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1453 y Fh(N)q(;\014)r(;h)p Fp(=0)994 1446 y Fm([)p Fl(!)r Fm(])8 b(\()o Fl(B)1109 1453 y Fh(\032)1130 1446 y Fm(\))13 b(=)g(1)602 b(\(1)p Fl(:)p Fm(10\))0 1567 y(where)702 1640 y Fl(B)736 1647 y Fh(\032)770 1640 y Fi(\021)947 1597 y Ff([)818 1696 y Fp(\()p Fh(\027;s)p Fp(\))p Fg(2)p Fh(I)-8 b(N)5 b Fg(\002f\000)p Fp(1)p Fh(;)p Fp(+1)p Fg(g)1135 1640 y Fl(B)1171 1621 y Fp(\()p Fh(\027;s)p Fp(\))1169 1651 y Fh(\032)1834 1640 y Fm(\(1)p Fl(:)p Fm(11\))0 1774 y(is)14 b(the)f(union)i(of)e(all)h(the) g(balls)g(app)q(earing)h(in)f(Theorem)f(1.)19 b(The)14 b(crucial)h (di\013erence)f(b)q(et)o(w)o(een)g(that)f(result)h(and)0 1847 y(our)i(new)g(one)g(is)h(that)e(this)i(time)f(w)o(e)g(can)g Fk(sele)n(ct)g(di\013er)n(ent)33 b Fm(limits)17 b(b)o(y)f(adding)h(an)f (arbitrarily)h(small)g(bias)f(in)0 1920 y(terms)10 b(of)h(the)f(magnetic)h (\014eld)i(aligned)f(to)e(one)h(of)f(the)h(patterns.)18 b(T)l(o)10 b(appreciate)i(the)f(di\013erence)h(b)q(et)o(w)o(een)f(these)0 1992 y(results,)19 b(notice)h(that)d(from)h(Theorem)h(1)f(it)h(follo)o(ws)f (in)i(particular)f(that)f(the)g(\014nite)i(dimensional)h(marginal)0 2065 y(distributions)g(p)q(ossess)e(limit)i(p)q(oin)o(ts)f(that)e(clearly)j (distinguish)g(the)f(selected)g(pattern;)h(to)d(b)q(e)i(precise,)h(let)0 2138 y Fl(I)26 b Fi(\032)c Fl(I)-6 b(N)25 b Fm(denote)c(some)g(\014nite)g (set)g(of)g(p)q(ositiv)o(e)g(in)o(tegers,)h(let)g Fl(I)-6 b(R)1188 2118 y Fh(I)1230 2138 y Fm(denote)21 b(the)g(\014nite)h(dimensional)h(space)0 2211 y(generated)15 b(b)o(y)h(the)f(v)o(ectors)g Fl(e)525 2194 y Fh(\026)549 2211 y Fm(,)g(with)h Fl(\026)e Fi(2)f Fl(I)t Fm(,)h(and)i(let)g(\005)1005 2218 y Fh(I)1041 2211 y Fm(b)q(e)g(the)g (orthogonal)e(pro)s(jector)h(from)f Fl(I)-6 b(R)1762 2191 y Fh(M)s Fp(\()p Fh(N)s Fp(\))1878 2211 y Fm(\(for)0 2283 y(an)o(y)13 b Fl(N)18 b Fm(suc)o(h)c(that)f Fl(I)j Fi(\032)d(f)p Fm(1)p Fl(;)8 b(:)g(:)g(:)t(;)g(M)d Fm(\()p Fl(N)g Fm(\))p Fi(g)p Fm(\))11 b(on)o(to)h Fl(I)-6 b(R)896 2264 y Fh(I)917 2283 y Fm(.)19 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Fm(,)h(the)h(limiting)h(marginal)f(is)g(concen)o(trated)f(on)h(on)f(a)g Fi(j)p Fl(I)t Fi(j)p Fm(-dimensional)i(ball)f(around)g(the)0 118 y(v)o(ector)d Fl(e)157 102 y Fh(\021)180 118 y Fm(.)20 b(If)15 b(w)o(e)g(had)g(only)h(\(1.10\),)d(w)o(e)i(w)o(ould)h(get)e(instead)i (of)f(\(1.13\))e(only)679 235 y(lim)682 265 y Fh(h)p Fg(#)p Fp(0)762 235 y Fm(lim)750 265 y Fh(N)s Fg("1)844 235 y Fi(Q)881 214 y Fh(\021)q(;I)881 249 y(N)q(;\014)r(;h)979 235 y Fm([)p Fl(!)r Fm(])8 b(\()n(\005)1093 242 y Fh(I)1114 235 y Fl(B)1148 242 y Fh(\032)1170 235 y Fm(\))k(=)h(1)563 b(\(1)p Fl(:)p Fm(14\))0 353 y(from)14 b(whic)o(h)h(it)g(is)h(not)e(p)q(ossible)i(to)e(conclude)j (that)d(there)g(exists)h(an)o(y)g(\014nite)g Fl(I)k Fm(for)14 b(whic)o(h)h(the)g(corresp)q(onding)0 425 y(marginal)h(distribution)g(is)g Fk(not)f Fm(concen)o(trated)g(on)g(a)g(ball)i(around)e(the)g(origin!)0 534 y Fn(Remark:)21 b Fm(In)13 b(the)f(discussion)j(ab)q(o)o(v)o(e)d(w)o(e)g 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b(w)o(e)g(will)i(sho)o(w)e(that)g(the)g(quan)o(tities)h Fl(F)907 2143 y Fp(\()p Fh(\021)q(;s)p Fp(\))901 2180 y Fh(N)q(;\014)r(;\032)1014 2167 y Fm(satisfy)f(a)g(strong)g(self-a)o(v)o(eraging)g(condition.)30 b(Note)0 2239 y(that)18 b(they)i(can)f(b)q(e)h(naturally)g(regarded)f(as)f (`lo)q(cal)i(free)g(energies',)g(asso)q(ciated)f(with)h(the)f(particular)h (state)0 2312 y(lab)q(elled)e(\()p Fl(\021)r(;)8 b(s)p Fm(\).)18 b(The)d(crucial)i(estimate)e(is)h(con)o(tained)f(in)h(the)g(follo)o(wing)0 2421 y Fb(Prop)r(osition)j(1.1:)i Fk(Assume)16 b(that)i Fm(lim)761 2399 y Fh(M)s Fp(\()p Fh(N)s Fp(\))p 761 2411 99 2 v 794 2437 a Fh(N)879 2421 y Fm(=)c Fl(\013)g(>)g Fm(0)p Fk(.)22 b(L)n(et)16 b Fl(\032)d(<)h Fm(1)p Fk(.)23 b(Then,)16 b(for)h(al)r(l)g Fl(n)d(<)g Fi(1)j Fk(ther)n(e)g(exists)0 2494 y Fl(\034)20 2501 y Fh(n)57 2494 y Fl(<)c Fi(1)p Fk(,)k(such)f(that)h(for)g(al)r(l)f Fl(\034)h Fi(\025)c Fl(\034)620 2501 y Fh(n)645 2494 y Fk(,)j(and)h(for)f Fl(N)21 b Fk(lar)n(ge)16 b(enough,)405 2631 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b(s)p Fm(\))k Fi(6)p Fm(=)h(\()p Fl(\021)r(;)8 b Fm(+1\))457 1072 y Fl(Z)491 1050 y Fh(\021)488 1086 y(N)q(;\014)r(;h)586 1072 y Fm([)p Fl(!)r Fm(])650 1022 y Ff(\020)676 1072 y Fl(B)712 1054 y Fp(\()p Fh(\026;s)p Fp(\))710 1084 y Fh(\032)793 1022 y Ff(\021)833 1072 y Fi(\024)13 b Fl(e)902 1054 y Fp(+)p Fh(\014)r(N)s(h)1006 1033 y Fg(p)p 1036 1033 V 21 x Fh(\032)1057 1072 y Fl(Z)1088 1079 y Fh(N)q(;\014)r(;h)p Fp(=0)1232 1072 y Fm([)p Fl(!)r Fm(])1296 1022 y Ff(\020)1321 1072 y Fl(B)1357 1054 y Fp(\()p Fh(\021)q(;)p Fp(+1\))1355 1084 y Fh(\032)1466 1022 y Ff(\021)1834 1072 y Fm(\(1)p Fl(:)p Fm(20\))0 1200 y(No)o(w)i(b)o(y)g(Prop)q(osition)g (1.1,)f(with)i(probabilit)o(y)g(greater)f(than,)f(sa)o(y)l(,)h(1)9 b Fi(\000)i Fl(N)1314 1183 y Fg(\000)p Fp(10)1380 1200 y Fm(,)k Fk(al)r(l)g Fm(of)g(the)g(quan)o(tities)0 1272 y Fl(Z)31 1279 y Fh(N)q(;\014)r(;h)p Fp(=0)175 1272 y Fm([)p Fl(!)r Fm(])239 1222 y Ff(\020)265 1272 y Fl(B)301 1249 y Fp(\()p Fh(\021)q(;)p Fp(+1\))299 1278 y Fh(\032)409 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b(mak)o(e)f(manifest)i(that,)f(b)o(y)g(symmetry)l(,)h (these)0 1623 y(a)o(v)o(eraged)e(quan)o(tities)h(do)f(not,)h(of)f(course,)i (dep)q(end)g(on)e(the)h(indices)h(\()p Fl(\026;)8 b(s)p Fm(\))19 b(if)h(the)g(magnetic)g(\014eld)g(is)g(zero.)0 1696 y(Ob)o(viously)l(,)c (again)g(with)f(probabilit)o(y)i(grater)d(than)h(1)10 b Fi(\000)g Fl(N)1040 1679 y Fg(\000)p Fp(10)1107 1696 y Fm(,)31 1863 y Fi(Q)68 1841 y Fh(\021)68 1877 y(N)q(;\014)r(;h)165 1863 y Fm([)p Fl(!)r Fm(])229 1813 y Ff(\020)255 1863 y Fl(B)291 1844 y Fp(\()p Fh(\021)q(;)p Fp(+1\))289 1875 y Fh(\032)399 1813 y Ff(\021)439 1863 y Fm(=)954 1816 y Fl(Z)988 1794 y Fh(\021)985 1830 y(N)q(;\014)r(;h)1090 1766 y Ff(\020)1117 1816 y Fl(B)1153 1792 y Fp(\()p Fh(\021)q(;)p Fp(+1\))1151 1821 y Fh(\032)1261 1766 y Ff(\021)p 493 1853 1257 2 v 493 1911 a Fl(Z)527 1889 y Fh(\021)524 1924 y(N)q(;\014)r(;h)629 1860 y Ff(\020)656 1911 y Fl(B)692 1887 y Fp(\()p Fh(\021)q(;)p Fp(+1\))690 1916 y Fh(\032)800 1860 y Ff(\021)837 1911 y Fm(+)883 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1215 y Fm(\()8 b(~)-31 b Fl(m)p Fm(\))9 b Fi(\000)i Fl(I)-8 b(E)r(F)824 1222 y Fh(N)q(;\014)r(;\032)919 1215 y Fm(\()9 b(~)-32 b Fl(m)p Fm(\))p Fi(j)12 b(\025)h Fl(\034)5 b Fm(\(ln)j Fl(N)d Fm(\))1216 1196 y Fp(3)p Fh(=)p Fp(2)1272 1215 y Fl(N)1314 1196 y Fg(\000)1348 1184 y Fc(1)p 1348 1189 16 2 v 1348 1207 a(2)1371 1164 y Ff(i)1405 1215 y Fi(\024)13 b Fl(N)1495 1196 y Fg(\000)p Fh(n)1857 1215 y Fm(\(2)p Fl(:)p Fm(2\))91 1411 y(Prop)q(osition)18 b(1.1)e(from)h(the)g(last)g(section)h(is)g (of)f(course)h(an)f(immediate)h(corollary)g(of)f(this)h(prop)q(osition,)0 1483 y(whic)o(h)e(is)g(in)g(fact)e(far)h(more)g(general.)20 b(Th)o(us)15 b(all)h(that)f(is)h(left)f(is)h(to)e(pro)o(v)o(e)h(Prop)q (osition)h(2.1.)0 1593 y Fa(Pro)q(of:)31 b Fm(\(of)19 b(Prop)q(osition)g (2.1\))f(F)l(or)g(tec)o(hnical)j(reasons)d(that)g(will)j(b)q(ecome)f(clear)f (later,)h(w)o(e)e(will)j(consider)0 1665 y(instead)11 b(of)g Fl(F)235 1643 y Fh(H)r(opf)229 1679 y(N)q(;\014)328 1665 y Fm(\()d(~)-31 b Fl(m)o Fm(\))10 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Fh(k)1281 215 y Fm(that)f(are)h(generated) g(b)o(y)g(the)g(random)0 288 y(v)m(ariables)j Fi(f)o Fl(\030)233 266 y Fh(\026)231 301 y(i)258 288 y Fi(g)280 264 y Fh(\026)p Fg(2)p Fh(I)-8 b(N)280 303 y(i)p Fg(\025)p Fh(k)385 288 y Fm(and)16 b(the)f(corresp)q(onding)h(martingale)f(di\013erence)i(sequence)531 387 y(~)521 399 y Fl(f)548 375 y Fp(\()p Fh(k)q Fp(\))543 412 y Fh(N)599 399 y Fm(\()9 b(~)-32 b Fl(m)p Fm(\))12 b Fi(\021)h Fl(I)-8 b(E)791 362 y Ff(\002)810 399 y Fl(f)832 406 y Fh(N)867 399 y Fm(\()8 b(~)-31 b Fl(m)o Fm(\))942 360 y Ff(\014)942 387 y(\014)957 399 y Fi(F)990 406 y Fh(k)1012 362 y Ff(\003)1041 399 y Fi(\000)11 b Fl(I)-8 b(E)1143 362 y Ff(\002)1162 399 y Fl(f)1184 406 y Fh(N)1218 399 y Fm(\()9 b(~)-32 b Fl(m)p Fm(\))1294 360 y Ff(\014)1294 387 y(\014)1309 399 y Fi(F)1342 406 y Fh(k)q Fp(+1)1410 362 y Ff(\003)1857 399 y Fm(\(2)p Fl(:)p Fm(5\))0 509 y(Notice)16 b(that)e(w)o(e)h(ha)o(v)o(e)g(the)g(iden)o(tit)o(y) 647 644 y Fl(f)669 651 y Fh(N)704 644 y Fm(\()8 b(~)-31 b 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b(E)1217 1442 y Ff(h)1238 1493 y Fl(e)1259 1476 y Fh(t)1280 1468 y Fp(~)1273 1476 y Fh(f)1295 1459 y Fc(\()p Fe(k)p Fc(\))1291 1488 y Fe(N)1339 1476 y Fp(\()7 b(~)-25 b Fh(m)p Fp(\))1402 1454 y Ff(\014)1402 1481 y(\014)1417 1493 y Fi(F)1450 1500 y Fh(k)q Fp(+1)1518 1442 y Ff(i)1552 1493 y Fi(\024)13 b(L)1631 1476 y Fp(\()p Fh(k)q Fp(\))1682 1493 y Fm(\()p Fl(t)p Fm(\),)e(uniformly)0 1566 y(in)16 b Fi(F)86 1573 y Fh(k)q Fp(+1)154 1566 y Fm(,)f(then)g(w)o(e)g (obtain)h(that)448 1702 y Fl(I)-8 b(P)503 1624 y Ff(")530 1622 y(\014)530 1649 y(\014)530 1677 y(\014)530 1704 y(\014)530 1731 y(\014)562 1645 y Fh(N)545 1658 y Ff(X)545 1756 y Fh(k)q Fp(=1)628 1690 y Fm(~)619 1702 y Fl(f)646 1678 y Fp(\()p Fh(k)q Fp(\))641 1715 y Fh(N)696 1702 y Fm(\()9 b(~)-32 b Fl(m)p Fm(\))772 1622 y Ff(\014)772 1649 y(\014)772 1677 y(\014)772 1704 y(\014)772 1731 y(\014)799 1702 y Fi(\025)13 b Fl(N)5 b(z)912 1624 y Ff(#)951 1702 y Fi(\024)13 b Fm(2)k(inf)1030 1731 y Fh(t)p Fg(2)p Fh(I)-8 b(R)1111 1702 y Fl(e)1132 1680 y Fg(\000j)p Fh(t)p Fg(j)p Fh(N)s(z)q Fp(+)1274 1649 y Ff(P)1323 1660 y Fe(N)1323 1697 y(k)p Fc(=1)1388 1680 y Fg(L)1413 1666 y Fc(\()p Fe(k)p Fc(\))1458 1680 y Fp(\()p Fh(t)p Fp(\))1857 1702 y Fm(\(2)p Fl(:)p Fm(8\))0 1835 y(T)l(o)15 b(b)q(ound)h(the)f(conditional)i(Laplace)g(transforms,)c(w)o(e)i(in)o(tro)q (duce)634 1945 y Fl(H)676 1922 y Fp(\()p Fh(k)q Fp(\))672 1959 y Fh(N)726 1945 y Fm(\()p Fl(\033)r Fm(\))c(=)i Fi(\000)911 1915 y Fm(1)p 889 1935 65 2 v 889 1977 a(2)p Fl(N)967 1902 y Ff(X)989 1998 y Fh(\026)1052 1902 y Ff(X)1067 1993 y Fe(i;j)1046 2019 y(i;j)q Fd(6)p Fc(=)p Fe(k)1137 1945 y Fl(\030)1159 1924 y Fh(\026)1157 1958 y(i)1184 1945 y Fl(\030)1206 1924 y Fh(\026)1204 1958 y(j)1230 1945 y Fl(\033)1256 1952 y Fh(i)1271 1945 y Fl(\033)1297 1952 y Fh(j)1857 1945 y Fm(\(2)p Fl(:)p Fm(9\))658 2131 y Fl(R)693 2107 y Fp(\()p Fh(k)q Fp(\))693 2144 y Fh(N)744 2131 y Fm(\()p Fl(\033)r Fm(\))f(=)h Fi(\000)918 2100 y Fm(1)p 908 2120 42 2 v 908 2162 a Fl(N)963 2087 y Ff(X)985 2183 y Fh(\026)1036 2087 y Ff(X)1063 2178 y Fe(i)1042 2204 y(i)p Fd(6)p Fc(=)p Fe(k)1109 2131 y Fl(\030)1131 2109 y Fh(\026)1129 2143 y(i)1156 2131 y Fl(\030)1178 2109 y Fh(\026)1176 2144 y(k)1202 2131 y Fl(\033)1228 2138 y Fh(i)1243 2131 y Fl(\033)1269 2138 y Fh(k)1834 2131 y Fm(\(2)p Fl(:)p Fm(10\))0 2274 y(and)664 2335 y(~)653 2346 y Fl(H)695 2323 y Fp(\()p Fh(k)q Fp(\))691 2360 y Fh(N)745 2346 y Fm(\()p Fl(\033)o(;)8 b(u)p Fm(\))j(=)i Fl(H)954 2323 y Fp(\()p Fh(k)q Fp(\))950 2360 y Fh(N)1004 2346 y Fm(\()p Fl(\033)r Fm(\))c(+)i Fl(uR)1184 2323 y Fp(\()p Fh(k)q Fp(\))1184 2360 y Fh(N)1234 2346 y Fm(\()p Fl(\033)r Fm(\))536 b(\(2)p Fl(:)p Fm(11\))0 2444 y(W)l(e)15 b(also)g(de\014ne)i(the)e Fl(M)5 b Fm(\()p Fl(N)g Fm(\)-dimensional)16 b(v)o(ectors)616 2604 y Fl(m)656 2580 y Fp(\()p Fh(k)q Fp(\))656 2617 y Fh(N)706 2604 y Fm(\()p Fl(\033)o(;)8 b(u)p Fm(\))j Fi(\021)888 2573 y Fm(1)p 879 2594 V 879 2635 a Fl(N)933 2499 y Ff(0)933 2579 y(B)933 2608 y(@)973 2561 y(X)1000 2652 y Fe(i)979 2678 y(i)p Fd(6)p Fc(=)p Fe(k)1046 2604 y Fl(\030)1066 2611 y Fh(i)1081 2604 y Fl(\033)1107 2611 y Fh(i)1133 2604 y Fm(+)f Fl(u\030)1224 2611 y Fh(k)1246 2604 y Fl(\033)1272 2611 y Fh(k)1295 2499 y Ff(1)1295 2579 y(C)1295 2608 y(A)1834 2604 y Fm(\(2)p Fl(:)p Fm(12\))964 2779 y(8)p eop %%Page: 9 9 bop 0 45 a Fm(Naturally)l(,)16 b(w)o(e)e(set)387 176 y Fl(Z)421 152 y Fp(\()p Fh(k)q Fp(\))418 189 y Fh(N)472 176 y Fm(\()8 b(~)-31 b Fl(m)o(;)8 b(u)p Fm(\))k Fi(\021)676 145 y Fm(1)p 659 165 58 2 v 659 207 a(2)682 194 y Fh(N)746 133 y Ff(X)729 230 y Fh(\033)q Fg(2S)797 235 y Fe(N)835 176 y Fl(e)856 157 y Fg(\000)p Fh(\014)916 149 y Fp(~)907 157 y Fh(H)939 140 y Fc(\()p Fe(k)p Fc(\))937 169 y Fe(N)984 157 y Fp(\()p Fh(\033)o(;u)p Fp(\))1067 176 y Fl(\037)1095 183 y Fh(\016)o(;\032)1151 125 y Ff(\020)1178 176 y Fi(k)p Fl(m)1241 152 y Fp(\()p Fh(k)q Fp(\))1241 189 y Fh(N)1291 176 y Fm(\()p Fl(\033)o(;)c(u)p Fm(\))h Fi(\000)19 b Fm(~)-32 b Fl(m)p Fi(k)1516 157 y Fp(2)1516 187 y(2)1536 125 y Ff(\021)1834 176 y Fm(\(2)p Fl(:)p Fm(13\))0 333 y(and)15 b(\014nally)480 409 y Fl(f)507 385 y Fp(\()p Fh(k)q Fp(\))502 422 y Fh(N)558 409 y Fm(\()8 b(~)-31 b Fl(m)o(;)8 b(u)p Fm(\))k(=)h Fi(\000)p Fl(\014)803 390 y Fg(\000)p Fp(1)860 359 y Ff(\020)887 409 y Fm(ln)8 b Fl(Z)966 385 y Fp(\()p Fh(k)q Fp(\))963 422 y Fh(N)1017 409 y Fm(\()h(~)-32 b Fl(m;)8 b(u)p Fm(\))h Fi(\000)h Fm(ln)f Fl(Z)1274 385 y Fp(\()p Fh(k)q Fp(\))1271 422 y Fh(N)1325 409 y Fm(\()f(~)-31 b Fl(m)o(;)8 b Fm(0\))1444 359 y Ff(\021)1834 409 y Fm(\(2)p Fl(:)p Fm(14\))0 517 y(Since)16 b(for)f(the)f(remainder)i(of)e(the)h(pro)q(of,)23 b(~)-32 b Fl(m)15 b Fm(as)f(w)o(ell)i(as)f Fl(N)k Fm(will)d(b)q(e)g(\014xed)f(v)m (alues,)h(to)e(simplify)j(our)d(notations)0 590 y(w)o(e)h(will)i(write)e Fl(f)291 597 y Fh(k)314 590 y Fm(\()p Fl(u)p Fm(\))d Fi(\021)h Fl(f)463 566 y Fp(\()p Fh(k)q Fp(\))458 603 y Fh(N)514 590 y Fm(\()8 b(~)-31 b Fl(m)o(;)8 b(u)p Fm(\).)19 b(Notice)d(that)576 708 y(~)566 720 y Fl(f)593 696 y Fp(\()p Fh(k)q Fp(\))588 733 y Fh(N)644 720 y Fm(\()8 b(~)-31 b Fl(m)o Fm(\))13 b(=)g Fl(I)-8 b(E)9 b Fm([)p Fl(f)870 727 y Fh(k)893 720 y Fm(\(1\))p Fi(jF)998 727 y Fh(k)1018 720 y Fm(])h Fi(\000)h Fl(I)-8 b(E)10 b Fm([)o Fl(f)1177 727 y Fh(k)1200 720 y Fm(\(1\))p Fi(jF)1305 727 y Fh(k)q Fp(+1)1372 720 y Fm(])449 b(\(2)p Fl(:)p Fm(15\))0 850 y(T)l(o)15 b(b)q(ound)h(the)f(Laplace)i(transform,)c(w)o(e)i(use)h(that,)e (for)g(all)i Fl(x)d Fi(2)g Fl(I)-6 b(R)o Fm(,)770 981 y Fl(e)791 962 y Fh(x)827 981 y Fi(\024)13 b Fm(1)c(+)i Fl(x)f Fm(+)1040 950 y(1)p 1040 970 23 2 v 1040 1012 a(2)1068 981 y Fl(x)1094 962 y Fp(2)1114 981 y Fl(e)1135 962 y Fg(j)p Fh(x)p Fg(j)1834 981 y Fm(\(2)p Fl(:)p Fm(16\))0 1111 y(so)15 b(that)361 1187 y Fl(I)-8 b(E)417 1137 y Ff(h)438 1187 y Fl(e)459 1168 y Fh(t)480 1160 y Fp(~)473 1168 y Fh(f)495 1152 y Fc(\()p Fe(k)p Fc(\))491 1180 y Fe(N)539 1168 y Fp(\()7 b(~)-25 b Fh(m)p Fp(\))602 1149 y Ff(\014)602 1176 y(\014)617 1187 y Fi(F)650 1194 y Fh(k)q Fp(+1)718 1137 y Ff(i)752 1187 y Fi(\024)13 b Fm(1)d(+)884 1156 y(1)p 884 1177 V 884 1218 a(2)912 1187 y Fl(t)928 1168 y Fp(2)949 1187 y Fl(I)-8 b(E)1005 1123 y Ff(\024)1029 1137 y(\020)1065 1175 y Fm(~)1056 1187 y Fl(f)1083 1164 y Fp(\()p Fh(k)q Fp(\))1078 1201 y Fh(N)1134 1187 y Fm(\()8 b(~)-31 b Fl(m)o Fm(\))1209 1137 y Ff(\021)1236 1146 y Fp(2)1264 1187 y Fl(e)1285 1168 y Fg(j)p Fh(t)1317 1160 y Fp(~)1310 1168 y Fh(f)1332 1152 y Fc(\()p Fe(k)p Fc(\))1328 1180 y Fe(N)1375 1168 y Fp(\()8 b(~)-26 b Fh(m)q Fp(\))p Fg(j)1449 1149 y Ff(\014)1449 1176 y(\014)1464 1187 y Fi(F)1497 1194 y Fh(k)q Fp(+1)1565 1123 y Ff(\025)1834 1187 y Fm(\(2)p Fl(:)p Fm(17\))0 1317 y(Our)16 b(strategy)e(will)i(b)q(e)g(to)f(use)g(a)g(rather)g(p)q(o)q(or)g Fk(uniform)h Fm(b)q(ound)g(on)1218 1305 y(~)1208 1317 y Fl(f)1235 1294 y Fp(\()p Fh(k)q Fp(\))1230 1331 y Fh(N)1286 1317 y Fm(\()8 b(~)-31 b Fl(m)p Fm(\))14 b(in)j(the)e(exp)q(onen)o(t)h(but)f(to)g(pro)o(v)o (e)0 1390 y(a)i(b)q(etter)g(estimate)g(on)g(the)g(remaining)i(conditioned)g (exp)q(ectation)e(of)g(the)g(square.)26 b(The)18 b(uniform)f(estimate)0 1463 y(has)e(b)q(een)h(obtained)g(in)g([BGP2].)j(W)l(e)c(rep)q(eat)g(its)h (deriv)m(ation)g(for)f(the)g(con)o(v)o(enience)i(of)e(the)g(reader.)91 1572 y(W)l(e)g(note)g(\014rst)g(that,)f(alternativ)o(ely)i(to)f(\(2.15\),)e (w)o(e)i(ha)o(v)o(e)f(also)471 1705 y(~)461 1717 y Fl(f)488 1693 y Fp(\()p Fh(k)q Fp(\))483 1730 y Fh(N)539 1717 y Fm(\()8 b(~)-31 b Fl(m)p Fm(\))12 b(=)675 1655 y Ff(Z)720 1666 y Fp(1)700 1758 y(0)748 1717 y Fl(du)806 1680 y Ff(\000)826 1717 y Fl(I)-8 b(E)882 1680 y Ff(\002)901 1717 y Fl(f)928 1698 y Fg(0)923 1728 y Fh(k)946 1717 y Fm(\()p Fl(u)p Fm(\))1008 1678 y Ff(\014)1008 1705 y(\014)1022 1717 y Fi(F)1055 1724 y Fh(k)1078 1680 y Ff(\003)1107 1717 y Fi(\000)10 b Fl(I)-8 b(E)1208 1680 y Ff(\002)1227 1717 y Fl(f)1254 1698 y Fg(0)1249 1728 y Fh(k)1271 1717 y Fm(\()p Fl(u)p Fm(\))1333 1678 y Ff(\014)1333 1705 y(\014)1348 1717 y Fi(F)1381 1724 y Fh(k)q Fp(+1)1449 1680 y Ff(\003\001)1834 1717 y Fm(\(2)p Fl(:)p Fm(18\))0 1847 y(But)54 2003 y Fl(f)81 1984 y Fg(0)76 2014 y Fh(k)98 2003 y Fm(\()p Fl(u)p Fm(\))12 b(=)h Fi(E)244 2010 y Fh(k)q(;u)306 1911 y Ff(0)306 1993 y(@)361 1972 y Fm(1)p 352 1992 42 2 v 352 2034 a Fl(N)406 1959 y Ff(X)428 2055 y Fh(\026)479 1959 y Ff(X)482 2057 y Fh(i)p Fg(6)p Fp(=)p Fh(k)553 2003 y Fl(\030)575 1981 y Fh(\026)573 2015 y(i)599 2003 y Fl(\030)621 1981 y Fh(\026)619 2016 y(k)645 2003 y Fl(\033)671 2010 y Fh(i)686 2003 y Fl(\033)712 2010 y Fh(k)745 2003 y Fm(+)819 1972 y(2)p 796 1992 70 2 v 796 2034 a Fl(\014)r(N)876 1966 y(\037)904 1949 y Fg(0)904 1979 y Fh(\032;\016)954 1966 y Fm(\()p Fi(k)p Fl(m)1035 1973 y Fh(N)1069 1966 y Fm(\()p Fl(\033)o(;)8 b(u)p Fm(\))h Fi(\000)19 b Fm(~)-32 b Fl(m)p Fi(k)1294 1949 y Fp(2)1294 1977 y(2)1314 1966 y Fm(\))p 876 1992 456 2 v 876 2034 a Fl(\037)904 2041 y Fh(\032;\016)954 2034 y Fm(\()p Fi(k)p Fl(m)1035 2041 y Fh(N)1069 2034 y Fm(\()p Fl(\033)o(;)8 b(u)p Fm(\))h Fi(\000)19 b Fm(~)-32 b Fl(m)p Fi(k)1294 2018 y Fp(2)1294 2046 y(2)1314 2034 y Fm(\))1345 1959 y Ff(X)1367 2055 y Fh(\026)1418 2003 y Fm(\()p Fl(m)1476 1981 y Fh(\026)1476 2016 y(N)1510 2003 y Fm(\()p Fl(\033)o(;)8 b(u)p Fm(\))h Fi(\000)19 b Fm(~)-32 b Fl(m)1712 1984 y Fh(\026)1737 2003 y Fm(\))7 b Fl(\030)1784 1981 y Fh(\026)1782 2016 y(k)1808 2003 y Fl(\033)1834 2010 y Fh(k)1857 1911 y Ff(1)1857 1993 y(A)1834 2112 y Fm(\(2)p Fl(:)p Fm(19\))0 2184 y(where)15 b Fi(E)155 2191 y Fh(k)q(;u)225 2184 y Fm(denotes)h(the)f(exp)q(ectation)h(w.r.t.)i(the)e(probabilit)o(y)g (measure)594 2289 y(1)p 502 2309 207 2 v 502 2362 a Fl(Z)536 2339 y Fp(\()p Fh(k)q Fp(\))533 2376 y Fh(N)587 2362 y Fm(\()9 b(~)-32 b Fl(m;)8 b(u)p Fm(\))714 2320 y Fl(\037)742 2327 y Fh(\032;\016)792 2320 y Fm(\()p Fi(k)p Fl(m)873 2327 y Fh(N)907 2320 y Fm(\()p Fl(\033)o(;)g(u)p Fm(\))h Fi(\000)19 b Fm(~)-32 b Fl(m)p Fi(k)1132 2301 y Fp(2)1132 2331 y(2)1152 2320 y Fm(\))p Fl(e)1191 2301 y Fg(\000)p Fh(\014)1251 2293 y Fp(~)1242 2301 y Fh(H)1274 2284 y Fc(\()p Fe(k)p Fc(\))1272 2313 y Fe(N)1319 2301 y Fp(\()p Fh(\033)o(;u)p Fp(\))1402 2320 y Fl(d\033)382 b Fm(\(2)p Fl(:)p Fm(20\))0 2472 y(By)15 b(some)g(trivial)h(manipulations)h (\(see)e([BGP2])f(w)o(e)h(get)240 2625 y Fi(j)o Fl(f)279 2606 y Fg(0)274 2636 y Fh(k)297 2625 y Fm(\()p Fl(u)p Fm(\))p Fi(j)d(\024)432 2546 y Ff(\014)432 2573 y(\014)432 2600 y(\014)432 2627 y(\014)432 2655 y(\014)447 2582 y(X)468 2677 y Fh(\026)529 2625 y Fm(~)-32 b Fl(m)560 2606 y Fh(\026)584 2625 y Fl(\030)606 2603 y Fh(\026)604 2639 y(k)631 2546 y Ff(\014)631 2573 y(\014)631 2600 y(\014)631 2627 y(\014)631 2655 y(\014)656 2625 y Fm(+)10 b Fi(E)725 2632 y Fh(k)q(;u)788 2547 y Ff(")814 2546 y(\014)814 2573 y(\014)814 2600 y(\014)814 2627 y(\014)814 2655 y(\014)829 2582 y(X)851 2677 y Fh(\026)902 2625 y Fm(\()p Fl(m)960 2603 y Fh(\026)960 2639 y(N)994 2625 y Fm(\()p Fl(\033)o(;)e(u)p Fm(\))h Fi(\000)19 b Fm(~)-31 b Fl(m)1197 2606 y Fh(\026)1221 2625 y Fm(\))7 b Fl(\030)1268 2603 y Fh(\026)1266 2639 y(k)1293 2625 y Fl(\033)1319 2632 y Fh(k)1341 2546 y Ff(\014)1341 2573 y(\014)1341 2600 y(\014)1341 2627 y(\014)1341 2655 y(\014)1356 2547 y(#)1390 2575 y(\020)1417 2625 y Fm(1)j(+)1528 2607 y Fp(2)p 1501 2614 73 2 v 1501 2641 a Fh(\014)r(N)s(\016)1579 2575 y Ff(\021)1616 2625 y Fm(+)1667 2607 y Fh(M)p 1667 2614 39 2 v 1670 2641 a(N)1834 2625 y Fm(\(2)p Fl(:)p Fm(21\))964 2779 y(9)p eop %%Page: 10 10 bop 0 45 a Fm(No)o(w)15 b(clearly)546 59 y Ff(\014)546 86 y(\014)546 113 y(\014)546 141 y(\014)546 168 y(\014)562 95 y(X)583 191 y Fh(\026)643 138 y Fm(~)-31 b Fl(m)675 120 y Fh(\026)699 138 y Fl(\030)721 117 y Fh(\026)719 152 y(k)745 59 y Ff(\014)745 86 y(\014)745 113 y(\014)745 141 y(\014)745 168 y(\014)773 138 y Fi(\024)821 95 y Ff(X)843 191 y Fh(\026)894 138 y Fi(j)9 b Fm(~)-32 b Fl(m)947 120 y Fh(\026)971 138 y Fi(j)13 b Fm(=)g Fi(k)8 b Fm(~)-31 b Fl(m)o Fi(k)1130 145 y Fp(1)1163 138 y Fi(\024)1211 98 y(p)p 1249 98 50 2 v 40 x Fl(M)5 b Fi(k)j Fm(~)-31 b Fl(m)o Fi(k)1383 145 y Fp(2)1834 138 y Fm(\(2)p Fl(:)p Fm(22\))0 269 y(Similarly)l(,)17 b(for)e Fl(\033)h Fm(in)h(the)e(supp)q(ort)g(of)g Fi(E)690 276 y Fh(k)q(;u)745 269 y Fm(,)313 342 y Ff(\014)313 369 y(\014)313 396 y(\014)313 423 y(\014)313 451 y(\014)328 378 y(X)350 473 y Fh(\026)402 421 y Fm(\()o Fl(m)459 399 y Fh(\026)459 434 y(N)494 421 y Fm(\()p Fl(\033)o(;)8 b(u)p Fm(\))h Fi(\000)19 b Fm(~)-32 b Fl(m)696 402 y Fh(\026)720 421 y Fm(\))8 b Fl(\030)768 399 y Fh(\026)766 435 y(k)792 421 y Fl(\033)818 428 y Fh(k)840 342 y Ff(\014)840 369 y(\014)840 396 y(\014)840 423 y(\014)840 451 y(\014)868 421 y Fi(\024)916 380 y(p)p 954 380 V 41 x Fl(M)d Fi(k)p Fl(m)1066 428 y Fh(N)1100 421 y Fm(\()p Fl(\033)o(;)j(u)p Fm(\))h Fi(\000)19 b Fm(~)-32 b Fl(m)p Fi(k)1325 428 y Fp(2)1358 421 y Fi(\024)1406 380 y Ff(p)p 1451 380 186 2 v 41 x Fl(M)5 b Fm(\()p Fl(\032)10 b Fm(+)g Fl(\016)r Fm(\))197 b(\(2)p Fl(:)p Fm(23\))0 577 y(F)l(rom)14 b(this)i(and)f(using)h(assumption)g (\(2\))e(on)h(the)g(function)h Fl(\037)p Fm(,)g(w)o(e)f(see)g(that)518 705 y Fi(j)p Fl(f)558 686 y Fg(0)553 716 y Fh(k)575 705 y Fm(\()p Fl(u)p Fm(\))p Fi(j)d(\024)h(k)8 b Fm(~)-31 b Fl(m)o Fi(k)795 712 y Fp(1)826 705 y Fm(+)871 664 y Fi(p)p 909 664 50 2 v 41 x Fl(M)958 664 y Ff(p)p 1003 664 101 2 v 41 x Fl(\032)10 b Fm(+)g Fl(\016)1112 654 y Ff(\020)1139 705 y Fm(1)g(+)1250 687 y Fp(2)p 1223 694 73 2 v 1223 720 a Fh(\014)r(N)s(\016)1300 654 y Ff(\021)1338 705 y Fm(+)1389 687 y Fh(M)p 1389 694 39 2 v 1392 720 a(N)1834 705 y Fm(\(2)p Fl(:)p Fm(24\))0 832 y(W)l(e)15 b(will)i(no)o(w)e(c)o(ho)q(ose)g Fl(\016)g Fm(=)509 814 y Fp(2)p 491 821 55 2 v 491 848 a Fh(\014)r(N)566 832 y Fm(so)g(that)f(w)o(e)h(get)g (e\013ectiv)o(ely)h(the)f(b)q(ound)728 959 y Fi(j)p Fl(f)768 941 y Fg(0)763 971 y Fh(k)785 959 y Fm(\()p Fl(u)p Fm(\))p Fi(j)d(\024)h(k)8 b Fm(~)-31 b Fl(m)p Fi(k)1006 966 y Fp(1)1036 959 y Fm(+)10 b(2)1104 919 y Ff(p)p 1149 919 73 2 v 40 x Fl(M)5 b(\032)612 b Fm(\(2)p Fl(:)p Fm(25\))0 1087 y(Using)16 b(this)f(b)q(ound)i (and)e(\(2.18\))e(to)i(estimate)824 1075 y(~)814 1087 y Fl(f)841 1063 y Fp(\()p Fh(k)q Fp(\))836 1100 y Fh(N)892 1087 y Fm(\()8 b(~)-31 b Fl(m)o Fm(\))15 b(and)h(inserting)g(the)f(result)h(in)g(\(2.17\),)d (w)o(e)i(get)f(that)287 1225 y Fl(I)-8 b(E)342 1175 y Ff(h)364 1225 y Fl(e)385 1206 y Fh(t)406 1198 y Fp(~)399 1206 y Fh(f)421 1190 y Fc(\()p Fe(k)p Fc(\))417 1218 y Fe(N)465 1206 y Fp(\()7 b(~)-25 b Fh(m)p Fp(\))528 1186 y Ff(\014)528 1214 y(\014)543 1225 y Fi(F)576 1232 y Fh(k)q Fp(+1)644 1175 y Ff(i)678 1225 y Fi(\024)13 b Fm(1)d(+)810 1194 y(1)p 810 1215 23 2 v 810 1256 a(2)838 1225 y Fl(t)854 1206 y Fp(2)875 1225 y Fl(e)896 1206 y Fp(2)p Fg(j)p Fh(t)p Fg(j)p Fp(\()p Fg(k)c Fp(~)-24 b Fh(m)o Fg(k)1033 1211 y Fc(1)1051 1206 y Fp(+2)1097 1176 y Fi(p)p 1134 1176 57 2 v 1134 1206 a Fh(M)s(\032)p Fp(\))1208 1225 y Fl(I)-8 b(E)1264 1161 y Ff(\024)1288 1175 y(\020)1325 1213 y Fm(~)1315 1225 y Fl(f)1342 1201 y Fp(\()p Fh(k)q Fp(\))1337 1238 y Fh(N)1393 1225 y Fm(\()8 b(~)-31 b Fl(m)o Fm(\))1468 1175 y Ff(\021)1495 1184 y Fp(2)1523 1186 y Ff(\014)1523 1214 y(\014)1538 1225 y Fi(F)1571 1232 y Fh(k)q Fp(+1)1639 1161 y Ff(\025)1834 1225 y Fm(\(2)p Fl(:)p Fm(26\))0 1362 y(Of)16 b(course)f(w)o(e)g(could)i(also)e(use)h(\(2.25\))e(to)h(b)q(ound)h(the)g(exp) q(ectation)g(of)f(the)h(sqare)f(in)h(\(2.26\),)d(but)j(due)g(to)f(the)0 1434 y(presence)k(of)f(the)321 1396 y Fi(p)p 359 1396 50 2 v 38 x Fl(M)23 b Fm(in)c(that)f(b)q(ound,)h(this)g(could)g(at)f(b)q(est)g(b)q (e)h(useful)h(for)d(v)o(ery)h(small)h Fl(\032)p Fm(.)28 b(W)l(e)19 b(will)g(need,)0 1507 y(ho)o(w)o(ev)o(er,)14 b(a)h(b)q(ound)h(for)f(\014nite) h Fl(\032)e Fm(and)i(th)o(us)f(will)i(ha)o(v)o(e)d(to)h(b)q(e)h(more)f (careful.)91 1616 y(W)l(e)g(will)i(no)o(w)e(use)g(\(2.15\))f(to)g(write)h (\(recall)h(that)f Fi(F)1002 1623 y Fh(k)1039 1616 y Fm(are)g(de\014ned)i(in) f(suc)o(h)f(a)g(w)o(a)o(y)f(that)h Fi(F)1689 1623 y Fh(k)1723 1616 y Fi(\033)e(F)1804 1623 y Fh(k)q Fp(+1)1872 1616 y Fm(\))261 1752 y Fl(I)-8 b(E)317 1688 y Ff(\024)341 1702 y(\020)378 1740 y Fm(~)369 1752 y Fl(f)396 1729 y Fp(\()p Fh(k)q Fp(\))391 1766 y Fh(N)446 1752 y Fm(\()9 b(~)-32 b Fl(m)p Fm(\))522 1702 y Ff(\021)549 1711 y Fp(2)577 1714 y Ff(\014)577 1741 y(\014)592 1752 y Fi(F)625 1759 y Fh(k)q Fp(+1)693 1688 y Ff(\025)730 1752 y Fm(=)13 b Fl(I)-8 b(E)834 1702 y Ff(h)855 1716 y(\000)876 1752 y Fl(I)g(E)932 1716 y Ff(\002)951 1752 y Fl(f)973 1759 y Fh(k)995 1752 y Fm(\(1\))10 b Fi(\000)g Fl(I)-8 b(E)r Fm([)p Fl(f)1192 1759 y Fh(k)1215 1752 y Fm(\(1\))p Fi(jF)1320 1759 y Fh(k)q Fp(+1)1386 1752 y Fm(])1407 1714 y Ff(\014)1407 1741 y(\014)1422 1752 y Fi(F)1455 1759 y Fh(k)1477 1716 y Ff(\003\001)1517 1725 y Fp(2)1545 1700 y Ff(\014)1545 1727 y(\014)1545 1755 y(\014)1560 1752 y Fi(F)1593 1759 y Fh(k)q Fp(+1)1661 1702 y Ff(i)730 1866 y Fi(\024)13 b Fl(I)-8 b(E)834 1816 y Ff(h)855 1866 y Fl(I)g(E)911 1816 y Ff(h)932 1866 y Fm(\()p Fl(f)972 1873 y Fh(k)995 1866 y Fm(\(1\))9 b Fi(\000)h Fl(I)-8 b(E)s Fm([)p Fl(f)1192 1873 y Fh(k)1214 1866 y Fm(\(1\))p Fi(jF)1319 1873 y Fh(k)q Fp(+1)1386 1866 y Fm(]\))1416 1843 y Fp(2)1452 1827 y Ff(\014)1452 1855 y(\014)1467 1866 y Fi(F)1500 1873 y Fh(k)1522 1816 y Ff(i)1551 1814 y(\014)1551 1841 y(\014)1551 1868 y(\014)1566 1866 y Fi(F)1599 1873 y Fh(k)q Fp(+1)1667 1816 y Ff(i)730 1966 y Fm(=)13 b Fl(I)-8 b(E)834 1916 y Ff(h)855 1966 y Fm(\()p Fl(f)895 1973 y Fh(k)917 1966 y Fm(\(1\))10 b Fi(\000)g Fl(I)-8 b(E)r Fm([)p Fl(f)1114 1973 y Fh(k)1137 1966 y Fm(\(1\))p Fi(jF)1242 1973 y Fh(k)q Fp(+1)1308 1966 y Fm(]\))1339 1943 y Fp(2)1374 1927 y Ff(\014)1374 1955 y(\014)1389 1966 y Fi(F)1422 1973 y Fh(k)q Fp(+1)1490 1916 y Ff(i)730 2066 y Fm(=)13 b Fl(I)-8 b(E)834 2016 y Ff(h)855 2066 y Fm(\()p Fl(f)895 2073 y Fh(k)917 2066 y Fm(\(1\)\))993 2043 y Fp(2)1021 2027 y Ff(\014)1021 2055 y(\014)1036 2066 y Fi(F)1069 2073 y Fh(k)q Fp(+1)1137 2016 y Ff(i)1169 2066 y Fi(\000)10 b Fm(\()p Fl(I)-8 b(E)r Fm([)p Fl(f)1315 2073 y Fh(k)1338 2066 y Fm(\(1\))p Fi(jF)1443 2073 y Fh(k)q Fp(+1)1509 2066 y Fm(]\))1540 2043 y Fp(2)730 2166 y Fi(\024)13 b Fl(I)-8 b(E)834 2116 y Ff(h)855 2166 y Fm(\()p Fl(f)895 2173 y Fh(k)917 2166 y Fm(\(1\)\))993 2143 y Fp(2)1029 2127 y Ff(\014)1029 2155 y(\014)1044 2166 y Fi(F)1077 2173 y Fh(k)q Fp(+1)1145 2116 y Ff(i)1834 1952 y Fm(\(2)p Fl(:)p Fm(27\))0 2289 y(A)o(t)16 b(this)h(p)q(oin)o(t)h(it)f(is)g (imp)q(ortan)o(t)f(to)g(notice)i(that)e Fl(f)903 2296 y Fh(k)925 2289 y Fm(\()p Fl(u)p Fm(\))g(is)i(a)e Fk(c)n(onc)n(ave)g Fm(function)h(of)g Fl(u)p Fm(.)24 b(Note)17 b(that)f(condition)0 2362 y(\(4\))f(w)o(as)g(imp)q (osed)i(on)f Fl(\037)433 2369 y Fh(\032;\016)499 2362 y Fm(to)f(ensure)h (this)h(fact.)k(Since)d(moreo)o(v)o(er)c Fl(f)1234 2369 y Fh(k)1257 2362 y Fm(\(0\))f(=)h(0,)h(this)i(implies)h(that)d Fi(j)p Fl(f)1808 2369 y Fh(k)1830 2362 y Fm(\(1\))p Fi(j)e(\024)0 2434 y Fm(max)7 b(\()p Fi(j)p Fl(f)150 2418 y Fg(0)145 2447 y Fh(k)167 2434 y Fm(\(0\))p Fi(j)p Fl(;)h Fi(j)p Fl(f)300 2418 y Fg(0)295 2447 y Fh(k)315 2434 y Fm(\(1\))p Fi(j)p Fm(\))o(.)18 b(Moreo)o(v)o(er,)9 b(from)h(\(2.19\))e(and)j(the)f(fact)g(that)g(neither)1369 2423 y(~)1357 2434 y Fl(H)1395 2441 y Fh(N)1429 2434 y Fm(\()p Fl(\033)o(;)e Fm(0\))h(nor)h Fl(m)1659 2441 y Fh(N)1694 2434 y Fm(\()p Fl(\033)o(;)e Fm(0\))g(dep)q(end)0 2507 y(on)15 b(the)g(spin)i Fl(\033)264 2514 y Fh(k)286 2507 y Fm(,)e(w)o(e)g(see)g(that)g Fl(f)580 2491 y Fg(0)575 2520 y Fh(k)597 2507 y Fm(\(0\))d(=)h(0,)h(so)h (that)g(w)o(e)g(get)f(simply)526 2645 y Fl(I)-8 b(E)582 2581 y Ff(\024)606 2595 y(\020)643 2633 y Fm(~)633 2645 y Fl(f)660 2621 y Fp(\()p Fh(k)q Fp(\))655 2658 y Fh(N)711 2645 y Fm(\()9 b(~)-32 b Fl(m)p Fm(\))787 2595 y Ff(\021)814 2604 y Fp(2)842 2606 y Ff(\014)842 2634 y(\014)857 2645 y Fi(F)890 2652 y Fh(k)q Fp(+1)958 2581 y Ff(\025)995 2645 y Fi(\024)13 b Fl(I)-8 b(E)1098 2595 y Ff(h)1120 2645 y Fm(\()p Fl(f)1165 2626 y Fg(0)1160 2656 y Fh(k)1182 2645 y Fm(\(1\)\))1258 2620 y Fp(2)1286 2606 y Ff(\014)1286 2634 y(\014)1301 2645 y Fi(F)1334 2652 y Fh(k)q Fp(+1)1402 2595 y Ff(i)1834 2645 y Fm(\(2)p Fl(:)p Fm(28\))952 2779 y(10)p eop %%Page: 11 11 bop 0 45 a Fm(Let)15 b(us)h(use)f(the)h(fact)e(that)h(\()p Fl(a)10 b Fm(+)g Fl(b)p Fm(\))620 29 y Fp(2)652 45 y Fi(\024)j Fm(2)p Fl(a)747 29 y Fp(2)777 45 y Fm(+)e(2)p Fl(b)866 29 y Fp(2)901 45 y Fm(and)k(\(2.19\))e(to)i(see)g(that)182 219 y(\()o Fl(f)226 200 y Fg(0)221 230 y Fh(k)244 219 y Fm(\(1\)\))320 194 y Fp(2)353 219 y Fi(\024)e Fm(2)432 128 y Ff(0)432 210 y(@)471 219 y Fi(E)495 226 y Fh(k)q(;)p Fp(1)554 128 y Ff(0)554 210 y(@)608 188 y Fm(1)p 599 209 42 2 v 599 250 a Fl(N)653 176 y Ff(X)675 271 y Fh(\026)727 176 y Ff(X)729 274 y Fh(i)p Fg(6)p Fp(=)p Fh(k)800 219 y Fl(\030)822 197 y Fh(\026)820 232 y(i)846 219 y Fl(\030)868 197 y Fh(\026)866 233 y(k)893 219 y Fl(\033)919 226 y Fh(i)934 219 y Fl(\033)960 226 y Fh(k)982 128 y Ff(1)982 210 y(A)1022 128 y(1)1022 210 y(A)1062 137 y Fp(2)350 397 y Fm(+)e(2)427 319 y Ff( )462 397 y Fi(E)486 404 y Fh(k)q(;)p Fp(1)545 319 y Ff( )610 366 y Fm(2)p 586 386 70 2 v 586 428 a Fl(\014)r(N)667 360 y(\037)695 343 y Fg(0)695 373 y Fh(\032;\016)745 360 y Fm(\()p Fi(k)p Fl(m)826 367 y Fh(N)860 360 y Fm(\()p Fl(\033)r Fm(\))e Fi(\000)19 b Fm(~)-32 b Fl(m)p Fi(k)1041 343 y Fp(2)1041 371 y(2)1061 360 y Fm(\))p 667 386 413 2 v 667 428 a Fl(\037)695 435 y Fh(\032;\016)745 428 y Fm(\()p Fi(k)p Fl(m)826 435 y Fh(N)860 428 y Fm(\()p Fl(\033)r Fm(\))9 b Fi(\000)19 b Fm(~)-32 b Fl(m)p Fi(k)1041 412 y Fp(2)1041 440 y(2)1061 428 y Fm(\))1092 353 y Ff(X)1114 449 y Fh(\026)1165 397 y Fm(\()p Fl(m)1223 375 y Fh(\026)1223 410 y(N)1257 397 y Fm(\()p Fl(\033)r Fm(\))9 b Fi(\000)19 b Fm(~)-31 b Fl(m)1416 378 y Fh(\026)1440 397 y Fm(\))7 b Fl(\030)1487 375 y Fh(\026)1485 410 y(k)1511 397 y Fl(\033)1537 404 y Fh(k)1560 319 y Ff(!!)1632 328 y Fp(2)1834 299 y Fm(\(2)p Fl(:)p Fm(29\))0 551 y(Bounding)16 b(the)g(square)f(of)g(the)g(exp)q(ectation)h(b)o(y)f(the)g (exp)q(ectation)h(of)f(the)g(square,)g(w)o(e)g(get)g(from)f(this)258 726 y Fl(I)-8 b(E)314 676 y Ff(h)336 726 y Fm(\()o Fl(f)380 708 y Fg(0)375 738 y Fh(k)398 726 y Fm(\(1\)\))474 701 y Fp(2)502 688 y Ff(\014)502 715 y(\014)517 726 y Fi(F)550 733 y Fh(k)q Fp(+1)618 676 y Ff(i)652 726 y Fi(\024)13 b Fm(2)p Fl(I)-8 b(E)779 621 y Ff(2)779 701 y(6)779 730 y(4)809 726 y Fi(E)833 733 y Fh(k)q(;)p Fp(1)892 635 y Ff(0)892 717 y(@)946 696 y Fm(1)p 937 716 42 2 v 937 757 a Fl(N)991 683 y Ff(X)1013 779 y Fh(\026)1065 683 y Ff(X)1067 781 y Fh(i)p Fg(6)p Fp(=)p Fh(k)1138 726 y Fl(\030)1160 704 y Fh(\026)1158 739 y(i)1184 726 y Fl(\030)1206 704 y Fh(\026)1204 740 y(k)1231 726 y Fl(\033)1257 733 y Fh(i)1272 726 y Fl(\033)1298 733 y Fh(k)1320 635 y Ff(1)1320 717 y(A)1360 644 y Fp(2)1388 674 y Ff(\014)1388 701 y(\014)1388 729 y(\014)1403 726 y Fi(F)1436 733 y Fh(k)q Fp(+1)1504 621 y Ff(3)1504 701 y(7)1504 730 y(5)649 922 y Fm(+)11 b(2)p Fl(I)-8 b(E)774 830 y Ff(2)774 912 y(4)804 922 y Fi(E)828 929 y Fh(k)q(;)p Fp(1)887 844 y Ff( )955 904 y Fp(2)p 928 911 73 2 v 928 937 a Fh(\014)r(N)s(\016)1013 879 y Ff(X)1035 974 y Fh(\026)1087 922 y Fm(\()o Fl(m)1144 900 y Fh(\026)1144 935 y(N)1179 922 y Fm(\()p Fl(\033)r Fm(\))9 b Fi(\000)19 b Fm(~)-32 b Fl(m)1337 903 y Fh(\026)1361 922 y Fm(\))8 b Fl(\030)1409 900 y Fh(\026)1407 935 y(k)1433 922 y Fl(\033)1459 929 y Fh(k)1481 844 y Ff(!)1517 853 y Fp(2)1545 870 y Ff(\014)1545 897 y(\014)1545 924 y(\014)1560 922 y Fi(F)1593 929 y Fh(k)q Fp(+1)1661 830 y Ff(3)1661 912 y(5)1834 817 y Fm(\(2)p Fl(:)p Fm(30\))0 1084 y(No)o(w)16 b(w)o(e)h(use)h(the)f(follo)o (wing)h(crucial)g(tric)o(k:)24 b Fi(E)829 1091 y Fh(k)q(;)p Fp(1)898 1084 y Fm(is)17 b(in)h(fact)f(indep)q(enden)o(t)i(of)e Fl(k)q Fm(,)g(and)h(therefore)f(the)g(exp)q(ec-)0 1156 y(tations)e (conditioned)j(on)e Fi(F)495 1163 y Fh(k)q Fp(+1)578 1156 y Fm(are)g(the)g(same)f(if)h(the)g(index)h Fl(k)g Fm(inside)g(it)f(is)h (replaced)g(b)o(y)e(an)o(y)h(of)f(the)h(indices)0 1229 y Fl(j)f Fi(2)e(f)p Fm(1)p Fl(;)8 b(:)g(:)g(:)t(;)g(k)q Fi(g)p Fm(.)19 b(This)d(allo)o(ws)f(us)g(to)g(replace)h(\(2.30\))e(b)o(y)29 1393 y Fl(I)-8 b(E)84 1343 y Ff(h)106 1393 y Fm(\()p Fl(f)151 1375 y Fg(0)146 1405 y Fh(k)168 1393 y Fm(\(1\)\))244 1368 y Fp(2)272 1355 y Ff(\014)272 1382 y(\014)287 1393 y Fi(F)320 1400 y Fh(k)q Fp(+1)388 1343 y Ff(i)422 1393 y Fi(\024)13 b Fm(2)p Fl(I)-8 b(E)549 1302 y Ff(2)549 1384 y(4)579 1393 y Fi(E)603 1400 y Fh(k)q(;)p Fp(1)662 1302 y Ff(0)662 1384 y(@)708 1363 y Fm(1)p 707 1383 26 2 v 707 1425 a Fl(k)768 1337 y Fh(k)745 1350 y Ff(X)747 1447 y Fh(j)r Fp(=1)844 1363 y Fm(1)p 824 1383 62 2 v 824 1425 a Fl(N)866 1411 y Fp(2)899 1350 y Ff(X)921 1446 y Fh(\026)972 1350 y Ff(X)995 1446 y Fh(\027)1045 1350 y Ff(X)1049 1448 y Fh(i)p Fg(6)p Fp(=)p Fh(j)1119 1350 y Ff(X)1123 1448 y Fh(l)p Fg(6)p Fp(=)p Fh(j)1192 1393 y Fl(\030)1214 1372 y Fh(\026)1212 1406 y(i)1238 1393 y Fl(\030)1260 1375 y Fh(\027)1258 1405 y(l)1283 1393 y Fl(\030)1305 1372 y Fh(\026)1303 1406 y(j)1329 1393 y Fl(\030)1351 1375 y Fh(\027)1349 1405 y(j)1374 1393 y Fl(\033)1400 1400 y Fh(i)1415 1393 y Fl(\033)1441 1400 y Fh(l)1455 1302 y Ff(1)1455 1384 y(A)1502 1341 y(\014)1502 1368 y(\014)1502 1396 y(\014)1517 1393 y Fi(F)1550 1400 y Fh(k)q Fp(+1)1618 1302 y Ff(3)1618 1384 y(5)420 1575 y Fm(+)10 b(2)p Fl(I)-8 b(E)544 1484 y Ff(2)544 1566 y(4)574 1525 y(\020)634 1557 y Fp(2)p 607 1564 73 2 v 607 1591 a Fh(\014)r(N)s(\016)684 1525 y Ff(\021)711 1534 y Fp(2)746 1544 y Fm(1)p 745 1565 26 2 v 745 1606 a Fl(k)806 1518 y Fh(k)783 1532 y Ff(X)784 1629 y Fh(j)r Fp(=1)856 1532 y Ff(X)878 1627 y Fh(\026)929 1532 y Ff(X)952 1627 y Fh(\027)1003 1575 y Fl(\030)1025 1553 y Fh(\026)1023 1588 y(j)1049 1575 y Fl(\030)1071 1556 y Fh(\027)1069 1586 y(j)1094 1575 y Fi(E)1118 1582 y Fh(k)q(;)p Fp(1)1176 1575 y Fm(\()p Fl(m)1234 1553 y Fh(\026)1234 1589 y(N)1269 1575 y Fm(\()p Fl(\033)r Fm(\))9 b Fi(\000)19 b Fm(~)-32 b Fl(m)1427 1556 y Fh(\026)1451 1575 y Fm(\))8 b(\()o Fl(m)1534 1556 y Fh(\027)1534 1586 y(N)1569 1575 y Fm(\()p Fl(\033)r Fm(\))h Fi(\000)19 b Fm(~)-32 b Fl(m)1727 1556 y Fh(\027)1750 1575 y Fm(\))1775 1523 y Ff(\014)1775 1550 y(\014)1775 1577 y(\014)1790 1575 y Fi(F)1823 1582 y Fh(k)q Fp(+1)1891 1484 y Ff(3)1891 1566 y(5)1834 1684 y Fm(\(2)p Fl(:)p Fm(31\))0 1757 y(Let)15 b(us)h(de\014ne)g(the)f(random)g Fl(M)g Fi(\002)c Fl(M)5 b Fm(-matrices)15 b Fl(B)903 1741 y Fp(\()p Fh(k)q Fp(\))970 1757 y Fm(with)g(elemen)o(ts)797 1904 y Fl(B)833 1885 y Fp(\()p Fh(k)q Fp(\))831 1915 y Fh(\026\027)897 1904 y Fi(\021)952 1873 y Fm(1)p 951 1894 V 951 1935 a Fl(k)1012 1847 y Fh(k)989 1861 y Ff(X)990 1958 y Fh(j)r Fp(=1)1062 1904 y Fl(\030)1084 1882 y Fh(\026)1082 1917 y(j)1108 1904 y Fl(\030)1130 1885 y Fh(\027)1128 1915 y(j)1834 1904 y Fm(\(2)p Fl(:)p Fm(32\))0 2062 y(Note)k(that)f(these)h(matrices)g(are)f(measurable)i(w.r.t.)29 b(the)19 b(sigma)g(algebra)g Fi(F)5 b(nF)1452 2069 y Fh(k)q Fp(+1)1519 2062 y Fm(.)31 b(W)l(e)19 b(will)h(write)f Fl(b)1874 2069 y Fh(k)1915 2062 y Fi(\021)0 2096 y Ff(\015)0 2124 y(\015)25 2135 y Fl(B)61 2119 y Fp(\()p Fh(k)q Fp(\))113 2096 y Ff(\015)113 2124 y(\015)153 2135 y Fm(for)c(the)g(norms)g(of)g(these)g(matrices.)91 2244 y(W)l(e)g(can)h(write)f(\(2.31\))e(in)j(the)f(form)80 2368 y Fl(I)-8 b(E)136 2317 y Ff(h)157 2368 y Fm(\()p Fl(f)202 2349 y Fg(0)197 2379 y Fh(k)219 2368 y Fm(\(1\)\))295 2342 y Fp(2)323 2329 y Ff(\014)323 2356 y(\014)338 2368 y Fi(F)371 2375 y Fh(k)q Fp(+1)439 2317 y Ff(i)92 2508 y Fi(\024)13 b Fm(2)p Fl(I)-8 b(E)219 2417 y Ff(2)219 2499 y(4)249 2465 y(X)255 2561 y Fh(\026;\027)329 2478 y Fm(1)p 328 2498 V 328 2540 a Fl(k)389 2452 y Fh(k)366 2465 y Ff(X)367 2562 y Fh(j)r Fp(=1)439 2508 y Fl(\030)461 2487 y Fh(\026)459 2521 y(j)486 2508 y Fl(\030)508 2490 y Fh(\027)506 2520 y(j)530 2508 y Fi(E)554 2515 y Fp(1)p Fh(;k)613 2431 y Ff( )664 2478 y Fm(1)p 655 2498 42 2 v 655 2540 a Fl(N)726 2452 y Fh(N)709 2465 y Ff(X)712 2562 y Fh(i)p Fp(=1)782 2508 y Fl(\030)804 2487 y Fh(\026)802 2521 y(i)829 2508 y Fl(\033)855 2515 y Fh(i)880 2508 y Fi(\000)940 2478 y Fm(1)p 931 2498 V 931 2540 a Fl(N)978 2508 y(\030)1000 2487 y Fh(\026)998 2521 y(j)1024 2508 y Fl(\033)1050 2515 y Fh(j)1069 2431 y Ff(!)8 b( )1164 2478 y Fm(1)p 1154 2498 V 1154 2540 a Fl(N)1225 2452 y Fh(N)1209 2465 y Ff(X)1213 2563 y Fh(l)p Fp(=1)1282 2508 y Fl(\030)1304 2490 y Fh(\027)1302 2520 y(l)1326 2508 y Fl(\033)1352 2515 y Fh(l)1377 2508 y Fi(\000)1437 2478 y Fm(1)p 1427 2498 V 1427 2540 a Fl(N)1474 2508 y(\030)1496 2490 y Fh(\027)1494 2520 y(j)1519 2508 y Fl(\033)1545 2515 y Fh(j)1564 2431 y Ff(!)1608 2456 y(\014)1608 2483 y(\014)1608 2511 y(\014)1623 2508 y Fi(F)1656 2515 y Fh(k)q Fp(+1)1724 2417 y Ff(3)1724 2499 y(5)90 2659 y Fm(+)i(2)166 2608 y Ff(\020)225 2641 y Fp(2)p 198 2648 73 2 v 198 2674 a Fh(\014)r(N)s(\016)276 2608 y Ff(\021)303 2618 y Fp(2)331 2659 y Fl(I)-8 b(E)387 2608 y Ff(h)408 2659 y Fi(E)432 2666 y Fh(k)q(;)p Fp(1)491 2608 y Ff(\020)518 2659 y Fm(\()p Fl(m)576 2666 y Fh(N)610 2659 y Fm(\()p Fl(\033)r Fm(\))9 b Fi(\000)19 b Fm(~)-31 b Fl(m)p Fm(\))p Fl(;)8 b(B)844 2640 y Fp(\()p Fh(k)q Fp(\))894 2659 y Fm(\()p Fl(m)952 2666 y Fh(N)986 2659 y Fm(\()p Fl(\033)r Fm(\))h Fi(\000)19 b Fm(~)-31 b Fl(m)p Fm(\))1163 2608 y Ff(\021)1197 2606 y(\014)1197 2634 y(\014)1197 2661 y(\014)1212 2659 y Fi(F)1245 2666 y Fh(k)q Fp(+1)1313 2608 y Ff(i)1834 2513 y Fm(\(2)p Fl(:)p Fm(33\))952 2779 y(11)p eop %%Page: 12 12 bop 0 45 a Fm(In)17 b(the)f(last)h(line)h(the)e(\()p Fi(\001)p Fl(;)8 b Fi(\001)p Fm(\))14 b(denotes)j(the)f(scalar)h(pro)q(duct)f(in)i Fl(I)-6 b(R)1138 26 y Fh(M)1178 45 y Fm(.)24 b(F)l(or)15 b(the)i(second)g (term)f(in)h(\(2.33\))d(w)o(e)i(get)0 118 y(immediately)h(the)e(b)q(ound)375 247 y(2)406 197 y Ff(\020)465 230 y Fp(2)p 438 237 73 2 v 438 263 a Fh(\014)r(N)s(\016)516 197 y Ff(\021)543 206 y Fp(2)571 247 y Fl(I)-8 b(E)627 197 y Ff(h)648 247 y Fi(E)672 254 y Fh(k)q(;)p Fp(1)731 197 y Ff(\020)758 247 y Fm(\()p Fl(m)816 254 y Fh(N)850 247 y Fm(\()p Fl(\033)r Fm(\))9 b Fi(\000)19 b Fm(~)-31 b Fl(m)p Fm(\))p Fl(;)8 b(B)1084 229 y Fp(\()p Fh(k)q Fp(\))1134 247 y Fm(\()p Fl(m)1192 254 y Fh(N)1226 247 y Fm(\()p Fl(\033)r Fm(\))h Fi(\000)19 b Fm(~)-31 b Fl(m)p Fm(\))1403 197 y Ff(\021)1437 195 y(\014)1437 222 y(\014)1437 250 y(\014)1452 247 y Fi(F)1485 254 y Fh(k)q Fp(+1)1553 197 y Ff(i)388 357 y Fi(\024)13 b Fm(2)p Fl(\032)491 306 y Ff(\020)549 339 y Fp(2)p 522 346 V 522 372 a Fh(\014)r(N)s(\016)600 306 y Ff(\021)627 316 y Fp(2)655 357 y Fl(I)-8 b(E)r(b)723 364 y Fh(k)1834 297 y Fm(\(2)p Fl(:)p Fm(34\))0 485 y(where)19 b(w)o(e)g(ha)o(v)o(e)g(used)h(that)e Fl(B)561 469 y Fp(\()p Fh(k)q Fp(\))632 485 y Fm(is)i(measurable)f(with)h (resp)q(ect)g(to)e Fi(F)5 b(nF)1344 492 y Fh(k)q Fp(+1)1430 485 y Fm(to)19 b(replace)h(the)f(conditional)0 558 y(exp)q(ectation)d(b)o(y)f (the)g(exp)q(ectation.)91 667 y(The)g(\014rst)g(term)g(in)h(\(2.33\))d(is)j (more)f(complicated.)21 b(W)l(e)15 b(rewrite)g(it)h(as)432 831 y(2)p Fl(I)-8 b(E)511 739 y Ff(2)511 821 y(4)541 787 y(X)548 883 y Fh(\026;\027)621 800 y Fm(1)p 620 820 26 2 v 620 862 a Fl(k)681 774 y Fh(k)658 787 y Ff(X)660 884 y Fh(j)r Fp(=1)731 831 y Fl(\030)753 809 y Fh(\026)751 843 y(j)778 831 y Fl(\030)800 812 y Fh(\027)798 842 y(j)822 831 y Fi(E)846 838 y Fp(1)p Fh(;k)905 831 y Fm(\()p Fl(m)963 809 y Fh(\026)963 844 y(N)997 831 y Fm(\()p Fl(\033)r Fm(\))p Fl(m)1101 812 y Fh(\027)1101 842 y(N)1135 831 y Fm(\()p Fl(\033)r Fm(\)\))1223 778 y Ff(\014)1223 806 y(\014)1223 833 y(\014)1238 831 y Fi(F)1271 838 y Fh(k)q Fp(+1)1339 739 y Ff(3)1339 821 y(5)443 1012 y Fm(+)10 b(2)p Fl(I)-8 b(E)567 921 y Ff(2)567 1003 y(4)597 969 y(X)603 1065 y Fh(\026;\027)677 982 y Fm(1)p 676 1002 V 676 1044 a Fl(k)737 956 y Fh(k)714 969 y Ff(X)715 1066 y Fh(j)r Fp(=1)787 1012 y Fl(\030)809 991 y Fh(\026)807 1025 y(j)833 1012 y Fl(\030)855 994 y Fh(\027)853 1024 y(j)903 982 y Fm(1)p 883 1002 62 2 v 883 1044 a Fl(N)925 1030 y Fp(2)951 1012 y Fl(\030)973 991 y Fh(\026)971 1025 y(j)997 1012 y Fl(\030)1019 994 y Fh(\027)1017 1024 y(j)1042 960 y Ff(\014)1042 987 y(\014)1042 1015 y(\014)1057 1012 y Fi(F)1090 1019 y Fh(k)q Fp(+1)1158 921 y Ff(3)1158 1003 y(5)443 1194 y Fi(\000)10 b Fm(4)p Fl(I)-8 b(E)567 1103 y Ff(2)567 1185 y(4)597 1151 y(X)603 1247 y Fh(\026;\027)677 1163 y Fm(1)p 676 1184 26 2 v 676 1225 a Fl(k)737 1137 y Fh(k)714 1151 y Ff(X)715 1248 y Fh(j)r Fp(=1)787 1194 y Fl(\030)809 1172 y Fh(\026)807 1207 y(j)833 1194 y Fl(\030)855 1175 y Fh(\027)853 1205 y(j)878 1194 y Fi(E)902 1201 y Fp(1)p Fh(;k)961 1117 y Ff( )1022 1163 y Fm(1)p 1002 1184 62 2 v 1002 1225 a Fl(N)1044 1212 y Fp(2)1094 1137 y Fh(N)1077 1151 y Ff(X)1080 1248 y Fh(i)p Fp(=1)1150 1194 y Fl(\030)1172 1172 y Fh(\026)1170 1207 y(i)1197 1194 y Fl(\033)1223 1201 y Fh(i)1238 1194 y Fl(\030)1260 1175 y Fh(\027)1258 1205 y(j)1282 1194 y Fl(\033)1308 1201 y Fh(j)1328 1117 y Ff(!)1371 1142 y(\014)1371 1169 y(\014)1371 1197 y(\014)1386 1194 y Fi(F)1419 1201 y Fh(k)q Fp(+1)1487 1103 y Ff(3)1487 1185 y(5)445 1322 y Fi(\021)13 b Fl(I)h Fm(+)c Fl(I)t(I)j Fi(\000)e Fl(I)t(I)t(I)1834 1049 y Fm(\(2)p Fl(:)p Fm(35\))0 1428 y(W)l(e)k(deal)h (with)g(the)f(three)g(terms)g(separately)l(.)20 b(First,)870 1563 y Fl(I)t(I)15 b Fm(=)e(2)1005 1533 y Fl(M)1054 1516 y Fp(2)p 1005 1553 70 2 v 1009 1594 a Fl(N)1051 1581 y Fp(2)1834 1563 y Fm(\(2)p Fl(:)p Fm(36\))0 1690 y(On)j(the)f(other)g(hand)444 1842 y Fl(I)t(I)t(I)g Fm(=)e(4)603 1812 y Fl(M)p 603 1832 50 2 v 607 1873 a(N)658 1842 y(I)-8 b(E)714 1751 y Ff(2)714 1833 y(4)744 1799 y(X)766 1895 y Fh(\026)845 1812 y Fm(1)p 823 1832 67 2 v 823 1873 a Fl(k)q(N)925 1785 y Fh(k)902 1799 y Ff(X)904 1896 y Fh(j)r Fp(=1)992 1785 y Fh(N)975 1799 y Ff(X)979 1896 y Fh(i)p Fp(=1)1049 1842 y Fi(E)1073 1849 y Fp(1)p Fh(;k)1132 1805 y Ff(\000)1152 1842 y Fl(\030)1174 1820 y Fh(\026)1172 1855 y(i)1199 1842 y Fl(\030)1221 1820 y Fh(\026)1219 1855 y(j)1245 1842 y Fl(\033)1271 1849 y Fh(i)1286 1842 y Fl(\033)1312 1849 y Fh(j)1331 1805 y Ff(\001)1360 1790 y(\014)1360 1817 y(\014)1360 1845 y(\014)1375 1842 y Fi(F)1408 1849 y Fh(k)q Fp(+1)1476 1751 y Ff(3)1476 1833 y(5)527 2024 y Fm(=)13 b(4)603 1993 y Fl(M)p 603 2014 50 2 v 607 2055 a(N)658 2024 y(I)-8 b(E)714 1933 y Ff(2)714 2015 y(4)744 2024 y Fi(E)768 2031 y Fp(1)p Fh(;k)827 1933 y Ff(0)827 2015 y(@)873 1993 y Fm(1)p 872 2014 26 2 v 872 2055 a Fl(k)933 1967 y Fh(k)910 1981 y Ff(X)912 2078 y Fh(j)r Fp(=1)983 2024 y Fl(\030)1003 2031 y Fh(j)1022 2024 y Fl(\033)1048 2031 y Fh(j)1075 2024 y Fl(;)1118 1993 y Fm(1)p 1108 2014 42 2 v 1108 2055 a Fl(N)1180 1967 y Fh(N)1163 1981 y Ff(X)1166 2078 y Fh(i)p Fp(=1)1236 2024 y Fl(\030)1256 2031 y Fh(i)1271 2024 y Fl(\033)1297 2031 y Fh(i)1312 1933 y Ff(1)1312 2015 y(A)1360 1972 y(\014)1360 1999 y(\014)1360 2026 y(\014)1375 2024 y Fi(F)1408 2031 y Fh(k)q Fp(+1)1476 1933 y Ff(3)1476 2015 y(5)1834 1933 y Fm(\(2)p Fl(:)p Fm(37\))0 2185 y(Using)16 b(the)f(Sc)o(h)o(w)o(arz)g(inequalit)o(y)l(,)h(one)g(obtains) f(therefore)g(that)452 2346 y Fi(j)p Fl(I)t(I)t(I)t Fi(j)c(\024)i Fm(4)637 2315 y Fl(M)p 637 2335 50 2 v 641 2377 a(N)692 2346 y(I)-8 b(E)748 2254 y Ff(2)748 2336 y(4)778 2346 y Fi(E)802 2353 y Fp(1)p Fh(;k)861 2253 y Ff(\015)861 2280 y(\015)861 2307 y(\015)861 2334 y(\015)861 2362 y(\015)861 2389 y(\015)893 2315 y Fm(1)p 891 2335 26 2 v 891 2377 a Fl(k)952 2289 y Fh(k)930 2303 y Ff(X)931 2399 y Fh(j)r Fp(=1)1003 2346 y Fl(\030)1023 2353 y Fh(j)1042 2346 y Fl(\033)1068 2353 y Fh(j)1087 2253 y Ff(\015)1087 2280 y(\015)1087 2307 y(\015)1087 2334 y(\015)1087 2362 y(\015)1087 2389 y(\015)1112 2419 y Fp(2)1140 2346 y Fi(k)p Fl(m)1203 2353 y Fh(N)1237 2346 y Fm(\()p Fl(\033)r Fm(\))p Fi(k)1323 2359 y Fp(2)1351 2307 y Ff(\014)1351 2334 y(\014)1366 2346 y Fi(F)1399 2353 y Fh(k)q Fp(+1)1467 2254 y Ff(3)1467 2336 y(5)1834 2346 y Fm(\(2)p Fl(:)p Fm(38\))0 2508 y(But)480 2522 y Ff(\015)480 2550 y(\015)480 2577 y(\015)480 2604 y(\015)480 2631 y(\015)480 2659 y(\015)512 2585 y Fm(1)p 511 2605 V 511 2647 a Fl(k)572 2559 y Fh(k)549 2572 y Ff(X)550 2669 y Fh(j)r Fp(=1)622 2615 y Fl(\030)642 2622 y Fh(j)661 2615 y Fl(\033)687 2622 y Fh(j)706 2522 y Ff(\015)706 2550 y(\015)706 2577 y(\015)706 2604 y(\015)706 2631 y(\015)706 2659 y(\015)732 2534 y Fp(2)732 2688 y(2)765 2615 y Fm(=)829 2585 y(1)p 818 2605 46 2 v 818 2647 a Fl(k)843 2633 y Fp(2)877 2572 y Ff(X)898 2668 y Fh(\026)973 2559 y(k)950 2572 y Ff(X)951 2669 y Fh(j)r Fp(=1)1046 2559 y Fh(k)1023 2572 y Ff(X)1027 2670 y Fh(l)p Fp(=1)1096 2615 y Fl(\033)1122 2622 y Fh(j)1141 2615 y Fl(\030)1163 2594 y Fh(\026)1161 2628 y(j)1188 2615 y Fl(\030)1210 2594 y Fh(\026)1208 2629 y(l)1234 2615 y Fl(\033)1260 2622 y Fh(l)1287 2615 y Fi(\024)1335 2563 y Ff(\015)1335 2590 y(\015)1335 2618 y(\015)1360 2615 y Fl(A)1394 2597 y Fp(\()p Fh(k)q Fp(\))1445 2563 y Ff(\015)1445 2590 y(\015)1445 2618 y(\015)1834 2615 y Fm(\(2)p Fl(:)p Fm(39\))952 2779 y(12)p eop %%Page: 13 13 bop 0 47 a Fm(where)19 b Fl(A)169 31 y Fp(\()p Fh(k)q Fp(\))239 47 y Fm(is)h(the)f Fl(k)14 b Fi(\002)f Fl(k)q Fm(-matrix)19 b(with)g(elemen)o(ts)h Fl(A)980 24 y Fp(\()p Fh(k)q Fp(\))980 61 y Fh(j)r(l)1050 47 y Fi(\021)1111 30 y Fp(1)p 1110 37 21 2 v 1110 63 a Fh(k)1143 13 y Ff(P)1191 61 y Fh(\026)1223 47 y Fl(\030)1245 26 y Fh(\026)1243 60 y(j)1269 47 y Fl(\030)1291 26 y Fh(\026)1289 61 y(l)1316 47 y Fm(.)31 b(It)19 b(is)h(easy)f(to)f(v)o (erify)h(b)o(y)g(simple)0 120 y(algebraic)d(manipulations)h(that)586 82 y Ff(\015)586 109 y(\015)611 120 y Fl(A)645 104 y Fp(\()p Fh(k)q Fp(\))696 82 y Ff(\015)696 109 y(\015)734 120 y Fm(=)782 82 y Ff(\015)782 109 y(\015)807 120 y Fl(B)843 104 y Fp(\()p Fh(k)q Fp(\))895 82 y Ff(\015)895 109 y(\015)920 120 y Fm(.)j(Since)d(on)e (the)g(other)g(hand)673 255 y Fi(E)697 262 y Fp(1)p Fh(;k)756 255 y Fi(k)o Fl(m)818 262 y Fh(N)853 255 y Fm(\()p Fl(\033)r Fm(\))p Fi(k)938 268 y Fp(2)971 255 y Fi(\024)e Fm(\()1037 224 y Fi(p)p 1075 224 24 2 v 31 x Fl(\032)c Fm(+)i Fi(k)d Fm(~)-31 b Fl(m)p Fi(k)1240 262 y Fp(2)1260 255 y Fm(\))556 b(\(2)p Fl(:)p Fm(40\))0 389 y(this)16 b(com)o(bines)g(to)342 523 y Fi(j)p Fl(I)t(I)t(I)t Fi(j)10 b(\024)j Fm(4)526 492 y Fl(M)p 526 513 50 2 v 530 554 a(N)581 523 y(I)-8 b(E)629 468 y Ff(q)p 675 468 139 2 v 675 484 a(\015)675 512 y(\015)700 523 y Fl(B)736 510 y Fp(\()p Fh(k)q Fp(\))788 484 y Ff(\015)788 512 y(\015)813 523 y Fm(\()831 493 y Fi(p)p 869 493 24 2 v 30 x Fl(\032)9 b Fm(+)i Fi(k)d Fm(~)-31 b Fl(m)o Fi(k)1033 530 y Fp(2)1053 523 y Fm(\))13 b Fi(\024)g Fm(4)1160 492 y Fl(M)p 1160 513 50 2 v 1164 554 a(N)1214 481 y Ff(p)p 1260 481 91 2 v 1260 523 a Fl(I)-8 b(E)r(b)1328 530 y Fh(k)1350 523 y Fm(\()1368 493 y Fi(p)p 1406 493 24 2 v 30 x Fl(\032)9 b Fm(+)i Fi(k)d Fm(~)-31 b Fl(m)o Fi(k)1570 530 y Fp(2)1590 523 y Fm(\))226 b(\(2)p Fl(:)p Fm(41\))0 657 y(W)l(e)15 b(are)g(left)h(with)f(the)h(term)e Fl(I)t Fm(.)20 b(W)l(e)15 b(write)316 817 y Fl(I)i Fm(=2)p Fl(I)-8 b(E)467 725 y Ff(2)467 807 y(4)497 774 y(X)503 869 y Fh(\026;\027)577 786 y Fm(1)p 576 806 26 2 v 576 848 a Fl(k)637 760 y Fh(k)614 774 y Ff(X)615 870 y Fh(j)r Fp(=1)687 817 y Fl(\030)709 795 y Fh(\026)707 829 y(j)733 817 y Fl(\030)755 798 y Fh(\027)753 828 y(j)778 817 y Fi(E)802 824 y Fp(1)p Fh(;k)861 817 y Fm(\()o(\()p Fl(m)936 795 y Fh(\026)936 830 y(N)971 817 y Fm(\()p Fl(\033)r Fm(\))9 b Fi(\000)19 b Fm(~)-32 b Fl(m)1129 798 y Fh(\026)1153 817 y Fm(\)\()p Fl(m)1229 798 y Fh(\027)1229 828 y(N)1263 817 y Fm(\()p Fl(\033)r Fm(\))9 b Fi(\000)19 b Fm(~)-31 b Fl(m)1422 798 y Fh(\027)1444 817 y Fm(\)\))1487 778 y Ff(\014)1487 805 y(\014)1502 817 y Fi(F)1535 824 y Fh(k)q Fp(+1)1603 725 y Ff(3)1603 807 y(5)398 999 y Fm(+)11 b(4)p Fl(I)-8 b(E)522 907 y Ff(2)522 989 y(4)553 955 y(X)559 1051 y Fh(\026;\027)632 968 y Fm(1)p 631 988 V 631 1030 a Fl(k)692 942 y Fh(k)669 955 y Ff(X)671 1052 y Fh(j)r Fp(=1)743 999 y Fl(\030)765 977 y Fh(\026)763 1011 y(j)789 999 y Fl(\030)811 980 y Fh(\027)809 1010 y(j)833 999 y Fi(E)857 1006 y Fp(1)p Fh(;k)916 999 y Fm(\(\()p Fl(m)992 977 y Fh(\026)992 1012 y(N)1026 999 y Fm(\()p Fl(\033)r Fm(\))9 b Fi(\000)19 b Fm(~)-31 b Fl(m)1185 980 y Fh(\026)1209 999 y Fm(\))8 b(~)-31 b Fl(m)1267 980 y Fh(\027)1289 999 y Fm(\))1314 960 y Ff(\014)1314 987 y(\014)1329 999 y Fi(F)1362 1006 y Fh(k)q Fp(+1)1431 907 y Ff(3)1431 989 y(5)398 1180 y Fm(+)11 b(2)p Fl(I)-8 b(E)522 1089 y Ff(2)522 1171 y(4)553 1137 y(X)559 1233 y Fh(\026;\027)632 1150 y Fm(1)p 631 1170 V 631 1212 a Fl(k)692 1124 y Fh(k)669 1137 y Ff(X)671 1234 y Fh(j)r Fp(=1)743 1180 y Fl(\030)765 1159 y Fh(\026)763 1193 y(j)789 1180 y Fl(\030)811 1162 y Fh(\027)809 1192 y(j)842 1180 y Fm(~)-32 b Fl(m)873 1162 y Fh(\026)906 1180 y Fm(~)h Fl(m)938 1162 y Fh(\027)960 1142 y Ff(\014)960 1169 y(\014)975 1180 y Fi(F)1008 1187 y Fh(k)q Fp(+1)1077 1089 y Ff(3)1077 1171 y(5)353 1321 y Fm(=2)p Fl(I)-8 b(E)467 1271 y Ff(h)488 1321 y Fi(E)512 1328 y Fp(1)p Fh(;k)571 1271 y Ff(\020)598 1321 y Fm(\()p Fl(m)656 1328 y Fh(N)690 1321 y Fm(\()p Fl(\033)r Fm(\))9 b Fi(\000)19 b Fm(~)-31 b Fl(m)p Fm(\))8 b Fl(;)14 b(B)938 1303 y Fp(\()p Fh(k)q Fp(\))989 1321 y Fm(\()p Fl(m)1047 1328 y Fh(N)1081 1321 y Fm(\()p Fl(\033)r Fm(\))9 b Fi(\000)19 b Fm(~)-31 b Fl(m)p Fm(\))1258 1271 y Ff(\021)1292 1283 y(\014)1292 1310 y(\014)1307 1321 y Fi(F)1340 1328 y Fh(k)q Fp(+1)1408 1271 y Ff(i)398 1421 y Fm(+)11 b(4)p Fl(I)-8 b(E)522 1371 y Ff(h)544 1421 y Fi(E)568 1428 y Fp(1)p Fh(;k)627 1371 y Ff(\020)654 1421 y Fm(\()p Fl(m)712 1428 y Fh(N)746 1421 y Fm(\()p Fl(\033)r Fm(\))9 b Fi(\000)19 b Fm(~)-32 b Fl(m)p Fm(\))8 b Fl(;)14 b(B)993 1403 y Fp(\()p Fh(k)q Fp(\))1053 1421 y Fm(~)-31 b Fl(m)1085 1371 y Ff(\021)1119 1383 y(\014)1119 1410 y(\014)1134 1421 y Fi(F)1167 1428 y Fh(k)q Fp(+1)1236 1371 y Ff(i)398 1562 y Fm(+)11 b(2)p Fl(I)-8 b(E)522 1471 y Ff(2)522 1553 y(4)559 1531 y Fm(1)p 558 1552 V 558 1593 a Fl(k)619 1505 y Fh(k)596 1519 y Ff(X)597 1616 y Fh(j)r Fp(=1)669 1484 y Ff( )705 1519 y(X)727 1614 y Fh(\026)779 1562 y Fl(\030)801 1540 y Fh(\026)799 1575 y(j)833 1562 y Fm(~)-31 b Fl(m)865 1543 y Fh(\026)889 1484 y Ff(!)925 1494 y Fp(2)946 1471 y Ff(3)946 1553 y(5)1834 1189 y Fm(\(2)p Fl(:)p Fm(42\))0 1733 y(Using)16 b(the)f(Sc)o(h)o(w)o(artz)f (inequalit)o(y)j(for)e(the)g(\014rst)g(t)o(w)o(o)f(terms,)g(w)o(e)h(obtain)g (from)g(\(2.42\))e(the)i(b)q(ound)650 1867 y Fl(I)h Fi(\024)d Fm(2)p Fl(\032I)-8 b(E)r(b)849 1874 y Fh(k)881 1867 y Fm(+)10 b(4)949 1837 y Fi(p)p 987 1837 24 2 v 30 x Fl(\032)o Fi(k)f Fm(~)-32 b Fl(m)p Fi(k)1096 1874 y Fp(2)1126 1867 y Fm(+)11 b(2)p Fi(k)d Fm(~)-31 b Fl(m)o Fi(k)1280 1848 y Fp(2)1280 1878 y(2)1834 1867 y Fm(\(2)p Fl(:)p Fm(43\))0 2002 y(Com)o(bining)16 b(all)g(these)g(estimates)f(with)g(\(2.34\))e(and)j(c)o(ho)q(osing)f(as)g(b)q (efore)h Fl(\016)e Fm(=)f(2)p Fl(=)p Fm(\()p Fl(\014)r(N)5 b Fm(\))14 b(giv)o(es)h(that)91 2136 y Fl(I)-8 b(E)147 2085 y Ff(h)169 2136 y Fm(\()p Fl(f)214 2117 y Fg(0)209 2147 y Fh(k)231 2136 y Fm(\(1\)\))307 2111 y Fp(2)335 2097 y Ff(\014)335 2124 y(\014)350 2136 y Fi(F)383 2143 y Fh(k)q Fp(+1)451 2085 y Ff(i)485 2136 y Fi(\024)13 b(k)8 b Fm(~)-31 b Fl(m)p Fi(k)619 2117 y Fp(2)619 2147 y(2)649 2136 y Fm(+)10 b(2)722 2118 y Fh(M)760 2104 y Fc(2)p 723 2125 56 2 v 726 2152 a Fh(N)758 2142 y Fc(2)794 2136 y Fm(+)g(4)862 2105 y Fi(p)p 900 2105 24 2 v 31 x Fl(\032)p Fi(k)e Fm(~)-31 b Fl(m)o Fi(k)1009 2143 y Fp(2)1039 2136 y Fm(+)11 b(4)p Fl(\032I)-8 b(E)q(b)1199 2143 y Fh(k)1231 2136 y Fm(+)11 b(4)1305 2118 y Fh(M)p 1305 2125 39 2 v 1308 2152 a(N)1349 2094 y Ff(p)p 1394 2094 91 2 v 42 x Fl(I)-8 b(E)r(b)1462 2143 y Fh(k)1484 2136 y Fm(\()p Fi(k)8 b Fm(~)-31 b Fl(m)p Fi(k)1588 2143 y Fp(2)1618 2136 y Fm(+)1663 2105 y Fi(p)p 1701 2105 24 2 v 31 x Fl(\032)p Fm(\))91 b(\(2)p Fl(:)p Fm(44\))0 2270 y(Collecting)17 b(our)e(b)q(ounds)h(and)f(inserting)h(them)f(in)o(to)h (\(2.26\))d(w)o(e)i(ha)o(v)o(e)41 2400 y Fl(I)-8 b(E)97 2350 y Ff(h)118 2400 y Fl(e)139 2381 y Fh(t)161 2373 y Fp(~)153 2381 y Fh(f)175 2365 y Fc(\()p Fe(k)p Fc(\))171 2393 y Fe(N)219 2381 y Fp(\()7 b(~)-25 b Fh(m)p Fp(\))282 2361 y Ff(\014)282 2389 y(\014)297 2400 y Fi(F)330 2407 y Fh(k)q Fp(+1)398 2350 y Ff(i)53 2501 y Fi(\024)13 b Fm(1)d(+)185 2483 y Fp(1)p 185 2490 19 2 v 185 2517 a(2)209 2501 y Fl(t)225 2482 y Fp(2)245 2501 y Fl(e)266 2482 y Fp(2)p Fg(j)p Fh(t)p Fg(j)p Fp(\()p Fg(k)d Fp(~)-25 b Fh(m)p Fg(k)404 2487 y Fc(1)422 2482 y Fp(+2)468 2451 y Fi(p)p 505 2451 57 2 v 505 2482 a Fh(M)s(\032)p Fp(\))586 2450 y Ff(\020)614 2501 y Fi(k)8 b Fm(~)-31 b Fl(m)o Fi(k)699 2482 y Fp(2)699 2512 y(2)729 2501 y Fm(+)11 b(2)803 2483 y Fh(M)841 2469 y Fc(2)p 803 2490 56 2 v 806 2517 a Fh(N)838 2508 y Fc(2)874 2501 y Fm(+)g(4)943 2470 y Fi(p)p 980 2470 24 2 v 980 2501 a Fl(\032)p Fi(k)d Fm(~)-31 b Fl(m)p Fi(k)1090 2508 y Fp(2)1120 2501 y Fm(+)10 b(4)p Fl(\032I)-8 b(E)r(b)1280 2508 y Fh(k)1312 2501 y Fm(+)10 b(4)1385 2483 y Fh(M)p 1385 2490 39 2 v 1388 2517 a(N)1429 2459 y Ff(p)p 1474 2459 91 2 v 42 x Fl(I)-8 b(E)s(b)1543 2508 y Fh(k)1565 2501 y Fm(\()p Fi(k)8 b Fm(~)-31 b Fl(m)o Fi(k)1668 2508 y Fp(2)1698 2501 y Fm(+)1744 2470 y Fi(p)p 1782 2470 24 2 v 31 x Fl(\032)o Fm(\))1823 2450 y Ff(\021)53 2602 y Fi(\024)13 b Fm(exp)178 2551 y Ff(n)214 2584 y Fp(1)p 214 2591 19 2 v 214 2617 a(2)238 2602 y Fl(t)254 2583 y Fp(2)275 2602 y Fl(e)296 2583 y Fp(2)p Fg(j)p Fh(t)p Fg(j)p Fp(\()p Fg(k)6 b Fp(~)-24 b Fh(m)o Fg(k)433 2588 y Fc(1)451 2583 y Fp(+2)497 2552 y Fi(p)p 534 2552 57 2 v 534 2583 a Fh(M)s(\032)p Fp(\))615 2551 y Ff(\020)643 2602 y Fi(k)8 b Fm(~)-31 b Fl(m)o Fi(k)728 2583 y Fp(2)728 2613 y(2)758 2602 y Fm(+)11 b(2)832 2584 y Fh(M)870 2570 y Fc(2)p 832 2591 56 2 v 835 2617 a Fh(N)867 2608 y Fc(2)903 2602 y Fm(+)g(4)972 2571 y Fi(p)p 1010 2571 24 2 v 31 x Fl(\032)o Fi(k)d Fm(~)-31 b Fl(m)p Fi(k)1119 2609 y Fp(2)1149 2602 y Fm(+)10 b(4)p Fl(\032I)-8 b(E)r(b)1309 2609 y Fh(k)1341 2602 y Fm(+)10 b(4)1414 2584 y Fh(M)p 1415 2591 39 2 v 1418 2617 a(N)1458 2560 y Ff(p)p 1504 2560 91 2 v 1504 2602 a Fl(I)-8 b(E)r(b)1572 2609 y Fh(k)1594 2602 y Fm(\()p Fi(k)8 b Fm(~)-31 b Fl(m)o Fi(k)1697 2609 y Fp(2)1727 2602 y Fm(+)1773 2571 y Fi(p)p 1811 2571 24 2 v 31 x Fl(\032)o Fm(\))1852 2551 y Ff(\021o)1834 2670 y Fm(\(2)p Fl(:)p Fm(45\))952 2779 y(13)p eop %%Page: 14 14 bop 0 45 a Fm(This)16 b(is)f(a)g(uniform)h(b)q(ound)g(on)f Fi(L)592 29 y Fp(\()p Fh(k)q Fp(\))643 45 y Fm(\()p Fl(t)p Fm(\))g(so)g(that)88 195 y Fl(I)-8 b(E)10 b Fm(exp)221 117 y Ff(\()257 195 y Fl(t)298 138 y Fh(N)281 152 y Ff(X)281 250 y Fh(k)q Fp(=1)365 183 y Fm(~)355 195 y Fl(f)382 171 y Fp(\()p Fh(k)q Fp(\))377 208 y Fh(N)433 195 y Fm(\()e(~)-31 b Fl(m)o Fm(\))508 117 y Ff(\))557 195 y Fi(\024)13 b Fm(exp)675 117 y Ff(\()717 177 y Fp(1)p 717 184 19 2 v 717 211 a(2)740 195 y Fl(t)756 176 y Fp(2)777 195 y Fl(e)798 176 y Fp(2)p Fg(j)p Fh(t)p Fg(j)p Fp(\()p Fg(k)7 b Fp(~)-25 b Fh(m)p Fg(k)936 181 y Fc(1)953 176 y Fp(+2)999 145 y Fi(p)p 1037 145 57 2 v 31 x Fh(M)s(\032)p Fp(\))1111 117 y Ff( )1147 195 y Fl(N)1196 144 y Ff(\020)1223 195 y Fi(k)8 b Fm(~)-31 b Fl(m)o Fi(k)1308 176 y Fp(2)1308 206 y(2)1339 195 y Fm(+)10 b(2)1412 177 y Fh(M)1450 163 y Fc(2)p 1412 184 56 2 v 1415 211 a Fh(N)1447 201 y Fc(2)1484 195 y Fm(+)g(4)1552 164 y Fi(p)p 1590 164 24 2 v 31 x Fl(\032)o Fi(k)f Fm(~)-32 b 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1146 y Fh(`)1102 1151 y Fe(n)1137 1139 y Fi(\000)d Fl(u)1242 1108 y Fm(1)p 1213 1129 80 2 v 1213 1138 a Fi(p)p 1252 1138 42 2 v 1252 1176 a Fl(N)1299 1139 y(J)1324 1146 y Fh(`)1339 1151 y Fe(k)1361 1139 y Fl(\033)1387 1146 y Fh(`)1402 1151 y Fe(k)1857 1139 y Fm(\(3)p Fl(:)p Fm(5\))0 1285 y(This)16 b(yields)h(that)d(this)i(time) 744 1357 y Fl(f)771 1339 y Fg(0)766 1369 y Fh(k)789 1357 y Fm(\()p Fl(u)p Fm(\))c(=)945 1327 y(1)p 916 1347 80 2 v 916 1356 a Fi(p)p 954 1356 42 2 v 38 x Fl(N)1001 1357 y(J)1026 1364 y Fh(`)1041 1369 y Fe(k)1064 1357 y Fi(E)1088 1364 y Fh(k)q(;u)1142 1357 y Fl(\033)1168 1364 y Fh(`)1183 1369 y Fe(k)1857 1357 y Fm(\(3)p Fl(:)p Fm(6\))0 1463 y(T)l(rivially)l(,)17 b(here)786 1536 y Fi(j)p Fl(f)826 1517 y Fg(0)821 1547 y Fh(k)843 1536 y Fm(\()p Fl(u)p Fm(\))p Fi(j)12 b(\024)1012 1505 y Fm(1)p 984 1525 80 2 v 984 1534 a Fi(p)p 1021 1534 42 2 v 1021 1573 a Fl(N)1076 1536 y Fi(j)p Fl(J)1114 1543 y Fh(`)1129 1548 y Fe(k)1151 1536 y Fi(j)693 b Fm(\(3)p Fl(:)p Fm(7\))0 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