Content-Type: multipart/mixed; boundary="-------------0003150531368" This is a multi-part message in MIME format. ---------------0003150531368 Content-Type: text/plain; name="00-113.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-113.keywords" 2d turbulence, stationary state, uniqueness ---------------0003150531368 Content-Type: application/postscript; name="NSuni.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="NSuni.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.78 Copyright 1998 Radical Eye Software (www.radicaleye.com) %%Title: NSuni.dvi %%Pages: 14 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips -o NSuni.ps NSuni.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2000.03.15:1212 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N 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Ff(\000)p Fh(1)2454 5726 y Fi(j)p Fj(k)p Fi(j)2540 5764 y Fn(;)1076 b Ft(\(31\))1831 6013 y(5)p eop %%Page: 6 6 6 5 bop -95 -3 a Ft(whic)m(h)29 b(follo)m(ws)h(from)f(\(26\))j(with)d Fn(s)c Ft(=)g(0,)31 b(in)e(\(20\),)k(w)m(e)d(conclude)g(that)299 218 y Fl(j)p Fn(!)381 233 y Fj(k)428 218 y Ft(\()p Fn(t)p Ft(\))p Fl(j)c(\024)f Fn(a\024)p Ft(\()p Fn(e)855 173 y Fi(\000)920 152 y Fh(1)p 921 158 35 4 v 921 194 a(2)966 173 y Fm(t)p Fj(k)1034 150 y Fh(2)1093 218 y Ft(+)20 b Fn(\025)p Ft(\(1)h Fl(\000)f Fn(e)1471 173 y Fi(\000)1536 152 y Fh(1)p 1536 158 V 1536 194 a(2)1581 173 y Fm(t)p Fj(k)1649 150 y Fh(2)1688 218 y Ft(\)\))p Fl(j)p Fo(k)p Fl(j)1863 180 y Fi(\000)p Fm(r)1957 218 y Fn(e)1999 180 y Fi(\000)p Fm(\024)p Fu(\()p Fm(t)p Fu(\))2174 157 y Ff(\000)p Fh(1)2258 180 y Fi(j)p Fj(k)p Fi(j)2369 218 y Fl(\024)25 b Fn(a\024)p Ft(\()p Fn(t)p Ft(\))p Fl(j)p Fo(k)p Fl(j)2773 180 y Fi(\000)p Fm(r)2868 218 y Fn(e)2910 180 y Fi(\000)p Fm(\024)p Fu(\()p Fm(t)p Fu(\))3085 157 y Ff(\000)p Fh(1)3169 180 y Fi(j)p Fj(k)p Fi(j)3255 218 y Fn(;)361 b Ft(\(32\))-95 420 y(since)873 623 y Fn(e)915 578 y Fi(\000)980 556 y Fh(1)p 980 563 V 980 599 a(2)1024 578 y Fm(t)p Fj(k)1092 555 y Fh(2)1152 623 y Ft(+)20 b Fn(\025)p Ft(\(1)h Fl(\000)e Fn(e)1529 578 y Fi(\000)1594 556 y Fh(1)p 1595 563 V 1595 599 a(2)1639 578 y Fm(t)p Fj(k)1707 555 y Fh(2)1746 623 y Ft(\))26 b Fl(\024)f Ft(\(1)c(+)f Fn(\021)s(t)15 b Ft(min)o(\(1)p Fn(;)g(\024)p Ft(\)\))2585 585 y Fi(\000)p Fu(1)2681 623 y Fn(;)935 b Ft(\(33\))-95 825 y(whic)m(h)29 b(holds,)g(since)h Fl(j)p Fo(k)p Fl(j)c(\025)f Ft(1,)31 b(for)f Fn(\025)h Ft(and)f Fn(\021)j Ft(small)c(enough)h(and)g(0)c Fl(\024)f Fn(t)g Fl(\024)g Ft(1.)42 b(Inequalit)m(y)29 b(\(32\))j(is)d(the)i (claim)e(of)-95 938 y(part)h(\(a\))i(of)e(the)h(Prop)s(osition)d (concerning)i(the)h Fn(\024)p Ft(\()p Fn(t)p Ft(\))g(norm)e(of)i(the)g (solution,)e(namely)g Fl(k)p Fn(!)s Ft(\()p Fn(t)p Ft(\))p Fl(k)3231 956 y Fm(\024)p Fu(\()p Fm(t)p Fu(\))3383 938 y Fl(\024)c Fn(a\024)p Ft(\()p 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Fk(X)748 2808 y Fj(0)p Fi(6)p Fu(=)p Fi(j)p Fj(l)p Fi(j\024)972 2774 y Ff(j)p Fd(k)p Ff(j)p 970 2793 75 4 v 992 2834 a Fh(2)1070 2679 y Ft(+)1271 2598 y Fk(X)1156 2808 y Fj(l)p Fi(6)p Fu(=)p Fj(k)p Fm(;)p Fi(j)p Fj(l)p Fi(j)p Fm(>)1425 2774 y Ff(j)p Fd(k)p Ff(j)p 1422 2793 V 1444 2834 a Fh(2)1507 2679 y Ft(\))p Fl(j)p Fn(!)1624 2694 y Fj(k)p Fi(\000)p Fj(l)1748 2679 y Ft(\()p Fn(s)p Ft(\))p Fl(jj)p Fn(!)1968 2694 y Fj(l)1996 2679 y Ft(\()p Fn(s)p Ft(\))p Fl(jj)p Fo(k)p Fl(jj)p Fo(l)p Fl(j)2318 2642 y Fi(\000)p Fu(1)2439 2679 y Fl(\021)e Ft(\006)2601 2693 y Fu(1)2660 2679 y Ft(+)20 b(\006)2817 2693 y Fu(2)2856 2679 y Fn(:)760 b Ft(\(36\))-95 3004 y(In)30 b(the)g(\014rst)g(sum,)g(w)m(e)g(b)s(ound,)f(using)g(Lemma)h (1,)1133 3206 y Fl(j)p Fn(!)1215 3221 y Fj(k)p Fi(\000)p Fj(l)1339 3206 y Ft(\()p Fn(s)p Ft(\))p Fl(j)c(\024)f Ft(2)p Fn(a\024)p Fl(j)p Fo(k)c Fl(\000)f Fo(l)p Fl(j)1990 3169 y Fi(\000)p Fm(r)2109 3206 y Fl(\024)25 b Fn(C)7 b(a\024)p Fl(j)p Fo(k)p Fl(j)2482 3169 y Fi(\000)p 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b(and)f(b)r(elo)n(w,)i(the) e(k)n(ernel)h Fa(AB)t Fg(\()p Fa(!)s(;)12 b(E)t Fg(\))25 b(is)h(de\014ned)f(in)g(the)g(ob)n(vious)h(w)n(a)n(y)g(b)n(y)2426 5623 y Fk(R)2494 5688 y Fa(A)p Fg(\()p Fa(!)s(;)12 b(d!)2757 5656 y Ff(0)2778 5688 y Fg(\))p Fa(B)t Fg(\()p Fa(!)2951 5656 y Ff(0)2973 5688 y Fa(;)h(E)t Fg(\).)1831 6013 y Ft(7)p eop %%Page: 8 8 8 7 bop -95 -3 a Fn(\024)p Ft(,)37 b(i.e.)56 b Fn(\024)36 b Ft(decreases\);)j(then,)d(if)f(the)g(\\kic)m(ks")i(of)e(the)h(noise)e (are)i(su\016cien)m(tly)e(small)g(\(but)h(not)h(to)s(o)g(small,)f(so) -95 110 y(that)c(this)e(ev)m(en)m(t)j(is)e(not)g(to)s(o)h (unprobable\),)e(the)h(solution)f(reac)m(hes)2300 87 y(\026)2284 110 y Fn(U)40 b Ft(in)29 b(a)i(\014nite)e(time)h(\(of)h (order)f(log)17 b Fn(R)31 b Ft(\).)46 223 y(Secondly)-8 b(,)39 b(w)m(e)f(sho)m(w)f(in)f(Lemma)i(3)g(that,)i(in)c(the)i(region) 2159 200 y(\026)2143 223 y Fn(U)10 b Ft(,)39 b(the)f(sto)s(c)m(hastic)g (dynamics)e(is)h(su\016cien)m(tly)-95 336 y(mixing;)f(this)e(is)g (again)h(due)f(to)i(the)f(fact)h(that)g(the)f(deterministic)e(Na)m (vier-Stok)m(es)k(ev)m(olution)e(increases)f(the)-95 449 y(domain)29 b(of)i(analyticit)m(y)f(of)g(the)h(solution.)46 561 y(Third,)h(the)h(fact)h(that)g Fn(Q)f Ft(is)f(not)i(a)f(b)s(ona)g (\014de)f(transition)g(probabilit)m(y)f(is)h(what)h(limits)e(the)j (Prop)s(osition)-95 674 y(to)28 b(\014nite)f(times.)39 b(F)-8 b(or)29 b(longer)e(times,)h(w)m(e)g(will)d(need)j(to)g(ha)m(v)m (e)h(some)f(estimate)g(on)g(the)g(probabilit)m(y)d(of)j(escaping)-95 787 y(the)i(region)f Fn(U)10 b Ft(,)29 b(whic)m(h)f(follo)m(ws)h(from)g (part)g(\(b\))h(of)f(the)h(Prop)s(osition.)38 b(Indeed,)29 b(the)g(latter)h(implies,)d(using)h(\(41,)-95 900 y(42\))k(and)d (taking)i Fn(\024)25 b Ft(=)g Fn(\024)744 867 y Fi(0)793 900 y Ft(=)g Fn(R)959 867 y Fm(p)1029 900 y Ft(that,)31 b(for)f(all)f Fn(!)g Fl(2)24 b Fn(U)10 b Ft(,)1218 1099 y Fn(P)j Ft(\()p Fn(!)s(;)i(U)1496 1062 y Fm(c)1532 1099 y Ft(\))25 b(=)g Fn(R)q Ft(\()p Fn(!)s(;)15 b Ft(\012\))26 b Fl(\024)f Fn(e)2158 1062 y Fi(\000)p Fm(cR)2297 1038 y Fe(q)2336 1099 y Fn(;)1280 b Ft(\(46\))-95 1298 y(with)29 b Fn(q)f Ft(=)287 1257 y Fu(2)p Fm(p)p 287 1277 71 4 v 300 1329 a(\013)388 1298 y Fl(\000)20 b Ft(1)26 b Fn(>)f Ft(0)30 b(\(remem)m(b)s(er)h(that)f Fn(p)25 b(>)1555 1262 y Fm(\013)p 1555 1277 46 4 v 1560 1329 a Fu(2)1611 1298 y Ft(\).)-95 1529 y Fo(Lemma)33 b(2.)40 b Fl(9)p Fn(c;)15 b(C)32 b(<)25 b Fl(1)p Fc(,)33 b(such)f(that)i Fl(8)p Fn(!)28 b Fl(2)d Fn(U)10 b Fc(,)1209 1728 y Fn(P)1280 1691 y Fm(T)1321 1700 y Fh(1)1360 1728 y Ft(\()p Fn(!)s(;)1511 1705 y Ft(\026)1495 1728 y Fn(U)g Ft(\))26 b Fl(\025)f Ft(exp\()p Fl(\000)p Fn(C)7 b Ft(\(log)16 b Fn(R)q Ft(\))2313 1691 y Fm(c)2348 1728 y Ft(\))1258 b(\(47\))-65 1927 y(with)29 b Fn(T)195 1941 y Fu(1)260 1927 y Ft(=)c Fn(C)d Ft(log)16 b Fn(R)q Ft(.)-95 2158 y Fo(Lemma)33 b(3.)40 b Fl(9)p Fn(c;)15 b(C)32 b(<)25 b Fl(1)p Fc(,)33 b(such)f(that,)i Fl(8)p Fn(!)s(;)15 b(!)1503 2125 y Fi(0)1551 2158 y Fl(2)1653 2135 y Ft(\026)1637 2158 y Fn(U)10 b Fc(,)33 b Fl(8)p Fn(B)c Fl(\032)2031 2135 y Ft(\026)2015 2158 y Fn(U)10 b Fc(,)964 2357 y Fn(P)j Ft(\()p Fn(!)s(;)i(B)5 b Ft(\))21 b(+)f Fn(P)13 b Ft(\()p Fn(!)1557 2319 y Fi(0)1580 2357 y Fn(;)1637 2334 y Ft(\026)1620 2357 y Fn(U)e Fl(n)p Fn(B)5 b Ft(\))25 b Fl(\025)g Ft(exp\()p Fl(\000)p Fn(C)7 b Ft(\(log)17 b Fn(R)q Ft(\))2558 2319 y Fm(c)2592 2357 y Ft(\))1014 b(\(48\))46 2787 y(Lemmas)37 b(2,)j(3)d(imply)e(that)j (there)f(exist)g Fn(\016)s Ft(\()p Fn(R)q Ft(\))h Fl(\021)e Ft(exp\()p Fl(\000)p Fn(C)7 b Ft(\(log)17 b Fn(R)q Ft(\))2450 2754 y Fm(c)2484 2787 y Ft(\))38 b(and)e Fn(T)50 b Fl(\021)36 b Fn(T)13 b Ft(\()p Fn(R)q Ft(\))37 b(=)f Fn(C)21 b Ft(log)c Fn(R)38 b Ft(with)-95 2900 y Fn(C)q(;)15 b(c)26 b(<)f Fl(1)p Ft(,)31 b(suc)m(h)f(that)h Fl(8)p Fn(!)s(;)15 b(!)932 2867 y Fi(0)980 2900 y Fl(2)25 b Fn(U)40 b Ft(and)30 b Fl(8)p Fn(B)f Fl(\032)1606 2877 y Ft(\026)1590 2900 y Fn(U)10 b Ft(,)1148 3099 y Fn(P)1219 3061 y Fm(T)1274 3099 y Ft(\()p Fn(!)s(;)15 b(B)5 b Ft(\))21 b(+)f Fn(P)1701 3061 y Fm(T)1756 3099 y Ft(\()p Fn(!)1851 3061 y Fi(0)1874 3099 y Fn(;)1930 3076 y Ft(\026)1914 3099 y Fn(U)11 b Fl(n)p Fn(B)5 b Ft(\))25 b Fl(\025)g Fn(\016)s Ft(\()p Fn(R)q Ft(\))1196 b(\(49\))-95 3298 y(whic)m(h)29 b(implies)f(in)h (turn,)g(since)1046 3275 y(\026)1030 3298 y Fn(U)35 b Fl(\032)25 b Fn(U)10 b Ft(,)31 b(that)g Fl(8)p Fn(!)s(;)15 b(!)1759 3265 y Fi(0)1807 3298 y Fl(2)25 b Fn(U)40 b Ft(and)30 b Fl(8)p Fn(B)f Fl(\032)c Fn(U)10 b Ft(,)1148 3497 y Fn(P)1219 3459 y Fm(T)1274 3497 y Ft(\()p Fn(!)s(;)15 b(B)5 b Ft(\))21 b(+)f Fn(P)1701 3459 y Fm(T)1756 3497 y Ft(\()p Fn(!)1851 3459 y Fi(0)1874 3497 y Fn(;)15 b(U)10 b Fl(n)p Fn(B)5 b Ft(\))26 b Fl(\025)f Fn(\016)s Ft(\()p Fn(R)q Ft(\))1196 b(\(50\))-95 3696 y(This)29 b(the)h(main)f(inequalit) m(y)g(that)i(w)m(e)g(shall)e(use)h(no)m(w.)-95 3938 y Fv(3.1)112 b(Pro)s(of)37 b(of)h(Prop)s(osition)e(2.)-95 4110 y Ft(W)-8 b(e)30 b(start)f(with)e(the)i(pro)s(of)f(of)h(part)g (\(a\),)h(where)e(w)m(e)h(shall)e(use)i(\(46\),)h(whic)m(h)e(is)g(a)h (consequence)g(of)g(part)f(\(b\),)i(to)-95 4223 y(b)s(e)g(pro)m(v)m(en) g(indep)s(enden)m(tly)e(b)s(elo)m(w.)46 4336 y(T)-8 b(o)31 b(get)h(\(43\))f(w)m(e)g(follo)m(w,)f(with)f(sligh)m(t)h(mo)s (di\014cations,)f(an)h(argumen)m(t)h(in)e([2],)i(p.)40 b(197{198.)k(Let)853 4535 y Fn(Q)p 853 4566 72 4 v Ft(\()p Fn(t;)15 b(E)5 b Ft(\))26 b(=)44 b(inf)1262 4594 y Fm(!)r Fi(2)p Fm(U)1426 4535 y Fn(Q)1498 4497 y Fm(t)1527 4535 y Ft(\()p Fn(!)s(;)15 b(E)5 b Ft(\))p Fn(;)p 1912 4463 V 118 w(Q)p Ft(\()p Fn(t;)15 b(E)5 b Ft(\))26 b(=)31 b(sup)2321 4612 y Fm(!)r Fi(2)p Fm(U)2485 4535 y Fn(Q)2557 4497 y Fm(t)2586 4535 y Ft(\()p Fn(!)s(;)15 b(E)5 b Ft(\))p Fn(:)-95 4788 y Ft(Fix)30 b Fn(!)s(;)15 b(!)228 4755 y Fi(0)277 4788 y Fl(2)24 b Fn(U)41 b Ft(and)29 b(consider)h(the)g (function)f(de\014ned)g(on)i(subsets)e Fn(E)i Fl(\032)25 b Ft(\012:)1173 4987 y Fn( )1232 5002 y Fm(!)r(;!)1344 4984 y Ff(0)1371 4987 y Ft(\()p Fn(E)5 b Ft(\))26 b(=)f Fn(Q)1707 4950 y Fm(T)1762 4987 y Ft(\()p Fn(!)s(;)15 b(E)5 b Ft(\))21 b Fl(\000)f Fn(Q)2188 4950 y Fm(T)2243 4987 y Ft(\()p Fn(!)2338 4950 y Fi(0)2362 4987 y Fn(;)15 b(E)5 b Ft(\))p Fn(:)-95 5186 y Ft(Let)32 b Fn(S)130 5153 y Fu(+)220 5186 y Ft(b)s(e)f(the)g(set)h(suc)m(h)f(that)h Fn( )1108 5201 y Fm(!)r(;!)1220 5182 y Ff(0)1247 5186 y Ft(\()p Fn(E)5 b Ft(\))27 b Fl(\025)g Ft(0)32 b(for)f Fn(E)h Fl(\032)26 b Fn(S)1988 5153 y Fu(+)2078 5186 y Ft(and)31 b Fn( )2315 5201 y Fm(!)r(;!)2427 5182 y Ff(0)2454 5186 y Ft(\()p Fn(E)5 b Ft(\))28 b Fl(\024)e Ft(0)32 b(for)f Fn(E)h Fl(\032)26 b Fn(U)10 b Fl(n)p Fn(S)3312 5153 y Fu(+)3399 5186 y Fl(\021)26 b Fn(S)3557 5153 y Fi(\000)3647 5186 y Ft(\()p Fn(S)3743 5153 y Fi(\006)-95 5299 y Ft(dep)s(end)33 b(on)i Fn(!)s(;)15 b(!)516 5266 y Fi(0)540 5299 y Ft(,)36 b(but)f(w)m(e)g(suppress)e(this)h(dep)s (endence\).)55 b(Observ)m(e)34 b(that)i(writing,)f(see)g(\(41\),)j Fn(P)47 b Ft(=)32 b Fn(Q)24 b Ft(+)f Fn(R)q Ft(,)-95 5412 y(and)30 b(using)f(\(46\),)j(w)m(e)f(ha)m(v)m(e,)h(for)e(an)m(y)g Fn(!)e Fl(2)d Fn(U)10 b Ft(,)31 b Fn(E)f Fl(\032)25 b Ft(\012,)31 b(that)424 5688 y Fl(j)p Fn(P)520 5651 y Fm(T)576 5688 y Ft(\()p Fn(!)s(;)15 b(E)5 b Ft(\))21 b Fl(\000)f Fn(Q)1002 5651 y Fm(T)1057 5688 y Ft(\()p Fn(!)s(;)15 b(E)5 b Ft(\))p Fl(j)27 b Ft(=)e Fl(j)1487 5583 y Fm(T)10 b Fi(\000)p Fu(1)1498 5608 y Fk(X)1500 5789 y Fm(t)p Fu(=0)1643 5688 y Fn(Q)1715 5651 y Fm(t)1745 5688 y Fn(R)q(P)1886 5651 y Fm(T)g Fi(\000)p Fm(t)p Fi(\000)p Fu(1)2111 5688 y Ft(\()p Fn(!)s(;)15 b(E)5 b Ft(\))p Fl(j)27 b(\024)e Fn(T)13 b(e)2609 5651 y Fi(\000)p Fm(cR)2748 5627 y Fe(q)2812 5688 y Fl(\021)2918 5641 y Fu(1)p 2918 5667 40 4 v 2918 5725 a(2)2967 5688 y Fn(\017)p Ft(\()p Fn(R)q Ft(\))p Fn(:)472 b Ft(\(51\))1831 6013 y(8)p eop %%Page: 9 9 9 8 bop -95 -3 a Ft(Then,)649 198 y Fl(j)p Fn( )733 213 y Fm(!)r(;!)845 194 y Ff(0)872 198 y Ft(\()p Fn(S)968 161 y Fu(+)1028 198 y Ft(\))20 b(+)g Fn( )1233 213 y Fm(!)r(;!)1345 194 y Ff(0)1372 198 y Ft(\()p Fn(S)1468 161 y Fi(\000)1527 198 y Ft(\))p Fl(j)26 b Ft(=)f Fl(j)p Fn(Q)1806 161 y Fm(T)1861 198 y Ft(\()p Fn(!)s(;)15 b Ft(\012\))21 b Fl(\000)f Fn(Q)2281 161 y Fm(T)2336 198 y Ft(\()p Fn(!)2431 161 y Fi(0)2455 198 y Fn(;)15 b Ft(\012\))p Fl(j)26 b(\024)e Fn(\017)p Ft(\()p Fn(R)q Ft(\))p Fn(:)697 b Ft(\(52\))-95 399 y(since)30 b Fn(P)199 366 y Fm(T)254 399 y Ft(\()p Fn(!)s(;)15 b Ft(\012\))26 b(=)f Fn(P)683 366 y Fm(T)738 399 y Ft(\()p Fn(!)833 366 y Fi(0)856 399 y Fn(;)15 b Ft(\012\))26 b(=)f(1.)41 b(Moreo)m(v)m(er,)33 b(using)28 b(\(51,)k(50\),)702 600 y Fn( )761 616 y Fm(!)r(;!)873 597 y Ff(0)900 600 y Ft(\()p Fn(S)996 563 y Fu(+)1056 600 y Ft(\))25 b(=)g Fn(Q)1284 563 y Fm(T)1339 600 y Ft(\()p Fn(!)s(;)15 b(S)1535 563 y Fu(+)1595 600 y Ft(\))20 b Fl(\000)g Fn(Q)1813 563 y Fm(T)1868 600 y Ft(\()p Fn(!)1963 563 y Fi(0)1986 600 y Fn(;)15 b(S)2087 563 y Fu(+)2147 600 y Ft(\))702 738 y Fl(\024)25 b Ft(1)c Fl(\000)f Ft(\()p Fn(P)1061 701 y Fm(T)1116 738 y Ft(\()p Fn(!)s(;)15 b(S)1312 701 y Fi(\000)1372 738 y Ft(\))20 b(+)g Fn(P)1589 701 y Fm(T)1644 738 y Ft(\()p Fn(!)1739 701 y Fi(0)1763 738 y Fn(;)15 b(S)1864 701 y Fu(+)1923 738 y Ft(\)\))21 b(+)f Fn(\017)p Ft(\()p Fn(R)q Ft(\))26 b Fl(\024)f Ft(1)20 b Fl(\000)g Fn(\016)s Ft(\()p Fn(R)q Ft(\))i(+)e Fn(\017)p Ft(\()p Fn(R)q Ft(\))p Fn(:)583 b Ft(\(53\))-95 939 y(Th)m(us,)p 373 1097 72 4 v 373 1169 a Fn(Q)p Ft(\()p Fn(t)20 b Ft(+)g Fn(T)8 b(;)15 b(E)5 b Ft(\))21 b Fl(\000)f Fn(Q)p 944 1200 V Ft(\()p Fn(t)g Ft(+)g Fn(T)8 b(;)15 b(E)5 b Ft(\))43 b(=)e(sup)1559 1247 y Fm(!)r(;!)1671 1228 y Ff(0)1710 1054 y Fk(Z)1793 1169 y Ft(\()p Fn(Q)1900 1131 y Fm(T)1955 1169 y Ft(\()p Fn(!)s(;)15 b(d!)2197 1131 y Fi(00)2240 1169 y Ft(\))21 b Fl(\000)f Fn(Q)2459 1131 y Fm(T)2514 1169 y Ft(\()p Fn(!)2609 1131 y Fi(0)2632 1169 y Fn(;)15 b(d!)2779 1131 y Fi(00)2822 1169 y Ft(\)\))p Fn(Q)2964 1131 y Fm(t)2994 1169 y Ft(\()p Fn(!)3089 1131 y Fi(00)3132 1169 y Fn(;)g(E)5 b Ft(\))1446 1416 y(=)41 b(sup)1559 1494 y Fm(!)r(;!)1671 1475 y Ff(0)1710 1301 y Fk(Z)1808 1416 y Fn( )1867 1431 y Fm(!)r(;!)1979 1413 y Ff(0)2006 1416 y Ft(\()p Fn(d!)2148 1379 y Fi(00)2191 1416 y Ft(\))p Fn(Q)2298 1379 y Fm(t)2328 1416 y Ft(\()p Fn(!)2423 1379 y Fi(00)2465 1416 y Fn(;)15 b(E)5 b Ft(\))1446 1627 y Fl(\024)41 b Ft(sup)1559 1706 y Fm(!)r(;!)1671 1687 y Ff(0)1695 1627 y Ft(\()p Fn( )1789 1643 y Fm(!)r(;!)1901 1624 y Ff(0)1928 1627 y Ft(\()p Fn(S)2024 1590 y Fu(+)2083 1627 y Ft(\))p 2118 1555 V Fn(Q)p Ft(\()p Fn(t;)15 b(E)5 b Ft(\))22 b(+)e Fn( )2577 1643 y Fm(!)r(;!)2689 1624 y Ff(0)2716 1627 y Ft(\()p Fn(S)2812 1590 y Fi(\000)2871 1627 y Ft(\))p Fn(Q)p 2906 1658 V Ft(\()p Fn(t;)15 b(E)5 b Ft(\)\))1446 1839 y Fl(\024)41 b Ft(\(1)21 b Fl(\000)f Fn(\016)s Ft(\()p Fn(R)q Ft(\))h(+)f Fn(\017)p Ft(\()p Fn(R)q Ft(\)\)\()p 2292 1767 V Fn(Q)q Ft(\()p Fn(t;)15 b(E)5 b Ft(\))21 b Fl(\000)f Fn(Q)p 2692 1870 V Ft(\()p Fn(t;)15 b(E)5 b Ft(\)\))22 b(+)e Fn(\017)p Ft(\()p Fn(R)q Ft(\))-95 2040 y(where,)27 b(to)g(get)h(the)f(last)f(inequalit)m(y)-8 b(,)27 b(w)m(e)g(write)f Fn( )1621 2055 y Fm(!)r(;!)1733 2036 y Ff(0)1760 2040 y Ft(\()p Fn(S)1856 2007 y Fi(\000)1915 2040 y Ft(\))g(=)f Fl(\000)p Fn( )2202 2055 y Fm(!)r(;!)2314 2036 y Ff(0)2340 2040 y Ft(\()p Fn(S)2436 2007 y Fu(+)2495 2040 y Ft(\))13 b(+)g Fn( )2686 2055 y Fm(!)r(;!)2798 2036 y Ff(0)2825 2040 y Ft(\()p Fn(S)2921 2007 y Fu(+)2980 2040 y Ft(\))g(+)g Fn( )3171 2055 y Fm(!)r(;!)3283 2036 y Ff(0)3309 2040 y Ft(\()p Fn(S)3405 2007 y Fi(\000)3465 2040 y Ft(\),)28 b(b)s(ound)-95 2153 y Fn( )-36 2168 y Fm(!)r(;!)76 2149 y Ff(0)103 2153 y Ft(\()p Fn(S)199 2120 y Fu(+)258 2153 y Ft(\))38 b(b)m(y)f(\(53\),)k Fl(j)p Fn( )774 2168 y Fm(!)r(;!)886 2149 y Ff(0)913 2153 y Ft(\()p Fn(S)1009 2120 y Fu(+)1068 2153 y Ft(\))25 b(+)g Fn( )1283 2168 y Fm(!)r(;!)1395 2149 y Ff(0)1422 2153 y Ft(\()p Fn(S)1518 2120 y Fi(\000)1577 2153 y Ft(\))p Fl(j)38 b Ft(b)m(y)f(\(52\))i(and)e(use)g Fn(Q)p 2355 2184 V Ft(\()p Fn(t;)15 b(E)5 b Ft(\))38 b Fl(\024)e Ft(1.)62 b(W)-8 b(e)38 b(conclude)f(that,)j(for)-95 2266 y Fn(\017)p Ft(\()p Fn(R)q Ft(\))26 b Fn(<)f(\016)s Ft(\()p Fn(R)q Ft(\),)64 2515 y Fl(j)p Fn(Q)161 2477 y Fm(nT)259 2515 y Ft(\()p Fn(!)s(;)15 b(E)5 b Ft(\))22 b Fl(\000)d Fn(Q)685 2477 y Fm(nT)783 2515 y Ft(\(0)p Fn(;)c(E)5 b Ft(\))p Fl(j)28 b(\024)p 1159 2443 V 25 w Fn(Q)o Ft(\()p Fn(nT)8 b(;)15 b(E)5 b Ft(\))21 b Fl(\000)f Fn(Q)p 1640 2546 V Ft(\()p Fn(nT)8 b(;)15 b(E)5 b Ft(\))26 b Fl(\024)f Ft(2\(1)c Fl(\000)f Fn(\016)s Ft(\()p Fn(R)q Ft(\))i(+)e Fn(\017)p Ft(\()p Fn(R)q Ft(\)\))2877 2477 y Fm(n)p Fi(\000)p Fu(1)3035 2515 y Ft(+)3283 2453 y Fn(\017)p Ft(\()p Fn(R)q Ft(\))p 3136 2494 473 4 v 3136 2577 a Fn(\016)s Ft(\()p Fn(R)q Ft(\))h Fl(\000)f Fn(\017)p Ft(\()p Fn(R)q Ft(\))3618 2515 y Fn(:)-95 2780 y Ft(Recall)29 b(that)h Fn(\016)s Ft(\()p Fn(R)q Ft(\))c(=)f(exp\()p Fl(\000)p Fn(C)7 b Ft(\(log)17 b Fn(R)q Ft(\))1269 2747 y Fm(c)1303 2780 y Ft(\))30 b(and)f(that)h Fn(T)38 b Ft(=)25 b Fn(T)13 b Ft(\()p Fn(R)q Ft(\))25 b(=)g Fn(C)d Ft(log)16 b Fn(R)q Ft(,)30 b(hence,)f(see)h(\(51\),)i Fn(\017)p Ft(\()p Fn(R)q Ft(\))25 b Fl(\024)g Fn(e)3577 2747 y Fi(\000)p Fm(c)3663 2724 y Ff(0)3685 2747 y Fm(R)3738 2724 y Fe(q)3777 2780 y Ft(;)-95 2893 y(so,)31 b(since)f(w)m(e)g(assume)h Fn(R)g Ft(to)g(b)s(e)e(large,)i(part)f(\(a\))i(of)e(the)h(Prop)s (osition)d(follo)m(ws.)46 3006 y(Let)j(us)f(no)m(w)g(pro)m(v)m(e)h (part)g(\(b\).)40 b(Using)30 b(\(9\))h(for)g Fn(n)24 b Ft(=)h(0,)31 b(w)m(e)g(ha)m(v)m(e)1116 3207 y Fl(k)p Fn(!)s Ft(\(1\))p Fl(k)1381 3223 y Fm(\024)1422 3204 y Ff(0)1475 3207 y Fl(\024)25 b(k)p Fn(F)13 b Ft(\()p Fn(!)s Ft(\(0\)\))p Fl(k)1977 3223 y Fm(\024)2018 3204 y Ff(0)2067 3207 y Ft(+)20 b Fl(k)p Fn(g)s Ft(\(1\))p Fl(k)2409 3223 y Fm(\024)2450 3204 y Ff(0)3641 3207 y Ft(\(54\))-95 3409 y(and)905 3610 y(\010\(1\))26 b(=)1218 3562 y Fu(1)p 1218 3589 40 4 v 1218 3646 a(2)1267 3610 y Fl(k)p Fn(!)s Ft(\(1\))p Fl(k)1532 3572 y Fu(2)1532 3636 y Fm(L)1580 3618 y Fh(2)1646 3610 y Fl(\024)f(k)p Fn(F)13 b Ft(\()p Fn(!)s Ft(\(0\)\))p Fl(k)2148 3572 y Fu(2)2148 3636 y Fm(L)2196 3618 y Fh(2)2258 3610 y Ft(+)20 b Fl(k)p Fn(g)s Ft(\(1\))p Fl(k)2600 3572 y Fu(2)2600 3636 y Fm(L)2648 3618 y Fh(2)3641 3610 y Ft(\(55\))-95 3837 y(and,)30 b(b)m(y)h(Prop)s(osition)e(1,)i(w)m(e)h(kno)m(w)e(that,) i(if)e Fn(!)s Ft(\(0\))d Fl(2)e Fn(U)1854 3851 y Fm(\024)1899 3837 y Ft(,)31 b Fn(F)13 b Ft(\()p Fn(!)s Ft(\(0\)\))28 b Fl(2)d Fn(U)2447 3855 y Fm(\024)p Fu(\(1\))2582 3837 y Ft(,)31 b(with)f Fn(\024)p Ft(\(1\))d(=)3190 3801 y Fm(\024)p 3147 3816 128 4 v 3147 3868 a Fu(1+)p Fm(\021)3315 3837 y Ft(\()p Fn(\024)g Fl(\025)e Fn(R)3595 3804 y Fm(p)3661 3837 y Fl(\025)g Ft(1)-95 3988 y(here\).)43 b(Then,)30 b(letting)g Fn(\020)j Ft(=)26 b(\(1)21 b(+)f Fn(\021)s Ft(\))1171 3943 y Fi(\000)1236 3922 y Fh(1)p 1237 3928 35 4 v 1237 3964 a(2)1312 3988 y Fn(<)26 b Ft(1,)31 b(w)m(e)h(get,)g (for)f Fn(\024)2014 3955 y Fi(0)2063 3988 y Fl(\025)26 b Fn(\020)7 b(\024)p Ft(,)31 b(that)h Fl(k)p Fn(F)13 b Ft(\()p Fn(!)s Ft(\(0\)\))p Fl(k)2919 4003 y Fm(\024)2960 3984 y Ff(0)3015 3988 y Fl(\024)26 b Ft(\(1)21 b(+)f Fn(\021)s Ft(\))3387 3943 y Fi(\000)3452 3922 y Fh(1)p 3453 3928 V 3453 3964 a(2)3502 3988 y Fn(a\024)3602 3955 y Fi(0)3656 3988 y Ft(and)-95 4114 y Fl(k)p Fn(F)13 b Ft(\()p Fn(!)s Ft(\(0\)\))p Fl(k)311 4081 y Fu(2)311 4145 y Fm(L)359 4126 y Fh(2)426 4114 y Fl(\024)25 b Ft(\(1)c(+)f Fn(\021)s Ft(\))797 4078 y Fi(\000)867 4051 y Fh(2)p 862 4063 41 4 v 862 4104 a Fe(\013)916 4114 y Fn('\024)1027 4078 y Fi(0)1062 4051 y Fh(2)p 1057 4063 V 1057 4104 a Fe(\013)1112 4114 y Ft(.)40 b(No)m(w,)32 b(assume)e(that)h Fn(g)s Ft(\(1\))h(satis\014es,)e Fl(8)p Fo(k)p Ft(,)1190 4340 y Fl(j)p Fn(g)1258 4355 y Fj(k)1306 4340 y Ft(\(1\))p Fl(j)d(\024)e Fn(\017\024)1658 4302 y Fi(0)1692 4275 y Fh(1)p 1687 4287 V 1687 4328 a Fe(\013)1741 4340 y Ft(\()p Fn(bR)q Ft(\))1920 4295 y Fi(\000)1985 4274 y Fh(1)p 1986 4280 35 4 v 1986 4316 a(2)2034 4340 y Fl(j)p Fo(k)p Fl(j)p Ft(\()p Fn(\015)2221 4355 y Fj(k)2270 4340 y Ft(\))2315 4274 y Fh(1)p 2315 4280 V 2315 4316 a(2)2364 4340 y Fn(;)1252 b Ft(\(56\))-95 4541 y(with)32 b Fn(\017)h Ft(small)f(\(dep)s(ending)f(on)j Fn(\021)i Ft(but)d(indep)s(enden)m(t)e (of)j Fn(R)g Ft(and)e Fn(\024)p Ft(\).)51 b(Then,)33 b(w)m(e)h(get)g(that)g Fn(!)s Ft(\(1\))e Fl(2)e Fn(U)3469 4556 y Fm(\024)3510 4537 y Ff(0)3536 4541 y Ft(,)k(using)-95 4654 y(the)d(upp)s(er)d(b)s(ound)g(in)h(\(6\))j(and)e(the)g(fact)h (that,)h(for)e Fn(R)h Ft(large,)g Fn(\024)2129 4621 y Fi(0)2177 4654 y Fl(\025)25 b Fn(\020)7 b(\024)26 b Fl(\025)e Fn(\020)7 b(R)2610 4621 y Fm(p)2680 4654 y Ft(is)29 b(m)m(uc)m(h)h (larger)h(than)f(2)p Fn(\024)3581 4668 y Fm(\015)3626 4654 y Ft(.)46 4767 y(Hence,)e(the)f(probabilit)m(y)d(in)h(\(44\))j(is) d(b)s(ounded)f(b)m(y)i(the)h(probabilit)m(y)d(that)j(at)g(least)g(one)f (of)h(the)f(inequalities)-95 4880 y(in)g(\(56\))j(is)d(violated.)39 b(Since)27 b(the)g Fn(g)1096 4895 y Fj(k)1143 4880 y Ft('s)h(are)f(Gaussian)g(random)f(v)-5 b(ariables)26 b(with)g(co)m(v)-5 b(ariance)29 b Fn(\015)3160 4895 y Fj(k)3207 4880 y Ft(,)f(this)e(ev)m(en)m(t)j(has)-95 4993 y(a)i(probabilit)m(y)d(less)h(than:)1023 5204 y(1)21 b Fl(\000)1180 5123 y Fk(Y)1211 5307 y Fj(k)1286 5204 y Ft(\(1)g Fl(\000)f Ft(exp)o(\()p Fl(\000)p Fn(C)7 b(\017)p Ft(\()p Fn(\024)1918 5166 y Fi(0)1942 5204 y Ft(\))1992 5139 y Fh(2)p 1987 5151 41 4 v 1987 5192 a Fe(\013)2041 5204 y Ft(\()p Fn(bR)q Ft(\))2220 5166 y Fi(\000)p Fu(1)2315 5204 y Fl(j)p Fo(k)p Fl(j)2420 5166 y Fu(2)2460 5204 y Ft(\)\))p Fn(;)1086 b Ft(\(57\))-95 5482 y(since,)32 b(for)g Fn(\024)348 5449 y Fi(0)399 5482 y Fl(\025)c Fn(\020)7 b(\024)28 b Fl(\025)g Fn(\020)7 b(R)841 5449 y Fm(p)880 5482 y Ft(,)32 b(with)f Fn(p)d(>)1328 5446 y Fm(\013)p 1328 5461 46 4 v 1333 5513 a Fu(2)1384 5482 y Ft(,)k(and)g Fn(R)g Ft(large,)h(eac)m(h)g(exp)s(onen)m(tial)e(in)g (the)h(pro)s(duct)f(is)g(small.)44 b(The)-95 5617 y(factor)36 b Fl(j)p Fo(k)p Fl(j)275 5584 y Fu(2)351 5617 y Ft(con)m(trols)f(the)h (sum)e(o)m(v)m(er)j Fo(k)e Ft(of)h(exp\()p Fl(\000)p Fn(C)7 b(\017)p Ft(\()p Fn(\024)1896 5584 y Fi(0)1919 5617 y Ft(\))1969 5554 y Fh(2)p 1964 5566 41 4 v 1964 5607 a Fe(\013)2019 5617 y Ft(\()p Fn(bR)q Ft(\))2198 5584 y Fi(\000)p Fu(1)2292 5617 y Fl(j)p Fo(k)p Fl(j)2397 5584 y Fu(2)2437 5617 y Ft(\),)38 b(sum)c(whic)m(h)g(is)g(small)g(for)h Fn(R)h Ft(large,)-95 5764 y(and)30 b(therefore)h(\(57\))h(is)d(b)s (ounded)f(from)i(ab)s(o)m(v)m(e)i(b)m(y)e(1)21 b Fl(\000)e Ft(\(1)i Fl(\000)f Ft(exp\()p Fl(\000)p Fn(c)2359 5728 y Fm(\024)2400 5700 y Ff(0)2433 5680 y Fh(2)p 2428 5689 V 2428 5721 a Fe(\013)p 2360 5743 124 4 v 2394 5795 a Fm(R)2493 5764 y Ft(\)\),)31 b(i.e.)40 b(b)m(y)31 b(the)f(RHS)g(of)h (\(44\).)172 b Fb(2)1831 6013 y Ft(9)p eop %%Page: 10 10 10 9 bop -95 -3 a Fv(3.2)112 b(Pro)s(ofs)38 b(of)f(Lemmas)h(2)f(and)h (3.)-95 169 y Fo(Pro)s(of)e(of)f(Lemma)e(2.)40 b Ft(Let)31 b Fn(!)s Ft(\(0\))26 b Fl(2)f Fn(U)1317 183 y Fm(\024)1387 169 y Fl(\022)g Fn(U)41 b Ft(and)29 b(consider)h(\(9\))h(for)f Fn(n)25 b Ft(=)g(0.)41 b(Cho)s(ose)30 b Fn(g)s Ft(\(1\))i(suc)m(h)e (that,)i Fl(8)p Fo(k)p Ft(,)1203 389 y Fl(j)p Fn(g)1271 404 y Fj(k)1319 389 y Ft(\(1\))p Fl(j)26 b(\024)f Fn(\017)1618 403 y Fu(1)1658 389 y Fn(e)1700 352 y Fm(\017)1729 361 y Fh(2)1763 352 y Fi(j)p Fj(k)p Fi(j)1849 389 y Ft(\()p Fn(bR)q Ft(\))2028 345 y Fi(\000)2093 323 y Fh(1)p 2094 330 35 4 v 2094 365 a(2)2142 389 y Ft(\()p Fn(\015)2224 404 y Fj(k)2272 389 y Ft(\))2317 323 y Fh(1)p 2317 330 V 2317 365 a(2)2366 389 y Fn(:)1250 b Ft(\(58\))-95 593 y(>F)-8 b(rom)31 b(Prop)s(osition)d(1,)j(\(58\))h(and)e(\(6\),)i(one)e (obtains,)846 825 y Fl(j)p Fn(!)928 840 y Fj(k)975 825 y Ft(\(1\))p Fl(j)d(\024)e Fn(a\024)p Ft(\(1\))p Fl(j)p Fo(k)p Fl(j)1558 787 y Fi(\000)p Fm(r)1653 825 y Fn(e)1695 787 y Fi(\000)p Fm(\024)p Fu(\(1\))1880 764 y Ff(\000)p Fh(1)1964 787 y Fi(j)p Fj(k)p Fi(j)2070 825 y Ft(+)20 b Fn(\017)2198 839 y Fu(1)2237 825 y Fn(e)2279 787 y Fm(\017)2308 796 y Fh(2)2343 787 y Fi(j)p Fj(k)p Fi(j)2429 825 y Fn(e)2471 787 y Fi(\000)p Fm(\024)2567 757 y Ff(\000)p Fh(1)2567 799 y Fe(\015)2660 753 y Ff(j)p Fd(k)p Ff(j)p 2660 772 75 4 v 2681 813 a Fh(2)3641 825 y Ft(\(59\))-95 1029 y(Then,)32 b(from)h(\(16\),)i(one)e(gets,)h(for)e(an)m(y)h Fn(\032)c(>)g Ft(0,)34 b(b)m(y)e(c)m(ho)s(osing)h Fn(\017)2124 1043 y Fu(1)2163 1029 y Fn(;)15 b(\017)2240 1043 y Fu(2)2313 1029 y Ft(small)31 b(enough,)i(that)g Fl(9)3148 1005 y Ft(\026)3144 1029 y Fn(\025)c(<)f Ft(1)33 b(suc)m(h)g(that,)-95 1142 y Fl(8)p Fo(k)p Ft(,)1248 1346 y Fl(j)p Fn(!)1330 1361 y Fj(k)1377 1346 y Ft(\(1\))p Fl(j)27 b(\024)e Fn(a\024)1740 1309 y Fi(0)1764 1346 y Fl(j)p Fo(k)p Fl(j)1869 1309 y Fi(\000)p Fm(r)1962 1346 y Fn(e)2004 1309 y Fi(\000)p Fu(\()p Fm(\024)2127 1285 y Ff(0)2150 1309 y Fu(\))2177 1285 y Ff(\000)p Fh(1)2260 1309 y Fi(j)p Fj(k)p Fi(j)3641 1346 y Ft(\(60\))-95 1550 y(with)k Fn(\024)164 1517 y Fi(0)213 1550 y Ft(=)c(max\()517 1526 y(\026)513 1550 y Fn(\025)q(\024;)15 b Ft(2)p Fn(\024)756 1564 y Fm(\015)822 1550 y Ft(+)20 b Fn(\032)p Ft(\).)46 1680 y(>F)-8 b(rom)35 b(\(29\),)j(\(55\),)g(\(56\),)f(one)e(also)g(easily)f(obtains)g(that)h (\010\(1\))f Fl(\024)e Fn(')p Ft(\()p Fn(\024)2601 1647 y Fi(0)2625 1680 y Ft(\))2675 1617 y Fh(2)p 2671 1629 41 4 v 2671 1670 a Fe(\013)2760 1680 y Ft(and)i(th)m(us)g(that)h Fn(!)s Ft(\(1\))f Fl(2)e Fn(U)3710 1696 y Fm(\024)3751 1677 y Ff(0)3777 1680 y Ft(.)-95 1810 y(Th)m(us,)e Fn(P)13 b Ft(\()p Fn(!)s(;)i(U)429 1825 y Fm(\024)470 1806 y Ff(0)496 1810 y Ft(\))26 b Fl(\025)653 1745 y Fk(Q)731 1832 y Fj(k)794 1810 y Fn(P)13 b Ft(\()p Fl(j)p Fn(g)968 1825 y Fj(k)1016 1810 y Ft(\(1\))p Fl(j)26 b(\024)f Fn(\017)1315 1824 y Fu(1)1355 1810 y Fn(e)1397 1777 y Fm(\017)1426 1786 y Fh(2)1460 1777 y Fi(j)p Fj(k)p Fi(j)1546 1810 y Ft(\()p Fn(bR)q Ft(\))1725 1765 y Fi(\000)1790 1743 y Fh(1)p 1791 1750 35 4 v 1791 1786 a(2)1839 1810 y Ft(\()p Fn(\015)1921 1825 y Fj(k)1969 1810 y Ft(\))2014 1743 y Fh(1)p 2014 1750 V 2014 1786 a(2)2063 1810 y Ft(\).)46 1922 y(No)m(w,)31 b(since)f(the)h Fn(g)701 1937 y Fj(k)748 1922 y Ft('s)g(are)f(Gaussian)g(random)g(v)-5 b(ariables)29 b(with)g(co)m(v)-5 b(ariance)31 b Fn(\015)2786 1937 y Fj(k)2833 1922 y Ft(,)g(w)m(e)g(ha)m(v)m(e)g(that,)558 2138 y Fn(P)13 b Ft(\()p Fl(j)p Fn(g)732 2153 y Fj(k)780 2138 y Ft(\(1\))p Fl(j)26 b(\024)f Fn(\017)1079 2152 y Fu(1)1119 2138 y Fn(e)1161 2100 y Fm(\017)1190 2109 y Fh(2)1224 2100 y Fi(j)p Fj(k)p Fi(j)1310 2138 y Ft(\()p Fn(bR)q Ft(\))1489 2093 y Fi(\000)1554 2072 y Fh(1)p 1555 2078 V 1555 2114 a(2)1603 2138 y Ft(\()p Fn(\015)1685 2153 y Fj(k)1733 2138 y Ft(\))1778 2072 y Fh(1)p 1778 2078 V 1778 2114 a(2)1827 2138 y Ft(\))g Fl(\025)g Ft(1)c Fl(\000)f Ft(exp\()p Fl(\000)p Fn(c\017)2461 2100 y Fu(2)2461 2160 y(1)2501 2138 y Fn(e)2543 2100 y Fu(2)p Fm(\017)2607 2109 y Fh(2)2641 2100 y Fi(j)p Fj(k)p Fi(j)2728 2138 y Ft(\()p Fn(bR)q Ft(\))2907 2100 y Fi(\000)p Fu(1)3001 2138 y Ft(\))605 b(\(61\))-95 2342 y(for)26 b Fl(j)p Fo(k)p Fl(j)g(\025)f Fn(C)d Ft(log)17 b Fn(R)q Ft(,)27 b(if)f Fn(C)33 b Ft(is)26 b(c)m(hosen)h(so)g(that)g Fn(bR)f Fl(\024)f Fn(e)1735 2309 y Fm(\017)1764 2318 y Fh(2)1798 2309 y Fi(j)p Fj(k)p Fi(j)1911 2342 y Ft(\(note)j(that)f(the)g(pro)s (duct)f(o)m(v)m(er)i(suc)m(h)e Fo(k)p Ft('s)h(of)g(the)g(RHS)-95 2455 y(of)k(\(61\))g(is)f(strictly)f(p)s(ositiv)m(e)h(uniformly)d(in)i Fn(R)q Ft(\),)i(while)d(for)i Fl(j)p Fo(k)p Fl(j)c(\024)f Fn(C)d Ft(log)16 b Fn(R)q Ft(,)349 2676 y Fn(P)d Ft(\()p Fl(j)p Fn(g)523 2691 y Fj(k)571 2676 y Ft(\(1\))p Fl(j)27 b(\024)e Fn(\017)871 2690 y Fu(1)910 2676 y Fn(e)952 2638 y Fm(\017)981 2647 y Fh(2)1016 2638 y Fi(j)p Fj(k)p Fi(j)1102 2676 y Ft(\()p Fn(bR)q Ft(\))1281 2631 y Fi(\000)1346 2610 y Fh(1)p 1346 2616 V 1346 2652 a(2)1395 2676 y Ft(\()p Fn(\015)1477 2691 y Fj(k)1525 2676 y Ft(\))1570 2610 y Fh(1)p 1570 2616 V 1570 2652 a(2)1619 2676 y Ft(\))g Fl(\025)g Fn(P)13 b Ft(\()p Fl(j)p Fn(g)1949 2691 y Fj(k)1997 2676 y Ft(\(1\))p Fl(j)27 b(\024)e Fn(\017)2297 2690 y Fu(1)2336 2676 y Ft(\()p Fn(bR)q Ft(\))2515 2631 y Fi(\000)2580 2610 y Fh(1)p 2581 2616 V 2581 2652 a(2)2629 2676 y Ft(\()p Fn(\015)2711 2691 y Fj(k)2759 2676 y Ft(\))2804 2610 y Fh(1)p 2804 2616 V 2804 2652 a(2)2853 2676 y Ft(\))g Fl(\025)g Fn(C)7 b(R)3151 2638 y Fi(\000)p Fu(1)3641 2676 y Ft(\(62\))-95 2880 y(whic)m(h)29 b(readily)g(implies)e(that)k Fl(9)p Fn(C)q(;)15 b(c)1167 2894 y Fu(1)1233 2880 y Fn(<)25 b Fl(1)30 b Ft(suc)m(h)g(that)h Fl(8)p Fn(!)d Fl(2)c Fn(U)2135 2894 y Fm(\024)2180 2880 y Ft(,)1204 3084 y Fn(P)13 b Ft(\()p Fn(!)s(;)i(U)1472 3099 y Fm(\024)1513 3081 y Ff(0)1540 3084 y Ft(\))25 b Fl(\025)g Ft(exp\()p Fl(\000)p Fn(C)7 b Ft(\(log)17 b Fn(R)q Ft(\))2286 3047 y Fm(c)2317 3056 y Fh(1)2355 3084 y Ft(\))1251 b(\(63\))-95 3288 y(Since)20 b Fn(U)35 b Ft(=)25 b Fn(U)388 3302 y Fm(R)441 3284 y Fe(p)482 3288 y Ft(,)e(and)d(since)g Fn(\024)i Ft(decreases)f(b)m(y)g(a)g(factor)1807 3264 y(\026)1803 3288 y Fn(\025)k(<)g Ft(1)d(at)f(eac)m(h)h(step,)h(as)e (long)g(as)2946 3264 y(\026)2942 3288 y Fn(\025\024)26 b Fl(\025)f Ft(2)p Fn(\024)3266 3302 y Fm(\015)3312 3288 y Ft(+)q Fn(\032)p Ft(,)e(one)e(ma)m(y)-95 3401 y(iterate)33 b(the)g(ab)s(o)m(v)m(e)h(argumen)m(t)f(and)f(reac)m(h)1461 3378 y(\026)1445 3401 y Fn(U)39 b Ft(=)29 b Fn(U)1708 3415 y Fu(2)p Fm(\024)1784 3423 y Fe(\015)1824 3415 y Fu(+)p Fm(\032)1919 3401 y Ft(,)k(see)g(\(45\),)i(in)d(a)h(time)f(less) g(than)h Fn(T)3180 3415 y Fu(1)3219 3401 y Ft(\()p Fn(R)q Ft(\))d(=)e Fn(C)22 b Ft(log)17 b Fn(R)q Ft(,)-95 3514 y Fl(8)p Fn(!)s Ft(\(0\))31 b Fl(2)f Fn(U)10 b Ft(.)50 b(Therefore,)34 b(the)f(claim)g(of)h(the)f(lemma)g(follo)m(ws)g(\(with) f(a)i(di\013eren)m(t)f Fn(C)40 b Ft(than)33 b(in)f(\(63\),)k(and)d (with)-95 3627 y Fn(c)26 b Ft(=)f Fn(c)105 3641 y Fu(1)165 3627 y Ft(+)19 b(1\).)3375 b Fb(2)-95 3858 y Fo(Pro)s(of)36 b(of)g(Lemma)d(3.)42 b Ft(Let)31 b Fn(!)1028 3872 y Fu(0)1093 3858 y Fl(2)1195 3835 y Ft(\026)1180 3858 y Fn(U)40 b Ft(and)30 b Fn(B)h Fl(\032)1671 3835 y Ft(\026)1656 3858 y Fn(U)9 b Ft(.)42 b(Since)30 b(the)h Fn(g)2232 3873 y Fj(k)2280 3858 y Ft('s)f(are)i(Gaussian)d(random)h(v)-5 b(ariables)30 b(with)-95 3971 y(co)m(v)-5 b(ariance)32 b Fn(\015)392 3986 y Fj(k)439 3971 y Ft(,)e(w)m(e)h(ha)m(v)m(e,)907 4203 y Fn(P)13 b Ft(\()p Fn(!)1070 4217 y Fu(0)1110 4203 y Fn(;)i(B)5 b Ft(\))25 b(=)1380 4088 y Fk(Z)1426 4277 y Fm(B)1502 4123 y Fk(Y)1534 4307 y Fj(k)1633 4157 y Fm(d)5 b Fu(\026)-40 b Fm(!)1713 4169 y Fd(k)1754 4157 y Fi(^)p Fm(d!)1881 4169 y Fd(k)p 1633 4182 293 4 v 1688 4239 a Fu(2)p Fm(\031)r(i\015)1826 4251 y Fd(k)1951 4203 y Ft(exp\()p Fl(\000)2206 4152 y Fi(j)p Fm(!)2270 4164 y Fd(k)2310 4152 y Fi(\000)p Fm(F)2410 4164 y Fd(k)2450 4152 y Fu(\()p Fm(!)2521 4161 y Fh(0)2556 4152 y Fu(\))p Fi(j)2603 4133 y Fh(2)p 2206 4182 436 4 v 2383 4239 a Fm(\015)2419 4251 y Fd(k)2652 4203 y Ft(\))954 b(\(64\))-95 4483 y(where)29 b(w)m(e)g(recall)g(from)g(\(9\))h(that)g Fn(F)13 b Ft(\()p Fn(!)1259 4497 y Fu(0)1298 4483 y Ft(\))30 b(denotes)g(the)f(v)-5 b(alue)29 b(at)h(time)f(1)g(of)h(the)f(solution) f(of)i(\(17\))g(with)e(initial)-95 4596 y(condition)h Fn(!)361 4610 y Fu(0)400 4596 y Ft(.)40 b(In)30 b(view)f(of)h(Prop)s (osition)e(1,)j(and)f(the)g(de\014nition)e(of)2337 4573 y(\026)2321 4596 y Fn(U)35 b Ft(=)25 b Fn(U)2576 4610 y Fu(2)p Fm(\024)2652 4618 y Fe(\015)2692 4610 y Fu(+)p Fm(\032)2787 4596 y Ft(,)30 b(w)m(e)h(can)f(b)s(ound,)f Fl(8)p Fn(!)3556 4610 y Fu(0)3620 4596 y Fl(2)3721 4573 y Ft(\026)3705 4596 y Fn(U)10 b Ft(,)1169 4853 y Fl(j)p Fn(F)1252 4868 y Fj(k)1300 4853 y Ft(\()p Fn(!)1392 4867 y Fu(0)1431 4853 y Ft(\))p Fl(j)26 b(\024)f Fn(C)7 b(ae)1775 4799 y Fi(\000)1856 4765 y Ff(j)p Fd(k)p Ff(j)p 1840 4784 107 4 v 1840 4825 a Fh(2)p Fe(\024)1906 4833 y(\015)1960 4853 y Fn(e)2002 4815 y Fi(\000)p Fm(\032)p Fi(j)p Fj(k)p Fi(j)2205 4853 y Fl(\021)25 b Fn(\017)2338 4868 y Fj(k)2385 4853 y Fn(;)1231 b Ft(\(65\))-95 5070 y(pro)m(vided)37 b(w)m(e)h(c)m(ho)s(ose)h Fn(\032)f Ft(su\016cien)m(tly)e(small)g(so)j (that)1850 5025 y Fu(1+)p Fm(\021)15 b Fu(min)n(\(1)p Fm(;)p Fu(2)p Fm(\024)2265 5033 y Fe(\015)2306 5025 y Fu(+)p Fm(\032)p Fu(\))p 1850 5049 574 4 v 2034 5102 a(2)p Fm(\024)2110 5110 y Fe(\015)2149 5102 y Fu(+)p Fm(\032)2472 5070 y Fl(\025)2630 5035 y Fu(1)p 2590 5050 116 4 v 2590 5102 a(2)p Fm(\024)2666 5110 y Fe(\015)2741 5070 y Ft(+)25 b Fn(\032)p Ft(.)63 b(Th)m(us,)39 b(w)m(e)f(can)h(b)s (ound)-95 5204 y Fl(j)p Fn(!)-13 5219 y Fj(k)59 5204 y Fl(\000)25 b Fn(F)213 5219 y Fj(k)261 5204 y Ft(\()p Fn(!)353 5218 y Fu(0)392 5204 y Ft(\))p Fl(j)452 5171 y Fu(2)530 5204 y Fl(\024)38 b Ft(\()p Fl(j)p Fn(!)756 5219 y Fj(k)803 5204 y Fl(j)26 b Ft(+)f Fn(\017)987 5219 y Fj(k)1034 5204 y Ft(\))1069 5171 y Fu(2)1109 5204 y Ft(;)42 b(this)37 b(giv)m(es)h(a)h(lo)m(w)m(er)f(b)s(ound)e(on)i (\(64\))i(indep)s(enden)m(t)c(of)i Fn(!)3233 5218 y Fu(0)3310 5204 y Ft(and)f(w)m(e)i(ma)m(y)-95 5317 y(use)32 b(this)f(b)s(ound)g (on)h(eac)m(h)h(term)g(of)f(the)h(LHS)f(of)g(\(48\),)j(with)c Fn(!)2143 5331 y Fu(0)2210 5317 y Ft(=)e Fn(!)s(;)15 b(!)2470 5284 y Fi(0)2493 5317 y Ft(.)47 b(W)-8 b(e)33 b(get)h(that)f(the)f(LHS)g(of)h(\(48\))h(is)-95 5430 y(b)s(ounded)28 b(from)i(b)s(elo)m(w)g(b)m(y)1192 5547 y Fk(Z)1250 5719 y Fu(\026)1238 5736 y Fm(U)1312 5581 y Fk(Y)1344 5766 y Fj(k)1443 5615 y Fm(d)5 b Fu(\026)-40 b Fm(!)1523 5627 y Fd(k)1564 5615 y Fi(^)p Fm(d!)1691 5627 y Fd(k)p 1443 5641 293 4 v 1498 5698 a Fu(2)p Fm(\031)r(i\015)1636 5710 y Fd(k)1761 5662 y Ft(exp\()p Fl(\000)2016 5611 y Fu(\()p Fi(j)p Fm(!)2107 5623 y Fd(k)2147 5611 y Fi(j)p Fu(+)p Fm(\017)2251 5623 y Fd(k)2291 5611 y Fu(\))2318 5592 y Fh(2)p 2016 5641 342 4 v 2146 5698 a Fm(\015)2182 5710 y Fd(k)2367 5662 y Ft(\))1239 b(\(66\))1808 6013 y(10)p eop %%Page: 11 11 11 10 bop -95 -3 a Ft(In)30 b(order)g(to)h(estimate)g(that)g(latter)f (in)m(tegral,)h(observ)m(e)g(that,)g(b)m(y)f(\(6\),)i Fn(!)c Fl(2)2564 -26 y Ft(\026)2548 -3 y Fn(U)35 b Ft(=)25 b Fn(U)2803 11 y Fu(2)p Fm(\024)2879 19 y Fe(\015)2919 11 y Fu(+)p Fm(\032)3044 -3 y Ft(pro)m(vided)k(that,)j Fl(8)p Fo(k)p Ft(,)1165 226 y Fl(j)p Fn(!)1247 241 y Fj(k)1293 226 y Fl(j)26 b(\024)f Fn(\017)1477 240 y Fu(1)1516 226 y Fn(e)1558 189 y Fm(\017)1587 198 y Fh(2)1622 189 y Fi(j)p Fj(k)p Fi(j)1708 226 y Ft(\()p Fn(bR)q Ft(\))1887 182 y Fi(\000)1952 160 y Fh(1)p 1952 167 35 4 v 1952 203 a(2)2001 226 y Ft(\()p Fn(\015)2083 241 y Fj(k)2131 226 y Ft(\))2176 160 y Fh(1)p 2176 167 V 2176 203 a(2)2250 226 y Fl(\021)h Ft(\026)-46 b Fn(\017)2383 241 y Fj(k)3641 226 y Ft(\(67\))-95 431 y(if)25 b(w)m(e)i(tak)m(e)h Fn(\017)344 445 y Fu(1)384 431 y Fn(;)15 b(\017)461 445 y Fu(2)527 431 y Ft(small)25 b(enough.)39 b(Th)m(us,)26 b(b)m(y)g(restricting)g (the)g(domain)g(of)g(in)m(tegration,)i(w)m(e)f(get)g(a)g(lo)m(w)m(er)g (b)s(ound)-95 544 y(on)j(\(66\):)1281 712 y Fk(Y)1312 896 y Fj(k)1387 793 y Ft([)1412 678 y Fk(Z)1496 704 y Fu(\026)-36 b Fm(\017)1524 716 y Fd(k)1458 866 y Fu(0)1593 746 y(2)p Fm(r)r(dr)p 1593 772 144 4 v 1625 829 a(\015)1661 841 y Fd(k)1762 793 y Ft(exp\()p Fl(\000)2017 742 y Fu(\()p Fm(r)r Fu(+)p Fm(\017)2162 754 y Fd(k)2202 742 y Fu(\))2229 723 y Fh(2)p 2017 772 252 4 v 2102 829 a Fm(\015)2138 841 y Fd(k)2278 793 y Ft(\))1328 b(\(68\))-95 1067 y(Eac)m(h)31 b(factor)g(is)f(b)s(ounded)e(from)i(b)s(elo)m(w)f(b)m(y)1122 1322 y(1)21 b Fl(\000)f Fn(C)1365 1261 y(\017)1402 1228 y Fu(2)1402 1287 y Fj(k)p 1360 1301 95 4 v 1360 1384 a Fn(\015)1407 1399 y Fj(k)1484 1322 y Fl(\000)g Ft(exp\()p Fl(\000)p Fn(c\017)1896 1284 y Fu(2)1896 1344 y(1)1936 1322 y Fn(e)1978 1284 y Fu(2)p Fm(\017)2042 1293 y Fh(2)2077 1284 y Fi(j)p Fj(k)p Fi(j)2163 1322 y Ft(\()p Fn(bR)q Ft(\))2342 1284 y Fi(\000)p Fu(1)2437 1322 y Ft(\))1169 b(\(69\))-95 1570 y(for)29 b Fl(j)p Fo(k)p Fl(j)d(\025)f Fn(C)c Ft(log)c Fn(R)q Ft(,)29 b(where)f Fn(C)36 b Ft(is)28 b(c)m(hosen)h(as)g(in)f(the)h(pro)s(of)f(of)h(Lemma)g(2.)41 b(T)-8 b(o)29 b(b)s(ound)e(the)i(pro)s(duct)f(o)m(v)m(er)i(those)-95 1683 y Fo(k)p Ft('s)f(of)g(the)g(factors)h(giv)m(en)f(b)m(y)g(\(69\))h (b)m(y)f(a)g(strictly)f(p)s(ositiv)m(e)g(constan)m(t,)j(indep)s(enden)m (t)26 b(of)j Fn(R)q Ft(,)g(observ)m(e)h(\014rst)e(that)-95 1796 y(the)36 b(last)g(term)g(is)f(summable)f(o)m(v)m(er)j Fo(k)p Ft(,)h(for)d Fl(j)p Fo(k)p Fl(j)g(\025)f Fn(C)22 b Ft(log)17 b Fn(R)36 b Ft(and)f(that)i(the)f(sum)f(is)g(small.)55 b(Moreo)m(v)m(er,)40 b(using)-95 1909 y(\(65\))32 b(and)e(the)g(lo)m(w) m(er)h(b)s(ound)d(in)h(\(6\),)1391 2107 y Fn(\017)1428 2074 y Fu(2)1428 2133 y Fj(k)p 1386 2147 V 1386 2231 a Fn(\015)1433 2246 y Fj(k)1516 2168 y Fl(\024)24 b Fn(C)7 b(ab)15 b Ft(exp\()p Fl(\000)p Fn(\032)p Fl(j)p Fo(k)p Fl(j)p Ft(\))1424 b(\(70\))-95 2416 y(if)34 b(w)m(e)h(c)m(ho)s(ose)i (in)c(\(6\))j Fn(\016)h Ft(=)c Fn(\032)p Ft(.)54 b(Then,)36 b(\(70\))g(is)f(also)f(summable)g(o)m(v)m(er)i Fo(k)p Ft(,)h(for)e Fl(j)p Fo(k)p Fl(j)e(\025)g Fn(C)22 b Ft(log)16 b Fn(R)36 b Ft(and)e(the)h(sum)g(is)-95 2529 y(also)d(small.)43 b(Finally)-8 b(,)30 b(for)i Fl(j)p Fo(k)p Fl(j)c(\024)f Fn(C)22 b Ft(log)16 b Fn(R)q Ft(,)32 b(eac)m(h)h(factor)g(in)d(\(68\))j (is)e(b)s(ounded)e(from)j(b)s(elo)m(w,)f(using)f(\(6,)j(67\),)h(b)m(y) -95 2642 y Fn(C)-8 2573 y Fk(R)48 2599 y Fu(\026)-36 b Fm(\017)76 2611 y Fd(k)31 2669 y Fu(0)145 2614 y Fh(2)p Fe(r)r(dr)p 145 2621 128 4 v 171 2649 a(\015)203 2666 y Fd(k)308 2642 y Fl(\025)25 b Fn(C)7 b(R)546 2609 y Fi(\000)p Fu(1)640 2642 y Ft(,)30 b(whic)m(h)f(yields)g(the)h(claim)g (of)g(the)h(Lemma.)1546 b Fb(2)-95 2928 y Fp(4)135 b(Pro)t(of)45 b(of)g(the)g(Theorem.)-95 3131 y Ft(W)-8 b(e)32 b(deduce)e(the)g (Theorem)g(from)g(Prop)s(osition)f(2.)46 3244 y(Let)37 b(us)e(c)m(ho)s(ose)i(a)f(n)m(um)m(b)s(er)f(\026)-45 b Fn(\017)35 b Ft(small)g(enough)g(and)h(a)g(time)g Fn(\034)10 b Ft(,)37 b(large)f(enough,)h(i.e)f Fn(\034)45 b Ft(=)34 b Fn(cm)3281 3211 y Fi(\000)p Fu(1)3375 3244 y Fn(R)3445 3211 y Fm(q)3519 3244 y Ft(so)i(that)-95 3357 y(\(43\))c(is)d(less)h (than)582 3321 y Fu(\026)-36 b Fm(\017)p 578 3336 36 4 v 578 3388 a Fu(2)623 3357 y Ft(.)41 b(Then,)30 b(for)g Fn(T)43 b Ft(an)30 b(in)m(teger)h(m)m(ultiple)d(of)i Fn(\034)10 b Ft(,)31 b(write)1291 3587 y Fn(P)1362 3549 y Fm(T)1417 3587 y Ft(\()p Fn(!)s(;)15 b(E)5 b Ft(\))27 b(=)d(\()p Fn(P)1887 3549 y Fm(\034)1931 3587 y Ft(\))1976 3522 y Fe(T)p 1976 3534 45 4 v 1981 3575 a(\034)2035 3587 y Ft(\()p Fn(!)s(;)15 b(E)5 b Ft(\))p Fn(:)1339 b Ft(\(71\))-95 3791 y(Next,)31 b(let)1256 3995 y Fn(\031)s Ft(\()p Fn(!)s(;)15 b(E)5 b Ft(\))26 b Fl(\021)f Fn(\031)s Ft(\()p Fn(E)5 b Ft(\))27 b(=)e Fn(Q)2067 3958 y Fm(\034)2110 3995 y Ft(\(0)p Fn(;)15 b(E)5 b Ft(\))1304 b(\(72\))-95 4199 y(and)785 4381 y(\026)766 4404 y Fn(R)p Ft(\()p Fn(!)s(;)15 b(E)5 b Ft(\))43 b(=)f Fn(P)1304 4366 y Fm(\034)1347 4404 y Ft(\()p Fn(!)s(;)15 b(E)5 b Ft(\))21 b Fl(\000)f Fn(Q)1773 4366 y Fm(\034)1816 4404 y Ft(\()p Fn(!)s(;)15 b(E)5 b Ft(\))742 4542 y Fn(R)812 4504 y Fi(0)835 4542 y Ft(\()p Fn(!)s(;)15 b(E)5 b Ft(\))43 b(=)f Fn(Q)1305 4504 y Fm(\034)1348 4542 y Ft(\()p Fn(!)s(;)15 b(E)5 b Ft(\))21 b Fl(\000)f Fn(Q)1774 4504 y Fm(\034)1817 4542 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5459 y Fk(X)1404 5644 y Fm(k)1441 5654 y Fe(i)1510 5540 y Fn(\031)1565 5503 y Fm(k)1602 5512 y Fh(1)1641 5540 y Fn(r)1685 5503 y Fm(k)1722 5512 y Fh(2)1775 5540 y Fn(:)15 b(:)g(:)h(\031)1951 5503 y Fm(k)1988 5515 y Fe(l)2037 5540 y Ft(+)k Fn(r)2181 5475 y Fe(T)p 2181 5487 V 2186 5529 a(\034)2265 5540 y Fl(\021)25 b Ft(\006)2427 5503 y Fu(1)2486 5540 y Ft(+)20 b Fn(r)2630 5475 y Fe(T)p 2631 5487 V 2636 5529 a(\034)3641 5540 y Ft(\(76\))1808 6013 y(11)p eop %%Page: 12 12 12 11 bop -95 -3 a Ft(where)29 b(the)h(sum)e(\006)580 -36 y Fu(1)649 -3 y Ft(runs)g(o)m(v)m(er)j Fn(k)1092 11 y Fm(i)1145 -3 y Fl(\025)25 b Ft(0,)1342 -67 y Fk(P)1444 -3 y Fn(k)1491 11 y Fm(i)1545 -3 y Ft(=)1651 -39 y Fm(T)p 1651 -24 51 4 v 1657 28 a(\034)1741 -3 y Ft(and)k(collects)h(all)f(the) g(terms)h(with)e(at)i(least)g(one)g(factor)g Fn(\031)s Ft(.)-95 110 y(No)m(w)j(observ)m(e)g(that,)h(using)e(\(75\))i(with)d Fn(T)42 b Ft(=)28 b Fn(\034)10 b Ft(,)34 b(w)m(e)f(ha)m(v)m(e)g(that)h Fn(r)s Ft(\()p Fn(!)s(;)15 b 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Fm(R)2247 2867 y Ft(\))25 b Fl(\024)g Fn(C)7 b Ft(\()p Fn(R)q Ft(\))p Fn(e)2657 2829 y Fi(\000)p Fm(mT)2825 2806 y Ff(0)3641 2867 y Ft(\(81\))-95 3150 y(whic)m(h)29 b(pro)m(v)m(es)i(the)g(existence)g(of)f(lim)1219 3164 y Fm(T)10 b Fi(!1)1431 3150 y Fn(P)1502 3117 y Fm(T)1557 3150 y Ft(\(0)p Fn(;)15 b(E)5 b Ft(\).)42 b(Finally)-8 b(,)29 b(the)i(b)s(ound)d(\(10\))k(follo)m(ws)e(from)g(\(44\).)229 b Fb(2)-95 3334 y Fo(Pro)s(of)43 b(of)f(Lemma)e(4.)59 b Ft(De\014ne,)39 b(for)d Fn(n)f Fl(\025)h Ft(0,)j Fn(U)10 b Ft(\()p Fn(n)p Ft(\))35 b Fl(\021)h Fn(U)2012 3353 y Fm(\020)2048 3334 y Ff(\000)p Fe(n)2138 3353 y Fm(R)2191 3334 y Fe(p)9 b Ft(,)38 b(\(so)f(that)g Fn(U)46 b Ft(=)35 b Fn(U)10 b Ft(\(0\)\))38 b(with)e Fn(\020)42 b(<)35 b Ft(1)i(as)g(in)-95 3447 y(Prop)s(osition)28 b(2,)j(and)f(de\014ne)g Fn(V)20 b Ft(\()p Fn(n)p Ft(\))31 b(b)m(y)f Fn(V)20 b Ft(\()p Fn(n)p Ft(\))26 b(=)f Fn(U)10 b Ft(\()p Fn(n)p Ft(\))p Fl(n)p Fn(U)g Ft(\()p Fn(n)20 b Fl(\000)g Ft(1\),)32 b(for)e Fn(n)25 b Fl(\025)g Ft(1,)31 b(and)e Fn(V)21 b Ft(\(0\))26 b(=)f Fn(U)10 b Ft(\(0\).)42 b(Next,)31 b(let)1329 3632 y Fn(\032)1376 3646 y Fm(mn)1511 3632 y Fl(\021)90 b Ft(sup)1607 3713 y Fm(!)r Fi(2)p Fm(V)16 b Fu(\()p Fm(m)p Fu(\))1890 3632 y Fl(j)p Fn(r)s Ft(\()p Fn(!)s(;)f(V)20 b Ft(\()p Fn(n)p Ft(\)\))p Fl(j)p Fn(;)-95 3887 y Ft(where)30 b Fn(r)j Ft(is)c(de\014ned)g(in)g(\(74\).)43 b(Observ)m(e)30 b(that)h(w)m(e)g(ha)m(v)m(e)g(the)g(follo)m(wing)e(b)s (ounds)f(on)i Fn(\032)2899 3901 y Fm(mn)3008 3887 y Ft(:)1293 4072 y Fn(\032)1340 4086 y Fu(00)1457 4072 y Fl(\024)42 b Ft(\026)-46 b Fn(\017)1259 4210 y(\032)1306 4224 y Fm(mn)1457 4210 y Fl(\024)41 b Ft(exp\()p Fl(\000)p Fn(c\030)1897 4172 y Fm(n)1944 4210 y Fn(R)2014 4172 y Fm(q)2052 4210 y Ft(\))76 b Fn(n)25 b Fl(\025)g Fn(m)1259 4347 y(\032)1306 4361 y Fm(mn)1457 4347 y Fl(\024)41 b Ft(4)76 b Fn(n)25 b(<)g(m)1695 b Ft(\(82\))-95 4546 y(where)31 b Fn(\030)g Fl(\021)c Fn(\020)385 4510 y Fi(\000)454 4483 y Fh(2)p 449 4495 41 4 v 449 4536 a 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y Ft(\()p Fn(!)s(;)15 b(E)5 b Ft(\))26 b(=)1453 5122 y Fk(Z)1551 5131 y Fm(N)7 b Fi(\000)p Fu(1)1575 5156 y Fk(Y)1571 5339 y Fm(i)p Fu(=0)1720 5237 y Fn(r)s Ft(\()p Fn(!)1856 5251 y Fm(i)1883 5237 y Fn(;)15 b(d!)2027 5251 y Fm(i)p Fu(+1)2146 5237 y Ft(\))p Fn(\037)p Ft(\()p Fn(!)2330 5251 y Fm(N)2423 5237 y Fl(2)25 b Fn(E)5 b Ft(\))1025 b(\(83\))-95 5491 y(with)29 b Fn(!)169 5505 y Fu(0)233 5491 y Ft(=)c Fn(!)s Ft(,)31 b(and)e(insert)h(a)h(decomp)s (osition)e(of)h(the)h(iden)m(tit)m(y)e(for)i(eac)m(h)g Fn(i)26 b Ft(=)f(1)p Fn(;)15 b(:)g(:)g(:)i(;)e(N)1396 5676 y Ft(1)25 b(=)1582 5595 y Fk(X)1562 5778 y Fm(n)1605 5788 y Fe(i)1631 5778 y Fi(\025)p Fu(0)1737 5676 y Fn(\037)p Ft(\()p Fn(!)1886 5690 y Fm(i)1939 5676 y Fl(2)g Fn(V)20 b Ft(\()p Fn(n)2188 5690 y Fm(i)2216 5676 y Ft(\)\))p Fn(:)1808 6013 y Ft(12)p eop %%Page: 13 13 13 12 bop -95 -3 a Ft(This)29 b(leads)g(to)728 256 y(sup)656 337 y Fm(!)r Fi(2)p Fm(U)7 b Fu(\()p Fm(n)874 346 y Fh(0)909 337 y Fu(\))951 256 y Ft(sup)992 333 y Fm(E)1103 256 y Fl(j)p Fn(r)1172 218 y Fm(N)1239 256 y Ft(\()p Fn(!)s(;)15 b(E)5 b Ft(\))p Fl(j)27 b(\024)1772 175 y Fk(X)1629 370 y Fu(\()p Fm(n)1699 380 y Fe(i)1726 370 y Fu(\))1753 348 y Fe(N)1753 394 y(i)p Fh(=1)1857 370 y Fm(;n)1920 380 y Fe(i)1945 370 y Fi(\025)p Fu(0)2051 150 y Fm(N)7 b Fi(\000)p Fu(1)2074 175 y Fk(Y)2070 357 y Fm(i)p Fu(=0)2219 256 y Fn(\032)2266 270 y Fm(n)2309 280 y Fe(i)2335 270 y Fm(n)2378 280 y Fe(i)p Fh(+1)2511 256 y Fl(\021)2607 175 y Fk(X)2645 349 y Fm(n)2742 256 y Fn(\032)2789 218 y Fm(N)2789 278 y(n)2832 287 y Fh(0)2866 278 y Fm(n)2913 256 y Fn(:)703 b Ft(\(84\))-95 565 y(Note)38 b(that)f(the)g(RHS)f (describ)s(es)f("random)i(w)m(alks")f(on)h(nonnegativ)m(e)g(in)m (tegers,)i(where)d(only)g(steps)h(strictly)-95 678 y(do)m(wn)31 b(\()p Fn(n)236 692 y Fm(i)p Fu(+1)382 678 y Fn(<)c(n)535 692 y Fm(i)563 678 y Ft(\))32 b(are)g(not)g(suppressed.)42 b(T)-8 b(o)32 b(estimate)h(it,)f(write)f Fn(\032)c Ft(=)g Fn(d)22 b Ft(+)e Fn(u)32 b Ft(where)f Fn(d)h Ft(is)f(the)h("do)m(wn")g (part)-95 791 y(of)d Fn(\032)p Ft(,)g(i.e.)41 b(the)29 b(matrix)f(whose)h(elemen)m(ts)g(are)g(giv)m(en)g(b)m(y)g Fn(\032)1909 805 y Fm(mn)2047 791 y Ft(with)f Fn(n)c(<)h(m)k Ft(and)f(zero)i(otherwise,)f(and)f(u)g(is)g(the)-95 904 y(rest)j(\("up"\).)41 b(W)-8 b(e)31 b(note)g(the)g(simple)d(estimates) 1370 1027 y Fk(X)1409 1201 y Fm(n)1505 1108 y Fn(d)1552 1070 y Fm(k)1552 1130 y(mn)1687 1108 y Fl(\024)d Fn(C)1855 1070 y Fm(k)1973 1108 y Fn(k)j Fl(\024)d Fn(m)1417 b Ft(\(85\))-95 1377 y(and)30 b(zero)g(otherwise)g(\(where)g(the)g (restriction)f Fn(k)g Fl(\024)c Fn(m)30 b Ft(comes)g(from)g(the)g(fact) h(that)g(the)f(indices)f(of)h Fn(d)3468 1391 y Fm(mn)3608 1377 y Ft(m)m(ust)-95 1490 y(b)s(e)g(p)s(ositiv)m(e)f(and,)h(whenev)m (er)g Fn(d)1013 1504 y Fm(mn)1148 1490 y Fl(6)p Ft(=)25 b(0,)31 b(m)m(ust)g(satisfy)e Fn(n)c(<)g(m)p Ft(\),)31 b(and)1341 1614 y Fk(X)1379 1788 y Fm(n)1461 1695 y Ft(\()p Fn(u)1548 1657 y Fm(k)1591 1695 y Fn(d)1638 1657 y Fm(l)1664 1695 y Ft(\))1699 1709 y Fm(mn)1834 1695 y Fl(\024)25 b Ft(\()p Fn(C)7 b Ft(\026)-45 b Fn(\017)p Ft(\))2109 1657 y Fm(k)r Fu(+)p Fm(l)2228 1695 y Fn(:)1388 b Ft(\(86\))-95 1964 y(Indeed,)30 b(\(85\))i(is)d(estimated)i(b)m(y)686 2148 y Fk(X)637 2323 y Fm(n)680 2332 y Fh(1)715 2323 y Fm(:::)o(n)817 2335 y Fe(k)870 2229 y Fn(d)917 2243 y Fm(mn)1022 2252 y Fh(1)1076 2229 y Fn(:)15 b(:)g(:)h(d)1244 2243 y Fm(n)1287 2255 y Fe(k)q Ff(\000)p Fh(1)1404 2243 y Fm(n)1447 2255 y Fe(k)1514 2229 y Fl(\024)1709 2148 y Fk(X)1610 2301 y(P)1698 2321 y Fe(k)1698 2388 y Fh(1)1748 2359 y Fm(p)1784 2369 y Fe(i)1810 2359 y Fi(\024)p Fm(m)1942 2229 y Ft(4)1987 2192 y Fm(k)2055 2229 y Ft(=)25 b(4)2196 2192 y Fm(k)2283 2123 y(m)2255 2148 y Fk(X)2257 2333 y Fm(l)q Fu(=)p Fm(k)2390 2110 y Fk(\022)2477 2173 y Fn(l)d Fl(\000)e Ft(1)2466 2286 y Fn(k)j Fl(\000)d Ft(1)2688 2110 y Fk(\023)2774 2229 y Fl(\024)25 b Ft(4)2915 2192 y Fm(k)2958 2229 y Ft(2)3003 2192 y Fm(m)-95 2575 y Ft(with)k Fn(p)158 2589 y Fm(i)212 2575 y Ft(=)c Fn(n)363 2589 y Fm(i)p Fi(\000)p Fu(1)501 2575 y Fl(\000)20 b Fn(n)647 2589 y Fm(i)700 2575 y Fl(\025)25 b Ft(1,)31 b Fn(n)952 2589 y Fu(0)1017 2575 y Ft(=)25 b Fn(m)p Ft(,)30 b(yielding)e(the)j (claim)f(since)g Fn(k)e Fl(\024)d Fn(m)p Ft(.)41 b(T)-8 b(o)31 b(pro)m(v)m(e)g(\(86\),)i(write)c(\()p Fn(u)3445 2542 y Fm(k)3488 2575 y Fn(d)3535 2542 y Fm(l)3562 2575 y Ft(\))3597 2589 y Fm(mn)3732 2575 y Ft(=)-95 2624 y Fk(P)-7 2711 y Fm(m)55 2692 y Ff(0)97 2688 y Fn(u)149 2655 y Fm(k)149 2715 y(mm)273 2696 y Ff(0)300 2688 y Fn(d)347 2655 y Fm(l)347 2715 y(m)409 2696 y Ff(0)432 2715 y Fm(n)510 2688 y Ft(and)g(note)i(that)g Fn(u)1137 2655 y Fm(k)1137 2715 y(mm)1261 2696 y Ff(0)1319 2688 y Ft(is)e(b)s(ounded)g(b)m(y)h(\()p Fn(C)7 b Ft(\026)-45 b Fn(\017)p Ft(\))2087 2655 y Fm(k)2160 2688 y Ft(if)29 b Fn(m)c Ft(=)g Fn(m)2524 2655 y Fi(0)2573 2688 y Ft(=)g(0)30 b(and)g(b)m(y)1423 2892 y(exp\()p Fl(\000)p Fn(c\030)1751 2855 y Fm(M)1830 2892 y Fn(R)1900 2855 y Fm(q)1938 2892 y Ft(\)\()p Fn(C)7 b Ft(\026)-45 b Fn(\017)p Ft(\))2152 2855 y Fm(k)r Fi(\000)p Fu(1)-95 3096 y Ft(for)42 b Fn(M)56 b Ft(=)45 b(max)q Fl(f)p Fn(m;)15 b(m)731 3063 y Fi(0)754 3096 y Fl(g)43 b Ft(otherwise.)77 b(The)42 b(b)s(ound)f(\(86\))j(follo) m(ws)e(b)m(y)g(com)m(bining)f(these)i(with)f(\(85\),)47 b(since)-95 3209 y Fn(l)27 b Fl(\024)e Fn(m)135 3176 y Fi(0)184 3209 y Fl(\024)f Fn(M)10 b Ft(,)29 b(w)m(e)g(can)f (therefore)h(use)e(the)i(factor)g(exp\()p Fl(\000)p Fn(c\030)2002 3176 y Fm(M)2081 3209 y Fn(R)2151 3176 y Fm(q)2189 3209 y Ft(\))f(and)f Fn(R)i Ft(large)f(to)h(obtain)f(the)g(factor)i(\026)-46 b Fn(\017)3583 3176 y Fm(l)q Fu(+1)3727 3209 y Ft(in)-95 3322 y(\(86\))32 b(\()q(\026)-46 b Fn(\017)30 b Ft(is)g(a)h(\014xed)e (small)g(n)m(um)m(b)s(er\).)46 3435 y(Inserting)g(\(85\),)j(\(86\))f (in)m(to)f Fn(\032)1067 3402 y Fm(N)1160 3435 y Ft(=)24 b(\()p Fn(u)c Ft(+)f Fn(d)p Ft(\))1534 3402 y Fm(N)1627 3435 y Ft(=)1723 3371 y Fk(P)1826 3435 y Fn(d)1873 3402 y Fm(l)1894 3411 y Fh(0)1933 3435 y Fn(u)1985 3402 y Fm(k)2022 3411 y Fh(1)2061 3435 y Fn(d)2108 3402 y Fm(l)2129 3411 y Fh(1)2183 3435 y Fn(:)c(:)g(:)h(u)2356 3402 y Fm(k)2393 3410 y Fe(s)2431 3435 y Fn(d)2478 3402 y Fm(l)2499 3410 y Fe(s)2566 3435 y Ft(where)30 b Fn(l)2856 3449 y Fm(i)2909 3435 y Fl(\025)25 b Ft(0,)31 b Fn(l)3133 3449 y Fm(i)3186 3435 y Fn(>)25 b Ft(0)31 b(for)e Fn(i)d Fl(6)p Ft(=)f(0)p Fn(;)15 b(s)p Ft(,)-95 3548 y(w)m(e)31 b(obtain)f(the)g(b)s(ound)1455 3580 y Fk(X)1493 3754 y Fm(n)1590 3661 y Fn(\032)1637 3623 y Fm(N)1637 3683 y(mn)1772 3661 y Fl(\024)24 b Fn(C)1939 3623 y Fm(N)2007 3661 y Ft(\026)-46 b Fn(\017)2043 3623 y Fm(N)7 b Fi(\000)p Fm(m)2228 3661 y Fn(:)-95 3904 y Ft(where)31 b(\026)-46 b Fn(\017)205 3871 y Fi(\000)p Fm(m)357 3904 y Ft(comes)31 b(from)f(the)g(fact)h(that)g(w)m(e)g(ha)m(v)m(e)h(no)f(\026)-46 b Fn(\017)30 b Ft(b)s(ound)f(on)h Fn(d)2363 3871 y Fm(l)2384 3880 y Fh(0)2423 3904 y Ft(,)h(but)e Fn(l)2672 3918 y Fu(0)2737 3904 y Fl(\024)c Fn(m)p Ft(.)46 4017 y(This)44 b(pro)m(v)m(es)i(the)g(Lemma,)j(if)c(w)m(e)h(c)m(ho)s(ose)g(in)e (\(79\))j Fn(m)k Ft(=)f Fl(\000)15 b Ft(log)h Fn(C)7 b Ft(\026)-45 b Fn(\017)p Ft(,)49 b(and,)g(for)c Fn(!)53 b Fl(2)d Fn(U)10 b Ft(\()p Fn(n)3395 4031 y Fu(0)3435 4017 y Ft(\),)50 b(c)m(ho)s(ose)-95 4142 y Fn(C)7 b Ft(\()p Fn(!)s Ft(\))30 b(=)f(\()p Fn(C)7 b Ft(\026)-45 b Fn(\017)p Ft(\))416 4106 y Fi(\000)481 4070 y Fe(n)519 4085 y Fh(0)p 481 4091 73 4 v 502 4132 a(2)568 4142 y Ft(;)35 b(indeed,)d(let,)i(for) f Fn(!)f Fl(2)d Fn(U)1487 4156 y Fm(\024)1532 4142 y Ft(,)34 b Fn(n)1646 4156 y Fu(0)1718 4142 y Ft(b)s(e)e(the)i(smallest)e (in)m(teger)h(suc)m(h)g(that)g Fn(\024)d Fl(\024)g Fn(\020)3296 4109 y Fi(\000)p Fm(n)3394 4118 y Fh(0)3432 4142 y Fn(R)3502 4109 y Fm(p)3541 4142 y Ft(;)35 b(then,)-95 4255 y Fn(n)-40 4269 y Fu(0)29 4255 y Fl(\024)29 b Fn(C)22 b Ft(log)16 b Fn(\024)34 b Ft(and)e Fn(C)7 b Ft(\()p Fn(!)s Ft(\))30 b Fl(\024)f Fn(C)7 b(\024)1069 4222 y Fm(c)1104 4255 y Ft(.)48 b(Moreo)m(v)m(er,)36 b(from)d(part)g(\(b\))g(of)g(Prop)s (osition)e(1,)j(w)m(e)g(kno)m(w)f(that,)h Fl(8)p Fn(!)e Fl(2)e Ft(\012,)-95 4368 y Fn(F)13 b Ft(\()p Fn(!)s Ft(\))26 b Fl(2)f Fn(U)280 4382 y Fm(\024)325 4368 y Ft(,)30 b(with)f Fn(\024)d Fl(\024)f Fn(C)7 b Ft(\()p Fl(jj)p Fn(!)s Fl(jj)1028 4335 y Fm(\013)1098 4368 y Ft(+)20 b(1\).)2441 b Fb(2)-95 4654 y Fp(References)-50 4857 y Ft([1])47 b(J.)i(Bricmon)m(t,)54 b(A.)49 b(Kupiainen,)i(R.)d(Lefev)m(ere:)79 b Fc(Pr)-5 b(ob)g(abilistic)51 b(estimates)g(for)f(the)g(two)h(dimensional)92 4970 y(sto)-5 b(chastic)34 b(Navier-Stokes)f(e)-5 b(quations)p Ft(.)32 b(Preprin)m(t.)-50 5158 y([2])47 b(J.L.)30 b(Do)s(ob:)41 b Fc(Sto)-5 b(chastic)35 b(Pr)-5 b(o)g(c)g(esses)p Ft(,)32 b(John)d(Wiley)-8 b(,)31 b(New-Y)-8 b(ork,)31 b(1953.)-50 5345 y([3])47 b(F.)36 b(Flandoli,)f(B.)i(Maslo)m(wski:)51 b Fc(Er)-5 b(go)g(dicity)39 b(of)e(the)h(2-D)f(Navier-Stokes)h(e)-5 b(quation)38 b(under)g(r)-5 b(andom)40 b(p)-5 b(er-)92 5458 y(turb)g(ations.)32 b Ft(Comm)m(un.)e(Math.)h(Ph)m(ys.)f Fo(171)h Ft(\(1995\),)i(119-141.)-50 5646 y([4])47 b(S.)24 b(Kuksin,)g(A.)g(Shiriky)m(an:)36 b Fc(Sto)-5 b(chastic)29 b(dissip)-5 b(ative)28 b(PDE's)e(and)i(Gibbs)f(me)-5 b(asur)g(es)p Ft(.)27 b(Preprin)m(t,)d(to)h(app)s(ear)92 5759 y(in)k(Comm)m(un.)h(Math.)h(Ph)m(ys.)1808 6013 y(13)p eop %%Page: 14 14 14 13 bop -50 -3 a Ft([5])47 b(J.)36 b(C.)g(Mattingly)-8 b(,)38 b(Y.)e(Sinai:)50 b Fc(A)n(n)37 b(elementary)i(pr)-5 b(o)g(of)40 b(of)e(the)g(existenc)-5 b(e)38 b(and)h(uniqueness)e(the)-5 b(or)g(em)40 b(for)92 110 y(the)33 b(Navier-Stokes)f(e)-5 b(quations)p Ft(.)32 b(Preprin)m(t.)-50 297 y([6])47 b(J.)d(C.)g(Mattingly:)69 b Fc(Er)-5 b(go)g(dicity)46 b(of)g(2D)f(Navier-Stokes)h(e)-5 b(quations)46 b(with)g(r)-5 b(andom)48 b(for)-5 b(cing)46 b(and)h(lar)-5 b(ge)92 410 y(visc)g(osity.)31 b Ft(Comm)m(un.)f(Math.)h(Ph)m(ys.)f Fo(206)h Ft(\(1999\),)i(273-288.)1808 6013 y(14)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF 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