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Ff(1995;)i(9:46)664 b Fm(4)p eop %%Page: 5 6 bop 0 45 a Fq(2.)27 b(The)20 b(basic)h(large)f(deviation)h(estimates)91 155 y Fm(In)c(this)g(section)g(w)o(e)g(recall)g(exp)q(onen)o(tial)h(upp)q(er) g(and)f(lo)o(w)o(er)f(b)q(ounds)h(that)f(ha)o(v)o(e)g(already)h(b)q(een)h (deriv)o(ed)0 227 y(in)e([BGP2].)j(They)c(pro)o(vide)h(the)f(starting)g(p)q (oin)o(t)g(of)g(our)g(analysis.)91 336 y(Let)g(us)h(consider)g(the)f(quan)o (tities)450 518 y Fl(Z)481 525 y Fh(N)q(;\014)r(;\032)576 518 y Fm([)p Fl(!)r Fm(]\()p Fl(m)p Fm(\))c Fi(\021)i Fl(\026)794 525 y Fh(N)q(;\014)860 518 y Fm([)p Fl(!)r Fm(])8 b(\()n Fi(k)p Fl(m)1003 525 y Fh(N)1038 518 y Fm(\()p Fl(\033)r Fm(\))h Fi(\000)h Fl(m)p Fi(k)1219 525 y Fp(2)1252 518 y Fi(\024)j Fl(\032)p Fm(\))6 b Fl(Z)1379 525 y Fh(N)q(;\014)1445 518 y Fm([)p Fl(!)r Fm(])356 b(\(2)p Fl(:)p Fm(1\))0 613 y(W)l(e)15 b(\014rst)g(pro)q(of)g(a)g (large)g(deviation)h Fk(upp)n(er)g Fm(b)q(ound.)0 722 y Fb(Lemma)h(2.1:)468 764 y Fm(1)p 444 784 70 2 v 444 826 a 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Fm(\()p Fl(m)p Fm(\))f(that)h(arise)h(from)e(con)o(v)o(exit)o(y)l(.)19 b(W)l(e)11 b(set)0 1822 y(\000)i Fi(\021)89 1785 y Fg(\010)116 1822 y Fl(m)f Fi(2)h Fl(I)-6 b(R)259 1802 y Fh(M)312 1822 y Fi(j)13 b(k)p Fl(t)377 1805 y Fd(\003)398 1822 y Fm(\()p Fl(m)p Fm(\))p Fi(k)497 1829 y Fp(2)529 1822 y Fl(<)g Fi(1)622 1785 y Fg(\011)659 1822 y Fm(where)d Fl(t)801 1805 y Fd(\003)822 1822 y Fm(\()p Fl(m)p Fm(\))g(is)g(de\014ned)i(in)f(Lemma)f(2.1,)g Fl(D)j Fi(\021)1486 1785 y Fg(\010)1513 1822 y Fl(m)g Fi(2)f Fl(I)-6 b(R)1657 1802 y Fh(M)1710 1822 y Fi(j)12 b Fm(\010)1768 1829 y Fh(N)q(;\014)1833 1822 y Fm(\()p Fl(m)p Fm(\))g Fl(>)h Fi(\0001)2049 1785 y Fg(\011)2077 1822 y Fm(,)0 1895 y(and)18 b(w)o(e)g(denote)g(b)o(y)f(in)o(t)q Fl(D)h Fm(the)g(in)o(terior)g(of)g Fl(D)q Fm(.)27 b(W)l(e)18 b(moreo)o(v)o(er)f(denote)h(b)o(y)f Fl(I)t Fm(\()p Fl(x)p Fm(\))f Fi(\021)i Fm(sup)1581 1906 y Fh(t)p Fd(2)p Fh(I)-8 b(R)1658 1895 y Fm(\()p Fl(tx)12 b Fi(\000)g Fm(ln)c(cosh)g Fl(t)p Fm(\))0 1967 y(the)16 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b(is)i(a)f(prop)q(er)h(con)o(v)o(ex)f(function)h(on)f Fl(I)-6 b(R)1434 888 y Fh(M)1474 908 y Fm(.)35 b(Denoting)21 b(b)o(y)f Fl(h)p Fm(\()p Fl(m)p Fm(\))h Fi(\021)0 981 y Fm(sup)68 992 y Fh(t)p Fd(2)p Fh(I)-8 b(R)143 981 y Fe(M)7 b Fi(f)p Fm(\()p Fl(m;)h(t)p Fm(\))g Fi(\000)i Fl(g)r Fm(\()p Fl(t)p Fm(\))p Fi(g)j Fm(its)i(Legendre)h(transform,)d(it)j(follo)o(ws)e(from)g(standard)h (results)g(of)g(con)o(v)o(ex)f(analysis)0 1054 y(that)g Fl(h)p Fm(\()p Fl(m)p Fm(\))h(is)h(a)f(prop)q(er)g(con)o(v)o(ex)g(function)h(on)f Fl(I)-6 b(R)883 1034 y Fh(M)938 1054 y Fm(and)16 b(that)617 1207 y Fl(h)p Fm(\()p Fl(m)p Fm(\))c(=)130 b(inf)779 1241 y Fh(y)q Fd(2)p Fh(I)-8 b(R)859 1230 y Fe(N)888 1241 y Fp(:)p Fh(m)930 1246 y Fe(N)960 1241 y Fp(\()p Fh(y)q Fp(\)=)p Fh(m)1105 1176 y Fm(1)p 1081 1197 70 2 v 1081 1238 a Fl(\014)r(N)1180 1150 y Fh(N)1164 1164 y Fg(X)1167 1261 y Fh(i)p Fp(=1)1237 1207 y Fl(I)t Fm(\()p Fl(y)1301 1214 y Fh(i)1316 1207 y Fm(\))500 b(\(2)p Fl(:)p Fm(16\))0 1366 y(where)20 b(for)f(eac)o(h)g Fl(m)h Fi(2)g Fl(I)-6 b(R)475 1346 y Fh(M)535 1366 y Fm(the)19 b(in\014m)o(um)i(is)f(either)g(attained)g(or)f(is)h(+)p Fi(1)p Fm(.)34 b(This)20 b(immediately)h(yields)g(i\).)0 1439 y(Denoting)c(b)o(y)g (dom)o Fl(h)f Fi(\021)441 1402 y Fg(\010)467 1439 y Fl(x)d Fi(2)g Fl(I)-6 b(R)597 1419 y Fh(M)650 1439 y Fi(j)12 b Fl(h)p Fm(\()p Fl(m)p Fm(\))g Fl(<)h Fm(+)p Fi(1)917 1402 y Fg(\011)961 1439 y Fm(the)k(e\013ectiv)o(e)g(domain)h(of)e Fl(h)p Fm(,)h(w)o(e)f(ha)o(v)o (e,)h(b)o(y)f(\(1.7\),)g(that)0 1511 y(dom)p Fl(h)h Fm(equals)h(the)f(righ)o (t)g(hand)h(side)g(of)f(\(2.14\))e(,)i(and)g(since)i Fi(k)p Fl(m)p Fi(k)1177 1495 y Fp(2)1177 1523 y(2)1212 1511 y Fi(\025)e Fm(0,)g(ii\))h(is)f(pro)o(v)o(en.)26 b(iii\))18 b(simply)h(follo)o(ws)0 1584 y(from)f(the)h(fact)g(that)f Fl(h)i Fm(b)q(eing)g(con)o(v)o(ex,)f(it)h (is)f(con)o(tin)o(uous)h(relativ)o(e)f(to)g(the)g(in)o(terior)g(of)g(dom)p Fl(h)p Fm(.)31 b(Finally)l(,)22 b(to)0 1657 y(pro)o(v)o(e)17 b(iv\),)i(w)o(e)f(will)h(mak)o(e)f(use)g(of)g(the)g(follo)o(wing)h(t)o(w)o(o) d(imp)q(ortan)o(t)i(results)g(of)g(con)o(v)o(ex)g(analysis.)29 b(First,)18 b(the)0 1730 y(subgradien)o(t)d(of)f Fl(h)h Fm(at)f Fl(m)p Fm(,)g Fl(@)s(h)p Fm(\()p Fl(m)p Fm(\),)g(is)h(a)f(non)h(empt)o(y)f (set)h(if)g(and)g(only)g(if)g Fl(m)g Fm(b)q(elongs)g(to)f(the)h(in)o(terior)g (of)f(dom)p Fl(h)p Fm(,)0 1802 y(i.e.,)f Fl(m)g Fi(2)g Fm(in)o(t)p Fl(D)q Fm(.)19 b Fl(@)s(h)p Fm(\()p Fl(m)p Fm(\))12 b(is)i(moreo)o(v)o(er)e (a)h(b)q(ounded)i(con)o(v)o(ex)e(set.)19 b(Next,)13 b(\()p Fl(m;)8 b(t)p Fm(\))e Fi(\000)g Fl(g)r Fm(\()p Fl(t)p Fm(\))12 b(ac)o(hiev)o(es)h(its)h(suprem)o(um)0 1875 y(at)i Fl(t)73 1859 y Fd(\003)110 1875 y Fi(\021)h Fl(t)178 1859 y Fd(\003)199 1875 y Fm(\()p Fl(m)p Fm(\))f(if)i(and)f(only)h(if)f Fl(t)586 1859 y Fd(\003)623 1875 y Fi(2)f Fl(@)s(h)p Fm(\()p Fl(m)p Fm(\).)25 b(T)l(o)16 b(pro)o(v)o(e)h(v\))g(w)o(e)f(only)i(ha)o(v)o(e)f(to)f (consider)i(the)f(case)g(where)h Fl(t)1929 1859 y Fd(\003)0 1948 y Fm(exists)d(and)g(consequen)o(tly)h Fi(j)p Fl(y)519 1931 y Fd(\003)517 1960 y Fh(i)539 1948 y Fi(j)d Fl(<)g Fm(1)h(for)g(all)i Fl(i)p Fm(.)j(Using)d(the)f(fact)f(that)g Fl(I)1247 1931 y Fd(0)1259 1948 y Fm(\()p Fl(x)p Fm(\))e(=)h(tanh)1472 1927 y Fd(\000)p Fp(1)1521 1948 y Fm(\()p Fl(x)p Fm(\))h(and)h(the)g(de\014nition) 0 2020 y(of)g Fl(I)t Fm(\()p Fl(x)p Fm(\))f(as)h(the)g(Legendre)h(transform)e (of)h(ln)8 b(cosh\()p Fl(t)p Fm(\),)15 b(form)o(ula)g(\(2.15\))e(follo)o(ws)i (from)g(a)g(simple)h(computation.)0 2093 y(This)g(concludes)h(the)e(pro)q(of) g(of)f(the)i(lemma.)k Fi(})91 2202 y Fm(W)l(e)11 b(see)g(that)f(as)g(long)i (as)e Fl(\032)g Fm(can)h(b)q(e)h(c)o(hosen)f(as)f(a)h(function)g(of)g Fl(N)k Fm(that)10 b(tends)h(to)g(zero)f(as)h Fl(N)k Fm(go)q(es)c(to)f (in\014nit)o(y)l(,)0 2275 y(Lemma)15 b(2.1)e(and)i(Lemma)g(2.2)f(seem)h(to)f (pro)o(vide)h(asymptotically)g(coinciding)i(upp)q(er)f(and)f(lo)o(w)o(er)f(b) q(ounds,)i(at)0 2348 y(least)f(for)f(suc)o(h)h Fl(m)g Fm(for)f(whic)o(h)h Fl(t)546 2331 y Fd(\003)568 2348 y Fm(\()p Fl(m)p Fm(\))f(is)h(b)q(ounded.)21 b(The)15 b(unpleasan)o(t)g(feature)g(in)g(these)g(b)q(ounds)h(is)f(that)f (\011)1884 2355 y Fh(N)q(;\014)0 2420 y Fm(is)i(a)f(rather)f(complicated)j (random)d(function)i(and)g(that)e(the)h(\010)1115 2427 y Fh(N)q(;\014)1196 2420 y Fm(is)g(de\014ned)i(through)e(an)g(in\014m)o(um)h(of)f(suc)o(h)0 2493 y(a)h(function.)24 b(In)17 b(the)f(next)h(section)g(w)o(e)f(analyse)g (this)h(problem)g(and)f(sho)o(w)g(that)g(this)h(function)g(is)f(essen)o (tially)0 2566 y(non-random.)0 2779 y Ff(31)p Fe(=mar)q(ch=)p Ff(1995;)h(9:46)664 b Fm(7)p eop %%Page: 8 9 bop 0 45 a Fq(3.)27 b(Self)20 b(a)n(v)n(eraging)91 155 y Fm(W)l(e)13 b(sho)o(w)g(no)o(w)g(that)g(the)h(random)f(upp)q(er)h(and)g(lo)o(w)o(er)f(b)q (ounds)h(deriv)o(ed)h(in)f(the)g(last)f(section)h(are)f(actually)0 227 y(with)j(large)f(probabilit)o(y)i(indep)q(enden)o(t)g(of)e(the)g (realization)i(of)e(the)g(randomness.)20 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Fl(s)e Fi(2)g(W)659 1921 y Fh(N)q(;M)756 1914 y Fm(suc)o(h)i(that)g Fi(k)p Fl(s)10 b Fi(\000)g Fl(t)p Fi(k)1095 1921 y Fp(2)1128 1914 y Fi(\024)1176 1881 y(p)p 1214 1881 30 2 v 33 x Fl(\013)p Fm(,)15 b(while)i(on)e(the)g(other)g(hand)716 2034 y Fi(j)o(W)773 2041 y Fh(N)q(;M)855 2034 y Fm(\()p Fl(R)p Fm(\))p Fi(j)c(\024)i Fl(e)1019 2015 y Fh(\013N)s Fp(\(ln\()p Fh(R=\013)p Fp(\)\))1834 2034 y Fm(\(3)p Fl(:)p Fm(10\))0 2190 y Fb(Lemma)k(3.3:)263 2321 y Fl(I)-8 b(P)318 2244 y Fg(")410 2321 y Fm(sup)344 2362 y Fh(t)p Fd(2W)418 2367 y Fe(N)q(;M)488 2362 y Fp(\()p Fh(R)p Fp(\))552 2321 y Fi(j)p Fm(\011\()p Fl(m;)8 b(t)p Fm(\))h Fi(\000)h Fl(I)-8 b(E)r Fm(\011\()p Fl(m;)8 b(t)p Fm(\))p Fi(j)k Fl(>)h(x)1062 2244 y Fg(#)1101 2321 y Fi(\024)g Fl(e)1170 2297 y Fd(\000)p Fh(N)1230 2264 y Fg(\000)1257 2285 y Fe(x)1275 2276 y Ff(2)p 1257 2290 36 2 v 1263 2308 a Fe(R)1298 2297 y Fp(\(1)p Fd(\000)1364 2285 y Ff(1)p 1364 2290 16 2 v 1364 2308 a(2)1385 2297 y Fh(e)1402 2283 y Fe(x=R)1463 2297 y Fp(\))p Fd(\000)p Fh(\013)6 b Fp(ln\()p Fh(R=\013)p Fp(\))1664 2264 y Fg(\001)1834 2321 y Fm(\(3)p Fl(:)p Fm(11\))0 2550 y Fa(Pro)q(of:)23 b Fm(Clearly)16 b(w)o(e)f(only)g(ha)o (v)o(e)g(to)g(pro)o(v)o(e)f(that)h(for)f(all)i Fl(t)e Fi(2)e(W)1120 2557 y Fh(N)q(;M)1202 2550 y Fm(\()p Fl(R)p Fm(\))494 2670 y Fl(I)-8 b(P)15 b Fm([)o Fi(j)p Fm(\011\()p Fl(m;)8 b(t)p Fm(\))h Fi(\000)i Fl(I)-8 b(E)r Fm(\011\()p Fl(m;)8 b(t)p Fm(\))p Fi(j)k Fl(>)h(x)p Fm(])f Fi(\024)h Fl(e)1167 2651 y Fd(\000)p Fh(N)1233 2639 y Fe(x)1251 2630 y Ff(2)p 1233 2644 36 2 v 1239 2662 a Fe(R)1274 2651 y Fp(\(1)p Fd(\000)1340 2639 y Ff(1)p 1340 2644 16 2 v 1340 2662 a(2)1361 2651 y Fh(e)1378 2638 y Fe(x=R)1439 2651 y Fp(\))1834 2670 y Fm(\(3)p Fl(:)p Fm(12\))0 2779 y Ff(31)p Fe(=mar)q(ch=)p Ff(1995;)k(9:46)664 b Fm(9)p eop %%Page: 10 11 bop 0 45 a Fm(T)l(o)14 b(do)h(this)f(w)o(e)h(write)f(\011\()p Fl(m;)8 b(t)p Fm(\))g Fi(\000)h Fl(I)-8 b(E)r Fm(\011\()p Fl(m;)8 b(t)p Fm(\))14 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Fi([)10 b(f)p Fl(\030)513 480 y Fh(\026)511 516 y(k)537 502 y Fi(g)560 478 y Fh(\026)p Fd(\025)p Fh(\024)634 502 y Fm(.)20 b(W)l(e)15 b(set)143 637 y(~)133 649 y Fl(f)160 626 y Fp(\()p Fh(k)q(;\024)p Fp(\))155 663 y Fh(N)255 649 y Fi(\021)e Fl(I)-8 b(E)359 571 y Fg(")386 649 y Fl(\014)414 630 y Fd(\000)p Fp(1)470 606 y Fg(X)497 703 y Fh(i)544 649 y Fm(ln)8 b(cosh\()p Fl(\014)r Fm(\()p Fl(\030)759 656 y Fh(i)774 649 y Fl(;)g(t)p Fm(\)\))847 611 y Fg(\014)847 638 y(\014)861 649 y Fi(F)894 656 y Fh(k)q(;\024)948 571 y Fg(#)985 649 y Fi(\000)i Fl(I)-8 b(E)1086 571 y Fg(")1112 649 y Fl(\014)1140 630 y Fd(\000)p Fp(1)1197 606 y Fg(X)1223 703 y Fh(i)1270 649 y Fm(ln)9 b(cosh\()p Fl(\014)r Fm(\()p Fl(\030)1486 656 y Fh(i)1500 649 y Fl(;)f(t)p Fm(\)\))1573 611 y Fg(\014)1573 638 y(\014)1587 649 y Fi(F)1625 630 y Fp(+)1620 663 y Fh(k)q(;\024)1674 571 y Fg(#)1834 649 y Fm(\(3)p Fl(:)p Fm(14\))0 786 y(where)15 b(for)g(notational)g(con)o(v)o(enience)i(w)o(e)e(ha)o(v)o(e)g(set)693 900 y Fi(F)731 881 y Fp(+)726 914 y Fh(k)q(;\024)793 900 y Fm(=)841 836 y Fg(\032)883 871 y Fi(F)916 878 y Fh(k)q(;\024)p Fp(+1)1016 871 y Fl(;)45 b Fm(if)15 b Fl(\024)e(<)g(M)883 926 y Fi(F)916 933 y Fh(k)q Fp(+1)p Fh(;)p Fp(1)1074 926 y Fm(if)i Fl(\024)e Fm(=)g Fl(M)1834 900 y Fm(\(3)p Fl(:)p Fm(15\))0 1022 y(Notice)j(that)e(w)o(e)h(ha)o(v)o(e)g(the)g(iden)o(tit)o(y)586 1160 y(\011\()p Fl(m;)8 b(t)p Fm(\))i Fi(\000)g Fl(I)-8 b(E)r Fm(\011\()p Fl(m;)8 b(t)p Fm(\))k Fi(\021)1060 1130 y Fm(1)p 1051 1150 42 2 v 1051 1191 a Fl(N)1122 1103 y Fh(N)1105 1117 y Fg(X)1105 1215 y Fh(k)q Fp(=1)1193 1103 y Fh(M)1180 1117 y Fg(X)1179 1213 y Fh(\024)p Fp(=1)1264 1148 y Fm(~)1254 1160 y Fl(f)1281 1137 y Fp(\()p Fh(k)q(;\024)p Fp(\))1276 1174 y Fh(N)1834 1160 y Fm(\(3)p Fl(:)p Fm(16\))0 1298 y(Our)19 b(aim)f(is)h(to)f (use)g(an)h(exp)q(onen)o(tial)g(Mark)o(o)o(v)e(inequalit)o(y)j(for)e (martingales.)29 b(This)19 b(requires)g(in)g(particular)0 1370 y(b)q(ounds)14 b(on)f(the)g(conditional)i(Laplace)f(transforms)e(of)h(the)g (martingale)g(di\013erences)i(\(see)e(e.g.)19 b([L)l(T]\).)12 b(Namely)l(,)0 1443 y(w)o(e)j(clearly)h(ha)o(v)o(e)f(that)141 1582 y Fl(I)-8 b(P)197 1504 y Fg(")223 1502 y(\014)223 1529 y(\014)223 1557 y(\014)223 1584 y(\014)223 1611 y(\014)255 1525 y Fh(N)239 1538 y Fg(X)238 1636 y Fh(k)q Fp(=1)327 1525 y Fh(M)313 1538 y Fg(X)312 1635 y Fh(\024)p Fp(=1)397 1570 y Fm(~)387 1582 y Fl(f)414 1558 y Fp(\()p Fh(k)q(;\024)p Fp(\))409 1595 y Fh(N)497 1502 y Fg(\014)497 1529 y(\014)497 1557 y(\014)497 1584 y(\014)497 1611 y(\014)525 1582 y Fi(\025)13 b Fl(N)5 b(x)641 1504 y Fg(#)679 1582 y Fi(\024)13 b Fm(2)21 b(inf)758 1611 y Fh(u)p Fd(2)p Fh(I)-8 b(R)847 1582 y Fl(e)868 1563 y Fd(\000j)p Fh(u)p Fd(j)p Fh(N)s(x)995 1582 y Fl(I)g(E)10 b Fm(exp)1128 1504 y Fg(\()1164 1582 y Fl(u)1215 1525 y Fh(N)1198 1538 y Fg(X)1198 1636 y Fh(k)q Fp(=1)1286 1525 y Fh(M)1273 1538 y Fg(X)1272 1635 y Fh(\024)p Fp(=1)1356 1570 y Fm(~)1347 1582 y Fl(f)1374 1558 y Fp(\()p Fh(k)q(;\024)p Fp(\))1369 1595 y Fh(N)1457 1504 y Fg(\))199 1712 y Fm(=)j(2)k(inf)278 1741 y Fh(u)p Fd(2)p Fh(R)359 1712 y Fl(e)380 1693 y Fd(\000j)p Fh(u)p Fd(j)p Fh(N)s(x)507 1712 y Fl(I)-8 b(E)563 1661 y Fg(h)584 1712 y Fl(I)g(E)640 1661 y Fg(h)661 1712 y Fl(:)8 b(:)g(:)e(I)-8 b(E)778 1661 y Fg(h)799 1712 y Fl(e)820 1693 y Fh(u)849 1684 y Fp(~)841 1693 y Fh(f)863 1676 y Ff(\(1)p Fe(;)p Ff(1\))859 1705 y Fe(N)934 1673 y Fg(\014)934 1700 y(\014)949 1712 y Fi(F)987 1692 y Fp(+)982 1724 y(1)p Fh(;)p Fp(1)1031 1661 y Fg(i)1060 1712 y Fl(e)1081 1693 y Fh(u)1110 1684 y Fp(~)1102 1693 y Fh(f)1124 1676 y Ff(\(1)p Fe(;)p Ff(2\))1120 1705 y Fe(N)1194 1673 y Fg(\014)1194 1700 y(\014)1210 1712 y Fi(F)1248 1692 y Fp(+)1243 1724 y(1)p Fh(;)p Fp(2)1292 1661 y Fg(i)1321 1712 y Fl(:)8 b(:)g(:)d(e)1402 1693 y Fh(u)1431 1684 y Fp(~)1423 1693 y Fh(f)1445 1676 y Ff(\()p Fe(N)q(;M)r Ff(\))1441 1705 y Fe(N)1542 1673 y Fg(\014)1542 1700 y(\014)1557 1712 y Fi(F)1595 1692 y Fp(+)1590 1725 y Fh(N)q(;M)1671 1661 y Fg(i)1834 1635 y Fm(\(3)p Fl(:)p Fm(17\))0 1826 y(No)o(w)15 b(notice)g(that)25 1908 y(~)15 1920 y Fl(f)42 1897 y Fp(\()p Fh(k)q(;\024)p Fp(\))37 1934 y Fh(N)138 1920 y Fm(=)d Fl(I)-8 b(E)s Fm([)p Fl(\014)275 1902 y Fd(\000)p Fp(1)331 1877 y Fg(X)357 1974 y Fh(i)404 1920 y Fm(ln)9 b(cosh\()p Fl(\014)r Fm(\()p Fl(\030)620 1927 y Fh(i)634 1920 y Fl(;)f(t)p Fm(\)\))p Fi(jF)753 1927 y Fh(k)q(;\024)806 1920 y Fm(])i Fi(\000)g Fl(I)-8 b(E)r Fm([)p Fl(\014)963 1902 y Fd(\000)p Fp(1)1020 1877 y Fg(X)1046 1974 y Fh(i)1093 1920 y Fm(ln)8 b(cosh\()p Fl(\014)r Fm(\()p Fl(\030)1308 1927 y Fh(i)1323 1920 y Fl(;)g(t)p Fm(\)\))p Fi(jF)1447 1901 y Fp(+)1442 1934 y Fh(k)q(;\024)1495 1920 y Fm(])138 2036 y(=)k Fl(I)-8 b(E)s Fm([)p Fl(\014)275 2017 y Fd(\000)p Fp(1)331 2036 y Fm(ln)8 b(cosh)q(\()p Fl(\014)r Fm(\()p Fl(\030)547 2043 y Fh(k)568 2036 y Fl(;)g(t)p Fm(\)\))p Fi(jF)687 2043 y Fh(k)q(;\024)740 2036 y Fm(])i Fi(\000)g Fl(I)-8 b(E)r Fm([)p Fl(\014)897 2017 y Fd(\000)p Fp(1)954 2036 y Fm(ln)8 b(cosh\()p Fl(\014)r Fm(\()p Fl(\030)1169 2043 y Fh(k)1191 2036 y Fl(;)g(t)p Fm(\)\))p Fi(jF)1315 2016 y Fp(+)1310 2050 y Fh(k)q(;\024)1363 2036 y Fm(])138 2167 y(=)k Fl(I)-8 b(E)s Fm([)p Fl(\014)275 2148 y Fd(\000)p Fp(1)331 2167 y Fm(ln)8 b(cosh)q(\()p Fl(\014)516 2076 y Fg(0)516 2158 y(@)558 2124 y(X)556 2222 y Fh(\026)p Fd(6)p Fp(=)p Fh(\024)635 2167 y Fl(\030)657 2145 y Fh(\026)655 2181 y(k)681 2167 y Fl(t)697 2174 y Fh(\026)732 2167 y Fm(+)i Fl(\030)799 2148 y Fh(\024)797 2178 y(k)823 2167 y Fl(t)839 2174 y Fh(\024)863 2076 y Fg(1)863 2158 y(A)902 2167 y Fm(\))p Fi(jF)966 2174 y Fh(k)q(;\024)1020 2167 y Fm(])g Fi(\000)g Fl(I)-8 b(E)r Fm([)p Fl(\014)1177 2148 y Fd(\000)p Fp(1)1234 2167 y Fm(ln)8 b(cosh\()p Fl(\014)1418 2076 y Fg(0)1418 2158 y(@)1461 2124 y(X)1458 2222 y Fh(\026)p Fd(6)p Fp(=)p Fh(\024)1537 2167 y Fl(\030)1559 2145 y Fh(\026)1557 2181 y(k)1583 2167 y Fl(t)1599 2174 y Fh(\026)1634 2167 y Fm(+)j Fl(\030)1702 2148 y Fh(\024)1700 2178 y(k)1725 2167 y Fl(t)1741 2174 y Fh(\024)1765 2076 y Fg(1)1765 2158 y(A)1805 2167 y Fm(\))p Fi(jF)1874 2148 y Fp(+)1869 2181 y Fh(k)q(;\024)1922 2167 y Fm(])138 2349 y(=)191 2318 y(1)p 191 2339 23 2 v 191 2380 a(2)219 2349 y Fl(\014)247 2330 y Fd(\000)p Fp(1)296 2349 y Fl(I)-8 b(E)352 2258 y Fg(2)352 2339 y(4)382 2349 y Fm(ln)9 b(cosh\()p Fl(\014)567 2258 y Fg(0)567 2339 y(@)610 2306 y(X)607 2404 y Fh(\026)p Fd(6)p Fp(=)p Fh(\024)686 2349 y Fl(\030)708 2327 y Fh(\026)706 2363 y(k)732 2349 y Fl(t)748 2356 y Fh(\026)783 2349 y Fm(+)i Fl(\030)851 2330 y Fh(\024)849 2360 y(k)874 2349 y Fl(t)890 2356 y Fh(\024)914 2258 y Fg(1)914 2339 y(A)954 2349 y Fm(\))f Fi(\000)g Fm(ln)e(cosh)q(\()p Fl(\014)1212 2258 y Fg(0)1212 2339 y(@)1255 2306 y(X)1252 2404 y Fh(\026)p Fd(6)p Fp(=)p Fh(\024)1331 2349 y Fl(\030)1353 2327 y Fh(\026)1351 2363 y(k)1377 2349 y Fl(t)1393 2356 y Fh(\026)1428 2349 y Fi(\000)i Fl(\030)1495 2330 y Fh(\024)1493 2360 y(k)1519 2349 y Fl(t)1535 2356 y Fh(\024)1559 2258 y Fg(1)1559 2339 y(A)1599 2349 y Fm(\))1617 2310 y Fg(\014)1617 2338 y(\014)1631 2349 y Fi(F)1664 2356 y Fh(k)q(;\024)1718 2258 y Fg(3)1718 2339 y(5)1834 2458 y Fm(\(3)p Fl(:)p Fm(18\))0 2531 y(No)o(w)15 b(w)o(e)f(use)i(the)f(fact)g(that)449 2615 y(cosh)q(\()p Fl(a)9 b Fm(+)i Fl(b)p Fm(\))p 449 2635 221 2 v 449 2677 a(cosh)q(\()p Fl(a)e Fi(\000)i Fl(b)p Fm(\))688 2646 y(=)741 2615 y(1)f(+)h(tanh)c Fl(a)h Fm(tanh)f Fl(b)p 741 2635 327 2 v 741 2677 a Fm(1)j Fi(\000)h Fm(tanh)c Fl(a)h Fm(tanh)f Fl(b)1086 2646 y Fi(\024)1139 2615 y Fm(1)j(+)h(tanh)c Fi(j)p Fl(b)p Fi(j)p 1139 2635 222 2 v 1139 2677 a Fm(1)j Fi(\000)h Fm(tanh)c Fi(j)p Fl(b)p Fi(j)1379 2646 y(\024)13 b Fl(e)1448 2627 y Fp(2)p Fd(j)p Fh(b)p Fd(j)1834 2646 y Fm(\(3)p Fl(:)p Fm(19\))0 2779 y Ff(31)p Fe(=mar)q(ch=)p Ff(1995;)k(9:46)652 b Fm(10)p eop %%Page: 11 12 bop 0 45 a Fm(to)15 b(deduce)h(from)f(\(3.18\))e(that)844 118 y Fi(j)867 106 y Fm(~)857 118 y Fl(f)884 95 y Fp(\()p Fh(k)q(;\024)p Fp(\))879 132 y Fh(N)967 118 y Fi(j)f(\024)h(j)p Fl(t)1069 125 y Fh(\024)1093 118 y Fi(j)728 b Fm(\(3)p Fl(:)p Fm(20\))0 223 y(Using)16 b(the)f(standard)g(inequalities)j Fl(e)656 206 y Fh(x)691 223 y Fi(\024)13 b Fm(1)d(+)h Fl(x)f Fm(+)905 205 y Fh(x)926 191 y Ff(2)p 905 212 39 2 v 915 238 a Fp(2)948 223 y Fl(e)969 206 y Fd(j)p Fh(x)p Fd(j)1029 223 y Fm(and)16 b(1)9 b(+)i Fl(y)j Fi(\024)f Fl(e)1301 206 y Fh(y)1339 223 y Fm(w)o(e)i(get)f (therefore)562 365 y Fl(I)-8 b(E)618 315 y Fg(h)639 365 y Fl(e)660 346 y Fh(u)689 338 y Fp(~)681 346 y Fh(f)703 330 y Ff(\()p Fe(k;\024)p Ff(\))699 358 y Fe(N)777 346 y Fp(\()7 b(~)-25 b Fh(m)p Fp(\))840 327 y Fg(\014)840 354 y(\014)855 365 y Fi(F)893 346 y Fp(+)888 379 y Fh(k)q(;\024)942 315 y Fg(i)976 365 y Fi(\024)13 b Fm(exp)1101 301 y Fg(\022)1140 334 y Fl(u)1166 318 y Fp(2)p 1140 355 47 2 v 1152 396 a Fm(2)1192 365 y Fl(t)1208 346 y Fp(2)1208 376 y Fh(\024)1232 365 y Fl(e)1253 346 y Fd(j)p Fh(u)p Fd(jj)p Fh(t)1321 351 y Fe(\024)1342 346 y Fd(j)1355 301 y Fg(\023)1834 365 y Fm(\(3)p Fl(:)p Fm(21\))0 502 y(F)l(rom)h(this)i (and)f(\(3.17\))f(w)o(e)h(get)f(no)o(w)288 627 y Fl(I)-8 b(P)14 b Fm([)p Fi(j)p Fm(\011\()p Fl(m;)8 b(t)p Fm(\))h Fi(\000)h Fl(I)-8 b(E)r Fm(\011\()p Fl(m;)8 b(t)p Fm(\))p Fi(j)k Fl(>)h(x)p Fm(])f Fi(\024)h Fm(2)8 b(inf)986 654 y Fh(u)1032 627 y Fl(e)1053 608 y Fd(\000)p Fh(uN)s(x)p Fp(+)1189 596 y Fe(u)1208 587 y Ff(2)p 1189 601 37 2 v 1200 619 a(2)1232 608 y Fh(N)s Fd(k)p Fh(t)p Fd(k)1316 594 y Ff(2)1316 617 y(2)1333 608 y Fh(e)1350 594 y Fc(j)p Fe(u)p Fc(jk)p Fe(t)p Fc(k)1436 599 y(1)891 763 y Fi(\024)939 685 y Fg(\()983 749 y Fm(2)p Fl(e)1027 713 y Fd(\000)p Fh(N)1107 701 y Fe(x)1125 692 y Ff(2)p 1093 706 65 2 v 1093 730 a Fc(k)p Fe(t)p Fc(k)1140 722 y Ff(2)1140 744 y(2)1163 716 y Fm(\()1180 713 y Fp(1)p Fd(\000)1232 701 y Ff(1)p 1232 706 16 2 v 1232 724 a(2)1253 713 y Fh(e)1270 699 y Fe(x=)p Fc(k)p Fe(t)p Fc(k)1352 706 y Ff(2)1372 716 y Fm(\))1391 749 y Fl(;)46 b Fm(if)15 b Fi(k)p Fl(t)p Fi(k)1553 756 y Fp(2)1586 749 y Fi(\025)e Fm(1)983 815 y(2)p Fl(e)1027 796 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1345 y Fp(\()p Fh(R)p Fp(\))770 1305 y Fi(j)p Fm(\011\()p Fl(m;)8 b(t)p Fm(\))h Fi(\000)i Fl(I)-8 b(E)r Fm(\011\()p Fl(m;)8 b(t)p Fm(\))p Fi(j)k(\024)h Fl(\013)1284 1286 y Fp(1)p Fh(=)p Fp(2)p Fd(\000)p Fh(\016)1834 1305 y Fm(\(3)p Fl(:)p Fm(23\))0 1449 y(By)i(Lemma)h(3.3,)357 1586 y Fl(I)-8 b(P)7 b Fm([\012)451 1593 y Fp(2)471 1586 y Fm(])12 b Fi(\025)h Fm(1)d Fi(\000)g Fm(exp)699 1522 y Fg(\022)733 1586 y Fi(\000)p Fl(N)815 1555 y(\013)844 1539 y Fp(1)p Fd(\000)p Fp(2)p Fh(\016)p 815 1575 114 2 v 855 1617 a Fl(R)935 1586 y Fm(\(1)f Fi(\000)1036 1555 y Fm(1)p 1036 1575 23 2 v 1036 1617 a(2)1064 1586 y Fl(e)1085 1567 y Fh(\013)1109 1553 y Ff(1)p Fe(=)p Ff(2)p Fc(\000)p Fe(\016)1199 1567 y Fh(=R)1248 1586 y Fm(\))g(+)i Fl(N)5 b(\013)i Fm(ln)q(\()p Fl(R=\013)p Fm(\))1560 1522 y Fg(\023)496 1688 y Fm(=)13 b(1)d Fi(\000)g Fm(exp)699 1651 y Fg(\000)720 1688 y Fi(\000)p Fl(N)5 b(O)q Fm(\()p Fl(\013)880 1669 y Fp(1)p Fd(\000)p Fp(2)p Fh(\016)965 1688 y Fl(=R)p Fm(\))1041 1651 y Fg(\001)1834 1622 y Fm(\(3)p Fl(:)p Fm(24\))0 1797 y(Com)o(bining)16 b(Lemma)g(3.2)e(with)h(\(3.23\))f (and)h(taking)h(in)o(to)f(accoun)o(t)g(the)g(remark)g(preceeding)i(Lemma)e (3.3,)f(w)o(e)0 1870 y(see)h(that)g(on)g(\012)268 1877 y Fp(1)298 1870 y Fi(\\)c Fm(\012)372 1877 y Fp(2)392 1870 y Fm(,)66 1997 y(sup)23 2037 y Fh(t)6 b Fp(:)p Fd(k)p Fh(t)p Fd(k)105 2042 y Ff(2)122 2037 y Fd(\024)p Fh(R)185 1997 y Fi(j)p Fm(\011\()p Fl(m;)i(t)p Fm(\))d Fi(\000)g Fl(I)-8 b(E)r Fm(\011\()p Fl(m;)8 b(t)p Fm(\))p Fi(j)k(\024)h Fl(\013)689 1978 y Fp(1)p Fh(=)p Fp(2)p Fd(\000)p Fh(\016)797 1997 y Fm(+)5 b(2)860 1962 y Fi(p)p 899 1962 30 2 v 899 1997 a Fl(\013)p Fm(\()p Fi(k)p Fl(m)p Fi(k)1032 2004 y Fp(2)1057 1997 y Fm(+)g(\(1)g(+)1183 1962 y Fi(p)p 1221 1962 V 35 x Fl(\013)q Fm(\)\(1)g(+)g Fl(\017)p Fm(\)\))12 b Fi(\024)h Fl(\013)1498 1978 y Fp(1)p Fh(=)p Fp(2)p Fd(\000)p Fh(\016)1601 1997 y Fm(\(6)5 b(+)g Fi(k)p Fl(m)p Fi(k)1773 2004 y Fp(2)1793 1997 y Fm(\))23 b(\(3)p Fl(:)p Fm(25\))0 2141 y(for)15 b Fl(\013)g Fm(small,)h(whic)o(h)g(pro)o(v)o(es)e(Prop)q (osition)i(3.1.)p Fi(})0 2779 y Ff(31)p Fe(=mar)q(ch=)p Ff(1995;)h(9:46)652 b Fm(11)p eop %%Page: 12 13 bop 0 45 a Fq(4.)27 b(Pro)r(of)19 b(of)h(the)h(Theorem)91 155 y Fm(The)16 b(results)g(of)g(Sections)g(2.1)f(and)h(3.1)f(can)h(no)o(w)f(b)q (e)i(com)o(bined)g(to)e(get)g(a)h(large)g(deviation)g(principle)j(in)0 227 y(the)e(pro)q(duct)h(top)q(ology)l(.)26 b(The)18 b(p)q(oin)o(t)g(here)f (is)h(that,)f(apart)g(from)f(the)i(p)q(ossibilit)o(y)h(that)e Fl(t)1588 211 y Fd(\003)1609 227 y Fm(\()p Fl(m)p Fm(\))g(ma)o(y)f(b)q(ecome) 0 300 y(un)o(b)q(ounded,)g(the)f(estimates)f(in)h(Lemma)g(2.1)f(and)h(Lemma)f (2.2)g(together)g(with)h(Prop)q(osition)g(3.1)e(tell)j(us)f(that)0 373 y(up)k(to)e(corrections)h(that)g(tend)g(to)g(zero)g(with)g Fl(N)5 b Fm(,)18 b(the)h(quan)o(tit)o(y)e(\()p Fl(\014)r(N)5 b Fm(\))1277 356 y Fd(\000)p Fp(1)1333 373 y Fm(ln)j Fl(Z)1409 387 y Fh(N)q(;\014)r(;)1484 360 y Fd(p)p 1513 360 42 2 v 1513 387 a Fp(2)p 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b(N)21 b Fk(and)16 b(for)h(al)r(l)25 b Fm(~)-31 b Fl(m)13 b Fi(2)g Fm([)p Fi(\000)p Fm(1)p Fl(;)8 b Fm(1])1734 684 y Fh(I)1770 700 y Fk(such)16 b(that)0 773 y(for)h(al)r(l)f Fl(\017)d(>)g Fm(0)j Fk(ther)n(e)g(exists)f Fl(c)e Fm(=)g Fl(c)p Fm(\()8 b(~)-31 b Fl(m)o(;)8 b(\017)p Fm(\))k Fl(<)h Fi(1)p Fk(,)k Fi(9)p Fl(N)906 780 y Fp(0)938 773 y Fi(\024)c(1)p Fk(,)p Fi(8)p Fl(N)18 b Fi(\025)13 b Fl(N)1210 780 y Fp(0)1230 773 y Fk(,)319 892 y Fm(sup)210 932 y Fh(m)p Fp(:)7 b Fd(k)p Fp(\005)305 937 y Fe(I)323 932 y Fh(m)p Fd(\000)h Fp(~)-26 b Fh(m)q Fd(k)435 937 y Ff(2)452 932 y Fd(\024)p Fh(\017)530 892 y Fm(inf)503 925 y Fh(t)p Fd(2)p Fh(I)-8 b(R)577 915 y Fe(M)619 892 y Fl(I)g(E)r Fm(\011)702 899 y Fh(N)q(;\014)768 892 y Fm(\()p Fl(m;)8 b(t)p Fm(\))j(=)122 b(sup)940 932 y Fh(m)p Fp(:)7 b Fd(k)p Fp(\005)1035 937 y Fe(I)1053 932 y Fh(m)p Fd(\000)h Fp(~)-26 b Fh(m)q Fd(k)1165 937 y Ff(2)1182 932 y Fd(\024)p Fh(\017)1325 892 y Fm(inf)1233 925 y Fh(t)p Fd(2)p Fh(I)-8 b(R)1307 915 y Fe(M)1341 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Fm(for)g(whic)o(h)h(it)g (is)g(satis\014ed)g(can)g(b)q(e)g(constructed)f(explicitly)0 1740 y(and)g(is)h(nothing)g(else)g(than)f Fl(D)531 1748 y Fd(j)p Fh(I)r Fd(j)573 1740 y Fm(.)0 1849 y Fa(Pro)q(of:)23 b Fm(W)l(e)15 b(will)i(\014rst)e(establish)h(an)f(upp)q(er)h(b)q(ound)g(for)f(the)g(quan)o (tit)o(y)427 1968 y Fl(Z)461 1949 y Fh(I)458 1979 y(N)q(;\014)r(;\017)549 1968 y Fm([)p Fl(!)r Fm(]\()8 b(~)-31 b Fl(m)o Fm(\))12 b Fi(\021)h Fl(\026)767 1975 y Fh(N)q(;\014)833 1968 y Fm([)p Fl(!)r Fm(])8 b(\()o Fi(k)p Fm(\005)971 1975 y Fh(I)991 1968 y Fl(m)1031 1975 y Fh(N)1066 1968 y Fm(\()p Fl(\033)r Fm(\))h Fi(\000)19 b Fm(~)-32 b Fl(m)p Fi(k)1247 1975 y Fp(2)1280 1968 y Fi(\024)13 b Fl(\017)p Fm(\))8 b Fl(Z)1403 1975 y Fh(N)q(;\014)1468 1968 y Fm([)p Fl(!)r Fm(])333 b(\(4)p Fl(:)p Fm(3\))0 2087 y(T)l(o)16 b(do)f(so,)g(notice)i(that)e(on)h(\012)531 2094 y Fp(1)551 2087 y Fm(,)f Fi(k)p Fl(m)642 2094 y Fh(N)677 2087 y Fm(\()p Fl(\033)r Fm(\))p Fi(k)764 2094 y Fp(2)796 2087 y Fi(\024)f 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y Fm(\(4)p Fl(:)p Fm(4\))0 2779 y Ff(31)p Fe(=mar)q(ch=)p Ff(1995;)17 b(9:46)652 b Fm(12)p eop %%Page: 13 14 bop 0 45 a Fm(As)14 b(long)h(as)f Fl(\013)f Fi(#)g Fm(0,)g(the)i(factor)e Fl(e)574 29 y Fh(\013N)s Fp(\(ln)6 b(2)p Fh(=\013)p Fp(\))772 45 y Fm(in)16 b(the)e(upp)q(er)i(b)q(ound)f(is)g(irrelev)m(an)o(t)h(for)e (the)g(exp)q(onen)o(tial)i(asymp-)0 118 y(totic,)e(as)f(is)i(the)f (di\013erence)h(b)q(et)o(w)o(een)g Fl(\017)f Fm(and)g Fl(\017)9 b Fi(\000)f Fl(\032)p Fm(.)19 b(Using)14 b(the)g(estimates)g(used)h(in)g(the) f(pro)q(of)g(of)f(Lemma)h(2.1,)0 191 y(w)o(e)h(can)g(replace)h Fl(Z)335 198 y Fh(N)q(;\014)r(;\032)431 191 y Fm([)p Fl(!)r Fm(]\()p Fl(m)p Fm(\))d(in)j(\(4.4\))e(b)o(y)h(its)g(upp)q(er)i(b)q(ound)f (in)g(terms)e(of)h(the)g(function)h(\011.)21 b(Namely)l(,)126 282 y(1)p 103 302 70 2 v 103 344 a Fl(\014)r(N)186 313 y Fm(ln)8 b Fl(Z)265 294 y Fh(I)262 324 y(N)q(;\014)r(;\017)354 313 y Fm([)p Fl(!)r Fm(]\()f(~)-30 b Fl(m)n Fm(\))13 b Fi(\024)89 b Fm(sup)575 346 y Fe(m)p 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Fl(;)g(\030)1512 2637 y Fh(i)1526 2630 y Fm(\)\))h Fi(\000)i Fl(m)1657 2611 y Fd(\003)1657 2641 y Fh(\027)1679 2552 y Fg(!)1715 2562 y Fp(2)1857 2359 y Fm(\(4)p Fl(:)p Fm(9\))0 2779 y Ff(31)p Fe(=mar)q(ch=)p Ff(1995;)17 b(9:46)652 b Fm(13)p eop %%Page: 14 15 bop 0 45 a Fm(The)16 b(\014rst)g(summand)g(in)g(\(4.7\))f(is)h(b)q(ounded)h (b)o(y)f Fl(\013)p Fm(,)g(and)g(w)o(e)g(ha)o(v)o(e)f(to)g(con)o(trol)h(the)g (second.)22 b(T)l(o)16 b(do)g(so)f(w)o(e)h(use)0 118 y(\(4.8\))e(to)g(write) 394 205 y Fg(X)417 300 y Fh(\027)467 170 y Fg( )518 217 y Fm(1)p 509 238 42 2 v 509 279 a Fl(N)563 205 y Fg(X)590 302 y Fh(i)636 248 y Fl(\030)658 229 y Fh(\027)656 259 y(i)689 248 y Fm(tanh\()p Fl(\014)r Fm(\()p Fl(t)860 229 y Fd(\003)880 248 y Fl(;)8 b(\030)921 255 y Fh(i)936 248 y Fm(\)\))h Fi(\000)h Fl(m)1066 229 y Fd(\003)1066 259 y Fh(\027)1089 170 y Fg(!)1125 180 y Fp(2)351 413 y Fm(=)394 370 y Fg(X)417 466 y Fh(\027)467 336 y Fg( )518 383 y Fm(1)p 509 403 V 509 445 a Fl(N)563 370 y Fg(X)590 467 y Fh(i)636 413 y Fl(\030)658 395 y Fh(\027)656 425 y(i)689 413 y Fm(tanh\()p Fl(\014)r Fm(\()p Fl(t)860 395 y Fd(\003)880 413 y Fl(;)e(\030)921 420 y Fh(i)936 413 y Fm(\)\))h Fi(\000)h Fl(I)-8 b(E)s(\030)1097 395 y Fh(\027)1095 425 y Fp(1)1127 413 y Fm(tanh\()p Fl(\014)r Fm(\()p Fl(\030)1302 420 y Fp(1)1322 413 y Fl(;)8 b(t)1359 395 y Fd(\003)1379 413 y Fm(\)\))1415 336 y Fg(!)1451 345 y Fp(2)351 579 y Fm(=)394 535 y Fg(X)417 631 y Fh(\027)467 501 y Fg( )518 548 y Fm(1)p 509 568 V 509 610 a Fl(N)563 535 y Fg(X)590 632 y Fh(i)636 579 y Fl(\030)658 560 y Fh(\027)656 590 y(i)689 579 y Fm(tanh\()p Fl(\014)r Fm(\()p Fl(t)860 560 y Fd(\003)880 579 y Fl(;)g(\030)921 586 y Fh(i)936 579 y Fm(\)\))h Fi(\000)h Fl(I)-8 b(E)1090 548 y Fm(1)p 1080 568 V 1080 610 a Fl(N)1135 535 y Fg(X)1161 632 y Fh(i)1208 579 y Fl(\030)1230 560 y Fh(\027)1228 590 y(i)1260 579 y Fm(tanh\()p Fl(\014)r Fm(\()p Fl(t)1431 560 y Fd(\003)1452 579 y Fl(;)8 b(\030)1493 586 y Fh(i)1507 579 y Fm(\)\))1543 501 y Fg(!)1578 510 y Fp(2)399 694 y Fi(\021)13 b Fl(G)483 701 y Fh(N)517 694 y Fm(\()p Fl(t)551 675 y Fd(\003)572 694 y Fm(\))1834 446 y(\(4)p Fl(:)p Fm(10\))0 773 y(W)l(e)e(will)i(no)o(w)d(pro)o(v)o(e,)h(in)h(analogy)f(to)f(Prop)q (osition)i(3.1,)e(that)g Fl(G)1095 780 y Fh(N)1130 773 y Fm(\()p Fl(t)p Fm(\))g(is)i(actually)g(small)f(with)h(large)f(probabilit)o(y)l(.)0 846 y(This)j(will)h(b)q(e)f(sligh)o(tly)g(more)f(complicated)i(than)e(in)h (Prop)q(osition)g(3.1)e(and)h(will,)i(in)f(fact)f(consist)h(of)e(t)o(w)o(o)g (steps.)0 919 y(The)18 b(\014rst)f(is)g(a)h(fairly)g(crude)g(b)q(ound)g(on)f Fl(G)770 926 y Fh(N)804 919 y Fm(\()p Fl(t)p Fm(\))g(that)g(in)h(a)f(second)h (step)g(will)h(b)q(e)f(used)g(to)e(obtain)i(a)f(re\014ned)0 992 y(one.)0 1101 y Fb(Lemma)g(4.2:)45 b Fk(F)m(or)15 b(al)r(l)i Fl(!)d Fi(2)f Fm(\012)606 1108 y Fp(1)626 1101 y Fk(,)845 1173 y Fl(G)881 1180 y Fh(N)915 1173 y Fm([)p Fl(!)r Fm(]\()p Fl(t)p Fm(\))f Fi(\024)h Fm(6)728 b(\(4)p Fl(:)p Fm(11\))0 1374 y Fa(Pro)q(of:)23 b Fm(Let)15 b(us)g(for)g(notational)g(simplicit)o(y)j(set)d Fl(T)897 1381 y Fh(i)924 1374 y Fi(\021)e Fm(tanh\()p Fl(\014)r Fm(\()p Fl(\030)1147 1381 y Fh(i)1161 1374 y Fl(;)8 b(t)p Fm(\)\).)19 b(W)l(e)c(ha)o(v)o(e)g(that)444 1508 y Fl(G)480 1515 y Fh(N)515 1508 y Fm(\()p Fl(t)p Fm(\))d Fi(\024)h Fm(2)672 1452 y Fh(M)658 1465 y Fg(X)658 1561 y Fh(\026)p Fp(=1)733 1417 y Fg(0)733 1499 y(@)773 1431 y(")814 1478 y Fm(1)p 805 1498 V 805 1540 a Fl(N)859 1465 y Fg(X)886 1562 y Fh(i)932 1508 y Fl(\030)954 1487 y Fh(\026)952 1521 y(i)979 1508 y Fl(T)1006 1515 y Fh(i)1020 1431 y Fg(#)1047 1440 y Fp(2)1077 1508 y Fm(+)1123 1431 y Fg(")1164 1478 y Fm(1)p 1155 1498 V 1155 1540 a Fl(N)1209 1465 y Fg(X)1236 1562 y Fh(i)1283 1508 y Fl(I)-8 b(E)r(\030)1353 1487 y Fh(\026)1351 1521 y(i)1377 1508 y Fl(T)1404 1515 y Fh(i)1419 1431 y Fg(#)1446 1440 y Fp(2)1466 1417 y Fg(1)1466 1499 y(A)579 1680 y Fm(=)652 1649 y(2)p 632 1670 62 2 v 632 1711 a Fl(N)674 1698 y Fp(2)722 1623 y Fh(M)708 1637 y Fg(X)707 1733 y Fh(\026)p Fp(=1)783 1637 y Fg(X)796 1734 y Fh(i;j)856 1643 y Fg(\000)877 1680 y Fl(\030)899 1658 y Fh(\026)897 1693 y(i)923 1680 y Fl(\030)945 1658 y Fh(\026)943 1693 y(j)970 1680 y Fl(T)997 1687 y Fh(i)1011 1680 y Fl(T)1038 1687 y Fh(j)1067 1680 y Fm(+)11 b Fl(I)-8 b(E)r Fm(\()p Fl(\030)1201 1658 y Fh(\026)1199 1693 y(i)1225 1680 y Fl(T)1252 1687 y Fh(i)1267 1680 y Fm(\))p Fl(I)g(E)r Fm(\()p Fl(\030)1373 1658 y Fh(\026)1371 1693 y(j)1397 1680 y Fl(T)1424 1687 y Fh(j)1443 1680 y Fm(\))1461 1643 y Fg(\001)1834 1591 y Fm(\(4)p Fl(:)p Fm(12\))0 1806 y(F)l(or)15 b(the)g(\014rst)g(term,)f (w)o(e)h(can)g(use)h(simply)g(that)450 1897 y(2)p 430 1917 V 430 1959 a Fl(N)472 1946 y Fp(2)520 1871 y Fh(M)507 1884 y Fg(X)505 1981 y Fh(\026)p Fp(=1)581 1884 y Fg(X)594 1981 y Fh(i;j)654 1928 y Fl(\030)676 1906 y Fh(\026)674 1940 y(i)701 1928 y Fl(\030)723 1906 y Fh(\026)721 1940 y(j)747 1928 y Fl(T)774 1935 y Fh(i)789 1928 y Fl(T)816 1935 y Fh(j)847 1928 y Fi(\024)d Fm(2)926 1862 y Fg(\015)926 1889 y(\015)926 1916 y(\015)926 1944 y(\015)956 1897 y Fl(\030)r(\030)1000 1880 y Fh(t)p 956 1917 60 2 v 965 1959 a Fl(N)1021 1862 y Fg(\015)1021 1889 y(\015)1021 1916 y(\015)1021 1944 y(\015)1054 1850 y( )1105 1897 y Fm(1)p 1096 1917 42 2 v 1096 1959 a Fl(N)1150 1884 y Fg(X)1177 1981 y Fh(i)1223 1928 y Fl(T)1256 1909 y Fp(2)1250 1939 y Fh(i)1277 1850 y Fg(!)1325 1928 y Fi(\024)g Fm(2)1404 1862 y Fg(\015)1404 1889 y(\015)1404 1916 y(\015)1404 1944 y(\015)1434 1897 y Fl(\030)r(\030)1478 1880 y Fh(t)p 1434 1917 60 2 v 1444 1959 a Fl(N)1500 1862 y Fg(\015)1500 1889 y(\015)1500 1916 y(\015)1500 1944 y(\015)1834 1928 y Fm(\(4)p Fl(:)p Fm(13\))0 2059 y(But)h(on)h(\012)185 2066 y Fp(1)205 2059 y Fm(,)f(the)g(norm)g(in)h(the)f(last)g(line)i(is)f(b)q (ounded)h(b)o(y)e(\(1)8 b(+)1110 2026 y Fi(p)p 1148 2026 30 2 v 33 x Fl(\013)p Fm(\))1195 2042 y Fp(2)1215 2059 y Fm(\(1)g(+)g Fl(\017)p Fm(\).)20 b(T)l(o)14 b(b)q(ound)h(the)g(second)f(term)g(in)0 2132 y(\(4.12\),)f(w)o(e)i(use)g(the)h(indep)q(endence)i(of)d(b)q(oth)g Fl(\030)830 2110 y Fh(\026)828 2144 y(i)870 2132 y Fm(and)g Fl(T)985 2139 y Fh(i)1015 2132 y Fm(for)g(di\013eren)o(t)g(indices)i Fl(i)e Fm(to)g(write)428 2226 y(2)p 408 2247 62 2 v 408 2288 a Fl(N)450 2275 y Fp(2)498 2200 y Fh(M)484 2214 y Fg(X)483 2310 y Fh(\026)p Fp(=1)559 2214 y Fg(X)571 2311 y Fh(i;j)632 2257 y Fl(I)-8 b(E)r Fm(\()p Fl(\030)720 2235 y Fh(\026)718 2270 y(i)744 2257 y Fl(T)771 2264 y Fh(i)786 2257 y Fm(\))p Fl(I)g(E)r Fm(\()p Fl(\030)892 2235 y Fh(\026)890 2270 y(j)916 2257 y Fl(T)943 2264 y Fh(j)962 2257 y Fm(\))12 b(=)1065 2226 y(2)p 1046 2247 V 1046 2288 a Fl(N)1088 2275 y Fp(2)1136 2200 y Fh(M)1122 2214 y Fg(X)1121 2310 y Fh(\026)p Fp(=1)1196 2214 y Fg(X)1209 2311 y Fh(i;j)1269 2257 y Fl(I)-8 b(E)1325 2220 y Fg(\000)1346 2257 y Fl(\030)1368 2235 y Fh(\026)1366 2270 y(i)1393 2257 y Fl(T)1420 2264 y Fh(i)1434 2257 y Fl(\030)1456 2235 y Fh(\026)1454 2270 y(j)1481 2257 y Fl(T)1508 2264 y Fh(j)1526 2220 y Fg(\001)413 2423 y Fm(+)483 2392 y(2)p 464 2412 V 464 2454 a Fl(N)506 2441 y Fp(2)554 2366 y Fh(M)540 2380 y Fg(X)539 2476 y Fh(\026)p Fp(=1)614 2380 y Fg(X)641 2476 y Fh(i)688 2386 y Fg(\000)708 2423 y Fm(\()p Fl(I)g(E)r(\030)796 2401 y Fh(\026)794 2435 y(i)821 2423 y Fl(T)848 2430 y Fh(i)863 2423 y Fm(\))881 2404 y Fp(2)911 2423 y Fi(\000)10 b Fl(I)-8 b(E)s Fm(\()p Fl(T)1050 2430 y Fh(i)1064 2423 y Fm(\))1082 2404 y Fp(2)1102 2386 y Fg(\001)415 2572 y Fi(\024)13 b Fm(2)p Fl(I)-8 b(E)542 2506 y Fg(\015)542 2533 y(\015)542 2560 y(\015)542 2588 y(\015)573 2541 y Fl(\030)r(\030)617 2524 y Fh(t)p 573 2561 60 2 v 582 2603 a Fl(N)638 2506 y Fg(\015)638 2533 y(\015)638 2560 y(\015)638 2588 y(\015)674 2572 y Fm(+)724 2541 y(2)p Fl(M)p 724 2561 72 2 v 740 2603 a(N)415 2672 y Fi(\024)13 b Fm(2)p Fl(\013)d Fm(+)h(2\(1)e(+)690 2637 y Fi(p)p 727 2637 30 2 v 727 2672 a Fl(\013)q Fm(\))775 2653 y Fp(2)795 2672 y Fm(\(1)g(+)i Fl(\017)909 2653 y Fd(0)922 2672 y Fm(\))1834 2442 y(\(4)p Fl(:)p Fm(14\))0 2779 y Ff(31)p Fe(=mar)q(ch=)p Ff(1995;)17 b(9:46)652 b Fm(14)p eop %%Page: 15 16 bop 0 45 a Fm(Com)o(bining)16 b(these)f(t)o(w)o(o)f(b)q(ounds)i(w)o(e)f(get)g (\(4.11\).)p Fi(})91 155 y Fm(Lemma)f(4.2)f(tells)i(us)f(that)f Fl(G)609 162 y Fh(N)644 155 y Fm(\()p Fl(t)p Fm(\))g(is)i(b)q(ounded,)g(but)f (not)g(y)o(et)f(that)h(it)g(is)g(small.)21 b(T)l(o)13 b(do)h(this,)g(w)o(e)g (observ)o(e)0 227 y(\014rst)h(that)f(its)i(mean)f(v)m(alue)h(is)g(small.)0 336 y Fb(Lemma)h(4.3:)803 417 y Fm(0)c Fi(\024)g Fl(I)-8 b(E)r(G)971 424 y Fh(N)1005 417 y Fm(\()p Fl(t)p Fm(\))12 b Fi(\024)h Fl(\013)688 b Fm(\(4)p Fl(:)p Fm(15\))0 638 y Fa(Pro)q(of:)125 794 y Fl(I)-8 b(E)r(G)209 801 y Fh(N)243 794 y Fm(\()p Fl(t)p Fm(\))13 b(=)370 737 y Fh(M)356 751 y Fg(X)356 847 y Fh(\027)r Fp(=1)430 794 y Fl(I)-8 b(E)485 716 y Fg(")527 763 y Fm(1)p 517 783 42 2 v 517 825 a Fl(N)572 751 y Fg(X)598 847 y Fh(i)645 794 y Fl(\030)667 775 y Fh(\027)665 805 y(i)697 794 y Fm(tanh\()p Fl(\014)r Fm(\()p Fl(t;)8 b(\030)909 801 y Fh(i)923 794 y Fm(\)\))i Fi(\000)g Fl(I)-8 b(E)1077 763 y Fm(1)p 1068 783 V 1068 825 a Fl(N)1123 751 y Fg(X)1149 847 y Fh(i)1196 794 y Fl(\030)1218 775 y Fh(\027)1216 805 y(i)1248 794 y Fm(tanh\()p Fl(\014)r Fm(\()p Fl(t;)8 b(\030)1460 801 y Fh(i)1474 794 y Fm(\)\))1510 716 y Fg(#)1536 725 y Fp(2)308 963 y Fm(=)370 906 y Fh(M)356 920 y Fg(X)356 1016 y Fh(\027)r Fp(=1)430 872 y Fg(0)430 954 y(@)469 963 y Fl(I)-8 b(E)525 886 y Fg(")567 933 y Fm(1)p 557 953 V 557 994 a Fl(N)612 920 y Fg(X)638 1017 y Fh(i)685 963 y Fl(\030)707 945 y Fh(\027)705 975 y(i)737 963 y Fm(tanh\()p Fl(\014)r Fm(\()p Fl(t;)8 b(\030)949 970 y Fh(i)963 963 y Fm(\)\))999 886 y Fg(#)1025 895 y Fp(2)1055 963 y Fi(\000)1101 886 y Fg(")1142 933 y Fm(1)p 1133 953 V 1133 994 a Fl(N)1187 920 y Fg(X)1214 1017 y Fh(i)1261 963 y Fl(I)-8 b(E)r(\030)1331 945 y Fh(\027)1329 975 y(i)1361 963 y Fm(tanh\()p Fl(\014)r Fm(\()p Fl(t;)8 b(\030)1573 970 y Fh(i)1587 963 y Fm(\)\))1623 886 y Fg(#)1649 895 y Fp(2)1669 872 y Fg(1)1669 954 y(A)308 1135 y Fm(=)370 1078 y Fh(M)356 1092 y Fg(X)356 1188 y Fh(\027)r Fp(=1)430 1057 y Fg( )491 1104 y Fm(1)p 471 1125 62 2 v 471 1166 a Fl(N)513 1153 y Fp(2)546 1092 y Fg(X)572 1189 y Fh(i)619 1135 y Fl(I)-8 b(E)10 b Fm(tanh)766 1115 y Fp(2)786 1135 y Fm(\()p Fl(\014)r Fm(\()p Fl(t;)e(\030)907 1142 y Fh(i)921 1135 y Fm(\)\))i Fi(\000)1037 1104 y Fm(1)p 1018 1125 V 1018 1166 a Fl(N)1060 1153 y Fp(2)1093 1092 y Fg(X)1119 1189 y Fh(i)1166 1135 y Fm([)p Fl(I)-8 b(E)r(\030)1249 1116 y Fh(\027)1247 1146 y(i)1279 1135 y Fm(tanh\()p Fl(\014)r Fm(\()p Fl(t;)8 b(\030)1491 1142 y Fh(i)1505 1135 y Fm(\)\)])1553 1112 y Fp(2)1573 1057 y Fg(!)308 1273 y Fi(\024)361 1242 y Fl(M)p 361 1263 50 2 v 365 1304 a(N)1834 1016 y Fm(\(4)p Fl(:)p Fm(16\))0 1403 y(where)15 b(w)o(e)g(ha)o(v)o(e)g(used)h(the)f(indep)q(endence)k(of)c (the)g(summands)g(for)g(di\013eren)o(t)g(indices)i Fl(i)p Fm(.)j Fi(})91 1512 y Fm(In)11 b(the)f(sequel)i(w)o(e)e(will)i(need)f(that)f(the)g (mean)h(v)m(alue)g(of)f Fl(G)1070 1519 y Fh(N)1104 1512 y Fm(\()p Fl(t)p Fm(\))g(do)q(es)h(not)f(di\013er)h(m)o(uc)o(h)f(from)g(its)g (conditional)0 1585 y(exp)q(ectation)16 b(relativ)o(e)g(to)e(\012)495 1592 y Fp(1)515 1585 y Fm(.)20 b(Namely)l(,)578 1720 y Fi(j)p Fl(I)-8 b(E)r(G)675 1727 y Fh(N)709 1720 y Fm(\()p Fl(t)p Fm(\))10 b Fi(\000)g Fl(I)-8 b(E)s Fm([)p Fl(G)914 1727 y Fh(N)948 1720 y Fm(\()p Fl(t)p Fm(\))p Fi(j)p Fm(\012)1046 1727 y Fp(1)1065 1720 y Fm(])p Fi(j)12 b(\024)h Fm(2)p Fl(M)5 b(e)1244 1701 y Fd(\000)p Fh(\017N)1319 1687 y Ff(1)p Fe(=)p Ff(6)1834 1720 y Fm(\(4)p Fl(:)p Fm(17\))0 1855 y(is)16 b(arbitrarily)g(small.)91 1964 y(Finally)l(,)24 b(w)o(e)d(will)i(sho)o(w)d(that)h(on)g(\012)752 1971 y Fp(1)772 1964 y Fm(,)i(with)e(large)g(probabilit)o(y)l(,)j Fl(G)1319 1971 y Fh(N)1353 1964 y Fm(\()p Fl(t)p Fm(\))d(di\013ers)h(only)g (little)g(from)f(its)0 2036 y(conditional)c(exp)q(ectation)f(relativ)o(e)g (to)e(\012)730 2043 y Fp(1)750 2036 y Fm(.)0 2145 y Fb(Lemma)j(4.4:)45 b Fk(Assume)15 b(that)i Fl(x)c Fi(\035)g Fm(\(ln)8 b Fl(N)d Fm(\))p Fl(=)843 2107 y Fi(p)p 880 2107 42 2 v 880 2145 a Fl(N)t Fk(.)21 b(Then,)514 2280 y Fl(I)-8 b(P)569 2244 y Fg(\002)587 2280 y Fi(j)p Fl(G)636 2287 y Fh(N)670 2280 y Fm(\()p Fl(t)p Fm(\))10 b Fi(\000)h Fl(I)-8 b(E)r Fm([)p Fl(G)875 2287 y Fh(N)909 2280 y Fm(\()p Fl(t)p Fm(\))p Fi(j)p Fm(\012)1007 2287 y Fp(1)1026 2280 y Fm(])p Fi(j)12 b(\025)h Fl(x)1138 2242 y Fg(\014)1138 2269 y(\014)1153 2280 y Fm(\012)1186 2287 y Fp(1)1206 2244 y Fg(\003)1238 2280 y Fi(\024)g Fl(e)1307 2262 y Fd(\000)p Fh(b)1351 2235 y Fd(p)p 1381 2235 33 2 v 27 x Fh(N)t(x)1834 2280 y Fm(\(4)p Fl(:)p Fm(18\))0 2415 y Fk(for)k(some)f(p)n(ositive)g(c)n (onstant)f Fl(b)p Fk(.)0 2525 y Fa(Pro)q(of:)24 b Fm(Basically)18 b(the)d(pro)q(of)h(of)f(this)i(lemma)f(relies)h(on)e(the)h(same)g(tec)o (hnique)h(as)e(that)h(of)f(Prop)q(osition)h(3.1.)0 2597 y(Ho)o(w)o(ev)o(er,)d (a)i(n)o(um)o(b)q(er)g(of)f(details)i(are)f(mo)q(di\014ed.)21 b(In)15 b(particular,)g(w)o(e)g(use)g(a)f(coarser)g(\014ltration)h(of)g Fi(F)k Fm(to)14 b(de\014ne)0 2670 y(our)i(martingale)g(di\013erences.)24 b(Namely)l(,)16 b(w)o(e)g(denote)g(b)o(y)g Fi(F)1040 2677 y Fh(k)1078 2670 y Fm(the)g(sigma)g(algebra)g(generated)g(b)o(y)g(the)g(random) 0 2779 y Ff(31)p Fe(=mar)q(ch=)p Ff(1995;)h(9:46)652 b Fm(15)p eop %%Page: 16 17 bop 0 45 a Fm(v)m(ariables)17 b Fi(f)p Fl(\030)234 24 y Fh(\026)232 58 y(i)258 45 y Fi(g)281 24 y Fh(\026)p Fd(2)p Fh(I)-8 b(N)281 59 y(i)p Fd(\025)p Fh(k)371 45 y Fm(.)21 b(W)l(e)16 b(also)g(in)o(tro)q(duce) h(the)f(trace)f(sigma)h(algebra)1274 34 y(~)1262 45 y Fi(F)i(\021)c(F)h(\\)c Fm(\012)1484 52 y Fp(1)1520 45 y Fm(and)16 b(b)o(y)1685 34 y(~)1672 45 y Fi(F)1705 52 y Fh(k)1741 45 y Fi(\021)e(F)1823 52 y Fh(k)1856 45 y Fi(\\)d Fm(\012)1930 52 y Fp(1)0 118 y Fm(the)k(corresp)q(onding)h(\014ltration)g(of)f(the)g(trace)g(sigma)g (algebra.)20 b(W)l(e)15 b(set)566 218 y Fl(f)593 194 y Fp(\()p Fh(k)q Fp(\))588 231 y Fh(N)657 218 y Fi(\021)e Fl(I)-8 b(E)761 167 y Fg(h)782 218 y Fl(G)818 225 y Fh(N)853 218 y Fm(\()p Fl(t)p Fm(\))p Fi(j)929 206 y Fm(~)918 218 y Fi(F)951 225 y Fh(k)972 167 y Fg(i)1004 218 y Fi(\000)10 b Fl(I)-8 b(E)1105 167 y Fg(h)1126 218 y Fl(G)1162 225 y Fh(N)1197 218 y Fm(\()p Fl(t)p Fm(\))p Fi(j)1273 206 y Fm(~)1262 218 y Fi(F)1295 225 y Fh(k)q Fp(+1)1362 167 y Fg(i)1834 218 y Fm(\(4)p Fl(:)p Fm(19\))0 317 y(Ob)o(viously)l(,)16 b(w)o(e)f(ha)o(v)o(e)g(for)g Fl(!)f Fi(2)f Fm(\012)582 324 y Fp(1)622 440 y Fl(G)658 447 y Fh(N)692 440 y Fm([)p Fl(!)r Fm(]\()p Fl(t)p Fm(\))c Fi(\000)i Fl(I)-8 b(E)r Fm([)p Fl(G)952 447 y Fh(N)986 440 y Fm(\()p Fl(t)p Fm(\))p Fi(j)p Fm(\012)1084 447 y Fp(1)1103 440 y Fm(])13 b(=)1194 384 y Fh(N)1177 397 y Fg(X)1177 495 y Fh(k)q Fp(=1)1250 440 y Fl(f)1277 417 y Fp(\()p Fh(k)q Fp(\))1272 454 y Fh(N)1834 440 y Fm(\(4)p Fl(:)p Fm(20\))0 565 y(Th)o(us)21 b(the)g(lemma)g(will)h(b)q (e)f(pro)o(v)o(en)g(if)g(w)o(e)f(can)h(pro)o(v)o(e)f(an)h(estimate)g(of)f 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Fl(T)1540 2498 y Fh(k)1562 2399 y Fg(3)1562 2481 y(5)1593 2409 y Fp(2)477 2657 y Fm(=)531 2626 y(2)554 2588 y Fi(p)p 591 2588 50 2 v 591 2626 a Fl(M)p 531 2647 110 2 v 565 2688 a(N)646 2597 y Fg(q)p 691 2597 182 2 v 60 x Fl(G)727 2633 y Fp(\()p Fh(k)q Fp(\))727 2670 y Fh(N)778 2657 y Fm(\()p Fl(t;)8 b(z)r Fm(\))1834 2451 y(\(4)p Fl(:)p Fm(28\))0 2779 y Ff(31)p Fe(=mar)q(ch=)p Ff(1995;)17 b(9:46)652 b Fm(16)p eop %%Page: 17 18 bop 0 47 a Fm(But)15 b(on)f(\012)185 54 y Fp(1)220 47 y Fm(it)h(is)g(trivial) h(to)e(c)o(hec)o(k)h(that)f Fl(G)756 24 y Fp(\()p Fh(k)q Fp(\))756 61 y Fh(N)806 47 y Fm(\()p Fl(t;)8 b(z)r Fm(\))14 b(satis\014es,)h(for)f Fl(z)g Fi(2)f Fm([0)p Fl(;)8 b Fm(1],)k(the)j(same)g(b)q(ound)g(as)f Fl(G)1783 54 y Fh(N)1818 47 y Fm(\()p Fl(t)p Fm(\).)19 b(So)0 120 y(that)14 b(on)i(\012)195 127 y Fp(1)215 120 y Fm(,)809 206 y Fi(j)p Fl(g)846 188 y Fd(0)844 218 y Fh(k)866 206 y Fm(\()p Fl(z)r Fm(\))p Fi(j)c(\024)1003 176 y Fm(12)1049 137 y Fi(p)p 1086 137 50 2 v 1086 176 a Fl(M)p 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Fm(+)11 b(2)669 1677 y Fh(M)655 1690 y Fg(X)656 1787 y Fh(\027)r Fp(=1)734 1703 y Fl(z)i Fi(\000)d Fm(1)p 734 1723 102 2 v 754 1765 a Fl(N)796 1752 y Fp(2)841 1734 y Fl(T)874 1715 y Fp(2)868 1745 y Fh(k)1834 1653 y Fm(\(4)p Fl(:)p Fm(33\))0 1852 y(Th)o(us)393 1949 y(\()o Fl(g)434 1930 y Fd(0)432 1960 y Fh(k)454 1949 y Fm(\()p Fl(z)r Fm(\)\))530 1924 y Fp(2)563 1949 y Fi(\024)j Fm(8)642 1871 y Fg( )692 1892 y Fh(M)678 1906 y Fg(X)678 2002 y Fh(\027)r Fp(=1)752 1871 y Fg(")793 1918 y Fm(1)p 784 1939 42 2 v 784 1980 a Fl(N)838 1906 y Fg(X)865 2003 y Fh(i)911 1949 y Fl(\030)933 1930 y Fh(\027)931 1960 y(i)956 1949 y Fl(T)983 1956 y Fh(i)1008 1949 y Fi(\000)d Fl(I)-8 b(E)1116 1918 y Fm(1)p 1107 1939 V 1107 1980 a Fl(N)1162 1906 y Fg(X)1188 2003 y Fh(i)1235 1949 y Fl(\030)1257 1930 y Fh(\027)1255 1960 y(i)1279 1949 y Fl(T)1306 1956 y Fh(i)1321 1871 y Fg(#)1370 1918 y Fm(1)p 1361 1939 V 1361 1980 a Fl(N)1408 1949 y(\030)1430 1930 y Fh(\027)1428 1960 y(k)1452 1949 y Fl(T)1479 1956 y Fh(k)1501 1871 y Fg(!)1537 1877 y Fp(2)561 2093 y Fm(+)10 b(8)p Fl(T)662 2075 y Fp(4)656 2105 y Fh(k)682 2093 y Fm(\()p Fl(z)i Fi(\000)f Fm(1\))820 2075 y Fp(2)845 2063 y Fl(M)894 2046 y Fp(2)p 845 2083 70 2 v 849 2124 a Fl(N)891 2111 y Fp(4)1834 2002 y Fm(\(4)p Fl(:)p Fm(34\))0 2191 y(Let)j(us)f(abbreviate)h(the)f(t)o(w)o(o)g(summands)g(in)h (\(4.34\))e(b)o(y)h(\(I\))g(and)h(\(I)q(I\).)f(The)h(term)f(\(I)q(I\))g(is)h (of)f(order)h Fl(\013)1753 2174 y Fp(2)1773 2191 y Fl(N)1815 2174 y Fd(\000)p Fp(2)1877 2191 y Fm(and)0 2264 y(th)o(us)j(can)g(simply)i(b) q(e)f(b)q(ounded)g(uniformly)l(.)27 b(W)l(e)18 b(ha)o(v)o(e)e(to)h(w)o(ork)f (a)h(little)i(more)e(to)f(con)o(trol)h(the)g(conditional)0 2336 y(exp)q(ectation)f(of)f(the)g(\014rst.)k(W)l(e)d(write)78 2441 y Fl(I)-8 b(E)134 2391 y Fg(h)155 2441 y Fm(\()p Fl(I)t Fm(\))215 2402 y Fg(\014)215 2430 y(\014)12 b Fm(~)229 2441 y Fi(F)262 2448 y Fh(k)q Fp(+1)330 2391 y Fg(i)91 2568 y Fm(=)164 2538 y(8)p 144 2558 62 2 v 144 2600 a Fl(N)186 2586 y Fp(2)211 2568 y Fl(I)-8 b(E)267 2491 y Fg(")294 2525 y(X)300 2621 y Fh(\026;\027)367 2568 y Fl(\030)389 2547 y Fh(\026)387 2582 y(k)413 2568 y Fl(\030)435 2550 y Fh(\027)433 2580 y(k)458 2568 y Fl(T)491 2550 y Fp(2)485 2580 y Fh(k)519 2491 y Fg(")560 2538 y Fm(1)p 551 2558 42 2 v 551 2600 a Fl(N)605 2525 y Fg(X)632 2622 y Fh(i)679 2568 y Fl(\030)701 2547 y Fh(\026)699 2581 y(i)725 2568 y Fl(T)752 2575 y Fh(i)777 2568 y Fi(\000)10 b Fl(I)-8 b(E)885 2538 y Fm(1)p 876 2558 V 876 2600 a Fl(N)930 2525 y Fg(X)957 2622 y Fh(i)1004 2568 y Fl(\030)1026 2547 y Fh(\026)1024 2581 y(i)1050 2568 y Fl(T)1077 2575 y Fh(i)1092 2491 y Fg(#)7 b(")1167 2538 y Fm(1)p 1158 2558 V 1158 2600 a Fl(N)1212 2525 y Fg(X)1239 2622 y Fh(i)1286 2568 y Fl(\030)1308 2550 y Fh(\027)1306 2580 y(i)1330 2568 y Fl(T)1357 2575 y Fh(i)1382 2568 y Fi(\000)j Fl(I)-8 b(E)1491 2538 y Fm(1)p 1481 2558 V 1481 2600 a Fl(N)1536 2525 y Fg(X)1562 2622 y Fh(i)1609 2568 y Fl(\030)1631 2550 y Fh(\027)1629 2580 y(i)1654 2568 y Fl(T)1681 2575 y Fh(i)1695 2491 y Fg(#)1729 2530 y(\014)1729 2557 y(\014)13 b Fm(~)1744 2568 y Fi(F)1777 2575 y Fh(k)q Fp(+1)1846 2491 y Fg(#)1834 2670 y Fm(\(4)p Fl(:)p Fm(35\))0 2779 y Ff(31)p Fe(=mar)q(ch=)p Ff(1995;)k(9:46)652 b Fm(17)p eop %%Page: 18 19 bop 0 45 a Fm(W)l(e)20 b(observ)o(e)f(that)g(under)i(the)e(exp)q(ectation)i (conditioned)g(on)1147 34 y(~)1135 45 y Fi(F)1168 52 y Fh(k)q Fp(+1)1256 45 y Fm(w)o(e)e(ma)o(y)g(in)o(terc)o(hange)h(the)g(indices)h(of)0 118 y(1)13 b Fi(\024)g Fl(j)i Fi(\024)f Fl(k)i Fm(and)g(use)g(this)f(to)g (symmetrize)h(the)f(expression)i(\(4.35\).\(Notice)c(that)i(this)h(is)g(the)f (reason)g(wh)o(y)g(w)o(e)0 191 y(separated)g(the)g Fl(z)r Fm(-dep)q(enden)o (t)i(con)o(tribution)f(in)g(\(4.34\)\).This)d(giv)o(es)32 298 y Fl(I)-8 b(E)88 247 y Fg(h)110 298 y Fm(\()p Fl(I)t Fm(\))170 259 y Fg(\014)170 286 y(\014)11 b Fm(~)184 298 y Fi(F)217 305 y Fh(k)q Fp(+1)285 247 y Fg(i)45 438 y Fm(=)118 408 y(8)p 98 428 62 2 v 98 470 a Fl(N)140 457 y Fp(2)165 438 y Fl(I)-8 b(E)221 347 y Fg(2)221 429 y(4)252 395 y(X)258 491 y Fh(\026;\027)348 382 y(k)325 395 y Fg(X)326 492 y Fh(j)r Fp(=1)404 402 y Fl(\030)426 380 y Fh(\026)424 415 y(j)450 402 y Fl(\030)472 385 y Fh(\027)470 414 y(j)p 404 428 91 2 v 437 470 a Fl(k)500 438 y(T)533 420 y Fp(2)527 450 y Fh(j)561 361 y Fg(")602 408 y Fm(1)p 593 428 42 2 v 593 470 a Fl(N)647 395 y Fg(X)674 492 y Fh(i)721 438 y Fl(\030)743 417 y Fh(\026)741 451 y(i)767 438 y Fl(T)794 445 y Fh(i)819 438 y Fi(\000)10 b Fl(I)-8 b(E)927 408 y Fm(1)p 918 428 V 918 470 a Fl(N)973 395 y Fg(X)999 492 y Fh(i)1046 438 y Fl(\030)1068 417 y Fh(\026)1066 451 y(i)1092 438 y Fl(T)1119 445 y Fh(i)1134 361 y Fg(#)7 b(")1209 408 y Fm(1)p 1200 428 V 1200 470 a Fl(N)1254 395 y Fg(X)1281 492 y Fh(i)1328 438 y Fl(\030)1350 420 y Fh(\027)1348 450 y(i)1372 438 y Fl(T)1399 445 y Fh(i)1424 438 y Fi(\000)j Fl(I)-8 b(E)1533 408 y Fm(1)p 1523 428 V 1523 470 a Fl(N)1578 395 y Fg(X)1604 492 y Fh(i)1651 438 y Fl(\030)1673 420 y Fh(\027)1671 450 y(i)1696 438 y Fl(T)1723 445 y Fh(i)1737 361 y Fg(#)1771 400 y(\014)1771 427 y(\014)13 b Fm(~)1787 438 y Fi(F)1820 445 y Fh(k)q Fp(+1)1888 347 y Fg(3)1888 429 y(5)45 620 y Fi(\024)118 590 y Fm(8)p 98 610 62 2 v 98 651 a Fl(N)140 638 y Fp(2)165 620 y Fl(I)-8 b(E)221 529 y Fg(2)221 611 y(4)252 527 y(\015)252 554 y(\015)252 582 y(\015)252 609 y(\015)252 636 y(\015)252 663 y(\015)300 563 y Fh(k)277 577 y Fg(X)278 674 y Fh(j)r Fp(=1)356 585 y Fl(\030)376 592 y Fh(j)395 585 y Fl(\030)417 568 y Fh(t)415 596 y(j)p 356 610 79 2 v 382 651 a Fl(k)439 620 y(T)472 602 y Fp(2)466 632 y Fh(j)492 527 y Fg(\015)492 554 y(\015)492 582 y(\015)492 609 y(\015)492 636 y(\015)492 663 y(\015)540 563 y Fh(M)527 577 y Fg(X)525 673 y Fh(\026)p Fp(=1)601 543 y Fg(")642 590 y Fm(1)p 633 610 42 2 v 633 651 a Fl(N)687 577 y Fg(X)714 674 y Fh(i)761 620 y Fl(\030)783 598 y Fh(\026)781 633 y(i)807 620 y Fl(T)834 627 y Fh(i)859 620 y Fi(\000)10 b Fl(I)-8 b(E)967 590 y Fm(1)p 958 610 V 958 651 a Fl(N)1013 577 y Fg(X)1039 674 y Fh(i)1086 620 y Fl(\030)1108 598 y Fh(\026)1106 633 y(i)1132 620 y Fl(T)1159 627 y Fh(i)1174 543 y Fg(#)1200 552 y Fp(2)1228 568 y Fg(\014)1228 595 y(\014)1228 623 y(\014)1256 609 y Fm(~)1244 620 y Fi(F)1277 627 y Fh(k)q Fp(+1)1345 529 y Fg(3)1345 611 y(5)1834 729 y Fm(\(4)p Fl(:)p Fm(36\))0 802 y(But)15 b(b)o(y)g(Lemma)h(4.2,)e(on)h(\012)498 809 y Fp(1)518 802 y Fm(,)509 886 y Fh(M)496 900 y Fg(X)494 996 y Fh(\026)p Fp(=1)570 865 y Fg(")611 912 y Fm(1)p 602 932 V 602 974 a Fl(N)656 900 y Fg(X)683 996 y Fh(i)730 943 y Fl(\030)752 921 y Fh(\026)750 955 y(i)776 943 y Fl(T)803 950 y Fh(i)828 943 y Fi(\000)10 b Fl(I)-8 b(E)936 912 y Fm(1)p 927 932 V 927 974 a Fl(N)982 900 y Fg(X)1008 996 y Fh(i)1055 943 y Fl(\030)1077 921 y Fh(\026)1075 955 y(i)1101 943 y Fl(T)1128 950 y Fh(i)1143 865 y Fg(#)1169 874 y Fp(2)1202 943 y Fm(=)13 b Fl(G)1286 950 y Fh(N)1321 943 y Fm(\()p Fl(t)p Fm(\))f Fi(\024)h Fm(6)378 b(\(4)p Fl(:)p Fm(37\))0 1081 y(and)15 b(since)604 1086 y Fg(\015)604 1113 y(\015)604 1140 y(\015)604 1168 y(\015)604 1195 y(\015)604 1222 y(\015)652 1122 y Fh(k)629 1136 y Fg(X)630 1233 y Fh(j)r Fp(=1)708 1143 y Fl(\030)728 1150 y Fh(j)747 1143 y Fl(\030)769 1127 y Fh(t)767 1155 y(j)p 708 1168 79 2 v 734 1210 a Fl(k)791 1179 y(T)824 1160 y Fp(2)818 1190 y Fh(j)844 1086 y Fg(\015)844 1113 y(\015)844 1140 y(\015)844 1168 y(\015)844 1195 y(\015)844 1222 y(\015)882 1179 y Fi(\024)930 1086 y Fg(\015)930 1113 y(\015)930 1140 y(\015)930 1168 y(\015)930 1195 y(\015)930 1222 y(\015)978 1122 y Fh(k)956 1136 y Fg(X)957 1233 y Fh(j)r Fp(=1)1034 1143 y Fl(\030)1054 1150 y Fh(j)1073 1143 y Fl(\030)1095 1127 y Fh(t)1093 1155 y(j)p 1034 1168 V 1061 1210 a Fl(k)1118 1086 y Fg(\015)1118 1113 y(\015)1118 1140 y(\015)1118 1168 y(\015)1118 1195 y(\015)1118 1222 y(\015)1156 1179 y Fi(\021)e(k)p Fl(B)r Fm(\()p Fl(k)q Fm(\))p Fi(k)487 b Fm(\(4)p Fl(:)p Fm(38\))0 1310 y(w)o(e)15 b(get)g(that)378 1383 y Fl(I)-8 b(E)434 1332 y Fg(h)456 1383 y Fm(\()p Fl(I)t Fm(\))516 1344 y Fg(\014)516 1371 y(\014)11 b Fm(~)530 1383 y Fi(F)563 1390 y Fh(k)q Fp(+1)631 1332 y Fg(i)665 1383 y Fi(\024)727 1352 y Fm(48)p 719 1372 62 2 v 719 1414 a Fl(N)761 1401 y Fp(2)786 1383 y Fl(I)-8 b(E)842 1346 y Fg(\002)861 1383 y Fi(k)p Fl(B)r Fm(\()p Fl(k)q Fm(\))p Fi(k)1012 1344 y Fg(\014)1012 1371 y(\014)1026 1383 y Fm(\012)1059 1390 y Fp(1)1079 1346 y Fg(\003)1111 1383 y Fi(\024)1173 1352 y Fm(48)p 1164 1372 V 1164 1414 a Fl(N)1206 1401 y Fp(2)1232 1383 y Fl(I)g(E)r Fi(k)p Fl(B)r Fm(\()p Fl(k)q Fm(\))p Fi(k)p Fl(=I)g(P)6 b Fm([\012)1539 1390 y Fp(1)1559 1383 y Fm(])262 b(\(4)p Fl(:)p Fm(39\))0 1480 y(It)15 b(is)h(easy)f(to)g(sho)o(w)f(that)h (\(see)g([B]\))f(that)689 1596 y Fl(I)-8 b(E)s Fi(k)p Fl(B)r Fm(\()p Fl(k)q Fm(\))p Fi(k)12 b(\024)h Fl(c)969 1546 y Fg(\020)995 1596 y Fm(1)d(+)1073 1555 y Fg(p)p 1119 1555 95 2 v 1119 1596 a Fl(M)r(=k)1213 1546 y Fg(\021)1240 1555 y Fp(2)1834 1596 y Fm(\(4)p Fl(:)p Fm(40\))0 1707 y(for)15 b(some)f(constan)o(t)h(2)d Fl(>)h(c)f(>)h Fm(1.)20 b(Collecting)d(our)e(estimates)g(and)g(using)h(that)e (1)c(+)h Fl(x)h Fi(\024)h Fl(e)1573 1691 y Fh(x)1611 1707 y Fm(w)o(e)i(arriv)o(e)g(at)255 1825 y Fl(I)-8 b(E)311 1775 y Fg(h)332 1825 y Fl(e)353 1806 y Fh(uf)396 1790 y Ff(\()p Fe(k)p Ff(\))392 1818 y Fe(N)443 1786 y Fg(\014)443 1814 y(\014)12 b Fm(~)458 1825 y Fi(F)491 1832 y Fh(k)q Fp(+1)559 1775 y Fg(i)593 1825 y Fi(\024)h Fm(exp)718 1761 y Fg(\022)757 1794 y Fm(1)p 757 1815 23 2 v 757 1856 a(2)785 1825 y Fl(u)811 1806 y Fp(2)832 1825 y Fl(e)853 1806 y Fd(j)p Fh(u)p Fd(j)p Fp(12)932 1779 y Fd(p)p 962 1779 39 2 v 27 x Fh(M)s(=)n(N)1051 1825 y Fl(N)1093 1806 y Fd(\000)p Fp(2)1141 1825 y Fm(4)1172 1775 y Fg(h)1193 1825 y Fm(8)p Fl(\013)1245 1806 y Fp(2)1276 1825 y Fm(+)d(76\(1)f(+)1463 1784 y Fg(p)p 1508 1784 95 2 v 41 x Fl(M)r(=k)q Fm(\))1620 1806 y Fp(2)1640 1775 y Fg(i)1662 1761 y(\023)1834 1825 y Fm(\(4)p Fl(:)p Fm(41\))0 1942 y(Since)326 1973 y Fh(N)309 1987 y Fg(X)309 2085 y Fh(k)q Fp(=1)375 2030 y Fm(\(1)g(+)471 1989 y Fg(p)p 516 1989 V 41 x Fl(M)r(=k)r Fm(\))629 2011 y Fp(2)661 2030 y Fm(=)k Fl(N)i Fm(+)10 b(4)829 1989 y Fi(p)p 867 1989 91 2 v 41 x Fl(M)5 b(N)14 b Fm(+)d Fl(M)i Fm(ln)8 b Fl(N)17 b Fm(=)c Fl(N)5 b Fm(\(1)k(+)i(4)1378 1995 y Fi(p)p 1415 1995 30 2 v 1415 2030 a Fl(\013)f Fm(+)h Fl(\013)d Fm(ln)g Fl(N)d Fm(\))192 b(\(4)p Fl(:)p Fm(42\))0 2150 y(this)16 b(yields)g(that)28 2285 y Fl(I)-8 b(P)84 2207 y Fg(")127 2228 y Fh(N)110 2242 y Fg(X)110 2340 y Fh(k)q Fp(=1)184 2285 y Fl(f)211 2261 y Fp(\()p Fh(k)q Fp(\))206 2298 y Fh(N)274 2285 y Fi(\025)13 b Fl(x)348 2246 y Fg(\014)348 2274 y(\014)364 2285 y Fm(\012)397 2292 y Fp(1)417 2207 y Fg(#)456 2285 y Fi(\024)g Fm(inf)521 2312 y Fh(u)567 2285 y Fm(exp)644 2221 y Fg(\022)677 2285 y Fi(\000)p Fl(ux)e Fm(+)834 2254 y Fl(u)860 2238 y Fp(2)p 826 2275 65 2 v 826 2316 a Fm(2)p Fl(N)895 2285 y(e)916 2266 y Fd(j)p Fh(u)p Fd(j)p Fp(12)995 2239 y Fd(p)p 1026 2239 39 2 v 1026 2266 a Fh(M)s(=)n(N)1115 2285 y Fm(4)1146 2248 y Fg(\002)1164 2285 y Fm(8)p 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Fl(e)388 2498 y Fh(\015)412 2491 y Fm(,)h Fl(\015)h Fm(=)d(1)p Fl(;)8 b(:)g(:)g(:)d(;)j Fm(2)679 2474 y Fh(M)734 2491 y Fm(form)16 b(a)g(complete)h(en)o(umeration)g(of)f(all)h(v)o(ectors)e(with)i(comp)q(onen) o(ts)0 2563 y Fi(\006)p Fm(1)e(in)h Fl(I)-6 b(R)175 2544 y Fh(M)215 2563 y Fm(.)20 b(They)15 b(can)h(b)q(e)f(con)o(v)o(enien)o(tly)i(c)o (hosen)e(as)811 2670 y Fl(e)832 2651 y Fh(\026)832 2681 y(\015)869 2670 y Fm(=)e(\()p Fi(\000)p Fm(1\))1011 2654 y([)1023 2650 y Fh(\015)r Fp(2)1062 2637 y Ff(1)p Fc(\000)p Fe(\026)1124 2653 y Fm(])1834 2670 y(\(4)p Fl(:)p Fm(51\))0 2779 y Ff(31)p Fe(=mar)q(ch=)p Ff(1995;)k(9:46)652 b Fm(19)p eop %%Page: 20 21 bop 0 45 a Fm(where)17 b([)p Fl(x)p Fm(])e(denotes)i(the)g(smaller)g(in)o (teger)g(greater)e(or)h(equal)i(to)e Fl(x)p Fm(.)23 b(Note)17 b(that)e Fl(I)-8 b(E)s Fm(\011)1504 52 y Fh(N)1539 45 y Fm(\()p Fl(m;)8 b(t)p Fm(\))15 b(dep)q(ends)j(on)f Fl(N)0 118 y Fm(only)f(through)f Fl(M)5 b Fm(\()p Fl(N)g Fm(\).)18 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Ff(\()p Fe(y)q Ff(\))p Fc(\000)6 b Ff(~)-21 b Fe(m)o Fc(k)1098 1937 y Ff(2)1116 1930 y Fc(\024)p Fe(\017)1168 1854 y Fm(\002)1203 1861 y Fh(p)1224 1854 y Fm(\()p Fl(y)r Fm(\))550 b(\(4.33\))0 2020 y(Therefore,)14 b(for)h Fl(y)f Fi(2)f(A)403 1999 y Fh(p)403 2034 y(d)425 2020 y Fm(,)h(\002)487 2027 y Fh(p)508 2020 y Fm(\()p Fl(y)r Fm(\))e(=)h(\002)663 2027 y Fh(d)685 2020 y Fm(\()p Fl(y)r Fm(\),)h(while)i(at)e(the)h(same)g (time)g(the)g(constrain)o(t)f(in)i(the)f(sup)g(is)g(satis\014ed)0 2093 y(sim)o(ultaneously)i(w.r.t.)h Fl(n)462 2100 y Fh(p)498 2093 y Fm(or)d Fl(n)581 2100 y Fh(d)603 2093 y Fm(,)f(as)h(so)q(on)g(as)g Fl(d)g Fm(is)h(large)f(enough)g(suc)o(h)h(that)e Fl(I)j Fi(\032)c(f)p Fm(1)p Fl(;)8 b(:)g(:)g(:)t(;)g(d)p Fi(g)p Fm(.)18 b(Therefore,)328 2226 y(\007)363 2233 y Fh(p;\017)411 2226 y Fm(\()8 b(~)-31 b Fl(m)o Fm(\))12 b Fi(\025)117 b Fm(sup)635 2266 y Fe(y)q Fc(2A)699 2258 y Fe(M)699 2281 y(d)552 2303 y Fc(k)p Ff(\005)591 2311 y Fe(I)609 2303 y(n)629 2308 y(p)648 2303 y 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2577 y Ff(2)788 2572 y Fd(\024)p Fh(\017)866 2532 y Fm(inf)839 2565 y Fh(t)p Fd(2)p Fh(I)-8 b(R)914 2555 y Fe(M)955 2532 y Fl(I)g(E)s Fm(\011)1039 2539 y Fh(N)1073 2532 y Fm(\()p Fl(m;)8 b(t)p Fm(\))k(=)25 b(lim)1246 2561 y Fh(N)s Fd("1)1341 2532 y Fm(\007)1376 2539 y Fh(p;\017)1423 2532 y Fm(\()8 b(~)-31 b Fl(m)o Fm(\))1198 2641 y(=)13 b(sup)1271 2677 y Fh(p)1322 2641 y Fm(\007)1357 2648 y Fh(p;\017)1404 2641 y Fm(\()c(~)-32 b Fl(m)p Fm(\))1834 2602 y(\(4)p Fl(:)p Fm(54\))0 2779 y Ff(31)p Fe(=mar)q(ch=)p Ff(1995;)17 b(9:46)652 b Fm(20)p eop %%Page: 21 22 bop 0 45 a Fm(It)10 b(remains)h(to)e(consider)i(the)g(limit)g Fl(\017)i Fi(#)g Fm(0.)k(It)11 b(is)f(clear)h(that)e(sup)1079 57 y Fh(p)1107 45 y Fm(\007)1142 52 y Fh(p;\017)1190 45 y Fm(\()f(~)-31 b Fl(m)o Fm(\))10 b(con)o(v)o(erges)g(to)f(a)h(lo)o(w)o(er-semicon)o(tin)o (uous)0 118 y(function)16 b(and)f(that)492 191 y(lim)498 221 y Fh(\017)p Fd(#)p Fp(0)563 191 y Fm(sup)588 227 y Fh(p)639 191 y Fm(\007)674 198 y Fh(p;\017)722 191 y Fm(\()8 b(~)-31 b Fl(m)o Fm(\))12 b(=)h(lim)863 221 y Fh(\017)p Fd(#)p Fp(0)1037 191 y Fm(sup)928 231 y Fh(m)p Fp(:)7 b Fd(k)p Fp(\005)1023 236 y Fe(I)1041 231 y Fh(m)p Fd(\000)g Fp(~)-25 b Fh(m)p Fd(k)1152 236 y Ff(2)1170 231 y Fd(\024)p Fh(\017)1221 191 y Fm(sup)1246 227 y Fh(p)1297 191 y Fm(\007)1332 198 y Fh(p;)p Fp(0)1382 191 y Fm(\()9 b(~)-32 b Fl(m)p Fm(\))376 b(\(4)p Fl(:)p Fm(55\))0 311 y(Th)o(us)15 b(if)h(sup)225 322 y Fh(p)254 311 y Fm(\007)289 318 y Fh(p;)p Fp(0)339 311 y Fm(\()8 b(~)-31 b Fl(m)p Fm(\))15 b(is)g(con)o(tin)o(uous)h(in)g(a)f(neigh)o(b)q(orho)q(o)q(d)h(of)24 b(~)-32 b Fl(m)p Fm(,)15 b(w)o(e)g(get)674 430 y(lim)680 460 y Fh(\017)p Fd(#)p Fp(0)745 430 y Fm(sup)770 466 y Fh(p)821 430 y Fm(\007)856 437 y Fh(p;\017)903 430 y Fm(\()9 b(~)-32 b Fl(m)p Fm(\))12 b(=)h(sup)1064 466 y Fh(p)1115 430 y Fm(\007)1150 437 y Fh(p;)p Fp(0)1200 430 y Fm(\()c(~)-32 b Fl(m)p Fm(\))558 b(\(4)p Fl(:)p Fm(56\))0 562 y(as)16 b(desired.)26 b(But,)17 b(as)f(has)h(b)q(een)h(sho)o(wn)e(in)i([BGP2],)d(from)h(the)h(explicit)i (form)d(of)g(\007)h(one)g(sho)o(ws)f(easily)i(that)0 635 y(sup)68 646 y Fh(p)97 635 y Fm(\007)132 642 y Fh(p;)p Fp(0)182 635 y Fm(\()9 b(~)-32 b Fl(m)p Fm(\))16 b(is)h(Lipshitz)h(con)o(tin)o(uous)e(in)h (the)g(in)o(terior)f(of)g(the)h(set)f(on)g(whic)o(h)h(it)g(is)f(b)q(ounded.) 25 b(This)17 b(pro)o(v)o(es)0 708 y(Theorem)e(4.1)f Fi(})91 817 y Fm(W)l(e)i(will)i(sho)o(w)e(next)g(that)f(a)h(su\016cien)o(t)h (condition)g(for)f(condition)h(\(4.1\))e(to)g(hold)i(is)g(that)24 b(~)-32 b Fl(m)17 b Fm(b)q(elongs)g(to)0 890 y Fl(D)38 898 y Fd(j)p Fh(I)r Fd(j)80 890 y Fm(.)24 b(While)18 b(this)f(app)q(ears)g(in)o (tuitiv)o(ely)i(`clear',)d(the)h(rigorous)f(pro)q(of)g(is)h(surprisingly)i (tedious.)25 b(Let)17 b(us)f(\014rst)0 962 y(in)o(tro)q(duce)g(some)f (notation)g(and)g(results.)91 1072 y(Let)k Fl(E)210 1079 y Fh(p)249 1072 y Fm(b)q(e)h(the)f Fl(p)13 b Fi(\002)f Fm(2)503 1055 y Fh(p)524 1072 y Fm(-matrix)19 b(whose)g(ro)o(ws)f(are)g(giv)o(en)i(b)o (y)e(the)h(v)o(ectors)g Fl(e)1468 1079 y Fh(\015)1491 1072 y Fm(,)h Fl(\015)g Fm(=)g(1)p Fl(;)8 b(:)g(:)g(:)t(;)g Fm(2)1770 1055 y Fh(p)1790 1072 y Fm(,)19 b(whic)o(h,)0 1144 y(for)c(con)o(v)o (enience,)j(are)d(ordered)h(accordingly)h(to)e(\(4.51\).)21 b(W)l(e)16 b(will)h(denote)f(b)o(y)g Fl(e)1421 1128 y Fh(\026)1446 1144 y Fm(,)f Fl(\026)f Fm(=)h(1)p Fl(;)8 b(:)g(:)g(:)t(;)g(p)15 b Fm(the)h(column)0 1217 y(v)o(ectors)e(of)h Fl(E)240 1224 y Fh(p)276 1217 y Fm(and)g(b)o(y)g Fl(E)464 1200 y Fh(t)461 1228 y(p)497 1217 y Fm(its)h(transp)q(ose.)j(It)c(can)h(easily)g(b)q(e)g(v)o (eri\014ed)g(that)687 1336 y(2)710 1318 y Fd(\000)p Fh(p)759 1336 y Fm(\()p Fl(e)798 1318 y Fh(\026)822 1336 y Fl(;)8 b(e)864 1318 y Fh(\027)886 1336 y Fm(\))k(=)964 1286 y Fg(n)1002 1314 y Fm(1)45 b(if)16 b Fl(\026)d Fm(=)g Fl(\027)1002 1368 y Fm(0)45 b(otherwise)1834 1336 y(\(4)p Fl(:)p Fm(57\))0 1456 y(Th)o(us,)19 b(the)g(2)237 1439 y Fh(p)270 1456 y Fi(\002)13 b Fm(2)341 1439 y Fh(p)362 1456 y Fm(-matrix)18 b(2)551 1439 y Fd(\000)p Fh(p)601 1456 y Fl(E)635 1463 y Fh(p)655 1456 y Fl(E)692 1439 y Fh(t)689 1467 y(p)728 1456 y Fm(is)i(a)e(pro)s(jector)g(that)g(pro)s(jects) g(on)g(the)h(subspace)h(spanned)f(b)o(y)g(the)0 1529 y(orthogonal)c(v)o (ectors)g Fi(f)p Fl(e)427 1512 y Fh(\026)451 1529 y Fi(g)474 1507 y Fh(p)474 1541 y(\026)p Fp(=1)544 1529 y Fm(,)g(and)h(2)684 1512 y Fd(\000)p Fh(p)733 1529 y Fl(E)770 1512 y Fh(t)767 1540 y(p)788 1529 y Fl(E)822 1536 y Fh(p)858 1529 y Fm(is)g(the)g(iden)o(tit)o(y)g (in)h Fl(I)-6 b(R)1253 1509 y Fh(p)1274 1529 y Fm(.)21 b(Giv)o(en)16 b(a)g(linear)h(transformation)d Fl(A)0 1601 y Fm(from)g Fl(R)142 1585 y Fh(p)178 1601 y Fm(to)h Fl(R)269 1585 y Fh(q)289 1601 y Fm(,)g(w)o(e)g(de\014ne)635 1674 y Fl(AC)i Fm(=)c Fi(f)o Fl(Ax)g Fi(j)f Fl(x)h Fi(2)g Fl(C)s Fi(g)38 b Fm(for)30 b Fl(C)16 b Fi(\032)d Fl(I)-6 b(R)1294 1654 y Fh(p)1834 1674 y Fm(\(4)p Fl(:)p Fm(58\))0 1775 y(With)14 b(this)g(notations)f(the)h(v)o(ector)f Fl(n)640 1782 y Fh(p)661 1775 y Fm(\()p Fl(y)r Fm(\))g(and)g(the)h(set)f Fl(D)1004 1782 y Fh(p)1025 1775 y Fm(,)h(de\014ned)h(in)f(\(1.7\))e(and)i (\(1.8\),)e(can)h(b)q(e)i(rewritten)e(as)745 1878 y Fl(n)772 1885 y Fh(p)793 1878 y Fm(\()p Fl(y)r Fm(\))f(=)h(2)936 1859 y Fd(\000)p Fh(p)985 1878 y Fl(E)1022 1859 y Fh(t)1019 1889 y(p)1040 1878 y Fl(y)794 1965 y(D)832 1972 y Fh(p)865 1965 y Fm(=)g(2)936 1946 y Fd(\000)p Fh(p)985 1965 y Fl(E)1022 1946 y Fh(t)1019 1976 y(p)1040 1965 y Fm([)p Fi(\000)p Fm(1)p Fl(;)8 b Fm(1])1168 1946 y Fp(2)1186 1932 y Fe(p)1834 1922 y Fm(\(4)p Fl(:)p Fm(59\))0 2067 y(Moreo)o(v)o(er,)13 b(for)i(an)o(y)g(set)g Fl(I)h Fi(\032)d(f)p Fm(1)p Fl(;)8 b(:)g(:)g(:)t(;)g(p)p Fi(g)p Fm(,)14 b(w)o(e)h(ha)o(v)o(e)f(the)i(follo)o(wing)g(prop)q(ert)o(y)l(,)848 2187 y(\005)882 2194 y Fh(I)903 2187 y Fl(D)941 2194 y Fh(p)974 2187 y Fm(=)d Fl(D)1060 2195 y Fd(j)p Fh(I)r Fd(j)1834 2187 y Fm(\(4)p Fl(:)p Fm(60\))91 2343 y(Finally)l(,)g(let)d(us)h(remark)f(that)f 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Fm(0)h(and)g(all)g Fl(p)p Fm(,)g(the)g(suprem)o(um)g(o)o(v)o (er)f Fl(y)j Fm(s.t.)i Fi(k)p Fm(\005)1609 52 y Fh(I)1630 45 y Fl(n)1657 52 y Fh(p)1678 45 y Fm(\()p Fl(y)r Fm(\))7 b Fi(\000)17 b Fm(~)-33 b Fl(m)q Fi(k)1851 52 y Fp(2)1884 45 y Fi(\024)13 b Fl(\017)0 118 y Fm(of)i(\002)87 125 y Fh(p)108 118 y Fm(\()p Fl(y)r Fm(\))g(is)g(tak)o(en)g(on)g(at)g(a)g(p)q(oin)o(t)g Fl(y)651 102 y Fd(\003)687 118 y Fm(suc)o(h)h(that)770 280 y(2)793 262 y Fd(\000)p Fh(p)865 223 y Fp(2)883 210 y Fe(p)850 237 y Fg(X)850 333 y Fh(\015)r Fp(=1)924 243 y Fg(\002)943 280 y Fl(I)967 262 y Fd(0)980 280 y Fm(\()p Fl(y)1022 262 y Fd(\003)1020 292 y Fh(\015)1043 280 y Fm(\))1061 243 y Fg(\003)1079 253 y Fp(2)1112 280 y Fi(\024)d Fl(c)654 b Fm(\(4)p Fl(:)p Fm(61\))0 439 y(W)l(e)15 b(set)496 517 y Fi(A)532 524 y Fh(\017)549 517 y Fm(\()9 b(~)-32 b Fl(m)p Fm(\))12 b Fi(\021)685 467 y Fg(n)716 517 y Fl(y)i Fi(2)f Fm([)p Fi(\000)p Fm(1)p Fl(;)8 b Fm(1])923 499 y Fp(2)941 485 y Fe(M)988 517 y Fm(:)20 b Fi(k)p 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Fk(wher)n(e)619 1127 y Fl(T)646 1134 y Fh(p)666 1127 y Fm(\()p Fl(y)708 1108 y Fd(\003)729 1127 y Fm(\))12 b Fi(\021)h Fm(2)830 1108 y Fd(\000)p Fh(p)902 1070 y Fp(2)920 1056 y Fe(p)888 1084 y Fg(X)887 1180 y Fh(\015)r Fp(=1)961 1090 y Fg(\002)980 1127 y Fl(I)1004 1108 y Fd(0)1017 1127 y Fm(\()p Fl(y)1059 1108 y Fd(\003)1057 1138 y Fh(\015)1080 1127 y Fm(\))1098 1090 y Fg(\003)1117 1100 y Fp(2)1150 1127 y Fi(\024)g Fl(c)p Fm(\()8 b(~)-31 b Fl(m)o(;)8 b(\017)p Fm(\))502 b(\(4)p Fl(:)p Fm(64\))0 1346 y Fa(Pro)q(of:)22 b Fm(The)14 b(pro)q(of)f(pro)q(ceeds)h(b)o(y)g(sho)o (wing)f(that)g(if)h Fl(y)i Fm(do)q(es)d(not)h(satisfy)f(condition)i (\(4.64\),)c(then)j(w)o(e)g(can)f(\014nd)0 1418 y(a)h Fl(\016)r(y)i Fm(suc)o(h)e(that)g Fl(y)c Fm(+)e Fl(\016)r(y)15 b Fi(2)e(A)509 1425 y Fh(\017)526 1418 y Fm(\()c(~)-32 b Fl(m)p Fm(\))14 b(and)g(\002)738 1425 y Fh(p)759 1418 y Fm(\()p Fl(y)c Fm(+)f Fl(\016)r(y)r Fm(\))j Fl(<)h Fm(\002)1012 1425 y Fh(p)1033 1418 y Fm(\()p Fl(y)r Fm(\),)g(so)h(that)g 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Fm(65\))0 1800 y(Using)13 b(the)g(prop)q(erties)h(of)e(the)h (matrix)g Fl(E)718 1807 y Fh(p)751 1800 y Fm(w)o(e)g(can)g(b)q(ound)h(the)f (di\013erence)h(of)e(the)h(quadratic)g(terms)f(as)h(follo)o(ws)356 1918 y Fi(k)p Fl(n)406 1925 y Fh(p)427 1918 y Fm(\()p Fl(y)f Fm(+)e Fl(\016)r(y)r Fm(\))p Fi(k)611 1899 y Fp(2)611 1929 y(2)641 1918 y Fi(\000)h(k)p Fl(n)737 1925 y Fh(p)758 1918 y Fm(\()p Fl(y)r Fm(\))p Fi(k)841 1899 y Fp(2)841 1929 y(2)873 1918 y Fm(=)p Fi(k)p Fl(n)958 1925 y Fh(p)979 1918 y Fm(\()p Fl(\016)r(y)r Fm(\))p Fi(k)1084 1899 y Fp(2)1084 1929 y(2)1113 1918 y Fm(+)g(2)1182 1899 y Fd(\000)p Fh(p)p Fp(+1)1277 1918 y Fm(\()p Fl(\016)r(y)r(;)d Fm(2)1385 1899 y Fd(\000)p Fh(p)1433 1918 y Fl(E)1467 1925 y Fh(p)1487 1918 y Fl(E)1524 1899 y Fh(T)1521 1929 y(p)1552 1918 y Fl(y)r Fm(\))873 2004 y Fi(\025)i(\000)g Fm(2)986 1985 y Fd(\000)p Fh(p=)p Fp(2+1)1118 2004 y Fi(k)p Fl(\016)r(y)r Fi(k)1210 2011 y Fp(2)1834 1960 y Fm(\(4)p Fl(:)p Fm(66\))0 2118 y(Th)o(us)15 b(w)o(e)g(can)g(sho)o(w)g(that)g(\002)512 2125 y Fh(p)533 2118 y Fm(\()p Fl(y)d Fm(+)e Fl(\016)r(y)r Fm(\))i Fl(<)h Fm(\002)789 2125 y Fh(p)811 2118 y Fm(\()p Fl(y)r Fm(\))h(holds)i(b)o(y)f(sho)o(wing)g(that)512 2280 y(2)535 2262 y Fd(\000)p Fh(p)584 2280 y Fl(\014)612 2262 y Fd(\000)p Fp(1)684 2224 y(2)702 2210 y Fe(p)670 2237 y Fg(X)669 2333 y Fh(\015)r Fp(=1)744 2280 y Fm([)o Fl(I)t Fm(\()p Fl(y)820 2287 y Fh(\015)843 2280 y Fm(\))10 b Fi(\000)g Fl(I)t Fm(\()p Fl(y)980 2287 y Fh(\015)1013 2280 y Fm(+)h Fl(\016)r(y)1103 2287 y Fh(\015)1126 2280 y Fm(\)])h Fl(>)h Fm(2)1240 2262 y Fd(\000)p Fh(p=)p Fp(2)1326 2280 y Fi(k)p Fl(\016)r(y)r Fi(k)1418 2287 y Fp(2)1834 2280 y Fm(\(4)p Fl(:)p Fm(67\))0 2456 y(De\014ne)j(the)f (map)g Fl(Y)26 b Fm(from)14 b([)p Fi(\000)p Fm(1)p Fl(;)8 b Fm(1])607 2439 y Fp(2)625 2426 y Fe(p)660 2456 y Fm(to)15 b([)p Fi(\000)p Fm(1)p Fl(;)8 b Fm(1])844 2439 y Fp(2)862 2426 y Fc(j)p Fe(I)r Fc(j)917 2456 y Fm(b)o(y)482 2623 y Fl(Y)508 2630 y Fh(\015)532 2623 y Fm(\()p Fl(y)r Fm(\))k Fi(\021)h Fm(2)675 2605 y Fd(\000)p Fh(p)p Fp(+)p Fd(j)p Fh(I)r Fd(j)799 2567 y Fp(2)817 2553 y Fe(p)p Fc(\000j)p Fe(I)r Fc(j)898 2567 y Fd(\000)p Fp(1)839 2580 y Fg(X)840 2677 y Fp(~)-19 b Fh(\015)r Fp(=0)952 2623 y Fl(y)974 2634 y Fh(\015)r Fp(+)q(~)g Fh(\015)r Fp(2)1062 2625 y Fc(j)p Fe(I)r Fc(j)1142 2623 y Fl(;)37 b(\015)15 b Fm(=)e(1)p Fl(;)8 b(:)g(:)g(:)d(;)j Fm(2)1427 2605 y Fd(j)p Fh(I)r Fd(j)1834 2623 y Fm(\(4)p Fl(:)p Fm(68\))0 2779 y Ff(31)p Fe(=mar)q(ch=)p Ff(1995;)17 b(9:46)652 b Fm(22)p eop %%Page: 23 24 bop 0 45 a Fm(Using)16 b(\(4.59\))d(w)o(e)i(get)g(that)162 209 y(\005)196 217 y Fd(j)p Fh(I)r Fd(j)238 209 y Fl(n)265 216 y Fh(p)287 209 y Fm(\()p Fl(y)r Fm(\))c(=)i(2)429 190 y Fd(\000j)p Fh(I)r Fd(j)500 209 y Fl(E)537 190 y Fh(t)534 221 y Fd(j)p Fh(I)r Fd(j)584 118 y Fg(0)584 199 y(@)623 209 y Fm(2)646 190 y Fd(\000)p Fh(p)p Fp(+)p Fd(j)p Fh(I)r Fd(j)771 152 y Fp(2)789 138 y Fe(p)p Fc(\000j)p Fe(I)r Fc(j)788 166 y Fg(X)787 262 y Fh(\015)r Fp(=1)877 209 y Fm(\005)911 220 y Fd(f)p Fp(\()p Fh(\015)r Fd(\000)p Fp(1\)2)1043 210 y Fc(j)p Fe(I)r Fc(j)1083 220 y Fp(+1)p Fh(;:::;\015)r Fp(2)1223 210 y Fc(j)p Fe(I)r Fc(j)1262 220 y Fd(g)1282 209 y Fl(y)1306 118 y Fg(1)1306 199 y(A)1359 209 y Fm(=)g(2)1430 190 y Fd(\000j)p Fh(I)r Fd(j)1500 209 y Fl(E)1537 190 y Fh(t)1534 221 y Fd(j)p Fh(I)r Fd(j)1576 209 y Fl(Y)e Fm(\()p Fl(y)r Fm(\))161 b(\(4)p Fl(:)p Fm(69\))0 373 y(Therefore,)15 b(the)g(prop)q(ert)o(y)g(that)f Fl(y)h Fi(2)e(A)697 380 y Fh(\017)715 373 y Fm(\()8 b(~)-31 b Fl(m)o Fm(\))15 b(dep)q(ends)i(only)e(on)h(the)f(quan)o(tit)o(y)g Fl(Y)10 b Fm(\()p Fl(y)r Fm(\).)91 482 y(Notice)i(that)e(if)20 b(~)-31 b Fl(m)12 b Fi(2)h Fl(D)495 490 y Fd(j)p Fh(I)r Fd(j)548 482 y Fm(and)e Fl(\017)j(>)f Fm(0,)e(then)g(there)h(exists)f Fl(X)16 b Fi(2)d Fm(\()p Fi(\000)p Fm(1)p Fl(;)8 b Fm(1\))1327 465 y Fp(2)1345 451 y Fc(j)p Fe(I)r Fc(j)1396 482 y Fm(suc)o(h)j(that)g Fi(k)p Fl(n)1639 489 y Fh(I)1659 482 y Fm(\()p Fl(X)t Fm(\))r Fi(\000)f Fm(~)-31 b Fl(m)o Fi(k)1838 489 y Fp(2)1871 482 y Fi(\024)13 b Fl(\017)p Fm(.)0 554 y(This)f(implies)h(that)e(for)f(an)o(y)h Fl(p)p Fm(,)h(the)f(v)o(ector)g Fl(x)h Fi(2)h Fl(I)-6 b(R)877 535 y Fp(2)895 521 y Fe(p)927 554 y Fm(with)12 b(comp)q(onen)o(ts)f Fl(x)1299 561 y Fh(\015)1335 554 y Fi(\021)i Fl(X)1421 565 y Fh(y)8 b Fj(mo)q(d)f Fp(2)1545 556 y Fc(j)p Fe(I)r Fc(j)1597 554 y Fm(lies)13 b(also)e(in)h Fi(A)1844 561 y Fh(\017)1862 554 y Fm(\()c(~)-31 b Fl(m)p Fm(\).)0 627 y(Moreo)o(v)o(er,)649 700 y(max)681 727 y Fh(\015)741 700 y Fi(j)p Fl(x)780 707 y Fh(\015)803 700 y Fi(j)13 b Fm(=)g(max)908 727 y Fh(\015)969 700 y Fi(j)p Fl(X)1020 707 y Fh(\015)1042 700 y Fi(j)g(\021)g Fm(1)c Fi(\000)i Fl(d)h(<)h Fm(1)533 b(\(4)p Fl(:)p Fm(70\))0 804 y(and)15 b(therefore)781 877 y Fl(T)808 884 y Fh(p)828 877 y Fm(\()p Fl(x)p Fm(\))d Fi(\024)h Fm([)p Fl(I)987 858 y Fd(0)999 877 y Fm(\(1)d Fi(\000)g Fl(d)p Fm(\)])1149 852 y Fp(2)1834 877 y Fm(\(4)p Fl(:)p Fm(71\))0 982 y(is)15 b(some)f(\014nite)i Fl(p)p Fm(-indep)q(enden)o(t)h(constan)o(t.)h(W)l(e)d(will)h(use)f(this)g (fact)f(to)g(construct)g(our)g Fl(\016)r(y)r Fm(.)19 b(W)l(e)c(ma)o(y)f(of)g (course)0 1054 y(c)o(ho)q(ose)19 b(an)f(optimal)h Fl(X)t Fm(,)f(i.e.)31 b(one)18 b(for)g(whic)o(h)i Fl(d)e Fm(is)h(maximal.)30 b(In)20 b(the)e(sequel)i Fl(X)i Fm(and)c Fl(x)h Fm(will)h(refer)f(to)f(this)0 1127 y(v)o(ector.)27 b(Let)18 b(no)o(w)f Fl(y)j Fm(b)q(e)e(a)g(v)o(ector)f (in)i Fi(A)721 1134 y Fh(\017)738 1127 y Fm(\()9 b(~)-32 b Fl(m)p Fm(\))17 b(for)h(whic)o(h)g Fl(T)1063 1134 y Fh(p)1084 1127 y Fm(\()p Fl(y)r Fm(\))e Fl(>)h(c)h Fm(for)f(some)g(large)h(constan)o(t) f Fl(c)p Fm(.)27 b(W)l(e)18 b(will)0 1200 y(sho)o(w)d(that)f(this)i(cannot)f (minimize)i(\002)678 1207 y Fh(p)700 1200 y Fm(.)i(W)l(e)d(will)h (distinguish)g(t)o(w)o(o)d(cases:)0 1309 y Fn(Case)k(1:)47 b Fm(Let)16 b(us)g(in)o(tro)q(duce)h(t)o(w)o(o)e(parameters,)g(0)f Fl(<)g(\021)i Fi(\034)e Fl(d)i Fm(and)g(0)e Fl(<)g(\025)g(<)g Fm(1,)i(that)f(will)j(b)q(e)e(appropriately)0 1382 y(c)o(hosen)f(later.)20 b(In)c(this)g(\014rst)f(case)g(w)o(e)g(assume)g(that)f Fl(y)j Fm(is)f(suc)o(h)g(that)685 1478 y Fp(2)703 1464 y Fe(p)670 1491 y Fg(X)670 1587 y Fh(\015)r Fp(=1)744 1534 y Fm(1)-11 b(I)772 1543 y Fd(fj)p Fh(y)820 1548 y Fe(\015)841 1543 y Fd(j\025)p Fp(1)p Fd(\000)p Fh(\021)q Fd(g)980 1534 y Fi(\025)13 b Fm(\(1)c Fi(\000)i Fl(\025)p Fm(\)2)1192 1516 y Fh(p)p Fd(\000j)p Fh(I)r Fd(j)1834 1534 y Fm(\(4)p Fl(:)p Fm(72\))0 1689 y(and)k(w)o(e)g(c)o(ho)q(ose) 840 1761 y Fl(\016)r(y)f Fi(\021)f Fl(\032)p Fm(\()p Fl(x)c Fi(\000)i Fl(y)r Fm(\))723 b(\(4)p Fl(:)p Fm(73\))0 1866 y(where)15 b(0)e Fl(<)g(\032)f(<)h Fm(1)i(will)h(b)q(e)g(determined)g(later.)k(It)15 b(then)h(trivially)g(follo)o(ws)g(from)e(the)h(de\014nition)i(of)e Fl(x)g Fm(and)g(the)0 1939 y(con)o(v)o(exit)o(y)f(of)g(the)h(set)f Fi(A)436 1946 y Fh(\017)468 1939 y Fm(that)g Fl(y)d Fm(+)e Fl(\032)p Fm(\()p Fl(x)f Fi(\000)h Fl(y)r Fm(\))j Fi(2)h(A)896 1946 y Fh(\017)928 1939 y Fm(and)i(that)e Fl(y)e Fm(+)e Fl(\032)p Fm(\()p Fl(x)f Fi(\000)h Fl(y)r Fm(\))j Fi(2)h Fm([)p Fi(\000)p Fm(1)c(+)g Fl(\032d;)f Fm(1)e Fi(\000)j Fl(\032d)p Fm(])1734 1922 y Fp(2)1752 1908 y Fe(p)1773 1939 y Fm(.)19 b(If)c(Th)o(us)0 2011 y(if)g(w)o(e)g(can)g(sho)o(w)f(that)g(with)h(this)g(c)o(hoice,)g(and)g (with)g(an)g Fl(\032)f Fm(suc)o(h)h(that)f Fl(\032d)e(>)h(\021)r Fm(,)h(\(4.67\))f(holds,)i(w)o(e)f(can)h(exclude)0 2084 y(that)f(the)i (in\014m)o(um)g(is)g(tak)o(en)e(on)h(for)g(suc)o(h)g(an)h Fl(y)r Fm(.)91 2193 y(Let)i(us)g(\014rst)f(consider)h(comp)q(onen)o(ts)g Fl(y)786 2200 y Fh(\015)827 2193 y Fm(suc)o(h)g(that)f Fi(j)p Fl(y)1068 2200 y Fh(\015)1092 2193 y Fi(j)f Fl(>)h Fm(1)11 b Fi(\000)i Fl(d)p Fm(.)26 b(Since)20 b Fi(j)p Fl(x)1479 2200 y Fh(\015)1501 2193 y Fi(j)d(\024)g Fm(1)11 b Fi(\000)h Fl(d)18 b Fm(w)o(e)f(ha)o(v)o(e,)h(for)0 2266 y(those)e(comp)q(onen)o(ts,)g(sign)p Fl(\016)r(y)506 2273 y Fh(\015)544 2266 y Fm(=)e Fi(\000)p Fm(sign)q Fl(y)729 2273 y Fh(\015)769 2266 y Fm(and)i(th)o(us)g Fl(I)t Fm(\()p Fl(y)1023 2273 y Fh(\015)1046 2266 y Fm(\))10 b Fi(\000)h Fl(I)t Fm(\(\()p Fl(y)h Fm(+)f Fl(\016)r(y)r Fm(\))1324 2273 y Fh(\015)1347 2266 y Fm(\))i Fl(>)i Fm(0.)22 b(This)17 b(fact)e(together)h(with)0 2339 y(\(4.72\))d(en)o(tails)74 2494 y(2)97 2475 y Fd(\000)p Fh(M)188 2437 y Fp(2)206 2423 y Fe(p)174 2451 y Fg(X)173 2547 y Fh(\015)r Fp(=1)240 2494 y Fm([)p Fl(I)t Fm(\()p Fl(y)317 2501 y Fh(\015)340 2494 y Fm(\))c Fi(\000)i Fl(I)t Fm(\(\()p Fl(y)g Fm(+)f Fl(\016)r(y)r Fm(\))615 2501 y Fh(\015)638 2494 y Fm(\)]1)-11 b(I)696 2502 y Fd(fj)p Fh(y)744 2507 y Fe(\015)765 2502 y Fd(j\025)p Fp(1)p Fd(\000)p Fh(d)p Fd(g)903 2494 y Fi(\025)p Fm(2)961 2475 y Fd(\000)p Fh(M)1052 2437 y Fp(2)1070 2423 y Fe(p)1038 2451 y Fg(X)1037 2547 y Fh(\015)r Fp(=1)1104 2494 y Fm([)p Fl(I)t Fm(\()p Fl(y)1181 2501 y Fh(\015)1204 2494 y Fm(\))10 b Fi(\000)g Fl(I)t Fm(\(\()p Fl(y)h Fm(+)f Fl(\016)r(y)r Fm(\))1479 2501 y Fh(\015)1502 2494 y Fm(\)]1)-11 b(I)1560 2502 y Fd(fj)p Fh(y)1608 2507 y Fe(\015)1629 2502 y Fd(j\025)p Fp(1)p Fd(\000)p Fh(\021)q Fd(g)903 2619 y Fi(\025)56 b Fm(inf)952 2643 y Fc(j)p Fe(y)979 2648 y(\015)1000 2643 y Fc(j\025)p Ff(1)p Fc(\000)p Fe(d)951 2675 y Fc(j)p Fe(x)979 2680 y(\015)1001 2675 y Fc(j\024)p Ff(1)p Fc(\000)p Fe(d)1098 2619 y Fm(\(1)9 b Fi(\000)i Fl(\025)p Fm(\)2)1262 2600 y Fd(\000j)p Fh(I)r Fd(j)1332 2619 y Fm([)p Fl(I)t Fm(\()p Fl(y)1409 2626 y Fh(\015)1431 2619 y Fm(\))f Fi(\000)g Fl(I)t Fm(\(\()p Fl(y)h Fm(+)g Fl(\016)r(y)r Fm(\))1707 2626 y Fh(\015)1730 2619 y Fm(\)])1834 2560 y(\(4)p Fl(:)p Fm(74\))0 2779 y Ff(31)p Fe(=mar)q(ch=)p Ff(1995;)17 b(9:46)652 b Fm(23)p eop %%Page: 24 25 bop 0 45 a Fm(Note)13 b(that)g Fl(I)t Fm(\()p Fl(z)r Fm(\))f(is)i(symmetric)g (with)g(resp)q(ect)g(to)f(zero)g(and)h(is)g(a)f(strictly)h(increasing)h (function)f(of)f Fl(z)j Fm(for)d Fl(z)i(>)e Fm(0.)0 118 y(Th)o(us)i Fl(I)t Fm(\(\()p Fl(y)c Fm(+)g Fl(\016)r(y)r Fm(\))318 125 y Fh(\015)340 118 y Fm(\))k(is)h(maximized)g(o)o(v)o(er)f Fi(j)p Fl(x)784 125 y Fh(\015)807 118 y Fi(j)d(\024)h Fm(1)d Fi(\000)g Fl(d)15 b Fm(for)g Fl(x)1093 125 y Fh(\015)1129 118 y Fm(=)e(\(1)c Fi(\000)h Fl(d)p Fm(\)sign)p Fl(y)1414 125 y Fh(\015)1438 118 y Fm(.)20 b(F)l(rom)14 b(this)i(w)o(e)f(get)356 240 y(inf)306 271 y Fd(j)p Fh(x)338 276 y Fe(\015)358 271 y Fd(j\024)p Fp(1)p Fd(\000)p Fh(d)462 240 y Fm([)p Fl(I)t Fm(\()p Fl(y)539 247 y Fh(\015)562 240 y Fm(\))10 b Fi(\000)g Fl(I)t Fm(\(\()p Fl(y)h Fm(+)g Fl(\016)r(y)r Fm(\))838 247 y Fh(\015)860 240 y Fm(\)])h Fi(\025)h Fm([)p Fl(I)t Fm(\()p Fl(y)1028 247 y Fh(\015)1051 240 y Fm(\))c Fi(\000)i Fl(I)t Fm(\()p Fi(j)p Fl(y)1201 247 y Fh(\015)1223 240 y Fi(j)f Fm(+)h Fl(\032)p Fm(\(\(1)d Fi(\000)j Fl(d)p Fm(\))e Fi(\000)i(j)p Fl(y)1561 247 y Fh(\015)1584 240 y Fi(j)p Fm(\)\)])188 b(\(4)p Fl(:)p Fm(75\))0 372 y(and)13 b(the)g(in\014m)o(um)g(o)o(v)o(er)f Fi(j)p Fl(y)467 379 y Fh(\015)490 372 y Fi(j)g(\025)h Fm(1)5 b Fi(\000)g Fl(\021)14 b Fm(in)g(the)e(r.h.s.)19 b(of)12 b(\(4.75\))f(is)i(easily)g(seen)h(to)e(b)q(e)h(tak)o(en)f(on)h(for)f Fi(j)p Fl(y)1749 379 y Fh(\015)1772 372 y Fi(j)g Fm(=)h(1)5 b Fi(\000)g Fl(\021)r Fm(.)0 445 y(Th)o(us)107 541 y(inf)64 566 y Fc(j)p Fe(y)91 571 y(\015)112 566 y Fc(j\025)p Ff(1)p Fc(\000)p Fe(d)64 597 y Fc(j)p Fe(x)92 602 y(\015)113 597 y Fc(j\024)p Ff(1)p Fc(\000)p Fe(d)210 541 y Fm(\(1)10 b Fi(\000)g Fl(\025)p Fm(\)2)374 522 y Fd(\000j)p Fh(I)r Fd(j)444 541 y Fm([)p Fl(I)t Fm(\()p Fl(y)521 548 y Fh(\015)544 541 y Fm(\))f Fi(\000)i Fl(I)t Fm(\(\()p Fl(y)g Fm(+)f Fl(\016)r(y)r Fm(\))819 548 y Fh(\015)842 541 y Fm(\)])i Fi(\025)p Fm(\(1)e Fi(\000)g Fl(\025)p Fm(\)2)1084 522 y Fd(\000j)p Fh(I)r Fd(j)1154 541 y Fm([)p Fl(I)t Fm(\(1)f Fi(\000)h Fl(\021)r Fm(\))f Fi(\000)i Fl(I)t Fm(\(1)e Fi(\000)h Fl(\021)h Fi(\000)g Fl(\032)p Fm(\()p Fl(d)e Fi(\000)h Fl(\021)r Fm(\)\)])885 672 y Fi(\025)p Fm(\(1)g Fi(\000)g Fl(\025)p Fm(\)2)1084 653 y Fd(\000j)p Fh(I)r Fd(j)1154 672 y Fl(\032)p Fm(\()p Fl(d)f Fi(\000)h Fl(\021)r Fm(\))p Fl(I)1341 653 y Fd(0)1353 672 y Fm(\(1)f Fi(\000)i Fl(\021)g Fi(\000)f Fl(\032)p Fm(\()p Fl(d)f Fi(\000)i Fl(\021)r Fm(\)\))885 766 y Fi(\025)p Fm(\(1)f Fi(\000)g Fl(\025)p Fm(\)2)1084 747 y Fd(\000j)p Fh(I)r Fd(j)1154 766 y Fl(\032)p Fm(\()p Fl(d)f Fi(\000)h Fl(\021)r Fm(\))1322 735 y(1)p 1322 755 23 2 v 1322 797 a(2)1350 766 y Fi(j)e Fm(ln\()p Fl(\021)j Fm(+)g Fl(\032)p Fm(\()p Fl(d)d Fi(\000)j Fl(\021)r Fm(\)\))p Fi(j)1834 659 y Fm(\(4)p Fl(:)p Fm(76\))0 889 y(where)k(w)o(e)g(ha)o(v)o(e)g(used)h(the)f (con)o(v)o(exit)o(y)g(of)g Fl(I)k Fm(and)c(the)g(b)q(ound,)h Fl(I)1121 872 y Fd(0)1133 889 y Fm(\(1)10 b Fi(\000)g Fl(x)p Fm(\))j Fi(\025)1339 871 y Fp(1)p 1339 878 19 2 v 1339 904 a(2)1363 889 y Fi(j)8 b Fm(ln)f Fl(x)p Fi(j)15 b Fm(for)g(0)d Fl(<)h(x)g(<)g Fm(1.)91 998 y(W)l(e)18 b(no)o(w)f(ha)o(v)o(e)g(to)g(consider) i(the)f(comp)q(onen)o(ts)g Fl(y)970 1005 y Fh(\015)1011 998 y Fm(with)g Fi(j)p Fl(y)1152 1005 y Fh(\015)1175 998 y Fi(j)f(\024)g Fm(1)12 b Fi(\000)g Fl(d)p Fm(.)27 b(Here)18 b(the)g(en)o(trop)o(y)f (di\013erence)0 1071 y Fl(I)t Fm(\()p Fl(y)64 1078 y Fh(\015)87 1071 y Fm(\))5 b Fi(\000)g Fl(I)t Fm(\(\()p Fl(y)i Fm(+)e Fl(\016)r(y)r Fm(\))343 1078 y Fh(\015)366 1071 y Fm(\))12 b(can)h(of)f(course)h(b)q(e)g (negativ)o(e.)19 b(T)l(o)13 b(get)f(a)h(lo)o(w)o(er)f(b)q(ound)i(on)e(this)h (di\013erence)h(w)o(e)f(use)g(\(4.75\))0 1143 y(and)i(p)q(erform)g(the)h(c)o (hange)f(of)g(v)m(ariable)h Fi(j)p Fl(y)746 1150 y Fh(\015)769 1143 y Fi(j)d Fm(=)g(\(1)c Fi(\000)i Fl(d)p Fm(\))e Fi(\000)i Fl(z)1057 1150 y Fh(\015)1095 1143 y Fm(to)k(write)169 1248 y(inf)126 1273 y Fc(j)p Fe(y)153 1278 y(\015)174 1273 y Fc(j\024)p Ff(1)p Fc(\000)p Fe(d)126 1304 y Fc(j)p Fe(x)154 1309 y(\015)175 1304 y Fc(j\024)p Ff(1)p Fc(\000)p Fe(d)272 1248 y Fm([)p Fl(I)t Fm(\()p Fl(y)349 1255 y Fh(\015)372 1248 y Fm(\))9 b Fi(\000)i Fl(I)t Fm(\(\()p Fl(y)g Fm(+)f Fl(\016)r(y)r Fm(\))647 1255 y Fh(\015)670 1248 y Fm(\)])i(=)70 b(inf)756 1278 y Fp(0)p Fd(\024)p Fh(z)819 1283 y Fe(\015)841 1278 y Fd(\024)p Fp(1)p Fd(\000)p Fh(d)942 1248 y Fl(I)t Fm(\(\(1)9 b Fi(\000)h Fl(d)p Fm(\))g Fi(\000)g Fl(z)1197 1255 y Fh(\015)1231 1248 y Fm(+)g Fl(\032z)1321 1255 y Fh(\015)1344 1248 y Fm(\))g Fi(\000)g Fl(I)t Fm(\(\(1)f Fi(\000)i Fl(d)p Fm(\))e Fi(\000)i Fl(z)1673 1255 y Fh(\015)1696 1248 y Fm(\))713 1375 y Fi(\025)70 b Fm(inf)756 1404 y Fp(0)p Fd(\024)p Fh(z)819 1409 y Fe(\015)841 1404 y Fd(\024)p Fp(1)p Fd(\000)p Fh(d)942 1375 y Fi(\000)p Fl(\032z)1022 1382 y Fh(\015)1045 1375 y Fl(I)1069 1356 y Fd(0)1082 1375 y Fm(\(\(1)9 b Fi(\000)h Fl(d)p Fm(\))g Fi(\000)g Fl(z)1313 1382 y Fh(\015)1347 1375 y Fm(+)g Fl(\032z)1437 1382 y Fh(\015)1460 1375 y Fm(\))713 1502 y(=)70 b(inf)756 1532 y Fp(0)p Fd(\024)p Fh(z)819 1537 y Fe(\015)841 1532 y Fd(\024)p Fp(1)p Fd(\000)p Fh(d)942 1502 y Fi(\000)p Fl(\032z)1022 1509 y Fh(\015)1051 1471 y Fm(1)p 1051 1492 23 2 v 1051 1533 a(2)1087 1502 y(ln)1132 1438 y Fg(\022)1171 1471 y Fm(2)10 b Fi(\000)g Fl(d)g Fi(\000)h Fl(z)1350 1478 y Fh(\015)1383 1471 y Fm(+)g Fl(\032z)1474 1478 y Fh(\015)p 1171 1492 326 2 v 1210 1533 a Fl(d)f Fm(+)g Fl(z)1310 1540 y Fh(\015)1344 1533 y Fi(\000)g Fl(\032z)1434 1540 y Fh(\015)1502 1438 y Fg(\023)713 1630 y Fi(\025)h(\000)f Fl(\032)p Fm(\(1)f Fi(\000)h Fl(d)p Fm(\))970 1600 y(1)p 970 1620 23 2 v 970 1661 a(2)1006 1630 y(ln)1051 1566 y Fg(\022)1090 1600 y Fm(2)g Fi(\000)h Fl(d)p 1090 1620 102 2 v 1130 1661 a(d)1198 1566 y Fg(\023)713 1752 y Fi(\025)g(\000)809 1721 y Fl(\032)p 809 1741 24 2 v 809 1783 a Fm(2)846 1752 y(ln)897 1721 y(2)p 897 1741 V 897 1783 a Fl(d)1834 1508 y Fm(\(4)p Fl(:)p Fm(77\))0 1870 y(and)k(putting)h (together)e(\(4.77\))g(and)h(\(4.72\))e(yields)251 2020 y(2)274 2001 y Fd(\000)p Fh(M)365 1963 y Fp(2)383 1950 y Fe(p)350 1977 y Fg(X)350 2073 y Fh(\015)r Fp(=1)417 2020 y Fm([)p Fl(I)t Fm(\()p Fl(y)494 2027 y Fh(\015)516 2020 y Fm(\))d Fi(\000)h Fl(I)t Fm(\(\()p Fl(y)f Fm(+)h Fl(\016)r(y)r Fm(\))792 2027 y Fh(\015)815 2020 y Fm(\)]1)-11 b(I)873 2028 y Fd(f)o(j)p Fh(y)920 2033 y Fe(\015)941 2028 y Fd(j)p Fh(<)p Fp(1)p Fd(\000)p Fh(d)p Fd(g)1080 2020 y Fi(\025)13 b(\000)p Fm(\(1)d Fi(\000)g Fm(\(1)f Fi(\000)i Fl(\025)p Fm(\)2)1423 2001 y Fd(\000j)p Fh(I)r Fd(j)1493 2020 y Fm(\))1516 1989 y Fl(\032)p 1516 2010 V 1516 2051 a Fm(2)1552 2020 y(ln)1598 1956 y Fg(\022)1637 1989 y Fm(2)p 1637 2010 V 1637 2051 a Fl(d)1666 1956 y Fg(\023)1834 2020 y Fm(\(4)p Fl(:)p Fm(78\))0 2171 y(Therefore,)k(\(4.78\))e(together)h (with)i(\(4.74\))d(and)j(\(4.76\))d(giv)o(e)174 2321 y Fl(\014)202 2302 y Fd(\000)p Fp(1)251 2321 y Fm(2)274 2302 y Fd(\000)p Fh(M)365 2264 y Fp(2)383 2251 y Fe(p)351 2278 y Fg(X)350 2374 y Fh(\015)r Fp(=1)417 2321 y Fm([)p Fl(I)t Fm(\()p Fl(y)494 2328 y Fh(\015)516 2321 y Fm(\))d Fi(\000)h Fl(I)t Fm(\(\()p Fl(y)g Fm(+)f Fl(\016)r(y)r Fm(\))792 2328 y Fh(\015)815 2321 y Fm(\)])139 2469 y Fi(\025)p Fl(\014)202 2450 y Fd(\000)p Fp(1)251 2469 y Fl(\032)283 2405 y Fg(\032)316 2469 y Fm(\(1)f Fi(\000)i Fl(\025)p Fm(\)2)480 2450 y Fd(\000j)p Fh(I)r Fd(j)550 2469 y Fm(\()p Fl(d)e Fi(\000)i Fl(\021)r Fm(\))695 2438 y(1)p 695 2459 23 2 v 695 2500 a(2)722 2469 y Fi(j)d Fm(ln\()p Fl(\021)j Fm(+)g Fl(\032)p Fm(\()p Fl(d)e Fi(\000)h Fl(\021)r Fm(\)\))p Fi(j)e(\000)j Fm(\(1)e Fi(\000)i Fm(\(1)e Fi(\000)i Fl(\025)p Fm(\)2)1386 2450 y Fd(\000j)p Fh(I)r Fd(j)1455 2469 y Fm(\))1478 2438 y(1)p 1478 2459 V 1478 2500 a(2)1514 2469 y(ln)1560 2405 y Fg(\022)1599 2438 y Fm(2)p 1599 2459 24 2 v 1599 2500 a Fl(d)1628 2405 y Fg(\023\033)1834 2386 y Fm(\(4)p Fl(:)p Fm(79\))0 2597 y(On)16 b(the)f(other)g(hand,)g(w)o(e)g(ha)o(v)o(e)803 2670 y(2)826 2651 y Fd(\000)p Fh(M)q(=)p Fp(2)929 2670 y Fi(k)p Fl(\016)r(y)r Fi(k)1021 2677 y Fp(2)1053 2670 y Fi(\024)e Fm(2)p Fl(\032)686 b Fm(\(4)p Fl(:)p Fm(80\))0 2779 y Ff(31)p Fe(=mar)q(ch=)p Ff(1995;)17 b(9:46)652 b Fm(24)p eop %%Page: 25 26 bop 0 45 a Fm(Consequen)o(tly)l(,)16 b(\(4.67\))d(holds)j(if)g(w)o(e)e(can)i (c)o(ho)q(ose)f Fl(\025)p Fm(,)f Fl(\021)r Fm(,)g(and)i Fl(\032)e Fm(so)h(that)g(the)g(follo)o(wing)h(inequalit)o(y)h(holds,)126 177 y Fl(\014)154 158 y Fd(\000)p Fp(1)211 113 y Fg(\032)245 177 y Fm(\(1)9 b Fi(\000)i Fl(\025)p Fm(\)2)409 158 y Fd(\000j)p Fh(I)r Fd(j)479 177 y Fm(\()p Fl(d)e Fi(\000)h Fl(\021)r Fm(\))623 146 y(1)p 623 166 23 2 v 623 208 a(2)651 177 y Fi(j)e Fm(ln\()p Fl(\021)j Fm(+)f Fl(\032)p Fm(\()p Fl(d)f Fi(\000)i Fl(\021)r Fm(\)\))p Fi(j)d(\000)j Fm(\(1)e Fi(\000)i Fm(\(1)e Fi(\000)i Fl(\025)p Fm(\)2)1315 158 y Fd(\000j)p Fh(I)r Fd(j)1384 177 y Fm(\))1407 146 y(1)p 1407 166 V 1407 208 a(2)1443 177 y(ln)1489 113 y Fg(\022)1528 146 y Fm(2)p 1528 166 24 2 v 1528 208 a Fl(d)1557 113 y Fg(\023\033)1637 177 y Fl(>)i Fm(2)126 b(\(4)p Fl(:)p Fm(81\))0 312 y(But)15 b(this)h(is)g(alw)o(a)o(ys)f(p)q(ossible)i(b)o (y)e(taking)h(e.g.)k Fl(\025)12 b(<)i Fm(1,)g Fl(\021)h Fi(\021)e Fl(\032d=)p Fm(2)h(and)i Fl(\032)c Fi(\021)h Fl(d)1364 296 y Fh(K)1415 312 y Fm(where)i Fl(K)h Fi(\021)d Fl(K)s Fm(\()p Fl(d;)8 b Fi(j)p Fl(I)t Fi(j)p Fl(;)g(\025)o Fm(\))i Fl(>)j Fm(1)0 385 y(is)j(c)o(hosen)f(su\016cien)o(tly)i(large)e(as)g(to)f(satisfy) 484 507 y(\(1)9 b Fi(\000)i Fl(\025)p Fm(\)2)648 489 y Fd(\000j)p Fh(I)r Fd(j)717 507 y Fl(d)p Fm(\(1)f Fi(\000)g Fl(d)861 489 y Fh(K)896 507 y Fl(=)p Fm(2\))965 477 y Fl(K)i Fm(+)f(1)p 965 497 121 2 v 1013 539 a(2)1090 507 y Fi(j)d Fm(ln)g Fl(d)p Fi(j)j Fl(>)i Fm(4)d(+)1336 477 y(1)p 1336 497 23 2 v 1336 539 a(2)1364 507 y Fi(j)e Fm(ln)g Fl(d)p Fi(j)367 b Fm(\(4)p Fl(:)p Fm(82\))0 666 y Fn(Case)17 b(2:)45 b Fm(W)l(e)15 b(will)i(assume)f (that)e Fl(\025)f(<)g Fm(1,)i(and)h(that)e Fl(\021)r Fm(,)h(and)g Fl(\032)g Fm(are)g(c)o(hosen)h(as)f(in)h(the)g(case)f(1.)21 b(W)l(e)15 b(can)h(then)0 739 y(assume)f(that)685 771 y Fp(2)703 758 y Fe(p)670 785 y Fg(X)670 881 y Fh(\015)r Fp(=1)744 828 y Fm(1)-11 b(I)772 837 y Fd(fj)p Fh(y)820 842 y Fe(\015)841 837 y Fd(j\025)p Fp(1)p Fd(\000)p Fh(\021)q Fd(g)980 828 y Fl(<)13 b Fm(\(1)c Fi(\000)i Fl(\025)p Fm(\)2)1192 810 y Fh(p)p Fd(\000j)p Fh(I)r Fd(j)1834 828 y Fm(\(4)p Fl(:)p Fm(83\))0 958 y(W)l(e)k(assume)g(further)g(that)881 1031 y Fl(T)908 1038 y Fh(p)929 1031 y Fm(\()p Fl(y)r Fm(\))d Fl(>)h(c)765 b Fm(\(4)p Fl(:)p Fm(84\))0 1133 y(for)15 b Fl(c)f Fm(su\016cien)o(tly)j(large)e(to)g(b) q(e)h(c)o(hosen)f(later.)20 b(Here)15 b(w)o(e)g(will)i(c)o(ho)q(ose)e Fl(\016)r(y)i Fm(suc)o(h)f(that)874 1255 y Fl(Y)11 b Fm(\()p Fl(\016)r(y)r Fm(\))h Fi(\021)h Fm(0)758 b(\(4)p Fl(:)p Fm(85\))0 1378 y(so)18 b(that)g(trivially)i Fl(y)14 b Fm(+)e Fl(\016)r(y)20 b Fi(2)e(A)567 1385 y Fh(\017)585 1378 y Fm(\()8 b(~)-31 b Fl(m)p Fm(\).)29 b(Let)18 b(us)h(in)o(tro)q(duce)g(a)f(parameter)g(0)f Fl(<)i(\020)i(<)d(\021)r Fm(,)g(that)g(w)o(e)g(will)i(c)o(ho)q(ose)0 1450 y(appropriately)c(later,)f(and)g(let)h(us)f(set,)g(for)f Fl(\015)h Fi(2)e(f)p Fm(1)p Fl(;)8 b(:)g(:)g(:)t(;)g Fm(2)1018 1434 y Fd(j)p Fh(I)r Fd(j)1059 1450 y Fi(g)p Fm(,)480 1573 y Fi(K)516 1554 y Fp(+)515 1584 y Fh(\015)559 1573 y Fi(\021)k(f)r Fm(~)-25 b Fl(\015)15 b Fi(2)e(f)p Fm(1)p Fl(;)8 b(:)g(:)g(:)t(;)g Fm(2)881 1554 y Fh(p)p Fd(\000j)p Fh(I)r Fd(j)970 1573 y Fi(g)1005 1534 y Fg(\014)1005 1561 y(\014)1033 1573 y Fi(j)p Fl(y)1068 1583 y Fh(\015)r Fp(+\()q(~)-19 b Fh(\015)r Fd(\000)p Fp(1\)2)1230 1574 y Fc(j)p Fe(I)r Fc(j)1272 1573 y Fi(j)12 b(\025)h Fm(1)d Fi(\000)h Fl(\020)s Fi(g)364 b Fm(\(4)p Fl(:)p Fm(86\))0 1695 y(and)480 1768 y Fi(K)516 1749 y Fd(\000)515 1779 y Fh(\015)558 1768 y Fi(\021)13 b(f)r Fm(~)-25 b Fl(\015)15 b Fi(2)e(f)p Fm(1)p Fl(;)8 b(:)g(:)g(:)t(;)g Fm(2)881 1749 y Fh(p)p Fd(\000j)p Fh(I)r Fd(j)969 1768 y Fi(g)1005 1729 y Fg(\014)1005 1756 y(\014)1033 1768 y Fi(j)p Fl(y)1068 1778 y Fh(\015)r Fp(+\()q(~)-19 b Fh(\015)r Fd(\000)p Fp(1\)2)1230 1769 y Fc(j)p Fe(I)r Fc(j)1272 1768 y Fi(j)12 b(\024)h Fm(1)d Fi(\000)g Fl(\021)r Fi(g)0 1870 y Fm(F)l(or)16 b(all)i(indices)h Fl(\015)g Fm(suc)o(h)f(that)e Fi(K)583 1853 y Fp(+)582 1881 y Fh(\015)628 1870 y Fm(=)g Fi(;)p Fm(,)h(w)o(e)g(simply)h(set)f Fl(\016)r(y)1065 1880 y Fh(\015)r Fp(+\()q(~)-19 b Fh(\015)r Fd(\000)p Fp(1\)2)1227 1871 y Fc(j)p Fe(I)r Fc(j)1285 1870 y Fi(\021)16 b Fm(0)h(for)f(all)k(~)-25 b Fl(\015)18 b Fi(2)e(f)p Fm(1)p Fl(;)8 b(:)g(:)g(:)t(;)g Fm(2)1770 1853 y Fh(p)p Fd(\000j)p Fh(I)r Fd(j)1859 1870 y Fi(g)p Fm(.)25 b(If)0 1943 y Fi(K)36 1926 y Fp(+)35 1954 y Fh(\015)83 1943 y Fm(w)o(ere)18 b(empt)o(y)g(for)f Fk(al)r(l)h Fl(\015)s Fm(,)f(then)h Fl(T)662 1950 y Fh(p)683 1943 y Fm(\()p Fl(y)r Fm(\))e Fi(\024)i Fm([)p Fl(I)849 1926 y Fd(0)861 1943 y Fm(\(1)11 b Fi(\000)h Fl(\020)s Fm(\)])1014 1926 y Fp(2)1052 1943 y Fm(whic)o(h)19 b(con)o(tradicts)e(our)h(assumption)g(\(4.84\),)f(for)0 2015 y(suitably)h(large)g Fl(c)f Fm(\(dep)q(ending)i(only)f(on)f Fl(\020)s Fm(\).)26 b(Th)o(us)18 b(w)o(e)f(consider)h(no)o(w)f(the)g (remaining)i(indices)g Fl(\015)g Fm(for)e(whic)o(h)0 2088 y Fi(K)36 2072 y Fp(+)35 2099 y Fh(\015)78 2088 y Fi(6)p Fm(=)c Fi(;)p Fm(.)91 2197 y(First)f(note)g(that)g(\(4.83\))e(implies)15 b(that)c Fi(jK)817 2181 y Fp(+)816 2209 y Fh(\015)846 2197 y Fi(j)i Fl(<)g Fm(\(1)t Fi(\000)t Fl(\025)p Fm(\)2)1072 2181 y Fh(p)p Fd(\000j)p Fh(I)r Fd(j)1173 2197 y Fm(and)g(that)e Fi(K)1390 2181 y Fd(\000)1389 2209 y Fh(\015)1433 2197 y Fl(>)i(\025)p Fm(2)1531 2181 y Fh(p)p Fd(\000j)p Fh(I)r Fd(j)1632 2197 y Fm(so)f(that)g(c)o(ho)q(osing)0 2270 y(1)h Fl(>)h(\025)e(>)178 2252 y Fp(1)p 178 2259 19 2 v 178 2286 a(2)202 2270 y Fm(,)j(w)o(e)g(ha)o(v)o (e)h Fi(jK)452 2253 y Fp(+)451 2281 y Fh(\015)481 2270 y Fi(j)d Fl(<)g Fi(jK)604 2253 y Fd(\000)603 2281 y Fh(\015)634 2270 y Fi(j)p Fm(.)20 b(Our)c(strategy)f(will)i(b)q(e)f(to)f(\014nd)h Fl(\016)r(y)i Fm(in)e(suc)o(h)g(a)f(w)o(a)o(y)g(as)g(to)g(decrease)h(the)0 2343 y(mo)q(duli)h(of)f(the)g(comp)q(onen)o(ts)f(in)i Fi(K)625 2326 y Fp(+)624 2354 y Fh(\015)670 2343 y Fm(at)e(the)h(exp)q(ense)h(of)e(p)q (ossibly)j(increasing)f(them)e(on)h Fi(K)1632 2326 y Fd(\000)1631 2354 y Fh(\015)1678 2343 y Fm(in)g(suc)o(h)h(a)e(w)o(a)o(y)0 2415 y(as)g(to)f(lea)o(v)o(e)i Fl(Y)10 b Fm(\()p Fl(y)i Fm(+)e Fl(\016)r(y)r Fm(\))j(=)f Fl(Y)f Fm(\()p Fl(y)r Fm(\).)91 2525 y(W)l(e)16 b(will)h(in)g(the)f(sequel)h(consider)f(the)g(case)g(where)g (there)g(is)g(only)g(one)g(index)h Fl(\015)s Fm(,)e(e.g.)21 b Fl(\015)16 b Fm(=)e(1,)h(for)g(whic)o(h)0 2597 y Fi(K)36 2581 y Fp(+)35 2609 y Fh(\015)81 2597 y Fm(is)i(nonempt)o(y)l(.)22 b(The)16 b(general)h(case)e(is)i(treated)e(essen)o(tially)j(b)o(y)e (iterating)g(the)g(same)f(pro)q(cedure.)23 b(W)l(e)16 b(will)0 2670 y(use)h(the)f(simpli\014ed)j(notation)d Fl(y)570 2680 y Fp(1+2)634 2671 y Fc(j)p Fe(I)r Fc(j)675 2680 y Fp(~)-19 b Fh(\015)711 2670 y Fi(\021)15 b Fl(y)784 2677 y Fp(~)-19 b Fh(\015)807 2670 y Fm(,)16 b Fl(\016)r(y)880 2680 y Fp(1+2)944 2671 y Fc(j)p Fe(I)r Fc(j)985 2680 y Fp(~)-19 b Fh(\015)1021 2670 y Fi(\021)15 b Fl(\016)r(y)1116 2677 y Fp(~)-19 b Fh(\015)1155 2670 y Fm(and)17 b(also)f(set)g Fi(K)1445 2651 y Fd(\006)1444 2682 y Fp(1)1490 2670 y Fi(\021)e(K)1575 2653 y Fd(\006)1605 2670 y Fm(.)24 b(W)l(e)16 b(will)i(assume)0 2779 y Ff(31)p Fe(=mar)q(ch=)p Ff(1995;)f(9:46)652 b Fm(25)p eop %%Page: 26 27 bop 0 45 a Fm(moreo)o(v)o(er)14 b(that)g(all)h(comp)q(onen)o(ts)g Fl(y)629 52 y Fp(~)-19 b Fh(\015)667 45 y Fm(are)14 b(p)q(ositiv)o(e,)h(as)g (this)g(is)g(the)g(w)o(orst)f(situation.)20 b(W)l(e)15 b(will)h(c)o(hose)f Fl(\016)r(y)i Fm(suc)o(h)0 118 y(that)g Fl(\016)r(y)146 125 y Fp(~)-19 b Fh(\015)186 118 y Fm(=)17 b(0,)h(if)i(~)-25 b Fl(\015)19 b Fi(2)f(fK)486 102 y Fp(+)527 118 y Fi([)12 b(K)605 102 y Fd(\000)635 118 y Fi(g)658 102 y Fh(c)694 118 y Fm(and)18 b Fl(\016)r(y)830 125 y Fp(~)-19 b Fh(\015)870 118 y Fl(<)17 b Fm(0)h(if)i(~)-25 b Fl(\015)19 b Fi(2)e(K)1133 102 y Fp(+)1163 118 y Fm(.)28 b(F)l(or)17 b(eac)o(h)j(~)-25 b Fl(\015)19 b Fi(2)e(K)1519 102 y Fp(+)1567 118 y Fm(w)o(e)g(will)j(c)o(ho)q(ose)e(in)g(a)0 191 y(unique)d(and)g(distinct)h(~)-24 b Fl(\015)424 174 y Fd(0)448 191 y Fi(2)13 b(K)527 174 y Fd(\000)571 191 y Fm(and)h(set)g Fl(\016)r(y)773 198 y Fp(~)-19 b Fh(\015)793 189 y Fc(0)820 191 y Fm(=)13 b Fi(\000)p Fl(\016)r(y)948 198 y Fp(~)-19 b Fh(\015)971 191 y Fm(.)19 b(This)c(ensures)f(that)g Fl(Y)c Fm(\()p Fl(\016)r(y)r Fm(\))i(=)h(0.)19 b(W)l(e)14 b(will)i(also)d(mak)o(e)0 264 y(sure)i(that)g(for)f(all)k(~)-24 b Fl(\015)r Fm(,)14 b Fi(j)p Fl(\016)r(y)s Fm(~)-24 b Fl(\015)r Fi(j)12 b(\024)h Fl(\021)r(=)p Fm(2)c Fi(\000)h Fl(\020)s Fm(.)91 373 y(W)l(e)16 b(ha)o(v)o(e)f(to)g(construct)g Fl(\016)r(y)576 380 y Fp(~)-19 b Fh(\015)614 373 y Fm(for)17 b(~)-25 b Fl(\015)15 b Fi(2)f(K)803 356 y Fp(+)833 373 y Fm(.)21 b(In)16 b(this)g(pro)q(cess)g(w)o(e)f(ha)o(v)o (e)g(to)g(consider)i(the)f(follo)o(wing)g(three)0 445 y(functionals:)10 555 y(\(1\))22 b(The)15 b(c)o(hange)h(in)g(the)f(quadratic)g(term)g(of)g (\002)865 562 y Fh(p)886 555 y Fm(.)20 b(This)c(is)f(b)q(ounded)i(b)o(y)718 683 y Fl(\016)r(E)s Fm(\()p Fl(\016)r(y)r Fm(\))11 b Fi(\021)i Fm(2)941 664 y Fd(\000)p Fh(p=)p Fp(2+1)1073 622 y Fg(s)p 1118 622 205 2 v 61 x Fm(2)1165 640 y Fg(X)1150 740 y Fp(~)-19 b Fh(\015)r Fd(2K)1222 731 y Ff(+)1255 683 y Fl(\016)r(y)1301 667 y Fp(2)1300 696 y(~)g Fh(\015)1834 683 y Fm(\(4)p Fl(:)p Fm(87\))10 878 y(\(2\))22 b(The)15 b(c)o(hange)h(in)g(the)f(en)o(trop)o(y)f (term,)438 988 y Fl(\016)r(I)t Fm(\()p Fl(\016)r(y)r Fm(\))d(=)i(2)648 970 y Fd(\000)p Fh(p)721 945 y Fg(X)706 1045 y Fp(~)-19 b Fh(\015)r Fd(2K)778 1036 y Ff(+)811 988 y Fm(\()p Fl(I)t Fm(\()p Fl(y)894 995 y Fp(~)g Fh(\015)926 988 y Fm(+)10 b Fl(\016)r(y)1016 995 y Fp(~)-19 b Fh(\015)1039 988 y Fm(\))10 b Fi(\000)g Fl(I)t Fm(\()p Fl(y)1177 995 y Fp(~)-19 b Fh(\015)1199 988 y Fm(\)\))575 1122 y(+)10 b(2)643 1103 y Fd(\000)p Fh(p)717 1079 y Fg(X)701 1179 y Fp(~)-19 b Fh(\015)r Fd(2K)773 1170 y Fc(\000)808 1122 y Fm(\()o Fl(I)t Fm(\()p Fl(y)890 1129 y Fp(~)g Fh(\015)922 1122 y Fm(+)11 b Fl(\016)r(y)1013 1129 y Fp(~)-19 b Fh(\015)1035 1122 y Fm(\))10 b Fi(\000)g Fl(I)t Fm(\()p Fl(y)1173 1129 y Fp(~)-19 b Fh(\015)1195 1122 y Fm(\)\))577 1255 y Fi(\025)13 b Fm(2)648 1237 y Fd(\000)p Fh(p)721 1212 y Fg(X)706 1312 y Fp(~)-19 b Fh(\015)r Fd(2K)778 1303 y Ff(+)811 1255 y Fi(j)p Fl(\016)r(y)869 1262 y Fp(~)g Fh(\015)892 1255 y Fi(j)p Fl(I)929 1237 y Fd(0)940 1255 y Fm(\()p Fl(y)981 1262 y Fp(~)g Fh(\015)1014 1255 y Fm(+)10 b Fl(\016)r(y)1104 1262 y Fp(~)-19 b Fh(\015)1127 1255 y Fm(\))10 b Fi(\000)g Fm(2)1223 1237 y Fd(\000)p Fh(p)1297 1212 y Fg(X)1281 1312 y Fp(~)-19 b Fh(\015)r Fd(2K)1353 1303 y Fc(\000)1388 1255 y Fl(\016)r(y)1433 1262 y Fp(~)g Fh(\015)1455 1255 y Fi(j)8 b Fm(ln)g Fl(\021)r Fi(j)p Fl(=)p Fm(2)577 1389 y(=)13 b(2)648 1370 y Fd(\000)p Fh(p)721 1346 y Fg(X)706 1446 y Fp(~)-19 b Fh(\015)r Fd(2K)778 1436 y Ff(+)811 1389 y Fi(j)p Fl(\016)r(y)869 1396 y Fp(~)g Fh(\015)892 1389 y Fi(j)8 b Fm(\()n Fl(I)953 1370 y Fd(0)966 1389 y Fm(\()p Fl(y)1007 1396 y Fp(~)-19 b Fh(\015)1039 1389 y Fm(+)11 b Fl(\016)r(y)1130 1396 y Fp(~)-19 b Fh(\015)1152 1389 y Fm(\))10 b Fi(\000)g(j)e Fm(ln)g Fl(\021)r Fi(j)p Fl(=)p Fm(2\))577 1522 y Fi(\025)13 b Fm(2)648 1503 y Fd(\000)p Fh(p)p Fd(\000)p Fp(1)768 1479 y Fg(X)753 1579 y Fp(~)-20 b Fh(\015)r Fd(2K)824 1570 y Ff(+)858 1522 y Fi(j)p Fl(\016)r(y)916 1529 y Fp(~)h Fh(\015)938 1522 y Fi(j)p Fl(I)975 1503 y Fd(0)987 1522 y Fm(\()p Fl(y)1028 1529 y Fp(~)g Fh(\015)1060 1522 y Fm(+)11 b Fl(\016)r(y)1151 1529 y Fp(~)-19 b Fh(\015)1173 1522 y Fm(\))1834 1277 y(\(4)p Fl(:)p Fm(88\))91 1675 y(where)18 b(w)o(e)f(ha)o(v)o(e)g(used)h(that)f(for)g(1)g Fi(\025)f(j)p Fl(x)p Fi(j)g(\035)h(j)p Fl(y)r Fi(j)f(\035)h Fm(0)p Fl(:)p Fm(9,)g Fl(I)t Fm(\()p Fl(x)p Fm(\))11 b Fi(\000)h Fl(I)t Fm(\()p Fl(y)r Fm(\))j Fi(\031)i(j)p Fl(x)11 b Fi(\000)h Fl(y)r Fi(jj)c Fm(ln\(1)j Fi(\000)h Fl(y)r Fm(\))p Fi(j)17 b Fm(and)h(that)91 1748 y(under)e(our)f(assumption)g(for)h(~)-24 b Fl(\015)14 b Fi(2)f(K)728 1731 y Fp(+)758 1748 y Fm(,)i Fl(y)809 1755 y Fp(~)-19 b Fh(\015)841 1748 y Fm(+)11 b Fl(\016)r(y)932 1755 y Fp(~)-19 b Fh(\015)967 1748 y Fi(\025)13 b Fm(1)d Fi(\000)g Fl(\021)r(=)p Fm(2.)10 1857 y(\(3\))22 b(Finally)l(,)16 b(w)o(e)f(ha)o(v)o(e) g(that)394 1967 y Fl(T)421 1974 y Fh(p)441 1967 y Fm(\()p Fl(y)d Fm(+)e Fl(\016)r(y)r Fm(\))i Fi(\024)h Fm(2)685 1948 y Fd(\000)p Fh(p)759 1924 y Fg(X)742 2024 y Fh(\015)r Fd(62K)815 2015 y Ff(+)841 1967 y Fm([)p Fl(I)878 1948 y Fd(0)890 1967 y Fm(\()p Fl(y)931 1974 y Fp(~)-19 b Fh(\015)963 1967 y Fm(+)11 b Fl(\016)r(y)1054 1974 y Fp(~)-19 b Fh(\015)1076 1967 y Fm(\)])1107 1948 y Fp(2)1137 1967 y Fm(+)11 b(2)1206 1948 y Fd(\000)p Fh(p)1279 1924 y Fg(X)1264 2024 y Fp(~)-20 b Fh(\015)r Fd(2K)1335 2015 y Ff(+)1361 1967 y Fm([)p Fl(I)1398 1948 y Fd(0)1410 1967 y Fm(\()p Fl(y)1451 1974 y Fp(~)h Fh(\015)1484 1967 y Fm(+)10 b Fl(\016)r(y)1574 1974 y Fp(~)-19 b Fh(\015)1597 1967 y Fm(\)])1628 1948 y Fp(2)614 2100 y Fi(\024)13 b Fm([)p Fl(I)699 2081 y Fd(0)711 2100 y Fm(\(1)d Fi(\000)g Fl(\021)r(=)p Fm(2\)])909 2081 y Fp(2)938 2100 y Fm(+)g(2)1006 2081 y Fd(\000)p Fh(p)1080 2057 y Fg(X)1064 2157 y Fp(~)-19 b Fh(\015)r Fd(2K)1136 2148 y Ff(+)1162 2100 y Fm([)p Fl(I)1199 2081 y Fd(0)1211 2100 y Fm(\()p Fl(y)1252 2107 y Fp(~)g Fh(\015)1284 2100 y Fm(+)11 b Fl(\016)r(y)1375 2107 y Fp(~)-19 b Fh(\015)1397 2100 y Fm(\)])1428 2081 y Fp(2)1834 2055 y Fm(\(4)p Fl(:)p Fm(89\))91 2291 y(Lo)q(oking)19 b(at)e(these)h(three)g (functionals)i(suggests)d(to)g(c)o(ho)q(ose)h Fl(\016)r(y)1230 2298 y Fp(~)-19 b Fh(\015)1271 2291 y Fm(for)19 b(~)-25 b Fl(\015)20 b Fi(2)d(K)1470 2275 y Fp(+)1518 2291 y Fm(as)g(the)h(solution)h(of)f(the)0 2364 y(equation)761 2437 y Fi(\000)p Fl(\016)r(y)841 2444 y Fp(~)-19 b Fh(\015)876 2437 y Fm(=)13 b Fl(\034)5 b(I)973 2418 y Fd(0)985 2437 y Fm(\()p Fl(y)1026 2444 y Fp(~)-19 b Fh(\015)1059 2437 y Fm(+)10 b Fl(\016)r(y)1149 2444 y Fp(~)-19 b Fh(\015)1172 2437 y Fm(\))644 b(\(4)p Fl(:)p Fm(90\))0 2538 y(The)15 b(p)q(oin)o(t)h(is)g (that)e(with)i(this)f(c)o(hoice)i(\(4.89\))c(yields)k(\(w)o(e)d(set)h(for)g (simplicit)o(y)i Fl(\016)r(E)s Fm(\()p Fl(\016)r(y)r Fm(\()p Fl(\034)5 b Fm(\)\))10 b Fi(\021)j Fl(\016)r(E)s Fm(\()p Fl(\034)5 b Fm(\),)14 b(etc.\))775 2657 y Fl(\016)r(I)t Fm(\()p Fl(\034)5 b Fm(\))12 b Fi(\025)960 2626 y Fm(1)p 947 2647 48 2 v 947 2688 a(8)p Fl(\034)1000 2657 y Fm(\()p Fl(\016)r(E)s Fm(\()p Fl(\034)5 b Fm(\)\))1156 2638 y Fp(2)1834 2657 y Fm(\(4)p Fl(:)p Fm(91\))0 2779 y Ff(31)p Fe(=mar)q(ch=)p Ff(1995;)17 b(9:46)652 b Fm(26)p eop %%Page: 27 28 bop 0 45 a Fm(while)630 118 y Fl(T)657 125 y Fh(p)677 118 y Fm(\()p Fl(\034)5 b Fm(\))12 b Fi(\024)h Fm([)p Fl(I)835 99 y Fd(0)847 118 y Fm(\(1)d Fi(\000)g Fl(\020)s Fm(\)])997 99 y Fp(2)1027 118 y Fm(+)g Fl(\034)1097 99 y Fd(\000)p Fp(2)1146 118 y Fm(\()p Fl(\016)r(E)s Fm(\()p Fl(\034)5 b Fm(\)\))1302 99 y Fp(2)1834 118 y Fm(\(4)p Fl(:)p Fm(92\))0 217 y(Th)o(us)22 b(w)o(e)f(can)h(ensure)h(that)e(the)h(en)o(trop)o(y)f(gain)h(dominates)g(the) g(p)q(oten)o(tial)h(loss)f(in)g(the)g(quadratic)g(term)0 290 y(pro)o(vided)17 b(w)o(e)f(can)h(c)o(ho)q(ose)f Fl(\034)k(<)15 b(\016)r(E)s Fm(\()p Fl(\034)5 b Fm(\))p Fl(=)p Fm(8.)22 b(Ho)o(w)o(ev)o(er,) 15 b(w)o(e)h(kno)o(w)g(that)g Fl(T)1289 297 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1869 y Fd(j)p Fh(I)r Fd(j)786 1861 y Fm(and)f(a)h(v)o(ector)e (whose)i(norm)f(go)q(es)g(to)g(zero)g(as)g Fl(N)18 b Fm(go)q(es)13 b(to)g(in\014nit)o(y)l(.)0 1934 y(Let)i Fl(e)102 1941 y Fh(\015)126 1934 y Fm(,)g Fl(\015)f Fm(=)f(1)p Fl(;)8 b(:)g(:)g(:)d(;)j Fm(2)388 1917 y Fd(j)p Fh(I)r Fd(j)429 1934 y Fm(,)15 b(b)q(e)h(the)f(column) h(v)o(ectors)f(of)f(the)i(matrix)f Fl(E)1226 1917 y Fh(t)1223 1948 y Fd(j)p Fh(I)r Fd(j)1265 1934 y Fm(.)k(W)l(e)d(set)561 2049 y Fl(v)583 2056 y Fh(\015)619 2049 y Fi(\021)667 2012 y Fg(\010)693 2049 y Fl(i)c Fi(2)h(f)p Fm(1)p Fl(;)8 b(:)g(:)g(:)d(;)j(N)d Fi(g)11 b(j)h Fl(\030)1035 2027 y Fh(\026)1033 2062 y(i)1072 2049 y Fm(=)h Fl(e)1141 2030 y Fh(\026)1141 2060 y(\015)1188 2049 y Fl(;)30 b Fi(8)p Fl(\026)13 b Fi(2)g Fl(I)1363 2012 y Fg(\011)1834 2049 y Fm(\(4)p Fl(:)p Fm(96\))0 2164 y(These)j(sets)f(are)g (random)f(sets,)h(dep)q(ending)i(on)f(the)f(realization)h(of)f(the)h(random)e (v)m(ariables)j Fl(\030)1655 2143 y Fh(\026)1653 2177 y(i)1679 2164 y 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y(in\014nit)o(y)l(,)g(suc)o(h)g(that)e (for)h(all)h(but)f(a)g(\014nite)h(n)o(um)o(b)q(er)g(of)f(indices,)702 2670 y Fi(j)p Fl(\025)742 2677 y Fh(\015)764 2670 y Fi(j)d Fl(<)h(\016)857 2677 y Fh(N)922 2670 y Fl(;)38 b(\015)14 b Fm(=)f(1)p Fl(;)8 b(:)g(:)g(:)d(;)j Fm(2)1207 2651 y Fd(j)p Fh(I)r Fd(j)1834 2670 y Fm(\(4)p Fl(:)p Fm(98\))0 2779 y Ff(31)p Fe(=mar)q(ch=)p Ff(1995;)17 b(9:46)652 b Fm(27)p eop %%Page: 28 29 bop 0 45 a Fm(This)16 b(fact)e(has)h(b)q(een)i(pro)o(v)o(en)e(in)h([G].)d (Using)j(\(4.96\),)d(\005)977 52 y Fh(I)998 45 y Fl(m)1038 52 y Fh(N)1072 45 y Fm(\()p Fl(\033)r Fm(\))i(can)g(b)q(e)h(rewritten)f(as) 603 173 y(\005)637 180 y Fh(I)658 173 y Fl(m)698 180 y Fh(N)732 173 y Fm(\()p Fl(\033)r Fm(\))d(=)h(2)879 154 y Fd(\000j)p Fh(I)r Fd(j)950 173 y Fl(E)987 154 y Fh(t)984 185 y Fd(j)p Fh(I)r Fd(j)1026 173 y Fm(\()p Fl(X)t Fm(\()p Fl(\033)r Fm(\))8 b(+)i Fl(\016)r(X)t Fm(\()p Fl(\033)r Fm(\)\))485 b(\(4)p Fl(:)p Fm(99\))0 300 y(where)15 b Fl(X)t Fm(\()p Fl(\033)r 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