338 3551 y Fm(N)1413 3565 y Fl(N)1481 3551 y Fx(\()p Fn(dx)p Fx(\))32 b(con)m(v)m(erges,)g(as)f Fn(N)k Fm(")25 b(1)p Fx(,)31 b(to)g(a)g(P)m(oisson)e(random)h(v)-5 b(ariable)50 3696 y(with)29 b(parameter)i Fn(e)734 3663 y Fl(E)795 3696 y Fx(.)41 b(In)30 b(particular,)97 4009 y(lim)73 4068 y Fl(N)7 b Fk("1)263 4009 y Fj(P)p Fx([)p Fm(j)p Fn(e)410 3971 y Fl(E)472 4009 y Fm(j)p Fn(T)13 b Fx(\()p Fn(E)5 b Fx(\))p Fm(j)22 b(\000)e Fx(1)p Fm(j)26 b Fn(>)f(\017)p Fx(])g(=)1218 3887 y Fl(e)1253 3860 y Fe(\000)p Fp(E)1355 3887 y Fv(\(1)p Fk(\000)p Fl(\017)p Fv(\))1311 3922 y Fi(X)1308 4115 y Fl(n)p Fv(=0)1561 3947 y Fn(e)1603 3914 y Fk(\000)p Fl(nE)p 1561 3988 206 4 v 1624 4071 a Fn(n)p Fx(!)1777 4009 y Fn(e)1819 3971 y Fk(\000)p Fl(e)1911 3944 y Fe(\000)p Fp(E)2037 4009 y Fx(+)2300 3895 y Fk(1)2271 3922 y Fi(X)2128 4123 y Fl(n)p Fv(=)p Fl(e)2264 4105 y Fe(\000)p Fp(E)2365 4123 y Fv(\(1+)p Fl(\017)p Fv(\))2571 3947 y Fn(e)2613 3914 y Fk(\000)p Fl(nE)p 2571 3988 V 2634 4071 a Fn(n)p Fx(!)2787 4009 y Fn(e)2829 3971 y Fk(\000)p Fl(e)2921 3944 y Fe(\000)p Fp(E)3052 4009 y Fm(\024)g Fn(C)7 b(e)3262 3971 y Fk(\000)p Fl(E)3380 4009 y Fn(e)3422 3971 y Fk(\000)p Fl(\017)3509 3944 y Fq(2)3545 3971 y Fl(e)3580 3944 y Fe(\000)p Fp(E)3432 4225 y Fx(\(4)p Fn(:)p Fx(131\))50 4371 y(Com)m(bining)28 b(these)j(t)m(w)m(o)g(observ)-5 b(ations)30 b(pro)m(v)m(es)h(the)g (lemma.)40 b Fm(})50 4589 y Fu(Remark:)k Fx(Note)30 b(that)e(w)m(e)h (actually)e(pro)m(v)m(e)i(that)g Fn(\034)1851 4603 y Fl(N)s(;E)2022 4589 y Fx(con)m(v)m(erges,)i(as)d Fn(N)35 b Fm(")26 b(1)p Fx(,)j(to)f(a)h(random)e(v)-5 b(ariable)50 4735 y Fn(\034)90 4749 y Fl(E)186 4735 y Fx(whic)m(h)34 b(in)g(turn,)i(as)f Fn(E)k Fm(#)34 b(\0001)p Fx(,)i(con)m(v)m(erges)h (to)f(a)g(constan)m(t.)56 b(This)34 b(latter)h(con)m(v)m(ergence)j(can) e(easily)50 4880 y(b)s(e)d(sho)m(wn)g(to)h(tak)m(e)h(place)f(almost)g (surely)-8 b(.)49 b(Ho)m(w)m(ev)m(er,)37 b(it)c(is)f(not)i(correct)h (that)f(the)g(join)m(t)f(con)m(v)m(ergence)50 5025 y(tak)m(es)39 b(place)f(almost)f(surely)-8 b(.)62 b(It)38 b(ma)m(y)g(b)s(e)f(p)s (ossible)e(to)k(sho)m(w)e(that)h(almost)g(sure)f(con)m(v)m(ergence)j (holds)50 5171 y(along)33 b(certain)g(diagonal)f(limits)f Fn(N)40 b Fm(")30 b(1)j Fx(with)e Fn(E)36 b Fx(=)29 b Fn(E)2014 5185 y Fl(N)2116 5171 y Fx(dep)s(ending)i(on)i Fn(N)42 b Fx(in)32 b(a)i(suitable)d(w)m(a)m(y)-8 b(.)50 b(Due)50 5316 y(to)30 b(the)g(generally)f(rather)g(slo)m(w)h(con)m(v)m (ergence)i(of)e(extremal)f(distributions,)e(pro)m(ving)i(suc)m(h)g(a)h (statemen)m(t)50 5462 y(rigorously)f(w)m(ould)g(require)g(a)i (considerable)d(extra)k(e\013ort)f(and)e(is)h(not)g(guaran)m(teed)i(to) f(succeed.)p eop %%Page: 45 45 45 44 bop 50 295 a Fo(A)l(ging)27 b(in)h(the)g(REM.)f(Part)h(2.)2661 b Fr(45)159 513 y Fx(The)27 b(next)h(lemma)f(is)f(an)i(immediate)e (application)g(of)i(the)g(w)m(eak)g(con)m(v)m(ergence)i(of)e(the)g(p)s (oin)m(t)e(pro)s(cess)50 658 y Fm(N)138 625 y Fk(\003)125 683 y Fl(N)s(;E)268 658 y Fx(:)50 876 y Fd(Lemma)31 b(4.16:)48 b Ft(L)-5 b(et)29 b Fn(g)i Ft(b)-5 b(e)28 b(a)h(b)-5 b(ounde)g(d)29 b(c)-5 b(ontinuous)29 b(function)f(on)h Fj(R)2474 843 y Fv(+)2540 876 y Ft(,)g(such)f(that)2980 799 y Fi(\014)2980 854 y(\014)3010 803 y(R)3071 826 y Fk(1)3053 909 y Fv(0)3249 841 y Fl(dx)p 3174 856 230 4 v 3174 911 a(x)3215 893 y Fq(1+1)p Fp(=\013)3414 876 y Fn(g)s Fx(\()p Fn(x)p Fx(\))3582 799 y Fi(\014)3582 854 y(\014)3639 876 y Fn(<)50 1022 y Fx(+)p Fm(1)p Ft(,)i(and)i(let)e Fn(X)645 1036 y Fl(N)745 1022 y Ft(b)-5 b(e)30 b(a)h(family)h(of)e(p)-5 b(ositive)32 b(r)-5 b(andom)33 b(variables)e(that)h(c)-5 b(onver)g(ge)31 b(in)g(distribution)g(to)g(the)50 1167 y(p)-5 b(ositive)33 b(r)-5 b(andom)35 b(variable)f Fn(X)7 b Ft(.)42 b(Then)33 b(for)g(any)g Fn(b)26 b(>)f Fx(0)p Ft(,)10 1385 y(\(i\))159 1312 y Fi(R)220 1335 y Fk(1)202 1418 y Fl(b)312 1385 y Fm(N)400 1352 y Fk(\003)387 1410 y Fl(N)s(;E)530 1385 y Fx(\()p Fn(dx)p Fx(\))p Fn(g)s Fx(\()p Fn(xX)907 1399 y Fl(N)978 1385 y Fx(\))33 b Ft(c)-5 b(onver)g(ges,)33 b(as)g Fn(N)i Fm(")26 b(1)p Ft(,)32 b(to)i(the)f(r)-5 b(andom)35 b(variable)2857 1312 y Fi(R)2918 1335 y Fk(1)2900 1418 y Fl(b)3010 1385 y Fm(N)3098 1352 y Fk(\003)3085 1410 y Fl(E)3146 1385 y Fx(\()p Fn(dx)p Fx(\))p Fn(g)s Fx(\()p Fn(xX)7 b Fx(\))p Ft(.)-18 1604 y(\(ii\))47 b(If)29 b Fn(X)326 1618 y Fl(E)416 1604 y Ft(is)g(a)g(family)g(of)g(r)-5 b(andom)32 b(variables)d(such)g(that,)i (as)e Fn(E)i Fm(#)26 b(\0001)p Ft(,)j Fn(X)2650 1618 y Fl(E)2737 1604 y Fm(!)c Fn(a)g Fm(2)g Fj(R)3071 1571 y Fv(+)3167 1604 y Ft(almost)30 b(sur)-5 b(ely,)159 1749 y(then)565 1903 y Fx(lim)516 1963 y Fl(E)s Fk(#\0001)755 1903 y Fn(e)797 1866 y Fv(+)p Fl(E)930 1780 y Fi(Z)1021 1802 y Fk(1)980 1986 y Fv(1)1113 1903 y Fm(N)1201 1866 y Fk(\003)1188 1926 y Fl(E)1249 1903 y Fx(\()p Fn(dx)p Fx(\))p Fn(g)s Fx(\()p Fn(xX)1626 1917 y Fl(E)1689 1903 y Fx(\))26 b(=)f Fn(\013)1904 1866 y Fk(\000)p Fv(1)2017 1780 y Fi(Z)2108 1802 y Fk(1)2068 1986 y Fv(1)2296 1842 y Fn(dx)p 2211 1883 270 4 v 2211 1967 a(x)2263 1941 y Fv(1+1)p Fl(=\013)2491 1903 y Fn(g)s Fx(\()p Fn(xa)p Fx(\))p Fn(;)110 b Ft(a.s.)451 b Fx(\(4)p Fn(:)p Fx(132\))-46 2213 y Ft(\(iii\))47 b(If)36 b Fn(g)j Ft(is)d(a)h(c)-5 b(omplex)37 b(value)-5 b(d)37 b(function)f(on)h Fj(C)18 b Ft(,)42 b(and)37 b(if)f(for)h(some)f(domain)i Fn(B)e Fm(\032)31 b Fj(C)18 b Ft(,)42 b(for)37 b(al)5 b(l)36 b Fn(x)c Fm(2)f Fj(R)3616 2180 y Fv(+)3682 2213 y Ft(,)159 2358 y Fn(z)e Fm(2)c Fn(B)5 b Ft(,)32 b Fn(g)s Fx(\()p Fn(z)t(x)p Fx(\))i Ft(is)f(b)-5 b(ounde)g(d,)34 b(and)f(for)h(al)5 b(l)33 b Fn(z)c Fm(2)c Fn(B)5 b Ft(,)1453 2508 y Fi(\014)1453 2563 y(\014)1453 2617 y(\014)1453 2672 y(\014)1483 2516 y(Z)1574 2539 y Fk(1)1534 2723 y Fv(0)1762 2579 y Fn(dx)p 1677 2619 V 1677 2704 a(x)1729 2678 y Fv(1+1)p Fl(=\013)1958 2640 y Fn(g)s Fx(\()p Fn(z)t(x)p Fx(\))2172 2508 y Fi(\014)2172 2563 y(\014)2172 2617 y(\014)2172 2672 y(\014)2229 2640 y Fn(<)25 b Fm(1)1016 b Fx(\(4)p Fn(:)p Fx(133\))159 2917 y Ft(holds,)34 b(then)317 3193 y Fx(lim)269 3253 y Fl(E)s Fk(#\0001)507 3193 y Fn(P)593 3065 y Fi(\024)665 3193 y Fx(lim)641 3253 y Fl(N)7 b Fk("1)834 3193 y Fx(sup)831 3270 y Fl(z)s Fk(2)p Fl(B)990 3061 y Fi(\014)990 3116 y(\014)990 3170 y(\014)990 3225 y(\014)1021 3193 y Fn(e)1063 3156 y Fl(E)1139 3069 y Fi(Z)1230 3092 y Fk(1)1190 3276 y Fv(1)1322 3193 y Fm(N)1410 3156 y Fk(\003)1397 3216 y Fl(E)1458 3193 y Fx(\()p Fn(dx)p Fx(\))p Fn(g)s Fx(\()p Fn(z)t(xX)1881 3207 y Fl(E)1945 3193 y Fx(\))21 b Fm(\000)f Fx(\()p Fn(az)t Fx(\))2256 3156 y Fv(1)p Fl(=\013)2382 3193 y Fn(\013)2440 3156 y Fk(\000)p Fv(1)2553 3069 y Fi(Z)2644 3092 y Fk(1)2604 3276 y Fl(az)2832 3132 y Fn(dx)p 2747 3172 V 2747 3257 a(x)2799 3231 y Fv(1+1)p Fl(=\013)3028 3193 y Fn(g)s Fx(\()p Fn(x)p Fx(\))3196 3061 y Fi(\014)3196 3116 y(\014)3196 3170 y(\014)3196 3225 y(\014)3252 3193 y Fn(>)25 b(\017)3385 3065 y Fi(\025)3459 3193 y Fx(=)g(0)3432 3361 y(\(4)p Fn(:)p Fx(134\))50 3724 y Fc(Pro)s(of:)44 b Fx(\(i\))24 b(is)f(a)h(standard)g(result)f(that)i(follo)m(w)e(from)h (the)g(equiv)-5 b(alence)24 b(of)g(con)m(v)m(ergence)j(in)22 b(distribution)50 3870 y(of)30 b(a)g(r.v.)41 b(and)29 b(almost)h(sure)f(con)m(v)m(ergence)k(of)d(a)g(sequence)h(of)f(r.v.)40 b(ha)m(ving)30 b(the)g(same)g(distribution.)37 b(T)-8 b(o)50 4015 y(pro)m(v)m(e)31 b(\(ii\),)f(recall)f(that)i(b)m(y)g (de\014nition)d(of)i(the)h(P)m(oisson)f(pro)s(cess)g Fm(N)2435 3982 y Fk(\003)2422 4040 y Fl(E)2483 4015 y Fx(,)1317 4195 y Fi(Z)1408 4217 y Fk(1)1368 4401 y Fv(1)1500 4319 y Fm(N)1588 4281 y Fk(\003)1575 4341 y Fl(E)1636 4319 y Fx(\()p Fn(dx)p Fx(\))p Fn(g)s Fx(\()p Fn(x)p Fx(\))e(=)2113 4204 y Fl(n)2158 4213 y Fp(E)2097 4232 y Fi(X)2103 4426 y Fl(i)p Fv(=1)2243 4319 y Fn(g)s Fx(\()p Fn(x)2376 4333 y Fl(i)2407 4319 y Fx(\))990 b(\(4)p Fn(:)p Fx(135\))50 4638 y(where)22 b Fn(n)360 4652 y Fl(E)443 4638 y Fx(is)f(a)i(P)m(oisson)f(random)f(v)-5 b(ariable)21 b(with)g(mean)h Fn(e)2047 4605 y Fl(E)2109 4638 y Fx(,)i(and)e Fn(x)2379 4652 y Fl(i)2409 4638 y Fn(;)15 b(i)26 b Fm(2)f Fj(N)35 b Fx(are)22 b(i.i.d.)37 b(random)21 b(v)-5 b(ariables)50 4784 y(suc)m(h)30 b(that)1327 4938 y Fn(P)13 b Fx([)p Fn(x)1475 4952 y Fl(i)1530 4938 y Fm(\024)25 b Fn(a)p Fx(])h Fm(\021)f Fn(\013)1879 4901 y Fk(\000)p Fv(1)1992 4815 y Fi(Z)2083 4837 y Fl(a)2042 5021 y Fv(1)2238 4877 y Fn(dx)p 2153 4917 V 2153 5002 a(x)2205 4976 y Fv(1+1)p Fl(=\013)3432 4938 y Fx(\(4)p Fn(:)p Fx(136\))50 5171 y(Note)k(that)g(\014rst)e(b)m(y)h(con)m(tin)m(uit)m(y)g Fn(g)s Fx(\()p Fn(xX)1406 5185 y Fl(E)1469 5171 y Fx(\))16 b Fm(\000)f Fn(g)s Fx(\()p Fn(xa)p Fx(\))29 b(con)m(v)m(erge)i(to)d (zero)h(and)f(since)f Fn(g)32 b Fx(is)27 b(in)m(tegrable)g(w.r.t.)50 5316 y(the)42 b(la)m(w)f(of)h Fn(x)560 5330 y Fl(i)590 5316 y Fx(,)i Fn(g)s Fx(\()p Fn(xX)867 5330 y Fl(E)930 5316 y Fx(\))28 b Fm(\000)f Fn(g)s Fx(\()p Fn(xa)p Fx(\))46 b Fm(#)e Fx(0)e(as)g(a)g(random)f(v)-5 b(ariable.)73 b(On)40 b(the)i(other)g(hand,)i(it)d(follo)m(ws)50 5462 y(from)27 b(our)h(assumptions)e(that)j Fn(g)s Fx(\()p Fn(x)1264 5476 y Fl(i)1295 5462 y Fx(\))f(are)h(b)s(ounded)d(random)h (v)-5 b(ariables.)39 b(In)27 b(particular,)g(their)g(momen)m(t)p eop %%Page: 46 46 46 45 bop 50 295 a Fr(46)3270 b Fo(Se)l(ction)27 b Fr(4)50 513 y Fx(generating)39 b(functions)e Fj(E)10 b Fn(e)1000 480 y Fl(\025g)s Fv(\()p Fl(x)1150 489 y Fp(i)1185 480 y Fv(\))1256 513 y Fx(is)38 b(\014nite)f(for)i(all)e Fn(\025)p Fx(.)65 b(Therefore)38 b(standard)g(argumen)m(ts)h(imply)d (that)50 658 y(there)30 b(exists)g(a)h(constan)m(t)h Fn(c)e Fx(suc)m(h)h(that)686 938 y Fn(P)772 782 y Fi(")825 938 y Fm(j)p Fn(n)905 899 y Fk(\000)p Fv(1)905 965 y Fl(E)1035 823 y(n)1080 832 y Fp(E)1018 851 y Fi(X)1025 1045 y Fl(i)p Fv(=1)1149 938 y Fx(\()p Fn(g)s Fx(\()p Fn(x)1317 952 y Fl(i)1349 938 y Fx(\))20 b Fm(\000)g Fj(E)11 b Fn(g)s Fx(\()q Fn(x)1684 952 y Fl(i)1720 938 y Fx(\)\))p Fm(j)26 b Fn(>)f(\017)1974 782 y Fi(#)2052 938 y Fm(\024)g Fx(2)p Fj(E)2248 952 y Fl(n)2293 961 y Fp(E)2372 938 y Fx(exp)2526 810 y Fi(\022)2593 938 y Fm(\000)2738 876 y Fn(\017)2775 843 y Fv(2)2816 876 y Fn(n)2871 890 y Fl(E)p 2674 917 322 4 v 2674 1000 a Fn(c)p Fx(v)-5 b(ar)2838 974 y Fv(2)2878 1000 y Fx(\()p Fn(g)s Fx(\))3007 810 y Fi(\023)3432 938 y Fx(\(4)p Fn(:)p Fx(137\))50 1238 y(where)30 b Fj(E)367 1252 y Fl(n)412 1261 y Fp(E)506 1238 y Fx(denotes)h(exp)s(ectation)f(with)f(resp)s(ect) i(to)g(the)g(P)m(oisson)f(v)-5 b(ariable)29 b Fn(n)2835 1252 y Fl(E)2926 1238 y Fx(and)713 1532 y(v)-5 b(ar)837 1494 y Fv(2)878 1532 y Fx(\()p Fn(g)s Fx(\))26 b Fm(\021)f Fn(\013)1174 1494 y Fk(\000)p Fv(1)1288 1408 y Fi(Z)1378 1430 y Fk(1)1338 1614 y Fv(1)1567 1470 y Fn(dx)p 1482 1511 270 4 v 1482 1596 a(x)1534 1569 y Fv(1+1)p Fl(=\013)1777 1403 y Fi(\022)1844 1532 y Fn(g)s Fx(\()p Fn(x)p Fx(\))c Fm(\000)f Fn(\013)2182 1494 y Fk(\000)p Fv(1)2295 1408 y Fi(Z)2386 1430 y Fk(1)2346 1614 y Fv(1)2574 1470 y Fn(dx)p 2489 1511 V 2489 1596 a(x)2541 1569 y Fv(1+1)p Fl(=\013)2770 1532 y Fn(g)s Fx(\()p Fn(x)p Fx(\))2938 1403 y Fi(\023)3006 1422 y Fv(2)3432 1532 y Fx(\(4)p Fn(:)p Fx(138\))50 1802 y(is,)34 b(b)m(y)h(our)e(assumptions)g(on)h Fn(g)s Fx(,)i(\014nite.)52 b(T)-8 b(ogether)35 b(with)e(the)i(exp)s (onen)m(tial)e(estimate)i(on)f(the)h(concen-)50 1947 y(tration)30 b(of)h(the)f(P)m(oisson)g(v)-5 b(ariable)29 b Fn(n)1337 1961 y Fl(E)1429 1947 y Fx(\(4.131\),)k(this)c(yields)50 2229 y Fn(P)136 2100 y Fi(\024)184 2097 y(\014)184 2151 y(\014)184 2206 y(\014)184 2260 y(\014)214 2229 y Fn(n)269 2190 y Fk(\000)p Fv(1)269 2255 y Fl(E)382 2105 y Fi(Z)472 2127 y Fk(1)432 2311 y Fv(1)565 2229 y Fm(N)653 2191 y Fk(\003)640 2251 y Fl(E)701 2229 y Fx(\()p Fn(dx)p Fx(\))p Fn(g)s Fx(\()p Fn(x)p Fx(\))22 b Fm(\000)e Fn(\013)1209 2191 y Fk(\000)p Fv(1)1322 2105 y Fi(Z)1413 2127 y Fk(1)1373 2311 y Fv(1)1601 2167 y Fn(dx)p 1516 2208 V 1516 2293 a(x)1568 2266 y Fv(1+1)p Fl(=\013)1796 2229 y Fn(g)s Fx(\()p Fn(x)p Fx(\))1964 2097 y Fi(\014)1964 2151 y(\014)1964 2206 y(\014)1964 2260 y(\014)2021 2229 y Fm(\025)25 b Fn(\017)2154 2100 y Fi(\025)2227 2229 y Fm(\024)g Fx(2)15 b(exp)2538 2100 y Fi(\022)2605 2229 y Fm(\000)2751 2167 y Fn(\017)2788 2134 y Fv(2)2829 2167 y Fn(e)2871 2134 y Fk(\000)p Fl(E)p 2687 2208 367 4 v 2687 2291 a Fx(2)p Fn(c)p Fx(v)-5 b(ar)2895 2265 y Fv(2)2936 2291 y Fx(\()p Fn(g)s Fx(\))3064 2100 y Fi(\023)3131 2229 y Fx(+)p Fn(C)7 b(e)3316 2191 y Fk(\000)p Fl(E)3434 2229 y Fn(e)3476 2191 y Fk(\000)p Fl(e)3568 2164 y Fe(\000)p Fp(E)r(=)p Fq(4)3432 2392 y Fx(\(4)p Fn(:)p Fx(139\))50 2538 y(F)-8 b(rom)29 b(this)f(\(ii\))g(follo)m(ws)g(immediately)-8 b(.)39 b(T)-8 b(o)29 b(pro)m(v)m(e)h(\(iii\),)e(note)i(that)f(\(ii\))f (also)h(holds)e(if)h Fn(g)33 b Fx(tak)m(es)d(complex)50 2683 y(v)-5 b(alues)42 b(b)m(y)g(simply)e(considering)h(real)h(and)g (imaginary)f(part)i(separately)-8 b(.)78 b(By)42 b(a)h(simple)e(c)m (hange)j(of)50 2829 y(v)-5 b(ariables)29 b(w)m(e)i(ha)m(v)m(e,)h(for)e Fn(s)25 b Fm(\024)f Fx(1,)1205 3095 y Fj(E)11 b Fn(g)s Fx(\()q Fn(s)p Fm(\001)p Fx(\))31 b(=)25 b Fn(s)1615 3058 y Fv(1)p Fl(=\013)1740 3095 y Fn(\013)1798 3058 y Fk(\000)p Fv(1)1911 2972 y Fi(Z)2002 2994 y Fk(1)1962 3178 y Fl(s)2190 3034 y Fn(dx)p 2105 3075 270 4 v 2105 3159 a(x)2157 3133 y Fv(1+1)p Fl(=\013)2385 3095 y Fn(g)s Fx(\()p Fn(x)p Fx(\))879 b(\(4)p Fn(:)p Fx(140\))50 3362 y(and)337 3633 y(v)-5 b(ar)461 3595 y Fv(2)502 3633 y Fx(\()p Fn(g)s Fx(\()p Fn(s)p Fm(\001)p Fx(\)\))27 b(=)e Fn(s)922 3595 y Fv(1)p Fl(=\013)1047 3633 y Fn(\013)1105 3595 y Fk(\000)p Fv(1)1218 3509 y Fi(Z)1309 3532 y Fk(1)1268 3716 y Fl(s)1497 3571 y Fn(dx)p 1412 3612 V 1412 3697 a(x)1464 3671 y Fv(1+1)p Fl(=\013)1707 3505 y Fi(\022)1774 3633 y Fn(g)s Fx(\()p Fn(x)p Fx(\))d Fm(\000)e Fn(s)2098 3595 y Fv(1)p Fl(=\013)2223 3633 y Fn(\013)2281 3595 y Fk(\000)p Fv(1)2394 3509 y Fi(Z)2485 3532 y Fk(1)2444 3716 y Fl(s)2673 3571 y Fn(dx)p 2588 3612 V 2588 3697 a(x)2640 3671 y Fv(1+1)p Fl(=\013)2868 3633 y Fn(g)s Fx(\()p Fn(x)p Fx(\))3036 3505 y Fi(\023)3104 3524 y Fv(2)3432 3633 y Fx(\(4)p Fn(:)p Fx(141\))50 3921 y(If)28 b(\(4.133\))j(holds,)d(this)f(implies)f(that)j Fj(E)11 b Fn(g)s Fx(\()q Fn(s)p Fm(\001)p Fx(\))31 b Fm(\024)25 b Fn(C)7 b(s)1866 3888 y Fv(1)p Fl(=\013)2019 3921 y Fx(and)28 b(v)-5 b(ar)2318 3888 y Fv(2)2359 3921 y Fx(\()p Fn(g)s Fx(\()p Fn(s)p Fm(\001)p Fx(\)\))27 b Fm(\024)e Fn(C)7 b(s)2851 3888 y Fv(1)p Fl(=\013)3004 3921 y Fx(for)28 b(small)f Fn(s)p Fx(.)40 b(Th)m(us,)50 4066 y(for)30 b Fn(s)g Fx(small,)f(w)m(e)i(get)g(from)f(\(4.139\))j(that)e(for)f (some)h(\014nite)f(constan)m(t)h Fn(C)2561 4080 y Fl(g)2634 4066 y Fx(dep)s(ending)d(on)i Fn(g)76 4351 y(P)162 4223 y Fi(\024)210 4219 y(\014)210 4273 y(\014)210 4328 y(\014)210 4383 y(\014)241 4351 y Fn(e)283 4313 y Fl(E)360 4227 y Fi(Z)451 4249 y Fk(1)410 4434 y Fv(1)543 4351 y Fm(N)631 4313 y Fk(\003)618 4373 y Fl(E)679 4351 y Fx(\()p Fn(dx)p Fx(\))p Fn(g)s Fx(\()p Fn(sx)p Fx(\))22 b Fm(\000)e Fn(\013)1230 4313 y Fk(\000)p Fv(1)1343 4227 y Fi(Z)1434 4249 y Fk(1)1393 4434 y Fv(1)1526 4351 y Fn(g)s Fx(\()p Fn(sx)p Fx(\))1737 4219 y Fi(\014)1737 4273 y(\014)1737 4328 y(\014)1737 4383 y(\014)1793 4351 y Fm(\025)25 b Fn(s)1932 4313 y Fv(1)p Fl(=)p Fv(\(2)p Fl(\013)p Fv(\))2150 4351 y Fn(\017)2187 4223 y Fi(\025)2261 4351 y Fm(\024)f Fx(2)15 b(exp)2571 4223 y Fi(\022)2638 4351 y Fm(\000)2720 4289 y Fn(\017)2757 4256 y Fv(2)2797 4289 y Fn(e)2839 4256 y Fk(\000)p Fl(E)p 2720 4330 239 4 v 2762 4413 a Fx(2)p Fn(C)2872 4427 y Fl(g)2969 4223 y Fi(\023)3056 4351 y Fx(+)20 b Fn(C)7 b(e)3261 4313 y Fk(\000)p Fl(E)3379 4351 y Fn(e)3421 4313 y Fk(\000)p Fl(e)3513 4286 y Fe(\000)p Fp(E)r(=)p Fq(4)3432 4516 y Fx(\(4)p Fn(:)p Fx(142\))50 4662 y Fu(Remark:)53 b Fx(This)32 b(means)h(that)h(\015uctuations)f(are)g(at)i(most)e(of)h (order)f Fn(s)2571 4629 y Fv(1)p Fl(=)p Fv(\(2)p Fl(\013)p Fv(\))2789 4662 y Fn(e)2831 4629 y Fl(E)s(=)p Fv(2)3000 4662 y Fx(whic)m(h)f(is)g(less)h(than)50 4807 y(the)f(mean)g(as)h(long) f(as)g Fn(s)c(>)g(e)1090 4774 y Fl(E)1152 4807 y Fx(.)46 b(This)30 b(should)g(b)s(e)i(tak)m(en)h(as)f(a)h(sign)e(that)i(on)f (time)g(scales)g(larger)g(than)50 4953 y Fn(e)92 4920 y Fk(\000)p Fl(E)241 4953 y Fx(self-a)m(v)m(eraging)f(no)f(longer)g (tak)m(es)i(place.)159 5171 y(The)k(uniformit)m(y)e(of)i(the)h(con)m(v) m(ergence)h(in)d Fn(z)41 b Fx(claimed)35 b(under)g(\(iii\))f(follo)m (ws)i(from)f(the)i(exp)s(onen)m(tial)50 5316 y(estimate)29 b(\(4.142\))i(and)d(the)h(con)m(tin)m(uit)m(y)g(of)g Fn(g)j Fx(b)m(y)c(standard)g(argumen)m(ts.)40 b(This)27 b(concludes)h(the)h(pro)s(of)f(of)50 5462 y(the)i(lemma.)p Fm(})p eop %%Page: 47 47 47 46 bop 50 295 a Fo(A)l(ging)27 b(in)h(the)g(REM.)f(Part)h(2.)2661 b Fr(47)159 513 y Fx(As)30 b(\014rst)g(and)g(main)f(application)g(of)h (this)f(lemma)h(w)m(e)h(obtain)f(the)50 731 y Fd(Corollary)38 b(4.17:)92 b Ft(Uniformly)33 b(in)g Fm(<)p Fx(\()p Fn(z)t Fx(\))26 b Fn(<)f Fx(max\()p Fm(j=)p Fx(\()p Fn(z)t Fx(\))p Fm(j)p Fn(;)15 b Fx(1)p Fn(=)p Fx(2\))p Ft(,)749 1081 y Fx(lim)700 1140 y Fl(E)s Fk(#\0001)963 1081 y Fx(lim)939 1140 y Fl(N)7 b Fk("1)1231 1019 y Fx(1)p 1124 1060 260 4 v 1124 1143 a Fm(j)p Fn(T)13 b Fx(\()p Fn(E)5 b Fx(\))p Fm(j)1474 995 y Fi(X)1409 1194 y Fl(\033)r Fk(2)p Fl(T)10 b Fv(\()p Fl(E)s Fv(\))1796 1019 y Fn(z)p 1695 1060 249 4 v 1695 1143 a(z)25 b Fm(\000)20 b Fn(z)1895 1157 y Fl(\033)1138 1384 y Fx(=)25 b Fn(\013)1292 1346 y Fk(\000)p Fv(1)1405 1260 y Fi(Z)1496 1283 y Fk(1)1456 1467 y Fv(1)1684 1322 y Fn(dx)p 1599 1363 270 4 v 1599 1448 a(x)1651 1422 y Fv(1+1)p Fl(=\013)1969 1322 y Fn(xz)t(\034)2107 1336 y Fk(1)p 1891 1363 372 4 v 1891 1446 a Fn(xz)t(\034)2029 1460 y Fk(1)2126 1446 y Fm(\000)20 b Fx(1)2273 1384 y Fn(;)108 b Ft(in)33 b(Pr)-5 b(ob)g(ability.)3432 1237 y Fx(\(4)p Fn(:)p Fx(143\))50 1737 y Ft(Mor)g(e)g(over,)33 b(on)h(the)f(same)g(set,)660 2091 y Fn(\013)718 2053 y Fk(\000)p Fv(1)831 1967 y Fi(Z)922 1989 y Fk(1)882 2174 y Fv(1)1110 2029 y Fn(dx)p 1025 2070 270 4 v 1025 2155 a(x)1077 2128 y Fv(1+1)p Fl(=\013)1394 2029 y Fn(xz)t(\034)1532 2043 y Fk(1)p 1316 2070 372 4 v 1316 2153 a Fn(xz)t(\034)1454 2167 y Fk(1)1551 2153 y Fm(\000)20 b Fx(1)1724 2091 y(=)25 b(\()p Fm(\000)p Fn(z)t(\034)2012 2105 y Fk(1)2089 2091 y Fx(\))2124 2053 y Fv(1)p Fl(=\013)2250 2091 y Fn(\031)s Ft(c)-5 b(ose)g(c)16 b Fx(\()p Fn(\031)s(=\013)p Fx(\))22 b(+)d Fn(O)s Fx(\()p Fm(j)p Fn(z)t Fm(j)p Fx(\))334 b(\(4)p Fn(:)p Fx(144\))50 2449 y Ft(for)33 b Fm(j)p Fn(z)t Fm(j)g Ft(smal)5 b(l.)50 2667 y Fc(Pro)s(of:)46 b Fx(T)-8 b(o)30 b(get)g(\(4.143\),)i(just)c(c)m(hec)m(k)j(that)e(the)h(h)m(yp)s (othesis)d(of)i(Lemma)g(4.16)i(are)e(satis\014ed.)40 b(T)-8 b(o)29 b(pro)m(v)m(e)50 2813 y(\(4.144\),)k(note)e(\014rst)f (that)1011 3053 y Fi(\014)1011 3108 y(\014)1011 3162 y(\014)1011 3217 y(\014)1041 3061 y(Z)1132 3084 y Fv(1)1091 3268 y(0)1284 3123 y Fn(dx)p 1199 3164 270 4 v 1199 3249 a(x)1251 3223 y Fv(1+1)p Fl(=\013)1568 3123 y Fn(xz)t(\034)1706 3137 y Fk(1)p 1490 3164 372 4 v 1490 3247 a Fn(xz)t(\034)1628 3261 y Fk(1)1725 3247 y Fm(\000)20 b Fx(1)1872 3053 y Fi(\014)1872 3108 y(\014)1872 3162 y(\014)1872 3217 y(\014)1928 3185 y Fm(\024)2024 3061 y Fi(Z)2115 3084 y Fv(1)2074 3268 y(0)2267 3123 y Fn(dx)p 2182 3164 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Fl(\033)2274 4146 y Fx(\()p Fn(u)p Fx(\))21 b Fm(\000)f Fx(1)2553 4045 y Fi(\021)441 4455 y Fx(+)608 4369 y Fi(X)532 4568 y Fl(\033)576 4550 y Fe(0)601 4568 y Fk(62)p Fl(T)10 b Fv(\()p Fl(E)s Fv(\))831 4455 y Fj(P)p Fx([)p Fn(\033)s Fx(\()p Fn(m)p Fx(\))27 b(=)e Fn(\033)1294 4418 y Fk(0)1320 4455 y Fn(;)15 b(\034)1410 4418 y Fl(\033)1400 4480 y(T)10 b Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)1678 4455 y Fn(>)25 b(m)p Fx(])1894 4354 y Fi(\020)1948 4455 y Fn(G)2020 4418 y Fl(\033)2064 4390 y Fe(0)2020 4480 y Fl(T)10 b Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)n(;T)g Fv(\()p Fl(E)s Fv(\))2457 4455 y Fx(\()p Fn(u)p Fx(\))21 b(+)e Fn(G)2762 4418 y Fl(\033)2806 4390 y Fe(0)2762 4480 y Fl(\033)n(;T)10 b Fv(\()p Fl(E)s Fv(\))2995 4455 y Fx(\()p Fn(u)p Fx(\))p Fn(G)3189 4418 y Fl(\033)3189 4480 y(T)g Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)3442 4455 y Fx(\()p Fn(u)p Fx(\))21 b Fm(\000)f Fx(1)3721 4354 y Fi(\021)3775 4300 y(#)3432 4670 y Fx(\(4)p Fn(:)p Fx(153\))50 4816 y(As)25 b(b)m(y)f(no)m(w)h(usual,)g(w)m(e)g(will)d(need)j(a)g(rather)f (crude)g(b)s(ound)f(for)i Fn(u)f Fx(a)m(w)m(a)m(y)j(from)d(the)h (origin)f(complemen)m(ted)50 4961 y(b)m(y)30 b(a)h(\014ner)e(estimate)i (for)f(v)m(ery)h(small)e(v)-5 b(alues)30 b(of)g Fm(j)p Fn(u)p Fm(j)p Fx(.)42 b(The)29 b(former)h(follo)m(ws)g(from)g(the)g (next)h(lemma.)50 5180 y Fd(Lemma)38 b(4.20:)51 b Ft(Assume)32 b(that)i Fm(<)p Fx(\()p Fn(u)p Fx(\))26 b Fm(\024)1588 5144 y Fv(1)p 1588 5159 37 4 v 1588 5211 a(2)1645 5157 y Fi(b)1635 5180 y Fx(\002)1706 5147 y Fk(\000)p Fv(1)1803 5180 y Ft(.)42 b(Then)763 5413 y Fm(j)p Fn(F)859 5376 y Fk(\003)846 5436 y Fl(\033)901 5413 y Fx(\()p Fn(m;)15 b(u)p Fx(\))p Fm(j)27 b(\024)1427 5352 y Fx(2)p 1302 5392 297 4 v 1302 5475 a Fm(j)p Fn(e)1369 5449 y Fl(u)1437 5475 y Fm(\000)20 b Fx(1)p Fm(j)1609 5413 y Fj(P)p Fx([)p Fn(\034)1739 5376 y Fl(\033)1729 5438 y(T)10 b Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)2008 5413 y Fn(>)25 b(m)p Fx(])2224 5312 y Fi(\020)2278 5309 y(\014)2278 5363 y(\014)2278 5418 y(\014)2309 5413 y Fn(G)2381 5376 y Fl(\033)2381 5438 y(T)10 b Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)2633 5413 y Fx(\()p Fn(u)p Fx(\))2755 5309 y Fi(\014)2755 5363 y(\014)2755 5418 y(\014)2806 5413 y Fx(+)20 b(2)2942 5312 y Fi(\021)3432 5413 y Fx(\(4)p Fn(:)p Fx(154\))p eop %%Page: 50 50 50 49 bop 50 295 a Fr(50)3270 b Fo(Se)l(ction)27 b Fr(4)50 731 y Fc(Pro)s(of:)76 b Fx(By)43 b(Lemma)g(3.4,)48 b(under)41 b(the)i(condition)f(on)h Fn(u)p Fx(,)2201 626 y Fi(\014)2201 681 y(\014)2201 735 y(\014)2231 731 y Fn(G)2303 698 y Fl(\033)2347 670 y Fe(0)2303 760 y Fl(\033)n(;T)10 b Fv(\()p Fl(E)s Fv(\))2536 731 y Fx(\()p Fn(u)p Fx(\))2658 626 y Fi(\014)2658 681 y(\014)2658 735 y(\014)2735 731 y Fx(=)2852 626 y Fi(\014)2852 681 y(\014)2852 735 y(\014)2882 731 y Fn(g)2928 698 y Fl(\033)2972 670 y Fe(0)2925 753 y Fl(\033)3002 731 y Fx(\()p Fn(u)p Fx(\))3124 626 y Fi(\014)3124 681 y(\014)3124 735 y(\014)3202 731 y Fm(\024)46 b Fx(2)d(\(in)f(fact)50 904 y Fm(\024)28 b Fx(2)p Fn(=)p Fx(\()p Fn(M)33 b Fm(\000)21 b Fx(1\)\).)48 b(Similarly)-8 b(,)1082 799 y Fi(\014)1082 854 y(\014)1082 908 y(\014)1113 904 y Fn(G)1185 871 y Fl(\033)1229 843 y Fe(0)1185 933 y Fl(T)10 b Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)n(;T)g Fv(\()p Fl(E)s Fv(\))1621 904 y Fx(\()p Fn(u)p Fx(\))1743 799 y Fi(\014)1743 854 y(\014)1743 908 y(\014)1803 904 y Fm(\024)28 b Fx(2.)47 b(Inserting)31 b(this)h(in)m(to)g(\(4.153\))j (and)d(noting)g(that)50 981 y Fi(P)146 1076 y Fl(\033)190 1058 y Fe(0)215 1076 y Fk(62)p Fl(T)10 b Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)532 1049 y Fj(P)p Fx([)p Fn(\033)s Fx(\()p Fn(m)p Fx(\))30 b(=)g Fn(\033)1003 1016 y Fk(0)1028 1049 y Fn(;)15 b(\034)1118 1016 y Fl(\033)1108 1078 y(T)10 b Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)1390 1049 y Fn(>)29 b(m)p Fx(])h(=)f Fj(P)p Fx([)p Fn(\034)1855 1016 y Fl(\033)1845 1078 y(T)10 b Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)2127 1049 y Fn(>)30 b(m)p Fx(])i(one)i(arriv)m(es)e(readily)f(at)j(the)f (claimed)50 1195 y(b)s(ound.)p Fm(})50 1446 y Fu(Bounds)j(for)f Fm(j)p Fn(u)p Fm(j)26 b(\034)f Fx(1)p Fu(.)159 1664 y Fx(As)33 b(w)m(as)g(the)h(case)g(for)f(the)g(resolv)m(en)m(t,)i(w)m(e)e (ha)m(v)m(e)i(to)f(iden)m(tify)d(more)j(precisely)d(the)j(leading)d (term)j(of)50 1809 y(the)29 b(inhomogeneous)g(term)g(for)g(the)h(con)m (tribution)e(to)i(the)f(in)m(v)m(ersion)f(in)m(tegral)h(for)g Fn(u)g Fx(v)m(ery)h(close)g(to)g(the)50 1955 y(origin.)39 b(W)-8 b(e)32 b(b)s(egin)d(with)g(the)h Fn(m)p Fx(-dep)s(enden)m(t)g (probabilities)d(in)i(\(4.153\).)50 2173 y Fd(Lemma)38 b(4.21:)91 b Ft(Ther)-5 b(e)34 b(is)e(a)h(\014nite)g(p)-5 b(ositive)33 b(c)-5 b(onstant)35 b Fn(C)k Ft(such)33 b(that,)g(with)h Fn(b)2906 2187 y Fl(\033)2987 2173 y Ft(as)g(in)e(\(3.8\),)358 2315 y Fi(\014)358 2369 y(\014)358 2424 y(\014)389 2419 y Fj(P)459 2319 y Fi(h)502 2419 y Fn(\034)552 2382 y Fl(\033)542 2444 y(T)10 b Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)820 2419 y Fn(>)25 b(m;)15 b(\033)s Fx(\()p Fn(m)p Fx(\))26 b(=)f Fn(\033)1418 2319 y Fi(i)1481 2419 y Fm(\000)20 b Fn(p)1618 2433 y Fl(N)1687 2419 y Fx(\()p Fn(\033)n(;)15 b(\033)s Fx(\))1902 2382 y Fl(m)1973 2315 y Fi(\014)1973 2369 y(\014)1973 2424 y(\014)2028 2419 y Fm(\024)25 b Fn(C)7 b(me)2318 2382 y Fk(\000)p Fl(\014)2419 2328 y Fk(p)p 2479 2328 65 4 v 54 x Fl(N)f(E)2596 2391 y Fp(\033)2644 2419 y Fn(e)2686 2377 y Fk(\000)p Fl(m=)p Fh(E)p Fl(\034)2923 2350 y Fp(\033)2916 2398 y(T)13 b Fe(n)p Fp(\033)3044 2377 y Fl(b)3076 2386 y Fp(\033)3432 2419 y Fx(\(4)p Fn(:)p Fx(155\))50 2884 y Fc(Pro)s(of:)44 b Fx(Note)27 b(that)e Fn(p)828 2898 y Fl(N)897 2884 y Fx(\()p Fn(\033)n(;)15 b(\033)s Fx(\))1112 2851 y Fl(m)1207 2884 y Fx(is)24 b(the)i(probabilit)m(y)c(of)j(the)h (ev)m(en)m(t)g(that)g Fn(\033)s Fx(\()p Fn(k)s Fx(\))g(remains)e(at)i Fn(\033)i Fx(during)23 b(the)50 3030 y(en)m(tire)30 b(p)s(erio)s(d)d (from)j(time)f(zero)i(to)f(time)g Fn(m)f Fx(whic)m(h)g(is)g(a)h(subset) f(of)h(the)g(ev)m(en)m(t)i Fm(f)p Fn(\034)2924 2997 y Fl(\033)2914 3059 y(T)10 b Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)3192 3030 y Fn(>)25 b(m;)15 b(\033)s Fx(\()p Fn(m)p Fx(\))26 b(=)50 3175 y Fn(\033)s Fm(g)p Fx(.)41 b(In)30 b(what)g(remains,)f(there)i(m)m(ust)f(b)s(e)g(a)h(\014rst)e (time)h(when)g Fn(\033)s Fx(\()p Fn(k)s Fx(\))c Fm(6)p Fx(=)f Fn(\033)s Fx(.)41 b(Th)m(us)398 3308 y Fi(\014)398 3363 y(\014)398 3417 y(\014)429 3413 y Fj(P)499 3312 y Fi(h)542 3413 y Fn(\034)592 3375 y Fl(\033)582 3437 y(T)10 b Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)860 3413 y Fn(>)25 b(m;)15 b(\033)s Fx(\()p Fn(m)p Fx(\))26 b(=)f Fn(\033)1458 3312 y Fi(i)1521 3413 y Fm(\000)20 b Fn(p)1658 3427 y Fl(N)1727 3413 y Fx(\()p Fn(\033)n(;)15 b(\033)s Fx(\))1942 3375 y Fl(m)2013 3308 y Fi(\014)2013 3363 y(\014)2013 3417 y(\014)409 3672 y Fm(\024)505 3558 y Fl(m)p Fk(\000)p Fv(1)518 3586 y Fi(X)517 3781 y Fl(k)r Fv(=1)677 3672 y Fn(p)723 3686 y Fl(N)792 3672 y Fx(\()p Fn(\033)n(;)g(\033)s Fx(\))1007 3634 y Fl(k)r Fk(\000)p Fv(1)1181 3586 y Fi(X)1161 3781 y Fl(\033)1205 3762 y Fe(0)1230 3781 y Fk(\030)p Fl(\033)1347 3672 y Fn(p)1393 3686 y Fl(N)1461 3672 y Fx(\()p Fn(\033)n(;)g(\033)1641 3634 y Fk(0)1668 3672 y Fx(\))p Fj(P)1773 3571 y Fi(h)1817 3672 y Fn(\034)1867 3634 y Fl(\033)1911 3607 y Fe(0)1857 3697 y Fl(T)10 b Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)2135 3672 y Fn(>)25 b(m)20 b Fm(\000)g Fn(k)s(;)15 b(\033)s Fx(\()p Fn(m)21 b Fm(\000)f Fn(k)s Fx(\))25 b(=)g Fn(\033)3055 3571 y Fi(i)409 3990 y Fm(\024)g Fx(\(1)c Fm(\000)e Fn(p)742 4004 y Fl(N)811 3990 y Fx(\()p Fn(\033)n(;)c(\033)s Fx(\)\))1076 3877 y Fl(m)p Fk(\000)p Fv(1)1091 3904 y Fi(X)1091 4100 y Fl(k)r Fv(=1)1251 3990 y Fn(p)1297 4004 y Fl(N)1366 3990 y Fx(\()p Fn(\033)n(;)g(\033)s Fx(\))1581 3953 y Fl(k)r Fk(\000)p Fv(1)1735 3990 y Fx(max)1735 4049 y Fl(\033)1779 4031 y Fe(0)1804 4049 y Fk(\030)p Fl(\033)1920 3990 y Fj(P)1990 3890 y Fi(h)2034 3990 y Fn(\034)2084 3953 y Fl(\033)2128 3926 y Fe(0)2074 4015 y Fl(T)10 b Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)2352 3990 y Fn(>)25 b(m)20 b Fm(\000)f Fn(k)2688 3890 y Fi(i)3432 3731 y Fx(\(4)p Fn(:)p Fx(156\))50 4282 y(The)i(probabilit)m(y)e(in)i(the)h (last)g(line)e(is)h(similar)e(to)j(the)g(probabilities)d(estimated)j (in)e(Corollary)h(3.3,)k(except)50 4428 y(that)k(the)f(starting)g(p)s (oin)m(t)g(is)f(no)m(w)h Fn(\033)1303 4395 y Fk(0)1357 4428 y Fx(instead)g(of)g Fn(\033)s Fx(.)40 b(Ho)m(w)m(ev)m(er)31 b(using)26 b(the)j(decomp)s(osition)e(\(4.152\),)k(one)50 4573 y(v)m(eri\014es)f(easily)h(that)h(follo)m(wing)d(the)j(same)g (lines)d(as)j(in)e(the)h(pro)s(of)g(of)g(that)h(corollary)-8 b(,)31 b(one)h(obtains)f(the)50 4719 y(estimate)1061 4864 y Fj(P)1131 4763 y Fi(h)1175 4864 y Fn(\034)1225 4827 y Fl(\033)1269 4799 y Fe(0)1215 4889 y Fl(T)10 b Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)1493 4864 y Fn(>)25 b(m)20 b Fm(\000)g Fn(k)1830 4763 y Fi(i)1898 4864 y Fm(\024)25 b Fn(C)7 b(e)2108 4822 y Fk(\000)p Fv(\()p Fl(m)p Fk(\000)p Fl(k)r Fv(\))p Fl(=)p Fh(E)p Fl(\034)2499 4794 y Fp(\033)2538 4776 y Fe(0)2492 4843 y Fp(T)12 b Fe(n)p Fp(\033)2618 4822 y Fl(b)2650 4831 y Fp(\033)3432 4864 y Fx(\(4)p Fn(:)p Fx(157\))50 5070 y(whic)m(h)28 b(is)g(all)g(w)m(e)h(will)e(need)i(here.)40 b(Inserting)28 b(this)g(estimate)i(in)m(to)f(\(4.156\))j(and)c(using)g(that,)i(b)m(y)f (Prop)s(o-)50 5215 y(sition)g(2.2)i(\(together)h(with)e(the)g(remark)g (that)h(follo)m(ws)f(it\),)768 5462 y Fn(p)814 5476 y Fl(N)883 5462 y Fx(\()p Fn(\033)n(;)15 b(\033)s Fx(\))1098 5424 y Fl(k)1169 5462 y Fx(=)25 b(\(1)c Fm(\000)f Fn(e)1499 5424 y Fk(\000)p Fl(\014)1600 5370 y Fk(p)p 1661 5370 V 1661 5424 a Fl(N)6 b(E)1778 5433 y Fp(\033)1826 5462 y Fx(\))1861 5424 y Fl(k)1932 5462 y Fm(\024)25 b Fn(e)2070 5424 y Fk(\000)p Fl(k)r(e)2203 5397 y Fe(\000)p Fp(\014)2290 5358 y Fe(p)p 2339 5358 54 3 v 2339 5397 a Fp(N)5 b(E)2439 5406 y(\033)2516 5462 y Fm(\024)25 b Fn(e)2654 5419 y Fk(\000)p Fl(k)r(=)p Fh(E)p Fl(\034)2867 5392 y Fp(\033)2860 5441 y(T)12 b Fe(n)p Fp(\033)3432 5462 y Fx(\(4)p Fn(:)p Fx(158\))p eop %%Page: 51 51 51 50 bop 50 295 a Fo(A)l(ging)27 b(in)h(the)g(REM.)f(Part)h(2.)2661 b Fr(51)50 513 y Fx(the)30 b(b)s(ound)f(\(4.155\))k(follo)m(ws)c (directly)-8 b(.)40 b Fm(})50 731 y Fu(Remark:)45 b Fx(Let)29 b(us)e(note)i(that)h(the)e(b)s(ound)f(\(4.155\))k(is)c(really)g (e\013ectiv)m(ely)j(smaller)d(than)h(the)h(dominan)m(t)50 876 y(term,)d(if)f Fn(E)434 890 y Fl(\033)508 876 y Fx(is)g(\\deep")h (within)c(the)k(top,)h(ev)m(en)f(though)f(w)m(e)h(concede)g(a)g(little) e(of)i(the)f(exp)s(onen)m(tial)g(deca)m(y)50 1022 y(when)36 b(replacing)f Fn(e)731 989 y Fl(\014)775 935 y Fk(p)p 836 935 65 4 v 836 989 a Fl(N)6 b(E)953 998 y Fp(\033)1038 1022 y Fx(b)m(y)36 b Fj(E)1225 989 y Fl(\033)1225 1051 y(T)10 b Fk(n)p Fl(\033)1370 1022 y Fx(.)60 b(The)36 b(p)s(oin)m(t)g(is)f(that)j(this)d(error)i(will)d(tend)i(to)h(zero,)j (while)35 b(the)50 1167 y(prefactor)f(of)f(the)h(exp)s(onen)m(tial)e (tends)h(to)i(zero)f(as)f(w)m(ell.)49 b(Since)33 b(it)g(will)d(b)s(e)j (the)h Fn(\033)i Fx(with)d(exceptionally)50 1313 y(large)d Fn(E)339 1327 y Fl(\033)419 1313 y Fx(that)h(con)m(tribute)f(to)h(the)f (long)g(time)g(b)s(eha)m(vior,)g(this)f(will)f(do)i(the)h(job.)50 1531 y Fd(Lemma)38 b(4.22:)51 b Ft(Ther)-5 b(e)33 b(exists)g(a)g (\014nite)g(p)-5 b(ositive)33 b(c)-5 b(onstant)35 b Fn(C)k Ft(such)33 b(that)10 1749 y(\(i\))47 b(If)32 b Fj(E)11 b Fn(\034)359 1716 y Fl(\033)349 1778 y(T)f Fk(n)p Fl(\033)519 1749 y Fn(>)625 1726 y Fi(b)615 1749 y Fx(\002)p Ft(,)32 b(then)759 2069 y Fj(P)829 1968 y Fi(h)873 2069 y Fn(\034)923 2032 y Fl(\033)913 2094 y(T)10 b Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)1191 2069 y Fn(>)25 b(m;)15 b(\033)s Fx(\()p Fn(m)p Fx(\))26 b Fm(62)f Fn(T)13 b Fx(\()p Fn(E)5 b Fx(\))1932 1968 y Fi(i)2001 2069 y Fm(\024)2144 2008 y Fn(e)2186 1975 y Fk(\000)p Fl(\014)2287 1920 y Fk(p)p 2347 1920 V 55 x Fl(N)h(E)2464 1984 y Fp(\033)2523 1985 y Fi(b)2513 2008 y Fx(\002)p 2107 2048 513 4 v 2107 2148 a(1)21 b Fm(\000)2274 2125 y Fi(b)2264 2148 y Fx(\002)p Fn(=)p Fj(E)11 b Fn(\034)2485 2116 y Fl(\033)2475 2179 y(T)f Fk(n)p Fl(\033)2630 2069 y Fn(e)2672 2027 y Fk(\000)p Fl(m=)p Fh(E)p Fl(\034)2909 1999 y Fp(\033)2902 2048 y(T)j Fe(n)p Fp(\033)3030 2027 y Fl(b)3062 2036 y Fp(\033)3432 2069 y Fx(\(4)p Fn(:)p Fx(159\))-18 2476 y Ft(\(ii\))47 b(If)32 b Fj(E)11 b Fn(\034)359 2443 y Fl(\033)349 2505 y(T)f Fk(n)p Fl(\033)519 2476 y Fm(\024)625 2453 y Fi(b)615 2476 y Fx(\002)p Ft(,)32 b(then)1084 2748 y Fj(P)1154 2647 y Fi(h)1198 2748 y Fn(\034)1248 2711 y Fl(\033)1238 2773 y(T)10 b Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)1516 2748 y Fn(>)25 b(m;)15 b(\033)s Fx(\()p Fn(m)p Fx(\))26 b Fm(62)f Fn(T)13 b Fx(\()p Fn(E)5 b Fx(\))2257 2647 y Fi(i)2326 2748 y Fm(\024)2432 2687 y Fn(m)p 2432 2727 80 4 v 2447 2804 a Fi(b)2437 2827 y Fx(\002)2523 2748 y Fn(e)2565 2711 y Fk(\000)p Fl(m=)2727 2707 y Fi(b)2724 2711 y Fv(\002)3432 2748 y Fx(\(4)p Fn(:)p Fx(160\))50 3223 y Fc(Pro)s(of:)52 b Fx(Note)33 b(that)g(if)e(the)i(ev)m(en)m(t)h Fm(f)p Fn(\034)1385 3190 y Fl(\033)1375 3252 y(T)10 b Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)1656 3223 y Fn(>)28 b(m;)15 b(\033)s Fx(\()p Fn(m)p Fx(\))30 b Fm(62)e Fn(T)13 b Fx(\()p Fn(E)5 b Fx(\))p Fm(g)33 b Fx(o)s(ccurs,)g(then)f(there)g (exists)g(a)h(last)50 3368 y(time)d Fn(m)20 b Fm(\000)g Fn(k)28 b(<)d(m)30 b Fx(when)g(the)g(pro)s(cess)g(visits)f(the)i Fn(\033)s Fx(.)40 b(This)29 b(giv)m(es)i(us)e(the)i(b)s(ound)681 3610 y Fj(P)751 3509 y Fi(h)794 3610 y Fn(\034)844 3573 y Fl(\033)834 3635 y(T)10 b Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)1112 3610 y Fn(>)25 b(m;)15 b(\033)s Fx(\()p Fn(m)p Fx(\))26 b Fm(62)f Fn(T)13 b Fx(\()p Fn(E)5 b Fx(\))1853 3509 y Fi(i)706 3869 y Fm(\024)802 3756 y Fl(m)p Fk(\000)p Fv(1)815 3783 y Fi(X)815 3979 y Fl(k)r Fv(=1)975 3869 y Fj(P)1045 3769 y Fi(h)1088 3869 y Fn(\034)1138 3832 y Fl(\033)1128 3894 y(T)10 b Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)1406 3869 y Fn(>)25 b(m)20 b Fm(\000)g Fn(k)s(;)1783 3769 y Fi(i)1861 3783 y(X)1842 3978 y Fl(\033)1886 3960 y Fe(0)1911 3978 y Fk(\030)p Fl(\033)2027 3869 y Fn(p)2073 3883 y Fl(N)2142 3869 y Fx(\()p Fn(\033)n(;)15 b(\033)2322 3832 y Fk(0)2348 3869 y Fx(\))p Fj(P)2453 3769 y Fi(h)2497 3869 y Fn(\034)2547 3832 y Fl(\033)2591 3804 y Fe(0)2537 3894 y Fl(T)10 b Fv(\()p Fl(E)s Fv(\))2733 3869 y Fn(>)25 b(k)f Fm(\000)19 b Fx(1)3035 3769 y Fi(i)706 4188 y Fm(\024)25 b Fx(\(1)c Fm(\000)f Fn(p)1040 4202 y Fl(N)1109 4188 y Fx(\()p Fn(\033)n(;)15 b(\033)s Fx(\)\))1374 4074 y Fl(m)p Fk(\000)p Fv(1)1389 4102 y Fi(X)1388 4297 y Fl(k)r Fv(=1)1548 4188 y Fn(e)1590 4146 y Fk(\000)p Fv(\()p Fl(m)p Fk(\000)p Fl(k)r Fv(\))p Fl(=)p Fh(E)p Fl(\034)1981 4118 y Fp(\033)1974 4167 y(T)e Fe(n)p Fp(\033)2101 4146 y Fk(\000)p Fv(\()p Fl(k)r Fk(\000)p Fv(1\))p Fl(=)2388 4142 y Fi(b)2385 4146 y Fv(\002\()p Fl(E)s Fv(\))3432 3929 y Fx(\(4)p Fn(:)p Fx(161\))50 4516 y(In)38 b(case)i(\(i\))f(w)m(e) g(can)g(extract)i Fn(e)1187 4474 y Fk(\000)p Fl(m=)p Fh(E)p Fl(\034)1424 4447 y Fp(\033)1417 4495 y(T)12 b Fe(n)p Fp(\033)1587 4516 y Fx(from)39 b(the)g(sum)f(and)g(o)m(v)m (ersum)i(the)f(remaining)e(geometric)50 4662 y(series)27 b(to)i(get)g(\(4.159\),)j(while)26 b(in)h(the)h(latter)h(case)g(w)m(e)f (simply)e(b)s(ound)g(the)j(exp)s(onen)m(tial)e(terms)h(b)m(y)g(their)50 4807 y(maxim)m(um)h(and)h(retain)g(that)h(there)f(are)h(only)f Fn(m)g Fx(terms)g(in)f(the)i(sum.)39 b(This)29 b(pro)m(v)m(es)i(the)g (lemma.)40 b Fm(})159 5025 y Fx(Next)34 b(w)m(e)f(w)m(an)m(t)h(to)g (deal)f(with)e(the)j(Laplace)f(transforms)f(app)s(earing)g(in)f (\(4.153\).)52 b(Concerning)32 b(the)50 5171 y(\014rst)21 b(line,)i(w)m(e)g(are)f(already)g(in)f(go)s(o)s(d)h(p)s(osition,)h (since)e(w)m(e)i(ha)m(v)m(e)h(the)e(estimates)h(needed)f(for)g Fn(G)3260 5138 y Fl(\033)3260 5200 y(T)10 b Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)3512 5171 y Fx(\()p Fn(u)p Fx(\))t Fm(\000)50 5316 y Fx(1)34 b(\(see)h(Prop)s(osition)d(3.6\).)53 b(The)34 b(second)g(term)g(has,)h(as)f(w)m(e)g(ha)m(v)m(e)i(seen,)f(a)f (prefactor)h(that)f(is)f(of)h(lo)m(w)m(er)50 5462 y(order)f(in)g(the)h Fn(m)f Fx(b)s(eha)m(vior,)h(but)g(w)m(e)g(ha)m(v)m(e)h(to)f(sho)m(w)g (that)h(the)f Fn(u)p Fx(-dep)s(enden)m(t)f(co)s(e\016cien)m(t)i(is)d (not)j(more)p eop %%Page: 52 52 52 51 bop 50 295 a Fr(52)3270 b Fo(Se)l(ction)27 b Fr(4)50 513 y Fx(singular)h(than)i(that)h(of)g(the)f(\014rst)g(term.)41 b(T)-8 b(o)30 b(this)g(end)f(w)m(e)i(rewrite)385 715 y Fn(G)457 677 y Fl(\033)501 650 y Fe(0)457 740 y Fl(T)10 b Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)n(;T)g Fv(\()p Fl(E)s Fv(\))893 715 y Fx(\()p Fn(u)p Fx(\))21 b(+)f Fn(G)1199 677 y Fl(\033)1243 650 y Fe(0)1199 740 y Fl(\033)n(;T)10 b Fv(\()p Fl(E)s Fv(\))1431 715 y Fx(\()p Fn(u)p Fx(\))p Fn(G)1625 677 y Fl(\033)1625 740 y(T)g Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)1878 715 y Fx(\()p Fn(u)p Fx(\))21 b Fm(\000)f Fx(1)410 897 y(=)25 b Fn(G)578 859 y Fl(\033)622 832 y Fe(0)578 921 y Fl(T)10 b Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)n(;T)g Fv(\()p Fl(E)s Fv(\))1014 897 y Fx(\()p Fn(u)p Fx(\))21 b(+)f Fn(G)1320 859 y Fl(\033)1364 832 y Fe(0)1320 921 y Fl(\033)n(;T)10 b Fv(\()p Fl(E)s Fv(\))1552 897 y Fx(\()p Fn(u)p Fx(\))1689 796 y Fi(\020)1745 897 y Fn(G)1817 859 y Fl(\033)1817 921 y(T)g Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)2069 897 y Fx(\()p Fn(u)p Fx(\))21 b Fm(\000)f Fx(1)2348 796 y Fi(\021)2422 897 y Fx(+)g Fn(G)2585 859 y Fl(\033)2629 832 y Fe(0)2585 921 y Fl(\033)n(;T)10 b Fv(\()p Fl(E)s Fv(\))2818 897 y Fx(\()p Fn(u)p Fx(\))21 b Fm(\000)f Fx(1)410 1097 y(=)25 b Fn(G)578 1059 y Fl(\033)622 1032 y Fe(0)578 1121 y Fl(T)10 b Fv(\()p Fl(E)s Fv(\))748 1097 y Fx(\()p Fn(u)p Fx(\))21 b Fm(\000)f Fx(1)h(+)f Fn(G)1211 1059 y Fl(\033)1255 1032 y Fe(0)1211 1121 y Fl(\033)n(;T)10 b Fv(\()p Fl(E)s Fv(\))1443 1097 y Fx(\()p Fn(u)p Fx(\))1580 996 y Fi(\020)1635 1097 y Fn(G)1707 1059 y Fl(\033)1707 1121 y(T)g Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)1959 1097 y Fx(\()p Fn(u)p Fx(\))21 b Fm(\000)f Fx(1)2238 996 y Fi(\021)3432 913 y Fx(\(4)p Fn(:)p Fx(162\))50 1339 y(It)30 b(will)e(su\016ce)i(to)h (use)f(that,)i(for)e Fm(<)p Fn(u)25 b(<)1473 1303 y Fv(1)p 1473 1318 37 4 v 1473 1370 a(2)1530 1316 y Fi(b)1520 1339 y Fx(\002)1591 1306 y Fk(\000)p Fv(1)1688 1339 y Fx(,)1446 1570 y Fm(j)p Fn(G)1543 1532 y Fl(\033)1587 1505 y Fe(0)1543 1594 y Fl(T)10 b Fv(\()p Fl(E)s Fv(\))1714 1570 y Fx(\()p Fn(u)p Fx(\))21 b Fm(\000)f Fx(1)p Fm(j)26 b(\024)f(j)p Fn(u)p Fm(j)2252 1547 y Fi(b)2242 1570 y Fx(\002)1119 b(\(4)p Fn(:)p Fx(163\))50 1800 y(and)30 b(that)h Fn(G)496 1767 y Fl(\033)540 1740 y Fe(0)496 1830 y Fl(\033)n(;T)10 b Fv(\()p Fl(E)s Fv(\))728 1800 y Fx(\()p Fn(u)p Fx(\))31 b(is)f(b)s(ounded)e(and)i(analytic.)50 1995 y Fu(4.6)35 b(Laplace)g(in)m(v)m(ersion)h(1.)47 b(The)34 b(error)i(terms.)159 2263 y Fx(After)g(this)e(preparation)h(w) m(e)h(are)g(no)m(w)f(ready)h(to)g(attac)m(k)i(the)e(Laplace)f(in)m(v)m (ersion)g(of)g(the)h(function)50 2408 y(\005)118 2375 y Fk(\003)160 2408 y Fx(\()p Fn(u;)15 b(m;)g(E)5 b Fx(\))32 b(giv)m(en)e(in)f(principle)e(b)m(y)k(\(4.5\).)42 b(Recall)30 b(that)h(w)m(e)g(are)g(in)m(terested)f(in)f(computing)726 2645 y(\005\()p Fn(n;)15 b(m;)g(E)5 b Fx(\))26 b Fm(\021)1391 2584 y Fx(1)p 1284 2624 260 4 v 1284 2708 a Fm(j)p Fn(T)13 b Fx(\()p Fn(E)5 b Fx(\))p Fm(j)1634 2559 y Fi(X)1569 2758 y Fl(\033)r Fk(2)p Fl(T)10 b Fv(\()p Fl(E)s Fv(\))1844 2645 y Fx(\005)1912 2659 y Fl(\033)1962 2645 y Fx(\()p Fn(n;)15 b(m;)g(E)5 b Fx(\))26 b Fm(\021)f Fx(\(1)-23 b(I)q Fn(;)15 b Fx(\005)p 2572 2660 69 4 v(\()p Fn(n;)g(m;)g(E)5 b Fx(\)\))400 b(\(4)p Fn(:)p Fx(164\))50 2940 y(Setting)1075 3085 y(\005)1143 3048 y Fv(0)1184 3085 y Fx(\()p Fn(n;)15 b(m;)g(E)5 b Fx(\))27 b Fm(\021)e Fx(\005\()p Fn(n;)15 b(m;)g(E)5 b Fx(\))21 b Fm(\000)f Fx(\(1)-23 b(I)q Fn(;)15 b(F)p 2332 3100 72 4 v 14 w Fx(\()p Fn(n;)g(m)p Fx(\)\))748 b(\(4)p Fn(:)p Fx(165\))50 3284 y(\(4.5\))32 b(and)e(the)g(in)m(v)m (ersion)f(form)m(ula)h(for)g(Laplace)h(transforms)e(w)m(e)i(can)g (write)852 3544 y(\005)920 3507 y Fv(0)961 3544 y Fx(\()p Fn(n;)15 b(m;)g(E)5 b Fx(\))26 b(=)1494 3483 y(1)p 1451 3523 132 4 v 1451 3606 a(2)p Fn(\031)s(i)1609 3420 y Fi(Z)1700 3443 y Fl(i\031)1660 3627 y Fk(\000)p Fl(i\031)1807 3544 y Fn(due)1948 3507 y Fk(\000)p Fl(un)2113 3544 y Fx(\(1)-23 b(I)q Fn(;)15 b(M)2342 3507 y Fk(\003)2332 3567 y Fl(E)2394 3544 y Fx(\()p Fn(u)p Fx(\))p Fn(F)p 2516 3559 72 4 v 2588 3504 a Fk(\003)2629 3544 y Fx(\()p Fn(m;)g(u)p Fx(\)\))526 b(\(4)p Fn(:)p Fx(166\))50 3797 y(The)38 b(notation)g(of)g(Section)g(4.2)i(\(see)f(\(4.31\)-\(4.34\)\)) k(are)c(again)f(brough)m(t)g(in)m(to)h(force)f(in)f(the)i(presen)m(t)50 3943 y(section;)31 b(recall)f(in)f(particular)g(that)i Fn(z)f Fx(=)1523 3920 y Fi(b)1513 3943 y Fx(\002)p Fn(u)p Fx(.)41 b(The)30 b(\014rst)g(step)g(of)h(the)g(analysis)e(consists)h (in)f(deforming)50 4088 y(the)h(con)m(tour)h(of)g(in)m(tegration)f(to)h (the)g(con)m(tour)g Fm(C)36 b Fx(consisting)29 b(of)h(the)h(three)f (parts)965 4319 y Fm(A)25 b(\021)1159 4218 y Fi(n)1219 4319 y Fn(u)h Fm(2)f Fj(C)64 b Fx(:)40 b Fm(<)p Fn(z)29 b Fx(=)c(1)p Fn(=)p Fx(2)p Fn(;)32 b Fm(j=)p Fn(z)t Fm(j)26 b(2)f Fx([1)p Fn(=)2368 4239 y Fm(p)p 2445 4239 98 4 v 2445 4319 a Fx(2)p Fn(\024)q(;)15 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2778 559 4 v 1574 2796 a Fi(p)p 1665 2796 468 4 v 81 x Fx(1)20 b Fm(\000)g Fn(O)s Fx(\()1941 2854 y(\026)1928 2877 y(\002)1999 2851 y Fk(\000)p Fv(1)2097 2877 y Fx(\))2623 2737 y(1)p 2154 2778 983 4 v 2154 2796 a Fi(q)p 2245 2796 892 4 v 115 x Fx(1)g(+)2411 2889 y Fi(b)2401 2911 y Fx(\002)2472 2885 y Fv(2)2513 2911 y Fx(2\(1)h Fm(\000)f Fx(cos)q(\()p Fn(s=)3005 2889 y Fi(b)2995 2911 y Fx(\002\)\))3432 2799 y(\(4)p Fn(:)p Fx(175\))50 3187 y(T)-8 b(o)28 b(b)s(ound)459 3082 y Fi(\015)459 3137 y(\015)459 3192 y(\015)510 3187 y Fn(F)p 510 3202 72 4 v 581 3147 a Fk(\003)623 3187 y Fx(\()p Fn(m;)15 b Fx(\(1)p Fn(=)p Fx(2)22 b(+)e Fn(is)p Fx(\))p Fn(=)1225 3164 y Fi(b)1215 3187 y Fx(\002)q(\))1322 3082 y Fi(\015)1322 3137 y(\015)1322 3192 y(\015)1373 3251 y Fk(1)1478 3187 y Fx(w)m(e)28 b(use)g(Lemma)g(4.20)i(together)g(with)c(the)j(fact)g (that)f(on)h Fm(A)p Fx(,)f(b)m(y)50 3332 y(the)i(estimates)h(of)g(Prop) s(osition)d(3.2,)k Fm(j)p Fn(G)1468 3299 y Fl(\033)1468 3362 y(T)10 b Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)1720 3332 y Fx(\()p Fn(u)p Fx(\))p Fm(j)27 b(\024)e Fn(c)p Fx(,)30 b(to)i(get)f(that)928 3525 y Fi(\015)928 3580 y(\015)928 3634 y(\015)979 3630 y Fn(F)p 979 3645 V 1050 3590 a Fk(\003)1092 3630 y Fx(\()p Fn(m;)15 b Fx(\(1)p Fn(=)p Fx(2)22 b(+)e Fn(is)p Fx(\))p Fn(=)1694 3607 y Fi(b)1684 3630 y Fx(\002)q(\))1791 3525 y Fi(\015)1791 3580 y(\015)1791 3634 y(\015)1841 3694 y Fk(1)1944 3630 y Fm(\024)25 b Fn(c)2094 3525 y Fi(\014)2094 3580 y(\014)2094 3634 y(\014)2124 3630 y Fn(e)2166 3592 y Fv(\(1)p Fl(=)p Fv(2+)p Fl(is)p Fv(\))p Fl(=)2487 3588 y Fi(b)2484 3592 y Fv(\002)2567 3630 y Fm(\000)20 b Fx(1)2703 3525 y Fi(\014)2703 3580 y(\014)2703 3634 y(\014)2734 3545 y Fk(\000)p Fv(1)3432 3630 y Fx(\(4)p Fn(:)p Fx(176\))50 3916 y(Set)35 b Fn(\032)f Fx(=)g(exp\(1)p Fn(=)p Fx(\(2)751 3893 y Fi(b)741 3916 y Fx(\002)q(\)\))i(and)f Fn(v)i Fx(=)d Fn(s=)1385 3893 y Fi(b)1375 3916 y Fx(\002)o(.)56 b(Then)1769 3838 y Fi(\014)1769 3893 y(\014)1799 3916 y Fn(\032e)1888 3883 y Fl(iv)1978 3916 y Fm(\000)20 b Fx(1)2114 3838 y Fi(\014)2114 3893 y(\014)2144 3861 y Fv(2)2219 3916 y Fx(=)33 b(\(1)25 b Fm(\000)e Fn(\032)p Fx(\))2604 3883 y Fv(2)2669 3916 y Fx(+)g(2)p Fn(\032)p Fx(\(1)i Fm(\000)e Fx(cos)16 b Fn(v)s Fx(\),)38 b(and)c(since)50 4061 y Fn(\032)25 b(>)g Fx(1)c(+)f(1)p Fn(=)p Fx(\(2)555 4038 y Fi(b)545 4061 y Fx(\002)q(\),)776 4255 y Fi(\015)776 4310 y(\015)776 4365 y(\015)827 4360 y Fn(F)p 827 4375 V 898 4320 a Fk(\003)940 4360 y Fx(\()p Fn(m;)15 b Fx(\(1)p Fn(=)p Fx(2)22 b(+)e Fn(is)p Fx(\))p Fn(=)1542 4337 y Fi(b)1532 4360 y Fx(\002)q(\))1639 4255 y Fi(\015)1639 4310 y(\015)1639 4365 y(\015)1689 4424 y Fk(1)1792 4360 y Fm(\024)2380 4299 y Fn(c)2429 4276 y Fi(b)2419 4299 y Fx(\002)p 1899 4339 1074 4 v 1899 4357 a Fi(q)p 1989 4357 983 4 v 1989 4473 a Fx(1)p Fn(=)p Fx(4)i(+)2247 4450 y Fi(b)2237 4473 y Fx(\002)2308 4446 y Fv(2)2348 4473 y Fx(2\(1)g Fm(\000)e Fx(cos)q(\()p Fn(s=)2841 4450 y Fi(b)2831 4473 y Fx(\002\)\))3432 4360 y(\(4)p Fn(:)p Fx(177\))50 4712 y(Inserting)29 b(\(4.175\))k(and)d (\(4.177\))j(in)c(\(4.173\))k(w)m(e)e(get)179 5032 y Fn(I)219 5046 y Fk(A)307 5032 y Fm(\024)25 b Fx(2)p Fn(ce)529 4994 y Fk(\000)p Fl(n=)p Fv(\(2)735 4990 y Fi(b)732 4994 y Fv(\002)q(\))837 4908 y Fi(Z)928 4931 y Fl(\031)976 4927 y Fi(b)973 4931 y Fv(\002)888 5115 y(1)p Fl(=)961 5062 y Fk(p)p 1021 5062 79 4 v 53 x Fv(2)p Fl(\024)1120 5032 y Fn(ds)2241 4970 y Fx(1)p 1236 5011 2057 4 v 1236 5029 a Fi(q)p 1327 5029 892 4 v 115 x Fx(1)20 b(+)1493 5121 y Fi(b)1483 5144 y Fx(\002)1554 5118 y Fv(2)1595 5144 y Fx(2\(1)h Fm(\000)f Fx(cos)q(\()p Fn(s=)2087 5121 y Fi(b)2077 5144 y Fx(\002\)\))2219 5029 y Fi(q)p 2309 5029 983 4 v 2309 5144 a Fx(1)p Fn(=)p Fx(4)i(+)2567 5121 y Fi(b)2557 5144 y Fx(\002)2628 5118 y Fv(2)2668 5144 y Fx(2\(1)g Fm(\000)e Fx(cos\()p Fn(s=)3160 5121 y Fi(b)3150 5144 y Fx(\002)q(\)\))307 5397 y Fm(\024)25 b Fx(2)p Fn(ce)529 5359 y Fk(\000)p Fl(n=)p Fv(\(2)735 5355 y Fi(b)732 5359 y Fv(\002)q(\))837 5273 y Fi(Z)928 5296 y Fl(\031)976 5292 y Fi(b)973 5296 y Fv(\002)888 5480 y(1)p Fl(=)961 5427 y Fk(p)p 1021 5427 79 4 v 53 x Fv(2)p Fl(\024)1120 5397 y Fn(ds)1704 5335 y Fx(1)p 1236 5376 983 4 v 1236 5475 a(1)p Fn(=)p Fx(4)c(+)1493 5452 y Fi(b)1483 5475 y Fx(\002)1554 5449 y Fv(2)1595 5475 y Fx(2\(1)g Fm(\000)f Fx(cos)q(\()p Fn(s=)2087 5452 y Fi(b)2077 5475 y Fx(\002\)\))3432 5202 y(\(4)p Fn(:)p Fx(178\))p eop %%Page: 55 55 55 54 bop 50 295 a Fo(A)l(ging)27 b(in)h(the)g(REM.)f(Part)h(2.)2661 b Fr(55)50 513 y Fx(T)-8 b(o)30 b(ev)-5 b(aluate)31 b(the)f(last)g(in)m (tegral)g(ab)s(o)m(v)m(e,)h(w)m(e)g(split)d(the)i(in)m(tegration)g(in)m (terv)-5 b(al)29 b(in)m(to)h([1)p Fn(=)3061 437 y Fm(p)p 3138 437 98 4 v 3138 513 a Fx(2)p Fn(\024)q(;)g(\031)3356 490 y Fi(b)3346 513 y Fx(\002)p Fn(=)p Fx(4])h(and)50 658 y([)p Fn(\031)140 635 y Fi(b)130 658 y Fx(\002)p Fn(=)p Fx(4)p Fn(;)g(\031)412 635 y Fi(b)402 658 y Fx(\002].)41 b(On)28 b(the)i(\014rst)f(of)g(these)h(in)m(terv)-5 b(als,)1790 635 y Fi(b)1780 658 y Fx(\002)1851 625 y Fv(2)1892 658 y Fx(2\(1)19 b Fm(\000)f Fx(cos)q(\()p Fn(s=)2380 635 y Fi(b)2370 658 y Fx(\002\)\))30 b(is)f(w)m(ell)f(appro)m(ximated)h(b)m (y)h Fn(s)3559 625 y Fv(2)3628 658 y Fx(so)50 804 y(that)728 860 y Fi(Z)819 883 y Fl(\031)867 879 y Fi(b)864 883 y Fv(\002)p Fl(=)p Fv(4)779 1067 y(1)p Fl(=)852 1014 y Fk(p)p 912 1014 79 4 v 53 x Fv(2)p Fl(\024)1013 984 y Fn(ds)1598 922 y Fx(1)p 1129 963 983 4 v 1129 1063 a(1)p Fn(=)p Fx(4)22 b(+)1387 1040 y Fi(b)1377 1063 y Fx(\002)1448 1036 y Fv(2)1488 1063 y Fx(2\(1)g Fm(\000)d Fx(cos)q(\()p Fn(s=)1980 1040 y Fi(b)1970 1063 y Fx(\002\)\))2148 984 y Fm(\024)25 b Fn(c)2298 860 y Fi(Z)2390 883 y Fl(\031)2438 879 y Fi(b)2435 883 y Fv(\002)o Fl(=)p Fv(4)2349 1067 y(1)p Fl(=)2422 1014 y Fk(p)p 2482 1014 79 4 v 53 x Fv(2)p Fl(\024)2669 922 y Fn(ds)p 2594 963 240 4 v 2594 1046 a Fx(1)c(+)f Fn(s)2794 1020 y Fv(2)2870 984 y Fm(\024)25 b Fn(c)3005 946 y Fk(0)3432 984 y Fx(\(4)p Fn(:)p Fx(179\))50 1241 y(W)-8 b(e)31 b(then)f(use)h(that)f(on)h(the)f(remaining)f(in)m (terv)-5 b(al)1813 1218 y Fi(b)1803 1241 y Fx(\002)1874 1208 y Fv(2)1915 1241 y Fx(2\(1)21 b Fm(\000)f Fx(cos)q(\()p Fn(s=)2407 1218 y Fi(b)2397 1241 y Fx(\002\)\))26 b Fn(>)2670 1218 y Fi(b)2660 1241 y Fx(\002)2731 1208 y Fv(2)2802 1241 y Fx(so)k(that)1090 1418 y Fi(Z)1181 1440 y Fl(\031)1229 1436 y Fi(b)1226 1440 y Fv(\002)1140 1624 y Fk(\032)1200 1620 y Fi(b)1197 1624 y Fv(\002)p Fl(=)p Fv(4)1346 1541 y Fn(ds)1931 1480 y Fx(1)p 1462 1520 983 4 v 1462 1620 a(1)p Fn(=)p Fx(4)22 b(+)1720 1597 y Fi(b)1710 1620 y Fx(\002)1781 1594 y Fv(2)1821 1620 y Fx(2\(1)g Fm(\000)e Fx(cos\()p Fn(s=)2313 1597 y Fi(b)2303 1620 y Fx(\002\)\))2481 1541 y Fm(\024)2604 1480 y Fn(c)p 2588 1520 71 4 v 2598 1597 a Fi(b)2588 1620 y Fx(\002)3432 1541 y(\(4)p Fn(:)p Fx(180\))50 1817 y(Inserting)29 b(\(4.179\))k(and)d(\(4.180\))j(in)c (\(4.178\))k(yields)c(the)h(claim)g(of)g(the)h(lemma.)40 b Fm(})50 2035 y Fd(Lemma)55 b(4.24:)85 b Ft(L)-5 b(et)46 b Fm(B)h Ft(b)-5 b(e)45 b(de\014ne)-5 b(d)46 b(in)f(\(4.168\).)81 b(If)45 b Fn(t)j Fx(=)g Fn(n=)2450 2012 y Fi(b)2440 2035 y Fx(\002)o(\()p Fn(E)5 b Fx(\))46 b Ft(and)2889 2018 y Fx(~)2888 2035 y Fn(t)h Fx(=)h Fn(t)3120 2002 y Fl(\021)3209 2035 y Ft(then,)h(for)c(al)5 b(l)50 2181 y Fx(0)26 b Fn(<)e(\021)29 b(<)c Fx(1)p Ft(,)888 2207 y Fi(\014)888 2262 y(\014)888 2316 y(\014)888 2371 y(\014)918 2215 y(Z)969 2422 y Fk(B)1038 2339 y Fn(due)1179 2302 y Fk(\000)p Fl(un)1344 2339 y Fx(\(1)-23 b(I)p Fn(;)15 b(M)1572 2302 y Fk(\003)1562 2362 y Fl(E)1624 2339 y Fx(\()p Fn(u)p Fx(\))p Fn(F)p 1746 2354 72 4 v 1818 2299 a Fk(\003)1860 2339 y Fx(\()p Fn(m;)g(u)p Fx(\)\))2138 2207 y Fi(\014)2138 2262 y(\014)2138 2316 y(\014)2138 2371 y(\014)2194 2339 y Fm(\024)25 b Fn(ct)2362 2302 y Fl(\021)2421 2339 y Fx(exp\()p Fm(\000)p Fn(t)2699 2302 y Fv(1)p Fk(\000)p Fl(\021)2837 2339 y Fx(\))560 b(\(4)p Fn(:)p Fx(181\))50 2761 y Fc(Pro)s(of:)68 b Fx(It)39 b(will)e(b)s(e)i(enough)g(to)h(use)f (norm)f(estimates,)43 b(that)d(is,)g(calling)e Fn(I)2782 2775 y Fk(B)2876 2761 y Fx(the)h(left)g(hand)f(side)h(of)50 2906 y(\(4.181\),)302 3159 y Fn(I)342 3173 y Fk(B)421 3159 y Fm(\024)517 3035 y Fi(Z)568 3242 y Fk(B)637 3159 y Fm(j)p Fn(du)p Fm(j)p Fn(e)828 3121 y Fk(\000)p Fl(n)p Fk(<)p Fl(u)1047 3159 y Fm(k)p Fn(M)1190 3121 y Fk(\003)1180 3181 y Fl(E)1242 3159 y Fx(\()p Fn(u)p Fx(\))p Fm(k)16 b(k)q Fn(F)p 1471 3174 V 1542 3119 a Fk(\003)1583 3159 y Fx(\()p Fn(m;)f(u)p Fx(\))p Fm(k)1872 3186 y Fk(1)421 3402 y Fx(=)528 3379 y Fi(b)517 3402 y Fx(\002)588 3364 y Fk(\000)p Fv(1)701 3278 y Fi(Z)754 3480 y(b)751 3484 y Fv(\002)807 3485 y Fk(B)876 3402 y Fm(j)p Fn(dz)t Fm(j)p Fn(e)1061 3364 y Fk(\000)p Fl(t)p Fk(<)p Fl(z)1257 3297 y Fi(\015)1257 3352 y(\015)1257 3406 y(\015)1308 3402 y Fn(M)1406 3364 y Fk(\003)1396 3424 y Fl(E)1457 3402 y Fx(\()p Fn(z)t(=)1593 3379 y Fi(b)1583 3402 y Fx(\002)q(\))1690 3297 y Fi(\015)1690 3352 y(\015)1690 3406 y(\015)1756 3297 y(\015)1756 3352 y(\015)1756 3406 y(\015)1806 3402 y Fn(F)p 1806 3417 V 1878 3362 a Fk(\003)1919 3402 y Fx(\()p Fn(m;)g(z)t(=)2175 3379 y Fi(b)2165 3402 y Fx(\002)q(\))2272 3297 y Fi(\015)2272 3352 y(\015)2272 3406 y(\015)2323 3466 y Fk(1)421 3650 y Fm(\024)25 b Fx(2)p Fn(c)611 3627 y Fi(b)601 3650 y Fx(\002)672 3612 y Fk(\000)p Fv(1)786 3526 y Fi(Z)877 3549 y Fl(s)911 3558 y Fp(A)836 3733 y Fl(s)870 3742 y Fp(B)982 3650 y Fn(ds)15 b(e)1129 3612 y Fk(\000)p Fl(\024s)1263 3585 y Fq(2)1299 3612 y Fl(t)1346 3545 y Fi(\015)1346 3600 y(\015)1346 3654 y(\015)1396 3650 y Fn(M)1494 3612 y Fk(\003)1484 3672 y Fl(E)1546 3650 y Fx(\(\()p Fn(\024s)1711 3612 y Fv(2)1773 3650 y Fx(+)20 b Fn(is)p Fx(\))p Fn(=)2028 3627 y Fi(b)2018 3650 y Fx(\002\))2124 3545 y Fi(\015)2124 3600 y(\015)2124 3654 y(\015)2190 3545 y(\015)2190 3600 y(\015)2190 3654 y(\015)2241 3650 y Fn(F)p 2241 3665 V 2312 3610 a Fk(\003)2354 3650 y Fx(\()p Fn(m;)15 b Fx(\()p Fn(\024s)2639 3612 y Fv(2)2700 3650 y Fx(+)20 b Fn(is)p Fx(\))p Fn(=)2955 3627 y Fi(b)2945 3650 y Fx(\002)q(\))3052 3545 y Fi(\015)3052 3600 y(\015)3052 3654 y(\015)3103 3713 y Fk(1)3432 3411 y Fx(\(4)p Fn(:)p Fx(182\))50 3916 y(As)30 b(in)f(the)i(pro)s(of)e(of)i (the)g(previous)d(lemma)i(w)m(e)h(use)f(\(4.48\))j(to)e(write)f(the)g (b)s(ound)286 4045 y Fi(\015)286 4099 y(\015)286 4154 y(\015)336 4149 y Fn(M)434 4112 y Fk(\003)424 4172 y Fl(E)486 4149 y Fx(\(\()p Fn(\024s)651 4112 y Fv(2)713 4149 y Fx(+)20 b Fn(is)p Fx(\))p Fn(=)968 4126 y Fi(b)958 4149 y Fx(\002\))1064 4045 y Fi(\015)1064 4099 y(\015)1064 4154 y(\015)200 4380 y Fm(\024)1658 4319 y Fx(1)h(+)f Fn(C)7 b(\015)1939 4286 y Fk(\000)p Fv(1)2036 4319 y Fn(\024s)2131 4286 y Fv(2)p 282 4359 3268 4 v 282 4378 a Fi(q)p 372 4378 1431 4 v 372 4493 a Fx(1)21 b(+)539 4470 y Fi(b)529 4493 y Fx(\002)600 4467 y Fv(2)641 4493 y Fx(2\(1)g Fm(\000)f Fx(cos)q(\()p Fn(s=)1133 4470 y Fi(b)1123 4493 y Fx(\002\)\)\(1)h Fm(\000)f Fn(O)s Fx(\()1576 4470 y(\026)1563 4493 y(\002)1634 4467 y Fk(\000)p Fv(1)1732 4493 y Fx(\)\))h Fm(\000)f Fx(1)g Fm(\000)2148 4457 y Fv(4)p 2081 4472 170 4 v 2081 4524 a Fl(M)7 b Fk(\000)p Fv(1)2261 4493 y Fx(\(1)21 b(+)f Fn(O)s Fx(\()p Fn(d=)-5 b(N)10 b Fx(\)\))22 b Fm(\000)e Fx(\()p Fn(C)27 b Fx(+)20 b Fn(C)3203 4467 y Fk(0)3228 4493 y Fx(\))p Fn(\015)3315 4467 y Fk(\000)p Fv(1)3413 4493 y Fn(\024s)3508 4467 y Fv(2)3432 4615 y Fx(\(4)p Fn(:)p Fx(183\))50 4760 y(Using)28 b(this)h(time)g(that)h(on)f(the)h(in)m(tegration)f(in)m(terv)-5 b(al,)1988 4737 y Fi(b)1978 4760 y Fx(\002)2049 4727 y Fv(2)2090 4760 y Fx(2\(1)19 b Fm(\000)f Fx(cos)q(\()p Fn(s=)2578 4737 y Fi(b)2568 4760 y Fx(\002\)\))26 b Fm(\025)f Fn(s)2874 4727 y Fv(2)2914 4760 y Fx(\(1)19 b Fm(\000)f Fx(1)p Fn(=)p Fx(\(6)p Fn(\024)3334 4737 y Fi(b)3324 4760 y Fx(\002)r(\))3432 4727 y Fv(2)3473 4760 y Fx(\),)30 b(and)50 4906 y(that)h(for)f(0)c Fn(<)g(x)f(<)h Fx(1,)829 4834 y Fm(p)p 905 4834 209 4 v 72 x Fx(1)20 b(+)g Fn(x)26 b Fm(\025)f Fx(1)c(+)f Fn(x=)p Fx(2,)32 b(w)m(e)f(get)h(\(for)e Fn(\024)h Fx(small)e(enough,)i Fn(t)f Fx(small)f(enough)i(compared)50 5051 y(with)e Fn(M)10 b Fx(,)32 b(and)e Fn(M)5 b(;)15 b(N)41 b Fx(large\))32 b(that)f(the)g(denominator)f(in)g(the)h(r.h.s.) 42 b(of)31 b(\(4.183\))j(is)29 b(greater)k(than)d Fn(s)3553 5018 y Fv(2)3594 5051 y Fn(=)p Fx(4.)50 5197 y(Since)f(the)i(n)m (umerator)f(is)g(b)s(ounded)e(ab)s(o)m(v)m(e)j(b)m(y)g(a)f(constan)m (t,)i(w)m(e)f(ma)m(y)g(write)1315 5335 y Fi(\015)1315 5389 y(\015)1315 5444 y(\015)1365 5439 y Fn(M)1463 5402 y Fk(\003)1453 5462 y Fl(E)1515 5439 y Fx(\(\()p Fn(\024s)1680 5402 y Fv(2)1742 5439 y Fx(+)20 b Fn(is)p Fx(\))p Fn(=)1997 5416 y Fi(b)1987 5439 y Fx(\002\))2093 5335 y Fi(\015)2093 5389 y(\015)2093 5444 y(\015)2169 5439 y Fm(\024)25 b Fn(cs)2347 5402 y Fk(\000)p Fv(2)3432 5439 y Fx(\(4)p Fn(:)p Fx(184\))p eop %%Page: 56 56 56 55 bop 50 295 a Fr(56)3270 b Fo(Se)l(ction)27 b Fr(4)50 526 y Fx(T)-8 b(urning)32 b(to)k(the)f(term)899 422 y Fi(\015)899 476 y(\015)899 531 y(\015)949 526 y Fn(F)p 949 541 72 4 v 1021 487 a Fk(\003)1062 526 y Fx(\()p Fn(m;)15 b Fx(\()p Fn(\024s)1347 493 y Fv(2)1409 526 y Fx(+)20 b Fn(is)p Fx(\))p Fn(=)1664 503 y Fi(b)1654 526 y Fx(\002)q(\))1761 422 y Fi(\015)1761 476 y(\015)1761 531 y(\015)1811 590 y Fk(1)1923 526 y Fx(observ)m(e)36 b(that,)g(pro)s(ceeding)e(as)h(w)m(e)h(did)d(to)i(deriv)m(e)50 672 y(\(4.176\))e(w)m(e)e(obtain,)927 835 y Fi(\015)927 889 y(\015)927 944 y(\015)978 939 y Fn(F)p 978 954 V 1049 899 a Fk(\003)1091 939 y Fx(\()p Fn(m;)15 b Fx(\()p Fn(\024s)1376 902 y Fv(2)1438 939 y Fx(+)20 b Fn(is)p Fx(\))p Fn(=)1693 916 y Fi(b)1683 939 y Fx(\002\))1789 835 y Fi(\015)1789 889 y(\015)1789 944 y(\015)1840 1003 y Fk(1)1942 939 y Fm(\024)25 b Fn(c)2092 835 y Fi(\014)2092 889 y(\014)2092 944 y(\014)2123 939 y Fn(e)2165 902 y Fv(\()p Fl(\024s)2270 874 y Fq(2)2306 902 y Fv(+)p Fl(is)p Fv(\))p Fl(=)2490 898 y Fi(b)2487 902 y Fv(\002)2568 939 y Fm(\000)20 b Fx(1)2704 835 y Fi(\014)2704 889 y(\014)2704 944 y(\014)2735 854 y Fk(\000)p Fv(1)3432 939 y Fx(\(4)p Fn(:)p Fx(185\))50 1217 y(No)m(w,)41 b(with)d Fn(\032)g Fx(=)h(exp\(\()p Fn(\024s)1007 1184 y Fv(2)1048 1217 y Fx(\))p Fn(=)1138 1194 y Fi(b)1128 1217 y Fx(\002)q(\))g Fn(>)g Fx(1)g(and)f Fn(v)k Fx(=)c Fn(s=)1946 1194 y Fi(b)1936 1217 y Fx(\002,)2073 1140 y Fi(\014)2073 1194 y(\014)2103 1217 y Fn(\032e)2192 1184 y Fl(iv)2282 1217 y Fm(\000)20 b Fx(1)2418 1140 y Fi(\014)2418 1194 y(\014)2448 1162 y Fv(2)2528 1217 y Fx(=)38 b(\(1)27 b Fm(\000)e Fn(\032)p Fx(\))2922 1184 y Fv(2)2989 1217 y Fx(+)g(2)p Fn(\032)p Fx(\(1)i Fm(\000)f Fx(cos)16 b Fn(v)s Fx(\))39 b Fm(\025)50 1363 y Fx(2\(1)12 b Fm(\000)g Fx(cos)17 b Fn(v)s Fx(\).)40 b(Com)m(bining)24 b(this)h(with)g(the)i(b)s(ound)d(established)h(on)h (the)g(line)f(follo)m(wing)g(\(4.183\),)30 b(\(4.185\))50 1508 y(b)s(ecomes)1199 1549 y Fi(\015)1199 1604 y(\015)1199 1658 y(\015)1250 1654 y Fn(F)p 1250 1669 V 1321 1614 a Fk(\003)1362 1654 y Fx(\()p Fn(m;)15 b Fx(\(1)p Fn(=)p Fx(2)23 b(+)d Fn(is)p Fx(\))p Fn(=)1965 1631 y Fi(b)1955 1654 y Fx(\002\))2061 1549 y Fi(\015)2061 1604 y(\015)2061 1658 y(\015)2112 1717 y Fk(1)2214 1654 y Fm(\024)25 b Fn(cs)2392 1616 y Fk(\000)p Fv(1)2500 1631 y Fi(b)2490 1654 y Fx(\002)871 b(\(4)p Fn(:)p Fx(186\))50 1863 y(Collecting)29 b(\(4.182\),)k(\(4.184\))h(and)29 b(\(4.186\),)34 b(w)m(e)c(arriv)m(e)h (at)169 2165 y Fn(I)209 2179 y Fk(B)289 2165 y Fm(\024)25 b Fn(c)439 2041 y Fi(Z)530 2064 y Fl(s)564 2073 y Fp(A)490 2248 y Fl(s)524 2257 y Fp(B)636 2165 y Fn(ds)15 b(e)783 2128 y Fk(\000)p Fl(\024s)917 2100 y Fq(2)952 2128 y Fl(t)984 2165 y Fn(s)1027 2128 y Fk(\000)p Fv(3)1150 2165 y Fm(\024)25 b Fn(c)1300 2041 y Fi(Z)1391 2064 y Fv(1)p Fl(=)1464 2011 y Fk(p)p 1524 2011 37 4 v 53 x Fv(2)1351 2248 y(1)p Fl(=)1424 2188 y Fk(p)p 1484 2188 28 4 v 1485 2236 a Fv(~)1484 2248 y Fl(t)1580 2165 y Fn(ds)15 b(e)1727 2128 y Fk(\000)p Fl(s)1818 2100 y Fq(2)1854 2128 y Fl(t)1886 2165 y Fn(s)1929 2128 y Fk(\000)p Fv(3)2052 2165 y Fm(\024)25 b Fn(ce)2229 2128 y Fk(\000)p Fl(t=)2351 2116 y Fv(~)2350 2128 y Fl(t)2398 2041 y Fi(Z)2489 2064 y Fv(1)p Fl(=)2562 2011 y Fk(p)p 2622 2011 37 4 v 53 x Fv(2)2448 2248 y(1)p Fl(=)2521 2188 y Fk(p)p 2582 2188 28 4 v 2583 2236 a Fv(~)2582 2248 y Fl(t)2678 2165 y Fn(ds)15 b(s)2826 2128 y Fk(\000)p Fv(3)2949 2165 y Fm(\024)25 b Fn(c)3085 2148 y Fx(~)3084 2165 y Fn(te)3159 2128 y Fk(\000)p Fl(t=)3281 2116 y Fv(~)3280 2128 y Fl(t)3432 2165 y Fx(\(4)p Fn(:)p Fx(187\))50 2460 y(Th)m(us,)k(c)m(ho)s(osing)676 2443 y(~)675 2460 y Fn(t)c Fx(=)g Fn(t)862 2427 y Fl(\021)906 2460 y Fx(,)30 b(0)c Fn(<)f(\021)k(<)c Fx(1,)31 b(concludes)e(the)i (pro)s(of)e(of)i(the)f(lemma.)41 b Fm(})159 2678 y Fx(W)-8 b(e)33 b(no)m(w)g(consider)e(the)i(error)f(term)g(resulting)f(from)g (the)i Fn(M)2318 2645 y Fk(\003)p Fv(\(1\))2453 2678 y Fx(\()p Fn(u)p Fx(\))g(part)f(of)h(the)f(resolv)m(en)m(t)h(on)g(the) 50 2823 y(part)d Fm(D)j Fx(of)e(the)f(in)m(tegration)g(con)m(tour.)50 3042 y Fd(Lemma)38 b(4.25:)51 b Ft(If)32 b Fn(t)25 b Fx(=)g Fn(n=)1101 3019 y Fi(b)1091 3042 y Fx(\002\()p Fn(E)5 b Fx(\))34 b Ft(then,)f(for)g(al)5 b(l)33 b Fx(0)26 b Fn(<)f(\016)k(<)c Fx(1)p Fn(=)p Fx(2)p Ft(,)637 3315 y Fx(lim)15 b(sup)664 3392 y Fl(E)s Fk(#\0001)930 3315 y Fx(lim)g(sup)982 3392 y Fl(N)7 b Fk("1)1224 3183 y Fi(\014)1224 3237 y(\014)1224 3292 y(\014)1224 3347 y(\014)1254 3191 y(Z)1304 3398 y Fk(D)1383 3315 y Fn(due)1524 3277 y Fk(\000)p Fl(nu)1689 3214 y Fi(\020)1743 3315 y Fx(1)-23 b(I)p Fn(;)15 b(M)1936 3277 y Fk(\003)p Fv(\(1\))2071 3315 y Fx(\()p Fn(u)p Fx(\))p Fn(F)p 2193 3330 72 4 v 2265 3275 a Fk(\003)2307 3315 y Fx(\()p Fn(m;)g(u)p Fx(\))2549 3214 y Fi(\021)2604 3183 y(\014)2604 3237 y(\014)2604 3292 y(\014)2604 3347 y(\014)647 3550 y Fm(\024)25 b Fn(ct)815 3513 y Fk(\000)p Fv(2\(1)p Fk(\000)p Fv(1)p Fl(=\013)p Fv(\))1203 3550 y Fx(+)1497 3489 y Fn(c)p 1305 3529 423 4 v 1305 3613 a Fx(2\(1)d Fm(\000)d Fx(1)p Fn(=\013)p Fx(\))1738 3550 y Fn(t)1771 3513 y Fk(\000)p Fv(2)p Fl(\016)r Fv(\(1)p Fk(\000)p Fv(1)p Fl(=\013)p Fv(\))2195 3550 y Fx(+)h Fn(ct)2358 3513 y Fk(\000)p Fv(\(1)p Fk(\000)p Fv(2)p Fl(\016)r Fv(\))2656 3550 y Fx(exp\()p Fm(\000)p Fn(t)2934 3513 y Fv(1)p Fk(\000)p Fv(2)p Fl(\016)3103 3550 y Fx(\))3432 3432 y(\(4)p Fn(:)p Fx(188\))50 3972 y Fc(Pro)s(of:)47 b Fx(Again,)31 b(it)f(will)d(b)s(e)j (enough)g(to)h(use)f(norm)g(estimates,)h(that)g(is)228 4108 y Fi(\014)228 4163 y(\014)228 4217 y(\014)228 4272 y(\014)258 4116 y(Z)309 4323 y Fk(D)387 4240 y Fn(due)528 4203 y Fk(\000)p Fl(nu)693 4139 y Fi(\020)747 4240 y Fx(1)-23 b(I)q Fn(;)15 b(M)941 4203 y Fk(\003)p Fv(\(1\))1076 4240 y Fx(\()p Fn(u)p Fx(\))p Fn(F)p 1198 4255 72 4 v 1270 4200 a Fk(\003)1312 4240 y Fx(\()p Fn(m;)g(u)p Fx(\))1554 4139 y Fi(\021)1609 4108 y(\014)1609 4163 y(\014)1609 4217 y(\014)1609 4272 y(\014)1665 4240 y Fm(\024)1761 4116 y Fi(Z)1811 4323 y Fk(D)1889 4240 y Fm(j)p Fn(du)p Fm(j)p Fn(e)2080 4203 y Fk(\000)p Fl(n)p Fk(<)p Fl(u)2284 4240 y Fm(k)p Fn(M)2427 4203 y Fk(\003)p Fv(\(1\))2562 4240 y Fx(\()p Fn(u)p Fx(\))p Fm(kk)p Fn(F)p 2774 4255 V 2847 4200 a Fk(\003)2889 4240 y Fx(\()p Fn(m;)g(u)p Fx(\))p Fm(k)3176 4254 y Fk(1)3432 4240 y Fx(\(4)p Fn(:)p Fx(189\))50 4514 y(T)-8 b(o)39 b(b)s(ound)e Fm(k)p Fn(F)p 526 4529 V 598 4475 a Fk(\003)639 4514 y Fx(\()p Fn(m;)15 b(u)p Fx(\))p Fm(k)926 4528 y Fk(1)1044 4514 y Fx(w)m(e)39 b(pro)s(ceed)g(as)g(in)f(the)h(previous)e(t)m(w)m(o)k(lemmata)e(and)f (use)h(Lemma)g(4.20)50 4660 y(together)c(with)e(the)h(fact)h(that)g(on) e Fm(D)s Fx(,)i(b)m(y)f(the)g(estimates)h(of)f(Prop)s(osition)e(3.2,)k Fm(j)p Fn(G)2996 4627 y Fl(\033)2996 4689 y(T)10 b Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)3248 4660 y Fx(\()p Fn(u)p Fx(\))p Fm(j)33 b(\024)e Fn(c)p Fx(,)k(to)50 4805 y(establish)29 b(that)1160 4951 y Fm(k)p Fn(F)p 1205 4966 V 1277 4911 a Fk(\003)1318 4951 y Fx(\()p Fn(m;)15 b(u)p Fx(\))p Fm(k)1605 4965 y Fk(1)1709 4951 y Fm(\024)25 b Fn(c)p Fm(j)p Fn(e)1911 4913 y Fl(u)1980 4951 y Fm(\000)20 b Fx(1)p Fm(j)2141 4913 y Fk(\000)p Fv(1)2264 4951 y Fm(\024)25 b Fn(c)p Fm(j)p Fn(u)p Fm(j)2501 4913 y Fk(\000)p Fv(1)3432 4951 y Fx(\(4)p Fn(:)p Fx(190\))50 5160 y(Hence)256 5292 y Fi(Z)306 5498 y Fk(D)384 5415 y Fm(j)p Fn(du)p Fm(j)p Fn(e)575 5378 y Fk(\000)p Fl(n)p Fk(<)p Fl(u)779 5415 y Fm(k)p Fn(M)922 5378 y Fk(\003)p Fv(\(1\))1057 5415 y Fx(\()p Fn(u)p Fx(\))p Fm(kk)p Fn(F)p 1269 5430 V 1342 5376 a Fk(\003)1384 5415 y Fx(\()p Fn(m;)15 b(u)p Fx(\))p Fm(k)1671 5429 y Fk(1)1774 5415 y Fm(\024)25 b Fn(c)1924 5292 y Fi(Z)1978 5494 y(b)1975 5498 y Fv(\002)p Fk(D)2109 5415 y Fm(j)p Fn(dz)t Fm(j)p Fn(e)2294 5378 y Fk(\000)p Fl(t)p Fk(<)p Fl(z)2475 5415 y Fm(k)p Fn(M)2618 5378 y Fk(\003)p Fv(\(1\))2753 5415 y Fx(\()p Fn(z)t(=)2889 5392 y Fi(b)2879 5415 y Fx(\002)q(\))p Fm(kj)p Fn(z)t Fm(j)3127 5378 y Fk(\000)p Fv(1)3432 5415 y Fx(\(4)p Fn(:)p Fx(191\))p eop %%Page: 57 57 57 56 bop 50 295 a Fo(A)l(ging)27 b(in)h(the)g(REM.)f(Part)h(2.)2661 b Fr(57)50 513 y Fx(and)197 480 y Fv(15)304 513 y Fx(b)m(y)30 b(Corollary)f(4.19)936 768 y(lim)15 b(sup)963 845 y Fl(E)s Fk(#\0001)1229 768 y Fx(lim)g(sup)1281 845 y Fl(N)7 b Fk("1)1523 644 y Fi(Z)1576 846 y(b)1573 850 y Fv(\002)1629 851 y Fk(D)1707 768 y Fm(j)p Fn(dz)t Fm(j)p Fn(e)1892 730 y Fk(\000)p Fl(t)p Fk(<)p Fl(z)2073 768 y Fm(k)p Fn(M)2216 730 y Fk(\003)p Fv(\(1\))2351 768 y Fx(\()p Fn(z)t(=)2487 745 y Fi(b)2477 768 y Fx(\002)q(\))p Fm(kj)p Fn(z)t Fm(j)2725 730 y Fk(\000)p Fv(1)1143 1026 y Fm(\024)p Fn(c)1268 903 y Fi(Z)1322 1105 y(b)1319 1109 y Fv(\002)p Fk(D)1453 1026 y Fm(j)p Fn(dz)t Fm(j)p Fn(e)1638 989 y Fk(\000)p Fl(t)p Fk(<)p Fl(z)1818 1026 y Fm(j)p Fn(z)t Fm(j)1914 989 y Fv(1)p Fk(\000)p Fv(2)p Fl(=\013)2159 1026 y Fm(\021)25 b Fn(cI)2337 1061 y Fi(b)2334 1065 y Fv(\002)p Fk(D)3432 899 y Fx(\(4)p Fn(:)p Fx(192\))50 1291 y(W)-8 b(e)31 b(no)m(w)g(decomp)s(ose)f Fn(I)896 1325 y Fi(b)893 1329 y Fv(\002)949 1330 y Fk(D)1043 1291 y Fx(as)g Fn(I)1197 1325 y Fi(b)1194 1329 y Fv(\002)1250 1330 y Fk(D)1338 1291 y Fx(=)25 b Fn(I)1477 1325 y Fi(b)1474 1329 y Fv(\002)1530 1330 y Fk(D)1587 1339 y Fq(1)1647 1291 y Fx(+)20 b Fn(I)1781 1325 y Fi(b)1778 1329 y Fv(\002)1834 1330 y Fk(D)1891 1339 y Fq(2)1960 1291 y Fx(according)31 b(to)g(\(4.170\).)43 b(Clearly)1051 1595 y Fn(I)1094 1629 y Fi(b)1091 1633 y Fv(\002)p Fk(D)1204 1642 y Fq(1)1269 1595 y Fm(\024)1365 1471 y Fi(Z)1456 1493 y Fv(2)p Fl(\031)1415 1677 y Fv(0)1556 1595 y Fn(d\022)18 b(t)1697 1557 y Fk(\000)p Fv(\(1)p Fk(\000)p Fv(2)p Fl(=\013)p Fv(\))p Fk(\000)p Fv(1)2147 1595 y Fx(=)25 b(2)p Fn(\031)s(t)2376 1557 y Fk(\000)p Fv(\(2)p Fk(\000)p Fv(2)p Fl(=\013)p Fv(\))3432 1595 y Fx(\(4)p Fn(:)p Fx(193\))50 1854 y(T)-8 b(o)31 b(b)s(ound)d Fn(I)507 1889 y Fi(b)504 1893 y Fv(\002)p Fk(D)617 1902 y Fq(2)687 1854 y Fx(w)m(e)i(\014rst)g(observ)m(e)h(that) 102 2169 y Fn(I)145 2204 y Fi(b)142 2208 y Fv(\002)o Fk(D)254 2217 y Fq(2)319 2169 y Fm(\024)415 2046 y Fi(Z)506 2068 y Fl(s)540 2077 y Fp(B)466 2252 y Fl(s)500 2261 y Fp(D)614 2169 y Fn(ds)15 b(e)761 2132 y Fk(\000)p Fl(s)852 2104 y Fq(2)888 2132 y Fl(\024t)962 2087 y Fi(p)p 1053 2087 409 4 v 82 x Fx(\(2)p Fn(\024s)p Fx(\))1263 2143 y Fv(2)1325 2169 y Fx(+)20 b(1)1477 2068 y Fi(\020)1531 2087 y(p)p 1622 2087 442 4 v 82 x Fx(\()p Fn(\024s)1752 2143 y Fv(2)1793 2169 y Fx(\))1828 2143 y Fv(2)1890 2169 y Fx(+)g Fn(s)2024 2143 y Fv(2)2064 2068 y Fi(\021)2118 2087 y Fv(1)p Fk(\000)p Fv(2)p Fl(=\013)2362 2169 y Fm(\024)25 b Fn(c)2512 2046 y Fi(Z)2603 2022 y Fk(p)p 2663 2022 43 4 v 46 x Fl(\024s)2740 2077 y Fp(B)2563 2206 y Fk(p)p 2623 2206 V 46 x Fl(\024)o(s)2699 2261 y Fp(D)2813 2169 y Fn(ds)15 b(e)2960 2132 y Fk(\000)p Fl(s)3051 2104 y Fq(2)3087 2132 y Fl(t)3119 2169 y Fn(s)3162 2132 y Fv(1)p Fk(\000)p Fv(2)p Fl(=\013)3432 2169 y Fx(\(4)p Fn(:)p Fx(194\))50 2448 y(and)30 b(since)f(for)h Fn(t)h Fx(large,)f Fn(s)942 2462 y Fl(D)1033 2448 y Fm(\031)25 b Fx(1)p Fn(=t)p Fx(,)1294 2727 y Fn(I)1337 2761 y Fi(b)1334 2765 y Fv(\002)1389 2766 y Fk(D)1446 2775 y Fq(2)1511 2727 y Fm(\024)g Fn(c)1661 2603 y Fi(Z)1753 2626 y Fv(1)1712 2763 y Fk(p)p 1772 2763 V 47 x Fl(\024=t)1899 2727 y Fn(ds)15 b(e)2046 2689 y Fk(\000)p Fl(s)2137 2662 y Fq(2)2173 2689 y Fl(t)2205 2727 y 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Fv(\))p Fk(n)p Fl(\033)1382 4969 y Fx(\()p Fn(u)p Fx(\))21 b Fm(\000)f Fx(1)p 1058 5029 V 1236 5112 a Fn(e)1278 5086 y Fl(u)1347 5112 y Fm(\000)f Fx(1)1698 5050 y(=)1804 5027 y Fi(b)1793 5050 y Fx(\002\()p Fn(E)5 b Fx(\))2120 4988 y(1)p 2017 5029 249 4 v 2017 5112 a Fn(z)2059 5126 y Fl(\033)2130 5112 y Fm(\000)19 b Fn(z)2293 5050 y Fx(\(1)i(+)f Fn(R)q Fx(\()p Fn(u)p Fx(\)\))720 b(\(4)p Fn(:)p Fx(204\))50 5316 y(where)25 b(the)h(remainder)e Fn(R)q Fx(\()p Fn(u)p Fx(\))h(is)g(of)h(the)f(same)h(t)m(yp)s(e)g(as)g(those)g(app)s(earing)e (in)g(the)i(pro)s(of)f(of)g(Lemma)h(4.14.)50 5462 y(Th)m(us)j(w)m(e)i (obtain)p eop %%Page: 59 59 59 58 bop 50 295 a Fo(A)l(ging)27 b(in)h(the)g(REM.)f(Part)h(2.)2661 b Fr(59)50 513 y Fd(Lemma)38 b(4.27:)51 b Ft(With)33 b(the)g(notation)h(of)f(L)-5 b(emma)34 b(4.14,)651 662 y Fi(\014)651 717 y(\014)651 771 y(\014)651 826 y(\014)651 880 y(\014)681 821 y Fn(p)727 835 y Fl(N)796 821 y Fx(\()p Fn(\033)n(;)15 b(\033)s Fx(\))1011 784 y Fl(m)1092 741 y Fn(G)1164 708 y Fl(\033)1164 770 y(T)10 b Fv(\()p Fl(E)s Fv(\))p Fk(n)p Fl(\033)1416 741 y Fx(\()p Fn(u)p Fx(\))21 b Fm(\000)f Fx(1)p 1092 800 604 4 v 1271 884 a Fn(e)1313 857 y Fl(u)1381 884 y Fm(\000)g Fx(1)1727 821 y Fm(\000)g Fn(e)1860 784 y Fk(\000)p Fl(me)2017 756 y Fe(\000)p Fp(\014)2104 717 y Fe(p)p 2153 717 54 3 v 2153 756 a Fp(N)5 b(E)2253 765 y(\033)2343 737 y Fi(b)2333 760 y Fx(\002\()p Fn(E)g Fx(\))p 2316 800 249 4 v 2316 884 a Fn(z)2358 898 y Fl(\033)2427 884 y Fm(\000)20 b Fn(z)2575 662 y Fi(\014)2575 717 y(\014)2575 771 y(\014)2575 826 y(\014)2575 880 y(\014)2631 821 y Fm(\024)25 b Fn(C)2809 798 y Fi(b)2799 821 y Fx(\002)o(\()p Fn(E)5 b Fx(\))p Fm(j)p Fn(z)t Fm(j)325 b Fx(\(4)p Fn(:)p Fx(205\))50 1308 y Fc(Pro)s(of:)47 b Fx(Essen)m(tially)29 b(con)m(tained)i(in)e (the)h(pro)s(of)g(of)g(Lemma)h(4.14.)p Fm(})159 1526 y Fx(Next)g(w)m(e)g(can)g(no)m(w)f(pro)m(v)m(e)h(the)g(analogue)g(of)f (Corollary)f(4.17.)50 1744 y Fd(Lemma)38 b(4.28:)91 b Ft(Set)33 b Fn(s)25 b Fm(\021)g Fn(m=)1237 1721 y Fi(b)1227 1744 y Fx(\002)p Ft(.)41 b(Then,)33 b(uniformly)h(on)f Fm(<)p Fn(z)c(<)c Fx(max\()p Fm(=)p Fn(z)t(;)15 b Fx(1)p Fn(=)p Fx(2\))p Ft(,)35 b(and)f Fm(<)p Fx(\()p Fn(u)p Fx(\))25 b Fm(\024)g(j=)p Fn(u)p Fm(j)p Ft(,)767 2018 y Fx(lim)719 2078 y Fl(E)s Fk(#\0001)981 2018 y Fx(lim)957 2078 y Fl(N)7 b Fk("1)1264 1957 y Fx(1)p 1157 1998 260 4 v 1157 2081 a Fm(j)p Fn(T)13 b Fx(\()p Fn(E)5 b Fx(\))p Fm(j)1507 1932 y Fi(X)1443 2131 y Fl(\033)r Fk(2)p Fl(T)10 b Fv(\()p Fl(E)s Fv(\))1718 2018 y Fn(e)1760 1981 y Fk(\000)p Fl(me)1917 1953 y Fe(\000)p Fp(\014)2004 1914 y Fe(p)p 2053 1914 54 3 v 2053 1953 a Fp(N)5 b(E)2153 1962 y(\033)2318 1957 y Fx(1)p 2216 1998 249 4 v 2216 2081 a Fn(z)2258 2095 y Fl(\033)2328 2081 y Fm(\000)19 b Fn(z)729 2321 y Fx(=)25 b Fn(\013)883 2284 y Fk(\000)p Fv(1)981 2321 y Fn(\034)1021 2335 y Fk(1)1113 2198 y Fi(Z)1204 2220 y Fk(1)1163 2404 y Fv(1)1296 2321 y Fn(e)1338 2284 y Fk(\000)p Fl(s=)p Fv(\()p Fl(x\034)1569 2293 y Fe(1)1636 2284 y Fv(\))1940 2260 y Fn(dx)p 1680 2300 620 4 v 1680 2385 a Fx(\(1)c Fm(\000)f Fn(z)t(x\034)2010 2399 y Fk(1)2087 2385 y Fx(\))p Fn(x)2174 2359 y Fv(1)p Fl(=\013)2403 2321 y Ft(in)33 b(Pr)-5 b(ob)g(ability.)3432 2176 y Fx(\(4)p Fn(:)p Fx(206\))50 2602 y Ft(Mor)g(e)g(over,)159 2880 y Fn(\013)217 2843 y Fk(\000)p Fv(1)315 2880 y Fn(\034)355 2894 y Fk(1)447 2756 y Fi(Z)538 2779 y Fk(1)497 2963 y Fv(1)630 2880 y Fn(e)672 2843 y Fk(\000)p Fl(s=)p Fv(\()p Fl(x\034)903 2852 y Fe(1)970 2843 y Fv(\))1274 2819 y Fn(dx)p 1014 2859 V 1014 2944 a Fx(\(1)21 b Fm(\000)f Fn(z)t(x\034)1344 2958 y Fk(1)1421 2944 y Fx(\))p Fn(x)1508 2918 y Fv(1)p Fl(=\013)277 3162 y Fx(=)25 b(\()p Fm(\000)p Fn(z)t(\034)565 3176 y Fk(1)642 3162 y Fx(\))677 3125 y Fv(1)p Fl(=\013)818 3007 y Fi( )890 3162 y Fn(z)936 3125 y Fk(\000)p Fv(1)1034 3162 y Fn(\031)s Ft(c)-5 b(ose)g(c)16 b Fx(\()p Fn(\031)s(=\013)p Fx(\))21 b Fm(\000)1644 3039 y Fi(Z)1735 3061 y Fk(1)1695 3245 y Fv(0)1827 3162 y Fn(dte)1949 3125 y Fl(z)s(t)2034 3039 y Fi(Z)2125 3061 y Fl(s=t)2085 3245 y Fv(0)2434 3101 y Fn(dx)p 2255 3141 457 4 v 2255 3226 a(x)2307 3200 y Fv(1)p Fl(=\013)2432 3226 y Fx(\(1)g(+)f Fn(x)p Fx(\))2722 3007 y Fi(!)2815 3162 y Fx(+)f Fn(O)s Fx(\()p Fn(e)3054 3125 y Fk(\000)p Fl(s=\034)3216 3134 y Fe(1)3288 3162 y Fx(\))3432 3036 y(\(4)p Fn(:)p Fx(207\))50 3626 y Fc(Pro)s(of:)57 b Fx(Observing)33 b(that,)j(b)m(y)e(\(4.125\),)39 b Fn(me)1644 3593 y Fk(\000)p Fl(\014)1745 3539 y Fk(p)p 1805 3539 65 4 v 54 x Fl(N)7 b(E)1923 3602 y Fp(\033)2003 3626 y Fx(=)32 b Fn(se)2191 3593 y Fk(\000)p Fl(\014)2292 3539 y Fk(p)p 2352 3539 V 54 x Fl(N)6 b(E)2469 3602 y Fp(\033)2527 3603 y Fi(b)2517 3626 y Fx(\002)32 b(=)g Fn(sz)2808 3640 y Fl(\033)2857 3626 y Fx(\(1)24 b Fm(\000)e Fx(1)p Fn(=)-5 b(M)10 b Fx(\))3272 3593 y Fk(\000)p Fv(1)3371 3626 y Fx(,)36 b(\(4.206\))50 3771 y(is)28 b(pro)m(v)m(en)i(lik)m(e)e(\(4.143\))33 b(of)c(Corollary)f(4.17.)42 b(T)-8 b(o)30 b(pro)m(v)m(e)g(\(4.207\),)i (it)d(will)e(b)s(e)i(con)m(v)m(enien)m(t)h(to)g(extend)g(the)50 3917 y(in)m(tegration)k(in)e(\(4.206\))k(all)d(the)h(w)m(a)m(y)h(to)f (zero,)i(as)e(in)e(the)i(pro)s(of)f(of)h(Corollary)e(4.17.)52 b(One)34 b(can)g(easily)50 4062 y(estimate)d(the)g(di\013erence,)f (namely)148 4222 y Fi(\014)148 4277 y(\014)148 4331 y(\014)148 4386 y(\014)179 4230 y(Z)270 4253 y Fv(1)229 4437 y(0)326 4354 y Fn(e)368 4316 y Fk(\000)p Fl(s=)p Fv(\()p Fl(x\034)599 4325 y Fe(1)666 4316 y Fv(\))970 4292 y Fn(dx)p 710 4333 620 4 v 710 4418 a Fx(\(1)21 b Fm(\000)f Fn(z)t(x\034)1040 4432 y Fk(1)1117 4418 y Fx(\))p Fn(x)1204 4392 y Fv(1)p Fl(=\013)1340 4222 y Fi(\014)1340 4277 y(\014)1340 4331 y(\014)1340 4386 y(\014)1396 4354 y Fm(\024)1540 4292 y Fx(1)p 1503 4333 122 4 v 1503 4351 a Fm(p)p 1578 4351 46 4 v 1578 4426 a Fx(2)1650 4230 y Fi(Z)1741 4253 y Fv(1)1700 4437 y(0)1797 4354 y Fn(e)1839 4316 y Fk(\000)p Fl(s=)p Fv(\()p Fl(x\034)2070 4325 y Fe(1)2137 4316 y Fv(\))2220 4292 y Fn(dx)p 2181 4333 178 4 v 2181 4418 a(x)2233 4392 y Fv(1)p Fl(=\013)2394 4354 y Fm(\024)2501 4292 y Fn(e)2543 4259 y Fk(\000)p Fl(s=\034)2705 4268 y Fe(1)p 2501 4333 276 4 v 2520 4426 a Fn(\034)2560 4440 y Fk(1)2636 4351 y Fm(p)p 2712 4351 46 4 v 75 x Fx(2)2802 4354 y(min)2969 4226 y Fi(\022)3036 4354 y Fn(s)3079 4316 y Fk(\000)p Fv(1)3176 4354 y Fn(;)3358 4292 y Fx(1)p 3227 4333 306 4 v 3227 4416 a(1)h Fm(\000)f Fx(1)p Fn(=\013)3544 4226 y Fi(\023)3432 4518 y Fx(\(4)p Fn(:)p Fx(208\))159 4736 y(In)32 b(the)h(extended)g(in)m(tegral)g(w)m(e)g(again)g(c)m (hange)i(v)-5 b(ariables)31 b(and)i(rotate)h(the)f(in)m(tegration)g (con)m(tour)h(to)50 4882 y(the)c(negativ)m(e)i(real)e(axis)g(to)h(get)g (that)320 5036 y Fi(Z)411 5058 y Fk(1)371 5242 y Fv(0)503 5160 y Fn(e)545 5122 y Fk(\000)p Fl(s=)p Fv(\()p Fl(x\034)776 5131 y Fe(1)843 5122 y Fv(\))1147 5098 y Fn(dx)p 887 5139 620 4 v 887 5224 a Fx(\(1)21 b Fm(\000)f Fn(z)t(x\034)1217 5238 y Fk(1)1294 5224 y Fx(\))p Fn(x)1381 5197 y Fv(1)p Fl(=\013)1543 5160 y Fx(=)25 b(\()p Fn(z)t(\034)1760 5174 y Fk(1)1837 5160 y Fx(\))1872 5122 y Fv(1)p Fl(=\013)p Fk(\000)p Fv(1)2106 5036 y Fi(Z)2197 5058 y Fl(z)s Fk(1)2157 5242 y Fv(0)2327 5160 y Fn(e)2369 5122 y Fk(\000)p Fl(sz)s(=x)2770 5098 y Fn(dx)p 2591 5139 457 4 v 2591 5224 a Fx(\(1)c Fm(\000)f Fn(x)p Fx(\))p Fn(x)2922 5197 y Fv(1)p Fl(=\013)1543 5411 y Fx(=)25 b Fm(\000)p Fx(\()p Fm(\000)p Fn(z)t(\034)1902 5425 y Fk(1)1979 5411 y Fx(\))2014 5374 y Fv(1)p Fl(=\013)p Fk(\000)p Fv(1)2248 5288 y Fi(Z)2338 5310 y Fk(1)2298 5494 y Fv(0)2431 5411 y Fn(e)2473 5374 y Fv(+)p Fl(sz)s(=x)2873 5350 y Fn(dx)p 2694 5391 V 2694 5475 a Fx(\(1)c(+)f Fn(x)p Fx(\))p Fn(x)3025 5449 y Fv(1)p Fl(=\013)3432 5287 y Fx(\(4)p Fn(:)p Fx(209\))p eop %%Page: 60 60 60 59 bop 50 295 a Fr(60)3225 b Fo(R)l(efer)l(enc)l(es)50 513 y Fx(According)30 b(to)h(whether)f Fm(<)p Fx(\()p Fn(z)t Fx(\))h(is)e(p)s(ositiv)m(e)g(or)i(negativ)m(e,)h(w)m(e)e(can)h (represen)m(t)755 772 y Fn(e)797 735 y Fv(+)p Fl(sz)s(=x)1033 772 y Fx(=)1129 648 y Fi(Z)1220 671 y Fl(z)s Fk(1)1180 855 y(\000)p Fl(z)s(s=x)1406 772 y Fn(e)1448 735 y Fk(\000)p Fl(t)1537 772 y Fn(dt)26 b Fx(=)f Fn(z)1800 648 y Fi(Z)1891 671 y Fv(+)p Fk(1)1851 855 y(\000)p Fl(s=x)2040 772 y Fn(e)2082 735 y Fk(\000)p Fl(z)s(t)2208 772 y Fn(dt)91 b Fx(or)755 1048 y Fn(e)797 1010 y Fv(+)p Fl(sz)s(=x)1033 1048 y Fx(=)1129 924 y Fi(Z)1220 947 y Fk(\000)p Fl(z)s Fk(1)1180 1131 y(\000)p Fl(z)s(s=x)1406 1048 y Fn(e)1448 1010 y Fk(\000)p Fl(t)1537 1048 y Fn(dt)26 b Fx(=)f Fm(\000)p Fn(z)1871 924 y Fi(Z)1962 947 y Fv(+)p Fk(1)1921 1131 y Fl(s=x)2110 1048 y Fn(e)2152 1010 y Fv(+)p Fl(z)s(t)2277 1048 y Fn(dt)91 b Fx(resp)s(ectiv)m(ely)3432 912 y(\(4)p Fn(:)p Fx(210\))50 1311 y(Inserting)34 b(these)j(represen)m(tation)f (in)m(to)f(\(4.209\))k(and)c(c)m(hanging)h(the)g(order)f(of)h(in)m (tegration)g(in)e(the)i(re-)50 1456 y(sulting)28 b(double)h(in)m (tegrals)h(giv)m(es)h(in)e(b)s(oth)h(cases)345 1575 y Fi(Z)436 1598 y Fk(1)395 1782 y Fv(0)528 1699 y Fn(e)570 1661 y Fk(\000)p Fl(s=)p Fv(\()p Fl(x\034)801 1670 y Fe(1)868 1661 y Fv(\))1172 1637 y Fn(dx)p 912 1678 620 4 v 912 1763 a Fx(\(1)21 b Fm(\000)f Fn(z)t(x\034)1242 1777 y Fk(1)1319 1763 y Fx(\))p Fn(x)1406 1737 y Fv(1)p Fl(=\013)355 1981 y Fx(=)25 b Fn(\034)501 1944 y Fk(\000)p Fv(1)491 2004 y Fk(1)599 1981 y Fx(\()p Fm(\000)p Fn(z)t(\034)791 1995 y Fk(1)868 1981 y Fx(\))903 1944 y Fv(1)p Fl(=\013)1043 1826 y Fi( )1115 1981 y Fx(\()p Fn(z)t Fx(\))1231 1944 y Fk(\000)p Fv(1)1330 1981 y Fn(\013\031)s Fx(cosec)17 b(\()p Fn(\031)s(=\013)p Fx(\))22 b Fm(\000)2002 1857 y Fi(Z)2093 1880 y Fk(1)2053 2064 y Fv(0)2185 1981 y Fn(dte)2307 1944 y Fl(z)s(t)2392 1857 y Fi(Z)2483 1880 y Fl(s=t)2443 2064 y Fv(0)2791 1920 y Fn(dx)p 2613 1960 457 4 v 2613 2045 a(x)2665 2019 y Fv(1)p Fl(=\013)2790 2045 y Fx(\(1)f(+)f Fn(x)p Fx(\))3080 1826 y Fi(!)3432 1855 y Fx(\(4)p Fn(:)p Fx(211\))50 2254 y Fm(})159 2472 y Fx(W)-8 b(e)39 b(can)g(no)m(w)f(com)m(bine)g(the)h(asymptotics)f(for) g(1)26 b Fm(\000)f Fn(\025)p Fx(\()p Fn(u)p Fx(\))39 b(obtained)f(in)f(Corollary)g(4.18)j(with)d(the)50 2618 y(preceding)29 b(result.)40 b(This)28 b(sho)m(ws)i(that)345 2882 y(lim)296 2941 y Fl(E)s Fk(#\0001)559 2882 y Fx(lim)535 2941 y Fl(N)7 b Fk("1)813 2820 y Fn(\025)p Fx(\()p Fn(u)p Fx(\))p 735 2861 333 4 v 735 2944 a(1)21 b Fm(\000)f Fn(\025)p Fx(\()p Fn(u)p Fx(\))1196 2820 y(1)p 1089 2861 260 4 v 1089 2944 a Fm(j)p Fn(T)13 b Fx(\()p Fn(E)5 b Fx(\))p Fm(j)1439 2795 y Fi(X)1375 2994 y Fl(\033)r Fk(2)p Fl(T)10 b Fv(\()p Fl(E)s Fv(\))1650 2882 y Fn(e)1692 2844 y Fk(\000)p Fl(me)1849 2817 y Fe(\000)1898 2777 y(p)p 1947 2777 54 3 v 1947 2817 a Fp(N)5 b(E)2047 2826 y(\033)2137 2797 y Fi(b)2127 2820 y Fx(\002\()p Fn(E)g Fx(\))p 2110 2861 249 4 v 2110 2944 a Fn(z)2152 2958 y Fl(\033)2221 2944 y Fm(\000)20 b Fn(z)210 3241 y Fx(=)p Fn(z)327 3203 y Fk(\000)p Fv(1)445 3241 y Fm(\000)547 3081 y Fi(R)608 3104 y Fk(1)590 3187 y Fv(0)700 3154 y Fn(dte)822 3121 y Fl(z)s(t)907 3081 y Fi(R)967 3104 y Fl(s=t)950 3187 y Fv(0)1228 3118 y Fl(dx)p 1097 3133 342 4 v 1097 3189 a(x)1138 3171 y Fq(1)p Fp(=\013)1249 3189 y Fv(\(1+)p Fl(x)p Fv(\))p 547 3220 903 4 v 747 3303 a Fn(\031)s Fx(cosec)d(\()p Fn(\031)s(=\013)p Fx(\))1476 3140 y Fi(\020)1530 3241 y Fx(1)k(+)f Fn(O)s Fx(\()p Fm(j)p Fn(z)t Fm(j)1890 3203 y Fv(1)p Fk(\000)p Fv(1)p Fl(=\013)2109 3241 y Fn(;)15 b Fm(j)p Fn(z)t Fm(j)2245 3203 y Fv(1)p Fl(=\013)2371 3241 y Fx(\))2406 3140 y Fi(\021)2481 3241 y Fx(+)20 b Fn(O)2659 3140 y Fi(\020)2714 3241 y Fn(z)2760 3203 y Fk(\000)p Fv(1)p Fl(=\013)2942 3241 y Fn(e)2984 3203 y Fk(\000)p Fl(s=\034)3146 3212 y Fe(1)3217 3140 y Fi(\021)3432 3051 y Fx(\(4)p Fn(:)p Fx(212\))50 3485 y(The)30 b(leading)f(term)h(is)g(readily)e(iden)m (ti\014ed)h(as)h(the)h(Laplace)g(transform)e(of)1305 3784 y Fn(H)1381 3798 y Fv(0)1421 3784 y Fx(\()p Fn(s=t)p Fx(\))d Fm(\021)f Fx(1)20 b Fm(\000)1901 3624 y Fi(R)1962 3646 y Fl(s=t)1944 3730 y Fv(0)2223 3661 y Fl(dx)p 2091 3676 342 4 v 2091 3732 a(x)2132 3714 y Fq(1)p Fp(=\013)2243 3732 y Fv(\(1+)p Fl(x)p Fv(\))p 1901 3763 543 4 v 1921 3846 a Fn(\031)s Fx(cosec)d(\()p Fn(\031)s(=\013)p Fx(\))3432 3784 y(\(4)p Fn(:)p Fx(213\))50 4028 y(whic)m(h)28 b(w)m(e)h(recognise) g(as)h(precisely)d(the)i(function)f(that)i(app)s(eared)e(as)h(the)g (leading)f(asymptotic)h(con)m(tri-)50 4173 y(bution)j(in)g(the)h(trap)g (mo)s(del)g(in)f(Theorem)h(1.1.)51 b(The)32 b(b)s(ounds)g(on)h(the)g (error)g(terms)h(then)f(follo)m(w)f(from)50 4319 y(simply)c(estimating) i(the)g(corrections)h(uniformly)c(on)k Fm(C)5 b Fx(.)40 b Fm(})159 4537 y Fx(The)24 b(last)g(step)h(b)s(efore)f(completing)g (the)h(pro)s(of)f(of)h(Theorem)f(1)h(is)f(no)m(w)g(to)i(consider)d(the) i(con)m(tribution)50 4682 y Fn(F)108 4696 y Fl(N)s(;E)252 4682 y Fx(\()p Fn(n)19 b Fx(+)g Fn(m)p Fx(\).)41 b(W)-8 b(e)31 b(lea)m(v)m(e)g(it)f(to)h(the)f(reader)g(to)g(sho)m(w)g(that)h (the)f(leading)f(asymptotics)h(of)g(this)f(term)h(is)50 4828 y(giv)m(en)g(b)m(y)459 5070 y Fn(\013)517 5033 y Fk(\000)p Fv(1)630 4947 y Fi(Z)721 4969 y Fk(1)680 5153 y Fv(1)813 5070 y Fn(dxe)954 5033 y Fk(\000)p Fv(\()p Fl(t)p Fv(+)p Fl(s)p Fv(\))p Fl(=x)1269 5070 y Fn(x)1321 5033 y Fk(\000)p Fv(1)p Fk(\000)p Fv(1)p Fl(=\013)1621 5070 y Fm(\024)1926 5009 y Fx(1)p 1728 5049 442 4 v 1728 5134 a Fn(\013)p Fx(\()p Fn(t)21 b Fx(+)f Fn(s)p Fx(\))2044 5108 y Fv(1)p Fl(=\013)2195 4947 y Fi(Z)2286 4969 y Fk(1)2246 5153 y Fv(0)2378 5070 y Fn(dxe)2519 5033 y Fk(\000)p Fv(1)p Fl(=x)2696 5070 y Fn(x)2748 5033 y Fk(\000)p Fv(1)p Fk(\000)p Fv(1)p Fl(=\013)3432 5070 y Fx(\(4)p Fn(:)p Fx(214\))50 5316 y(whic)m(h)28 b(is)g(sub-dominan)m(t)g(as)i Fn(s)f Fx(and)g Fn(t)g Fx(tend)g(to)h(in\014nit)m(y)-8 b(.)38 b(Collecting)29 b(all)f(the)i(estimates)g(of)f(this)g(section)50 5462 y(concludes)g(the)i(pro)s(of)f(of)g(the)h(main)e(theorem.)p Fm(}})p eop %%Page: 61 61 61 60 bop 50 295 a Fo(A)l(ging)27 b(in)h(the)g(REM.)f(Part)h(2.)2661 b Fr(61)50 513 y Fu(Ac)m(kno)m(wledgemen)m(ts:)43 b Fx(W)-8 b(e)27 b(thank)e(the)g(W)-8 b(eierstrass)26 b(Institute)e(and)h(the)g (Mathematics)h(departmen)m(t)50 658 y(of)k(the)h(EPFL)f(for)g (\014nancial)f(supp)s(ort)g(and)g(m)m(utual)h(hospitalit)m(y)-8 b(.)50 880 y Fw(References)-1 1098 y Fx([B])46 b(J.P)-8 b(.)36 b(Bouc)m(haud,)g(\\W)-8 b(eak)38 b(ergo)s(dicit)m(y)d(breaking)g (and)g(aging)h(in)e(disordered)g(systems",)i Ft(J.)h(Phys.)h(I)159 1244 y(\(F)-7 b(r)i(anc)g(e\))p Fx(,)32 b Fu(2)p Fx(,)f(1705)h (\(1992\).)-183 1462 y([BBG1])48 b(G.)22 b(Ben)g(Arous,)f(A.)h(Bo)m (vier,)h(and)d(V.)i(Ga)m(yrard,)h(\\Glaub)s(er)d(dynamics)g(of)i(the)g (random)f(energy)h(mo)s(del.)159 1607 y(I.)30 b(Metastable)i(motion)e (on)g(extreme)i(states",)g(\(2001\).)-221 1825 y([BCKM])46 b(J.P)-8 b(.)30 b(Bouc)m(haud,)g(L.)f(Cugliandolo,)e(J.)j(Kurc)m(han,)e (M.)i(M)m(\023)-43 b(ezard,)32 b(\\Out-of-equilibrium)25 b(dynamics)j(in)159 1971 y(spi-glasses)23 b(and)h(other)h(glassy)f (systems",)h(in)f Ft(Spin-glasses)k(and)g(r)-5 b(andom)29 b(\014elds)d Fx(\(A.P)-8 b(.)25 b(Y)-8 b(oung,)26 b(Ed.\).)159 2116 y(W)-8 b(orld)30 b(Scien)m(ti\014c,)g(Singap)s(ore)e(\(1998\).)-71 2335 y([BD])47 b(J.P)-8 b(.)31 b(Bouc)m(haud)f(and)g(D.)h(Dean,\\Aging) h(on)e(P)m(arisi's)f(tree")j Ft(J.)g(Phys.)h(I)f(\(F)-7 b(r)i(anc)g(e\))p Fx(,)33 b Fu(5)p Fx(,)e(265)g(\(1995\).)-142 2553 y([BDG])47 b(G.)40 b(Ben)g(Arous,)f(A.)h(Dem)m(b)s(o)g(and)f(A.)h (Guionnet,)g(\\Aging)f(of)h(spherical)e(spin)g(glasses",)i(Probab.)159 2698 y(Theor.)30 b(Rel.)g(Fields)f Fu(120)p Fx(,)j(1-67)f(\(2001\).) -251 2916 y([BEGK1])47 b(A.)35 b(Bo)m(vier,)h(M.)f(Ec)m(kho\013,)h(V.)g (Ga)m(yrard)f(and)f(M.)i(Klein,)d(\\Metastabilit)m(y)j(in)d(sto)s(c)m (hastic)j(dynamics)159 3062 y(of)30 b(disordered)f(mean)h(\014eld)f(mo) s(dels",)h Ft(Pr)-5 b(ob.)33 b(The)-5 b(or.)34 b(R)-5 b(el.)33 b(Fields)e Fu(119)p Fx(,)h(99-161)h(\(2001\).)-251 3280 y([BEGK2])47 b(A.)30 b(Bo)m(vier,)h(M.)g(Ec)m(kho\013,)g(V.)f(Ga)m (yrard)g(and)g(M.)g(Klein,)f(\\Metastabilit)m(y)h(and)g(lo)m(w)g(lying) e(sp)s(ectra)i(in)159 3425 y(rev)m(ersible)f(Mark)m(o)m(v)j(c)m (hains",)e(to)i(app)s(ear)d(in)g Ft(Commun.)34 b(Math.)f(Phys.)e Fx(\(2001\).)-129 3644 y([BKL])46 b(A.)39 b(Bo)m(vier,)h(I.)f(Kurk)m(o) m(v)-5 b(a,)40 b(M.)f(L\177)-45 b(ow)m(e,)40 b(\\Fluctuations)f(of)g (the)g(free)g(energy)g(in)f(the)h(REM)g(and)g(the)159 3789 y Fn(p)p Fx(-spin)28 b(SK)i(mo)s(dels",)g(to)h(app)s(ear)f(in)f (Ann.)g(Probab.)h(\(2001\).)-85 4007 y([BM])47 b(J.P)-8 b(.)22 b(Bouc)m(haud,)g(C.)g(Mon)m(th)m(us,)h(\\Mo)s(dels)e(of)h(traps) g(and)f(glass)h(phenomenology",)g Ft(J.)i(Phys.)h(A-Math.)159 4153 y(Gen.)30 b Fu(29)p Fx(,)h(\(14\))h(3847-3869)j(\(1996\).)-152 4371 y([BMR])47 b(J.P)-8 b(.)41 b(Bouc)m(haud,)g(P)-8 b(.)41 b(Maass,)g(B.)g(Rinn,)e(\\Multiple)g(scaling)g(regimes)h(in)f (simple)g(aging)h(mo)s(dels",)159 4516 y Ft(Phys.)33 b(R)-5 b(ev.)32 b(L)-5 b(etts.)31 b Fu(84)p Fx(,)h(\(23\))f(5403-5406)k (\(2000\).)-72 4735 y([CD])46 b(L.)34 b(Cugliandolo)e(and)h(D.)i(Dean,) g(\\F)-8 b(ull)34 b(dynamical)e(solution)h(for)h(a)g(spherical)f (spin-glass)f(mo)s(del",)159 4880 y Ft(J.)g(Phys.)h(A)f(-)g(Math.)h (Gen.)e Fu(28)p Fx(,)g(\(15\))h(4213-4234)i(\(1995\).)-52 5098 y([D1])47 b(B.)33 b(Derrida,)f(Random)g(energy)h(mo)s(del:)e (\\Limit)g(of)i(a)g(family)e(of)h(disordered)f(mo)s(dels",)h Ft(Phys.)j(R)-5 b(ev.)159 5244 y(L)g(etts.)31 b Fu(45)p Fx(,)g(79-82)h(\(1980\).)-52 5462 y([D2])47 b(B.)35 b(Derrida,)g (Random)f(energy)h(mo)s(del:)f(\\An)h(exactly)g(solv)-5 b(able)34 b(mo)s(del)g(of)h(disordered)e(disordered)p eop %%Page: 62 62 62 61 bop 50 295 a Fr(62)3225 b Fo(R)l(efer)l(enc)l(es)159 513 y Fx(systems",)31 b Ft(Phys.)i(R)-5 b(ev.)33 b(B)d Fu(24)p Fx(,)h(2613-2626)j(\(1981\).)-95 731 y([Do)s(e])47 b(G.)26 b(Do)s(etsc)m(h,)h Ft(Handbuch)h(der)h(L)-5 b(aplac)g(e-T)e(r)i (ansformation)p Fx(,)31 b(V)-8 b(ol)26 b(I)s(I,)e(Lehrb)s(\177)-48 b(uc)m(her)24 b(und)g(Monographien)159 876 y(aus)30 b(dem)f(Gebiete)i (der)f(exakten)h(Wissensc)m(haften,)f(Mathematisc)m(he)i(Reihe)d(Band)h (15,)h(Birkh\177)-45 b(auser)159 1022 y(V)-8 b(erlag,)31 b(Basel,)g(1955.)-29 1240 y([F)-8 b(e])47 b(W.)36 b(F)-8 b(eller,)34 b Ft(A)n(n)j(intr)-5 b(o)g(duction)39 b(to)e(pr)-5 b(ob)g(ability)40 b(the)-5 b(ory)38 b(and)g(its)f(applic)-5 b(ations)p Fx(,)38 b(V)-8 b(ol)36 b(I)s(I,)e(Wiley)g(series)159 1385 y(in)29 b(probabilit)m(y)f(and)i(mathematical)g(statistics,)h(New) f(Y)-8 b(ork,)32 b(1971.)-19 1604 y([Li])45 b(T.M.)31 b(Liggett,)g Ft(Inter)-5 b(acting)34 b(p)-5 b(article)34 b(systems)p Fx(,)e(Springer,)c(Berlin,)h(1985.)-118 1822 y([LLR])46 b(M.R.)34 b(Leadb)s(etter,)f(G.)h(Lindgren,)d(H.)j(Ro)s(otz) m(\023)-43 b(en,)35 b Ft(Extr)-5 b(emes)36 b(and)g(R)-5 b(elate)g(d)37 b(Pr)-5 b(op)g(erties)37 b(of)e(R)-5 b(andom)159 1967 y(Se)g(quenc)g(es)32 b(and)i(Pr)-5 b(o)g(c)g(esses)p Fx(,)32 b(Springer,)c(Berlin-Heidelb)s(erg-New)g(Y)-8 b(ork,)32 b(1983.)-54 2185 y([Ru])45 b(D.)h(Ruelle,)f(\\A)h (mathematical)g(reform)m(ulation)e(of)i(Derrida's)f(REM)g(and)g(GREM",) i Ft(Commun.)159 2331 y(Math.)33 b(Phys.)e Fu(108)p Fx(,)g(225-239)i (\(1987\).)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF