information : If you have any problems with printing PostScript file , I can send LaTeX file. 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b(\))i(imply)h(for)f Fk(j)p Fl(x)955 1867 y Fi(1)984 1860 y Fk(\000)c Fl(x)1056 1867 y Fi(2)1075 1860 y Fk(j)1088 1867 y Fi(1)1120 1860 y Fl(>)i(s)g Fn(:)249 2001 y(Co)o(v)o(\()p Fl(\037)p Fn(\()p Fl(x)417 2008 y Fi(1)437 2001 y Fl(;)8 b(dx)508 2008 y Fi(1)526 2001 y Fl(;)g(s)p Fn(\))p Fl(\037)p Fn(\()p Fl(x)658 2008 y Fi(2)677 2001 y Fl(;)g(dx)748 2008 y Fi(2)767 2001 y Fl(;)g(s)p Fn(\)\))j(=)904 1954 y Fh(\020)929 1948 y Fj(N)t Fg(\000)p Fi(2)938 1961 y Fh(X)929 2051 y Fj(m)p Fi(=0)1018 1970 y Fn(\()p Fk(\000)p Fn(1\))1112 1954 y Fj(m)p 1018 1991 127 2 v 1056 2032 a Fl(m)p Fn(!)1158 1944 y Fh(Z)1207 2001 y Fl(\032)1231 2008 y Fi(2+)p Fj(m;)p Fi(2)1336 2001 y Fn(\()p Fl(x)1380 2008 y Fi(1)1399 2001 y Fl(;)d(x)1446 2008 y Fi(2)1465 2001 y Fn(;)g Fl(:)g(:)g(:)d(x)1572 2008 y Fj(m)p Fi(+2)1651 2001 y Fn(\))p Fl(dx)1719 2008 y Fi(3)1745 2001 y Fl(:)j(:)g(:)e(dx)1856 2008 y Fj(m)p Fi(+2)1934 2001 y Fn(+)p Fk(R)2008 1954 y Fh(\021)2032 2001 y Fl(dx)2082 2008 y Fi(1)2102 2001 y Fl(dx)2152 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2403 y Fl(;)8 b(dx)1906 2410 y Fi(2)1924 2403 y Fl(;)g(s)p Fn(\)\))2001 2356 y Fh(\021)1123 2578 y Fn(14)p eop %%Page: 15 15 bop 489 454 a Fn(=)42 b(I)-9 b(E)626 388 y Fj(N)612 396 y Fh(Z)615 517 y Fi(0)751 388 y Fj(N)737 396 y Fh(Z)739 517 y Fi(0)657 547 y Fg(j)p Fj(x)687 552 y Ff(1)704 547 y Fg(\000)p Fj(x)751 552 y Ff(2)769 547 y Fg(j)779 552 y Ff(1)796 547 y Fj(>s)854 454 y Fn(\()p Fl(\037)p Fn(\()p Fl(x)944 461 y Fi(1)963 454 y Fl(;)8 b(dx)1034 461 y Fi(1)1053 454 y Fl(;)g(s)p Fn(\))h Fk(\000)h Fn(I)-9 b(E)q Fl(\037)p Fn(\()p Fl(x)1278 461 y Fi(1)1297 454 y Fl(;)8 b(dx)1368 461 y Fi(1)1387 454 y Fl(;)g(s)p Fn(\)\)\()p Fl(\037)p Fn(\()p Fl(x)1555 461 y Fi(2)1573 454 y Fl(;)g(dx)1644 461 y Fi(2)1663 454 y Fl(;)g(s)p Fn(\))h Fk(\000)i Fn(I)-9 b(E)p Fl(\037)p Fn(\()p Fl(x)1888 461 y Fi(2)1908 454 y Fl(;)8 b(dx)1979 461 y Fi(2)1997 454 y Fl(;)g(s)p Fn(\)\))489 668 y(+)42 b(I)-9 b(E)626 603 y Fj(N)612 611 y Fh(Z)615 732 y Fi(0)773 603 y Fj(N)759 611 y Fh(Z)762 732 y Fi(0)657 761 y(0)p 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Fh(G)1107 2281 y Fi(1)1147 2230 y Fk(A)1183 2237 y Fj(q)1230 2230 y Fn(=)28 b Fk(f)p Fn(1)p Fl(;)8 b(:)g(:)g(:)t Fn(2)p Fl(k)q Fk(g)249 2369 y Fn(and)14 b(denoting)42 b(\()p Fl(x)593 2376 y Fj(i)607 2369 y Fn(\))625 2376 y Fj(i)p Fg(2A)689 2380 y Fd(q)749 2369 y Fn(b)o(y)j(\026)-26 b Fl(x)p Fn(\()p Fk(A)919 2376 y Fj(q)938 2369 y Fn(\))41 b(w)o(e)14 b(obtain)g(the)g (represen)o(tation)g(of)41 b Fl(\032)1678 2376 y Fi(2)p Fj(k)q Fi(+)p Fj(m;)p Fi(2)p Fj(k)1822 2369 y Fn(\()s(\026)-26 b Fl(x)o Fn(\))42 b(as)13 b(the)1123 2578 y(19)p eop %%Page: 20 20 bop 249 403 a Fn(sum)15 b(of)g(pro)q(ducts)783 449 y Fh(X)734 543 y Fi(\()p Fg(A)776 548 y Ff(1)793 543 y Fj(;:::)p Fg(A)861 547 y Fd(p)879 543 y Fi(\))691 582 y Fg(jA)729 586 y Fd(q)746 582 y Fg(j\025)p Fi(2)h Fj(q)q Fi(=1)p Fj(;:::)o(p)996 434 y(p)978 449 y Fh(Y)996 540 y Fi(1)1073 490 y Fn(~)-27 b Fl(\032)1140 474 y Fh(P)1123 541 y Fd(j)q Fp(2A)1183 545 y Fd(q)1206 503 y Fj(n)p Fi(\()p Fj(x)1261 508 y Fd(j)1277 503 y Fi(\))p Fj(;)15 b 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Fh(\020)506 1163 y Fj(N)t Fg(\000jA)603 1167 y Fd(q)620 1163 y Fg(j)538 1181 y Fh(X)530 1272 y Fj(m)p Fi(=0)643 1191 y Fn(\()p Fk(\000)p Fn(1\))737 1174 y Fj(m)p 643 1211 127 2 v 680 1253 a Fl(m)p Fn(!)914 1164 y Fh(Z)773 1288 y Fi(\()p Fg(t)811 1294 y Fd(j)q Fp(2A)871 1298 y Fd(q)889 1288 y Fi([)p Fj(x)919 1293 y Fd(j)935 1288 y Fj(;x)965 1293 y Fd(j)981 1288 y Fi(+)p Fj(s)p Fi(]\))1048 1276 y Fd(m)1144 1222 y Fn(~)-27 b Fl(\032)1164 1231 y Fg(jA)1202 1235 y Fd(q)1218 1231 y Fg(j)p Fi(+)p Fj(m;)p Fg(jA)1334 1235 y Fd(q)1352 1231 y Fg(j)1364 1222 y Fn(\()s(\026)h Fl(x)o Fn(\()p Fk(A)1461 1229 y Fj(q)1480 1222 y Fn(\))p Fl(;)8 b(y)1541 1229 y Fi(1)1560 1222 y Fl(;)g(:)g(:)g(:)d(y)1663 1229 y Fj(m)1697 1222 y Fn(\))p Fl(dy)1761 1229 y Fi(1)1787 1222 y Fl(:)j(:)g(:)e(y)1870 1229 y Fj(m)1919 1174 y Fh(\021)389 1396 y Fn(+)57 b Fl(O)q Fn(\(\()558 1365 y Fl(const)p Fn(\()p Fl(s)p Fn(\))8 b(log)787 1347 y Fj(k)816 1365 y Fl(N)p 558 1385 301 2 v 687 1427 a(N)863 1396 y Fn(\))881 1377 y Fj(N)924 1396 y Fk(\001)i Fl(N)989 1377 y Fi(2)p Fj(k)1027 1396 y Fn(\))1060 1349 y Fh(\021)1962 1396 y Fn(\(39\))249 1537 y(where)20 b(the)g(sum)613 1496 y Fg(\003\003)608 1505 y Fh(P)692 1537 y Fn(is)g(o)o(v)o(er)f(all)i(partitions)f(of)59 b Fk(f)p Fn(1)p Fl(;)8 b(:)g(:)g(:)t Fn(2)p Fl(k)q Fk(g)59 b Fn(in)o(to)20 b(t)o(w)o(o-)e(and)i(more)g(elemen)o(t)249 1603 y(subsets.)249 1729 y(Since)51 b(~)-27 b Fl(\032)422 1736 y Fi(2+)p Fj(m;)p Fi(2)527 1729 y Fn(\()s(\026)h Fl(x)p Fn(\))12 b(=)h Fl(\032)673 1736 y Fi(2+)p Fj(m;)p Fi(2)779 1729 y Fn(\()s(\026)-26 b Fl(x)o Fn(\))249 1795 y(the)15 b(sum)h(o)o(v)o(er)e(partitions)h(in)o(to)h (the)f(t)o(w)o(o-elemen)o(t)g(subsets)g(is)h(exactly)749 1867 y Fh(X)700 1958 y Fj(par)q(titions)698 1991 y(into)g(pair)q(s)908 1867 y Fh(Y)901 1961 y Fi(\()p Fj(i;j)r Fi(\))990 1907 y Fn(Co)o(v)e(\()p Fl(\037)p Fn(\()p Fl(x)1173 1914 y Fj(i)1187 1907 y Fl(;)8 b(dx)1258 1914 y Fj(i)1271 1907 y Fl(;)g(s)p Fn(\))p Fl(;)g(\037)p Fn(\()p Fl(x)1424 1914 y Fj(j)1441 1907 y Fl(;)g(dx)1512 1914 y Fj(j)1529 1907 y Fl(;)g(s)p Fn(\)\))355 b(\(40\))249 2142 y(T)l(o)15 b(estimate)g(the)g(remaining)i(part,)d(let)i(us)f(in)o(tro)q(duce) h(the)f(notation)491 2254 y Fl(r)527 2264 y Fg(jA)565 2268 y Fd(q)582 2264 y Fg(j)609 2254 y Fn(\()s(\026)-26 b Fl(x)p Fn(\()p Fk(A)707 2261 y Fj(q)726 2254 y Fn(\))p Fl(;)8 b(s)p Fn(\))j(:=)414 2376 y(=)491 2318 y Fj(N)t Fg(\000jA)588 2322 y Fd(q)605 2318 y Fg(j)523 2336 y Fh(X)515 2427 y Fj(m)p Fi(=0)628 2346 y Fn(\()p Fk(\000)p Fn(1\))722 2329 y Fj(m)p 628 2366 127 2 v 665 2408 a Fl(m)p Fn(!)898 2319 y Fh(Z)758 2443 y Fi(\()p Fg(t)796 2449 y Fd(j)q Fp(2A)856 2453 y Fd(q)874 2443 y Fi([)p Fj(x)904 2448 y Fd(j)920 2443 y Fj(;x)950 2448 y Fd(j)966 2443 y Fi(+)p Fj(s)p Fi(]\))1033 2431 y Fd(m)1129 2376 y Fn(~)-28 b Fl(\032)1163 2386 y Fg(jA)1201 2390 y Fd(q)1218 2386 y Fg(j)p Fi(+)p Fj(m;)p Fg(jA)1334 2390 y Fd(q)1352 2386 y Fg(j)1379 2376 y Fn(\()s(\026)i Fl(x)o Fn(\()p Fk(A)1476 2383 y Fj(q)1495 2376 y Fn(\))p Fl(;)8 b(y)1556 2383 y Fi(1)1575 2376 y Fl(;)g(:)g(:)g(:)d(y) 1678 2383 y Fj(m)1712 2376 y Fn(\))p Fl(dy)1776 2383 y Fi(1)1803 2376 y Fl(:)j(:)g(:)d(y)1885 2383 y Fj(m)1962 2376 y Fn(\(41\))1123 2578 y(20)p eop %%Page: 21 21 bop 249 403 a Fn(The)16 b(next)f(lemma,together)f(with)31 b(\(39\))e(clearly) 16 b(implies)33 b(\(30\),)28 b(\(31\))i(:)249 519 y Fc(Lemm)o(a)14 b(2.2)388 599 y Fh(Z)354 727 y Fi([0)p Fj(;N)t Fi(])434 715 y Fd(l)253 765 y Fi(min)312 771 y Fd(i)p Fp(6)p Ff(=)p Fd(j)368 765 y Fg(j)p Fj(x)398 770 y Fd(i)411 765 y Fg(\000)p Fj(x)458 770 y Fd(j)475 765 y Fg(j)485 770 y Ff(1)502 765 y Fj(>s)616 656 y Fl(r)637 663 y Fj(l)650 656 y Fn(\()p Fl(x)694 663 y Fi(1)713 656 y Fl(;)8 b(:)g(:)g(:)d(x)820 663 y Fj(l)833 656 y Fl(;)j(s)p Fn(\))p Fl(dx)943 663 y Fi(1)969 656 y Fl(:)g(:)g(:)e(dx)1080 663 y Fj(l)1105 656 y Fn(=)1153 570 y Fh(8)1153 607 y(<)1153 682 y(:)1238 574 y(R)1207 652 y Fg(j)p Fj(x)p Fg(j)p Fj(>s)1318 608 y Fl(g)1340 615 y Fj(N)1373 608 y Fn(\()p Fl(s;)i(x)p Fn(\))p Fl(dx)g Fk(\001)i Fl(N)15 b Fn(+)10 b Fl(o)p Fn(\()p Fl(N)1737 592 y Fj(")1755 608 y Fn(\))41 b Fl(if)c(l)13 b Fn(=)g(2)j Fl(;)1211 704 y(o)p Fn(\()p Fl(N)1293 687 y Fi(1+)p Fj(")1355 704 y Fn(\))441 b Fl(if)37 b(l)13 b(>)g Fn(2)1962 818 y(\(42\))1715 884 y Fb(for)k(any)49 b Fl(")12 b(>)h Fn(0)j Fb(.)249 1077 y(Pr)n(o)n(of:)249 1144 y Fn(The)g(case)45 b Fl(l)13 b Fn(=)g(2)45 b(w)o(as)15 b(considered)h(ab)q(o)o(v)o(e)f(when)h(w)o(e)f(calculated)h(the)g (v)m(ariance)g(of)45 b Fl(\021)1813 1151 y Fj(N)1846 1144 y Fn(\()p Fl(s)p Fn(\))30 b(.)249 1210 y(Assume)16 b(no)o(w)45 b Fl(l)13 b(>)g Fn(2)45 b(and)15 b(denote)h(b)o(y)45 b Fl(r)1035 1217 y Fj(l;m)1134 1210 y Fn(the)15 b(m-th)g(term)g(in)h(\(41\).)249 1277 y(W)l(e)j(are)f(lo)q(oking)h(for)f(the)g(estimates)h(on)59 b(~)-27 b Fl(\032)1057 1284 y Fj(l)p Fi(+)p Fj(m;l)1204 1277 y Fn(similar)20 b(to)54 b(\(21\))17 b(,)37 b(\(22\).)e(Since)61 b(~)-27 b Fl(\032)1914 1284 y Fj(l)p Fi(+)p Fj(m;l)249 1343 y Fn(can)19 b(b)q(e)h(obtained)g(b)o(y)g(addition,)h(substraction)e(and)g(m)o (ultiplication)j(the)d(\014nite)i(n)o(um)o(b)q(er)e(of)g(times)249 1410 y(\(dep)q(ending)e(only)f(on)30 b Fl(l)16 b Fn(\))f(the)g(determinan)o (ts)g(of)g(the)g(form)882 1536 y(det)945 1489 y Fh(\020)1044 1505 y Fn(sin)9 b Fl(\031)r Fn(\()p Fl(x)1180 1512 y Fj(i)1203 1505 y Fk(\000)h Fl(x)1274 1512 y Fj(j)1293 1505 y Fn(\))p 975 1525 405 2 v 975 1567 a Fl(N)i Fn(sin)q(\()p Fl(\031)r Fn(\()p Fl(x)1170 1574 y Fj(i)1193 1567 y Fk(\000)e Fl(x)1264 1574 y Fj(j)1283 1567 y Fn(\))p Fl(=)m(N)5 b Fn(\))1385 1489 y Fh(\021)249 1664 y Fn(\(20\))44 b(pro)o(vides)16 b(the)f(estimates)795 1776 y Fk(j)t Fn(~)-27 b Fl(\032)832 1783 y Fj(l)p Fi(+)p Fj(m;l)924 1776 y Fn(\()p Fl(x)968 1783 y Fi(1)987 1776 y Fl(;)8 b(:)g(:)g(:)d(x)1094 1783 y Fj(l)1107 1776 y Fl(;)j(:)g(:)g(:)d(x)1214 1783 y Fj(l)p Fi(+)p Fj(m)1285 1776 y Fn(\))p Fk(j)12 b Fl(<)h(const)1482 1783 y Fj(l)1962 1776 y Fn(\(43\))249 1888 y(and)632 1897 y Fh(Z)598 2023 y Fi([0)p 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Fj(is)g(connected)1252 2301 y Fn(\()p Fk(\000)p Fn(1\))1346 2282 y Fj(\033)1376 2248 y(l)p Fi(+)p Fj(m)1384 2261 y Fh(Y)1382 2352 y Fj(i)p Fi(=1)1527 2266 y Fn(sin)9 b Fl(\031)r Fn(\()p Fl(x)1663 2273 y Fj(i)1686 2266 y Fk(\000)i Fl(x)1758 2275 y Fj(\033)q Fi(\()p Fj(i)p Fi(\))1820 2266 y Fn(\))p 1458 2290 450 2 v 1458 2332 a Fl(N)h Fn(sin)q(\()p Fl(\031)r Fn(\()p Fl(x)1653 2339 y Fj(i)1676 2332 y Fk(\000)f Fl(x)1748 2341 y Fj(\033)q Fi(\()p Fj(i)p Fi(\))1810 2332 y Fn(\))p Fl(=)m(N)5 b Fn(\))1123 2578 y(21)p eop %%Page: 22 22 bop 249 403 a Fn(Consider)16 b(an)f(arbitary)g(term)f(from)h(this)h(sum.)j (Our)d(goal)f(is)h(to)e(estimate)h(the)h(absolute)f(v)m(alue)i(of)878 474 y Fj(l)p Fi(+)p Fj(m)886 486 y Fh(Y)884 578 y Fj(i)p Fi(=1)1029 492 y Fn(sin)8 b Fl(\031)r Fn(\()p Fl(x)1164 499 y Fj(i)1188 492 y Fk(\000)i Fl(x)1259 501 y Fj(\033)q Fi(\()p Fj(i)p Fi(\))1322 492 y Fn(\))p 960 516 450 2 v 960 558 a Fl(N)i Fn(sin)q(\()p Fl(\031)r Fn(\()p Fl(x)1155 565 y Fj(i)1178 558 y Fk(\000)e 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1536 y Fj(m)249 1626 y Fn(eac)o(h)29 b Fl(\033)391 1633 y Fj(p)453 1626 y Fl(;)8 b(p)k Fn(=)h(1)p Fl(;)8 b(:)g(:)g(:)t(m)43 b Fn(determines)15 b(some)f(cyclic)i(excursion)g(on)e(the)g(graph)28 b Fk(J)9 b Fn(\()p Fl(\033)r Fn(\))k(,)h(with)g(ev)o(ery)249 1692 y(step)h(of)g(the)g(excursion)h(corresp)q(onding)h(to)d(some)h(term)45 b(2)8 b Fl(=)p Fn(\(1)h(+)h Fk(j)p Fl(x)1465 1699 y Fj(i)1489 1692 y Fk(\000)g Fl(x)1560 1702 y Fj(\033)q Fi(\()p Fj(i)p Fi(\))1623 1692 y Fk(j)p Fn(\))45 b(in)16 b(our)f(estimate.)249 1759 y(Since)k(the)e(graph)52 b Fk(J)8 b Fn(\()p Fl(\033)r Fn(\))51 b(is)18 b(connected,)g(the)g(path)f(of)g(ev)o(ery)g(excursion)i(in)o (tersects)e(the)h(path)f(of)249 1825 y(some)e(other)g(excursion,)h(and)f (after)f(sev)o(eral)i(switc)o(hes)f(w)o(e)g(can)h(go)e(from)h(one)g(path)g (to)g(another)249 1892 y(\()g(otherwise)g(w)o(e)g(w)o(ould)h(ha)o(v)o(e)f(a)f (non)o(trivial)j(maximal)e(connected)h(comp)q(onen)o(t)g(of)30 b Fk(J)38 b Fn(\).)249 1958 y(Therefore)15 b(w)o(e)g(can)g(com)o(bine)h (these)g(paths)f(in)o(to)g(one)g(big)h(cyclic)h(path)e(\()g(with)g(p)q (ossible)i(sel\014n)o(tersec-)249 2025 y(tions)e(\),)e(going)i(along)f(whic)o (h)h(w)o(e)f(will)i(visit)f(all)g(v)o(ertices)g(of)28 b Fk(J)36 b Fn(.)20 b(Again,)14 b(to)g(eac)o(h)g(step)h(of)e(the)i(path,)249 2091 y Fl(j)g Fk(\000)-7 b(!)13 b Fl(l)46 b Fn(there)15 b(corresp)q(onds)h(a) e(term)712 2164 y(2)p 579 2185 289 2 v 579 2226 a(1)c(+)g Fk(j)p Fl(x)696 2233 y Fj(i)720 2226 y Fk(\000)g Fl(x)791 2236 y Fj(\033)q Fi(\()p Fj(i)p Fi(\))854 2226 y Fk(j)930 2195 y Fn(:)58 b Fl(x)1027 2202 y Fj(i)1053 2195 y Fk(2)13 b Fn([)p Fl(x)1135 2202 y Fj(j)1153 2195 y Fl(;)8 b(x)1200 2202 y Fj(j)1228 2195 y Fn(+)i Fl(s)p Fn(])p Fl(;)38 b(x)1384 2204 y Fj(\033)q Fi(\()p Fj(i)p Fi(\))1474 2195 y Fk(2)13 b Fn([)p Fl(x)1556 2202 y Fj(l)1569 2195 y Fl(;)8 b(x)1616 2202 y Fj(l)1638 2195 y Fn(+)j Fl(s)p Fn(])249 2309 y(in)16 b(our)f(estimate.)249 2376 y(The)20 b(whole)g(n)o(um)o(b)q(er)g(of)g (steps)f(of)h(the)f(constructed)h(path)g(is)g(at)f(most)1551 2359 y Fi(3)1610 2376 y Fl(l)14 b Fn(+)f Fl(m)20 b Fn(.)33 b(No)o(w)19 b(w)o(e)h(will)p 249 2411 718 2 v 301 2437 a Ff(3)318 2453 y Fo(See)14 b(Remark)f(after)g(the)g(end)h(of)e(the)h(Lemma's)g(pro)q (of)1123 2578 y Fn(22)p eop %%Page: 23 23 bop 249 403 a Fn(eliminate)18 b(all)f(p)q(ossible)h(sel\014n)o(tersections.) 23 b(If)33 b Fl(j)h Fn(is)17 b(the)f(curren)o(t)g(p)q(osition)h(of)f(our)f(w) o(alk,)32 b Fl(n)h Fn(is)16 b(the)249 469 y(previous)k(one)e(and)38 b Fl(l)g Fn(is)19 b(the)g(next)f(one,)i(and)f(v)o(ector)36 b Fl(j)k Fn(has)18 b(b)q(een)i(already)f(visited)h(b)q(efore,)g(w)o(e)249 536 y(replace)c(t)o(w)o(o)e(steps)46 b Fl(n)12 b Fk(\000)-7 b(!)13 b Fl(j)18 b(;)37 b(j)15 b Fk(\000)-7 b(!)13 b Fl(l)46 b Fn(b)o(y)15 b(one)46 b Fl(n)13 b Fk(\000)-8 b(!)14 b Fl(l)45 b Fn(in)17 b(the)e(new)g(,mo)q(di\014ed)h(w)o(alk.)249 602 y(Using)g(the)f(inequalities)689 701 y(2)p Fl(=)p Fn(\(1)9 b(+)h Fk(j)p Fl(x)869 708 y Fj(i)893 701 y Fk(\000)g Fl(x)964 710 y Fj(\033)q Fi(\()p Fj(i)p Fi(\))1027 701 y Fk(j)p Fn(\))i Fk(\024)h Fl(const)p Fn(\()p Fl(s)p Fn(\))j(2)p Fl(=)p Fn(\(1)9 b(+)h Fk(j)p Fl(x)1477 708 y Fj(j)1505 701 y Fk(\000)h Fl(x)1577 708 y Fj(l)1589 701 y Fk(j)p Fn(\))689 780 y(2)p Fl(=)p Fn(\(1)e(+)h Fk(j)p Fl(x)g Fk(\000)g Fl(y)r Fk(j)p Fn(\))15 b(2)p Fl(=)p Fn(\(1)9 b(+)h Fk(j)p Fl(y)i Fk(\000)e Fl(z)r Fk(j)p Fn(\))i Fk(\024)h Fn(2)d Fk(\001)g Fn(2)p Fl(=)p Fn(\(1)f(+)h Fk(j)p Fl(x)g Fk(\000)g Fl(z)r Fk(j)p Fn(\))249 938 y(and)15 b(subsequen)o(tly)h (eliminating)h(from)d(the)h(path)g(sites)g(visited)h(b)q(efore,)f(w)o(e)f (\014nally)j(obtain)e(the)f(path)249 1004 y(without)h(sel\014n)o (tersections,giv)o(en)i(b)o(y)e(some)g(cyclic)i(p)q(erm)o(utation)45 b Fl(\034)18 b Fk(2)28 b Fl(S)1576 1011 y Fj(l)1619 1004 y Fl(:)249 1070 y Fn(This)16 b(leads)g(to)29 b(\(45\))h(and)15 b(the)g(inequalit)o(y)708 1141 y Fh(Z)572 1269 y Fi(\()p Fg(t)610 1257 y Fd(l)610 1281 y(j)q Ff(=1)679 1269 y Fi([)p Fj(x)709 1274 y Fd(j)725 1269 y Fj(;x)755 1274 y Fd(j)771 1269 y Fi(+)p Fj(s)p Fi(])h(\))854 1259 y Fd(m)930 1125 y Fh(\014)930 1150 y(\014)930 1175 y(\014)930 1200 y(\014)930 1224 y(\014)944 1146 y Fj(l)p Fi(+)p Fj(m)952 1158 y Fh(Y)950 1249 y Fj(i)p Fi(=1)1125 1164 y Fn(sin)9 b Fl(\031)r Fn(\()p Fl(x)1261 1171 y Fj(i)1284 1164 y Fk(\000)h Fl(x)1355 1173 y Fj(\033)q Fi(\()p Fj(i)p Fi(\))1418 1164 y Fn(\))p 1056 1188 450 2 v 1056 1230 a Fl(N)i Fn(sin)q(\()p Fl(\031)r Fn(\()p Fl(x)1251 1237 y Fj(i)1274 1230 y Fk(\000)e Fl(x)1345 1239 y Fj(\033)q Fi(\()p Fj(i)p Fi(\))1408 1230 y Fn(\))p Fl(=)m(N)5 b Fn(\))1510 1125 y Fh(\014)1510 1150 y(\014)1510 1175 y(\014)1510 1200 y(\014)1510 1224 y(\014)1547 1198 y Fl(dx)1597 1205 y Fj(l)p Fi(+1)1662 1198 y Fl(:)j(:)g(:)e(dx)1773 1205 y Fj(l)p Fi(+)p Fj(m)489 1386 y Fk(\024)57 b Fn(\()p Fl(l)10 b Fk(\001)g Fl(s)p Fn(\))685 1367 y Fj(m)718 1386 y Fn(\()p Fl(const)p Fn(\()p Fl(s)p Fn(\)\))917 1364 y Fj(l)p Fi(+)p Fj(m)1020 1346 y Fh(X)1011 1438 y Fj(\034)t Fg(2)p Fj(S)1076 1444 y Fd(l)1147 1333 y Fj(l)1126 1346 y Fh(Y)1143 1436 y Fi(1)1327 1356 y Fn(2)p 1191 1376 295 2 v 1191 1417 a(1)g(+)h Fk(j)p Fl(x)1309 1424 y Fj(j)1337 1417 y Fk(\000)f Fl(x)1408 1427 y Fj(\034)t Fi(\()p Fj(j)r Fi(\))1473 1417 y Fk(j)249 1516 y Fn(The)16 b(inequalit)o(y)430 1588 y Fj(N)416 1596 y Fh(Z)419 1717 y Fi(0)480 1654 y Fn(2)p Fl(=)p Fn(\(1)9 b(+)i Fk(j)p Fl(x)e Fk(\000)i Fl(y)r Fk(j)p Fn(\))e Fk(\001)h Fn(2)p Fl(=)p Fn(\(1)f(+)h Fk(j)p Fl(y)i Fk(\000)e Fl(z)r Fk(j)p Fn(\))15 b Fl(dy)f Fk(\024)28 b Fl(const)c Fn(log\()p Fl(N)15 b Fn(+)10 b(1\))15 b(2)p Fl(=)p Fn(\(1)9 b(+)h Fk(j)p Fl(x)g Fk(\000)g Fl(z)r Fk(j)p Fn(\))249 1789 y(implies)509 1825 y Fh(Z)476 1951 y Fi([0)p Fj(;N)t Fi(])556 1941 y Fd(l)639 1830 y Fj(l)618 1842 y Fh(Y)613 1933 y Fj(j)r Fi(=1)697 1882 y Fn(2)p Fl(=)p Fn(\(1)f(+)i Fk(j)p Fl(x)878 1889 y Fj(j)906 1882 y Fk(\000)f Fl(x)977 1892 y Fj(\034)t Fi(\()p Fj(j)r Fi(\))1042 1882 y Fk(j)p Fn(\))15 b Fl(dx)1138 1889 y Fi(1)1165 1882 y Fl(:)8 b(:)g(:)d(dx)1275 1889 y Fj(l)1301 1882 y Fk(\024)28 b Fl(const)16 b(N)c Fn(log)1593 1864 y Fj(l)p Fg(\000)p Fi(2)1652 1882 y Fn(\()p Fl(N)i Fn(+)c(1\))249 2019 y(and)272 2118 y Fk(j)309 2061 y Fh(Z)275 2187 y Fi([0)p Fj(;N)t Fi(])355 2177 y Fd(l)407 2118 y Fl(r)428 2125 y Fj(l;m)494 2118 y Fn(\()p Fl(x)538 2125 y Fi(1)558 2118 y Fl(;)e(:)g(:)g(:)t(x)664 2125 y Fj(l)677 2118 y Fn(\))k Fl(dx)757 2125 y Fi(1)784 2118 y Fl(:)c(:)g(:)e(dx)895 2125 y Fj(l)908 2118 y Fk(j)24 b(\024)1025 2087 y Fn(1)p 1011 2108 53 2 v 1011 2149 a Fl(m)p Fn(!)1068 2118 y Fl(l)q Fn(!\()p Fl(l)5 b Fn(+)g Fl(m)p Fn(\)!\()p Fl(l)10 b(s)p Fn(\))1325 2099 y Fj(m)1359 2118 y Fn(\()p Fl(const)p Fn(\()p Fl(s)p Fn(\)\))1558 2099 y Fj(l)p Fi(+)p Fj(m)1629 2118 y Fl(N)i Fn(log)1737 2099 y Fj(l)p Fg(\000)p Fi(2)1795 2118 y Fn(\()p Fl(N)d Fn(+)c(1\))22 b(\(46\))249 2270 y(Finally)l(,to)16 b(get)f(an)g(estimate)30 b(\(42\))f(of)448 2389 y Fk(j)485 2332 y Fh(Z)451 2457 y Fi([0)p Fj(;N)t Fi(])531 2448 y Fd(l)589 2389 y Fl(r)610 2396 y Fj(l)638 2389 y Fn(\()p Fl(x)682 2396 y Fi(1)701 2389 y Fl(;)8 b(:)g(:)g(:)d(x)808 2396 y Fj(l)821 2389 y Fn(\))15 b Fl(dx)904 2396 y Fi(1)931 2389 y Fl(:)8 b(:)g(:)d(dx)1041 2396 y Fj(l)1054 2389 y Fk(j)27 b(\024)1155 2336 y Fg(1)1142 2349 y Fh(X)1163 2439 y Fi(0)1250 2332 y Fh(Z)1216 2457 y Fi([0)p Fj(;N)t Fi(])1296 2448 y Fd(l)1354 2389 y Fk(j)p Fl(r)1388 2396 y Fj(l)1415 2389 y Fn(\()p Fl(x)1459 2396 y Fi(1)1478 2389 y Fl(;)8 b(:)g(:)g(:)d(x)1585 2396 y Fj(l)1598 2389 y Fn(\))p Fk(j)14 b Fl(dx)1693 2396 y Fi(1)1720 2389 y Fl(:)8 b(:)g(:)e(dx)1831 2396 y Fj(l)1123 2578 y Fn(23)p eop %%Page: 24 24 bop 249 403 a Fn(one)15 b(can)h(use)46 b(\(46\))e(for)h Fl(m)12 b(<)28 b(const)p Fn(\()p Fl(s;)8 b(l)q(;)g(")p Fn(\))g(log)o(\()p Fl(N)14 b Fn(+)d(1\))249 469 y(where)46 b Fl(const)p Fn(\()p Fl(s;)8 b(l)q(;)g(")p Fn(\))44 b(is)16 b(small)g(enough)30 b(,)45 b(and)h(\(44\))e(for)h Fl(m)13 b Fk(\025)28 b Fl(const)p Fn(\()p Fl(s;)8 b(l)q(;)g(")p Fn(\))g(log)o(\()p Fl(N)14 b Fn(+)d(1\))p 249 540 21 30 v 249 661 a(\(39\))44 b(,\(40\))g(,\(42\))g(giv)o (e)16 b(us)f(the)g(follo)o(wing)h(result)g(:)578 802 y(I)-9 b(E)770 745 y Fh(Z)732 873 y Fi([0)p Fj(;N)t Fi(])812 861 y Ff(2)p Fd(k)635 911 y Fi(min)694 917 y Fd(i)p Fp(6)p Ff(=)p Fd(j)750 911 y Fg(j)p Fj(x)780 916 y Fd(i)793 911 y Fg(\000)p Fj(x)840 916 y Fd(j)857 911 y Fg(j)867 916 y Ff(1)884 911 y Fj(>s)957 749 y Fi(2)p Fj(k)949 762 y Fh(Y)967 852 y Fi(1)1002 802 y Fn(\()p Fl(\037)p Fn(\()p Fl(x)1092 809 y Fj(i)1106 802 y Fl(;)8 b(dx)1177 809 y Fj(i)1190 802 y Fl(;)g(s)p Fn(\))h Fk(\000)i Fn(I)-9 b(E)p Fl(\037)p Fn(\()p Fl(x)1415 809 y Fj(i)1429 802 y Fl(;)8 b(dx)1500 809 y Fj(i)1513 802 y Fl(;)g(s)p Fn(\)\))501 1000 y(=)57 b(\(2)p Fl(k)11 b Fk(\000)f Fn(1\)!!)k(\()849 943 y Fh(Z)819 1067 y Fg(j)p Fj(x)p Fg(j)p Fj(>s)934 1000 y Fl(g)956 1007 y Fj(N)989 1000 y Fn(\()p Fl(s;)8 b(x)p Fn(\))p Fl(dx)14 b Fn(\))1174 978 y Fj(k)1211 1000 y Fl(N)1253 981 y Fj(k)1299 1000 y Fn(+)26 b Fl(O)1411 953 y Fh(\020)1451 1000 y Fl(const)p Fn(\()p Fl(s;)8 b(")p Fn(\))p Fl(N)1698 981 y Fj(k)q Fg(\000)p Fi(1+)p Fj(")1807 953 y Fh(\021)1962 1000 y Fn(\(47\))249 1222 y Fc(Remark)75 b Fn(The)19 b(trivial)h(b)q(ound)g(for)e(the)h(n)o(um)o(b)q (er)g(of)f(steps)h(of)f(the)h(path)g(\(at)e(most)37 b Fl(l)13 b Fn(+)g Fl(m)18 b Fn(\))h(is)249 1289 y(enough)e(for)f(our)g(purp)q(oses)h (no)o(w.)24 b(Ho)o(w)o(ev)o(er)15 b(in)j(the)e(pro)q(of)g(of)g(Corollary)h (1.5)e(w)o(e)i(need)g(an)f(estimate)249 1355 y(that)f(the)g(n)o(um)o(b)q(er)g (of)g(steps)g(is)h(b)q(ounded)h(b)o(y)e(some)g(n)o(um)o(b)q(er,)g(dep)q (ending)i(only)f(on)30 b Fl(l)16 b(:)249 1421 y Fn(T)l(o)d(acomplish)i(this)e (w)o(e)h(ha)o(v)o(e)e(to)h(eliminate)i(some)e(lo)q(ops)h(of)f(the)h(path)f (\(i.e.)19 b(replace)14 b(the)g(corresp)q(ond-)249 1488 y(ing)k(m)o (ultipliers)h(b)o(y)e(1)f(\).)25 b(Namely)18 b(w)o(e)e(eliminate)j(a)e(lo)q (op,)g(if)h(after)e(thro)o(wing)h(it)g(out,)g(w)o(e)f(still)j(ha)o(v)o(e)249 1554 y(a)e(closed)h(path,visiting)g(all)h(v)o(ertices)e(of)g Fk(J)26 b Fl(:)16 b Fn(After)h(suc)o(h)h(pro)q(cedure)g(completed)g(w)o(e)f (arriv)o(e)h(at)e(the)249 1621 y(path)f(with)h(at)e(most)656 1588 y Fh(P)700 1601 y Fj(l)p Fg(\000)p Fi(1)700 1633 y(1)765 1621 y Fl(j)28 b Fn(+)10 b Fl(l)h Fk(\000)f Fn(1)j(=)g Fl(l)q Fn(\()p Fl(l)c Fn(+)i(1\))p Fl(=)p Fn(2)24 b Fk(\000)10 b Fn(1)46 b(steps.)291 1812 y(In)15 b(the)f(second)h(part)f(of)f(the)i(pro)q(of)f(of)f (Theorem)h(1.1)f(w)o(e)h(tak)o(e)g(in)o(to)g(accoun)o(t)g(the)g(con)o (tributions)h(to)249 1879 y(I)-9 b(E)31 b(\()o Fl(\021)358 1886 y Fj(N)392 1879 y Fn(\()p Fl(s)p Fn(\))9 b Fk(\000)i Fn(I)-9 b(E)15 b Fl(\021)580 1886 y Fj(N)613 1879 y Fn(\()p Fl(s)p Fn(\)\))688 1857 y Fi(2)p Fj(k)772 1879 y Fn(from)g(the)g(diagonal)h(\()45 b Fl(x)1229 1886 y Fj(i)1255 1879 y Fn(=)13 b Fl(x)1329 1886 y Fj(j)1393 1879 y Fn(\))i(and)g("near-diagonal")249 1945 y(\()45 b(0)13 b Fl(<)g Fk(j)p Fl(x)435 1952 y Fj(i)458 1945 y Fk(\000)e Fl(x)530 1952 y Fj(j)548 1945 y Fk(j)561 1952 y Fi(1)593 1945 y Fk(\024)i Fl(s)31 b Fn(\))14 b(terms.)249 2012 y(W)l(e)f(in)o(tro)q(duce)i (an)e(equiv)m(alence)j(relation)e(on)f(the)g(set)h(of)e(particles)41 b Fk(f)p Fl(x)1502 2019 y Fi(1)1522 2012 y Fl(;)8 b(:)g(:)g(:)t(x)1628 2019 y Fi(2)p Fj(k)1667 2012 y Fk(g)40 b Fn(,)13 b(calling)28 b Fl(x)1937 2019 y Fj(i)1951 2012 y Fl(;)21 b(x)2011 2019 y Fj(j)249 2078 y Fn(the)15 b("neigh)o(b)q(ors",)g(if)h(there)f(is)h(a)f (sequence)h(of)f(particles)906 2190 y Fl(x)932 2197 y Fj(i)944 2202 y Ff(0)976 2190 y Fn(=)e Fl(x)1050 2197 y Fj(i)1064 2190 y Fl(;)8 b(x)1111 2197 y Fj(i)1123 2202 y Ff(1)1142 2190 y Fl(;)g(:)g(:)g(:)t(x)1248 2197 y Fj(i)1260 2201 y Fd(p)1293 2190 y Fn(=)13 b Fl(x)1367 2197 y Fj(j)249 2302 y Fn(suc)o(h)j(that)914 2369 y(max)875 2397 y Fj(r)q Fi(=)p Fj(o;:::)o(p)p Fg(\000)p Fi(1)1076 2369 y Fk(j)p Fl(x)1115 2376 y Fj(i)1127 2381 y Fd(r)q Ff(+1)1195 2369 y Fk(\000)10 b Fl(x)1266 2376 y Fj(i)1278 2380 y Fd(r)1297 2369 y Fk(j)1310 2376 y Fi(1)1342 2369 y Fk(\024)j Fl(s:)538 b Fn(\(48\))1123 2578 y(24)p eop %%Page: 25 25 bop 249 406 a Fn(W)l(e)14 b(claim)h(that)e(the)h(con)o(tributions)g(to)41 b(I)-9 b(E)28 b(\()p Fl(\021)1084 413 y Fj(N)1117 406 y Fn(\()p Fl(s)p Fn(\))7 b Fk(\000)g Fn(I)-9 b(E)14 b Fl(\021)1298 413 y Fj(N)1332 406 y Fn(\()p Fl(s)p Fn(\)\))1406 384 y Fi(2)p Fj(k)1486 406 y Fn(of)g(order)f(of)27 b Fl(N)1760 390 y Fj(k)1809 406 y Fn(app)q(ear)14 b(only)249 473 y(when)i(eac)o(h)f(equiv)m(alence)j (class)e(of)29 b(\(48\))h(con)o(tains)15 b(one)g(or)g(t)o(w)o(o)f(particles.) 249 539 y(Assume)i(that)e(w)o(e)h(ha)o(v)o(e)30 b Fl(l)h Fn(t)o(w)o(o-elemen) o(t)14 b(equiv)m(alence)k(classes,)e(sa)o(y)915 631 y Fk(f)p Fl(x)964 638 y Fi(1)983 631 y Fl(;)8 b(x)1030 638 y Fi(2)1049 631 y Fk(g)p Fl(;)g(:)g(:)g(:)d Fk(f)p Fl(x)1202 638 y Fi(2)p Fj(l)p Fg(\000)p Fi(1)1277 631 y Fl(;)j(x)1324 638 y Fi(2)p Fj(l)1354 631 y Fk(g)249 723 y Fn(and)31 b(2)p Fl(k)11 b Fk(\000)f Fn(2)p Fl(l)46 b Fn(one-elemen)o(t)16 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y Fj(l)p Fi(=0)566 2360 y Fn(\(2)p Fl(k)q Fn(\)!)p 477 2380 274 2 v 477 2422 a(\(2)p Fl(l)q Fn(\)!\(2)p Fl(k)8 b Fk(\000)j Fn(2)p Fl(l)q Fn(\)!)760 2360 y(\(2)p Fl(l)q Fn(\)!)p 760 2380 86 2 v 764 2422 a Fl(l)q Fn(!)j(2)829 2409 y Fj(l)858 2344 y Fh(\020)883 2391 y Fl(F)912 2398 y Fj(N)946 2391 y Fn(\()p Fl(s)p Fn(\))9 b Fk(\000)i Fn(2)p Fl(sF)1137 2372 y Fi(2)1131 2402 y Fj(N)1165 2391 y Fn(\()p Fl(s)p Fn(\))1222 2344 y Fh(\021)1246 2355 y Fj(l)1298 2360 y Fn(\(2)p Fl(k)f Fk(\000)h Fn(2)p Fl(l)q Fn(\)!)p 1272 2380 241 2 v 1272 2422 a(\()p Fl(k)g Fk(\000)f Fl(l)q Fn(\)!)k(2)1453 2409 y Fj(k)q Fg(\000)p Fj(l)1517 2344 y Fh(\020)1563 2333 y(Z)1533 2457 y Fg(j)p Fj(x)p Fg(j)p Fj(>s)1633 2391 y Fl(g)1655 2398 y Fj(N)1688 2391 y Fn(\()p Fl(s;)8 b(x)p Fn(\))p Fl(dx)1842 2344 y Fh(\021)1866 2355 y Fj(k)q Fg(\000)p Fj(l)1935 2391 y Fk(\001)i Fl(N)2000 2372 y Fj(k)1123 2578 y Fn(25)p eop %%Page: 26 26 bop 312 403 a Fn(+)42 b Fl(O)q Fn(\()p Fl(const)p Fn(\()p Fl(s;)8 b(")p Fn(\))p Fl(N)690 384 y Fj(k)q Fg(\000)p Fi(1+)p Fj(")799 403 y Fn(\))312 539 y(=)42 b(\(2)p Fl(k)11 b Fk(\000)f Fn(1\)!!)585 443 y Fh(0)583 516 y(B)583 542 y(@)620 539 y Fl(F)649 546 y Fj(N)682 539 y Fn(\()p Fl(s)p Fn(\))g Fk(\000)h Fn(2)p Fl(sF)874 521 y Fi(2)868 551 y Fj(N)902 539 y Fn(\()p Fl(s)p Fn(\))e(+)1035 482 y Fh(Z)1004 606 y Fg(j)p Fj(x)p Fg(j)p Fj(>s)1105 539 y Fl(g)1127 546 y Fj(N)1160 539 y Fn(\()p Fl(s;)f(x)p Fn(\))p Fl(dx)1314 443 y Fh(1)1314 516 y(C)1314 542 y(A)1349 454 y Fj(k)1385 539 y Fl(N)1427 521 y Fj(k)312 677 y Fn(+)42 b Fl(O)q Fn(\()p Fl(const)p Fn(\()p Fl(s;)8 b(")p Fn(\))p Fl(N)690 658 y Fj(k)q Fg(\000)p Fi(1+)p Fj(")799 677 y Fn(\))1145 b(\(49\))249 848 y(If)16 b(some)e(equiv)m(alence)k(classes)e(ha)o(v)o(e)f(three)g(elemen)o (ts)h(or)f(more,)f(the)h(con)o(tributing)h(terms)f(to)249 931 y(I)-9 b(E)21 b(\()o Fl(\021)348 938 y Fj(N)381 931 y Fn(\()p Fl(s)p Fn(\))p Fk(\000)p Fn(I)-9 b(E)11 b Fl(\021)545 938 y Fj(N)578 931 y Fn(\()p Fl(s)p Fn(\)\))652 908 y Fi(2)p Fj(k)722 931 y 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Fn(\(1\))253 b(\(50\))249 1797 y(Similar)17 b(calculations)g(yield)651 1914 y(I)-9 b(E)712 1867 y Fh(\020)737 1914 y Fn(\()o Fl(\021)777 1921 y Fj(N)811 1914 y Fn(\()p Fl(s)p Fn(\))9 b Fk(\000)i Fn(I)-9 b(E)15 b Fl(\021)999 1921 y Fj(N)1032 1914 y Fn(\()p Fl(s)p Fn(\)\))7 b Fl(=)m(N)1176 1896 y Fi(1)p Fj(=)p Fi(2)1231 1867 y Fh(\021)1256 1879 y Fi(2)p Fj(k)q Fi(+1)1352 1914 y Fn(=)29 b Fl(o)p Fn(\()p Fl(N)1498 1896 y Fg(\000)p Fi(1)p Fj(=)p Fi(2+)p Fj(")1623 1914 y Fn(\))249 2026 y(The)16 b(con)o(v)o(ergence)f(of)g(the)g(mixed)h(momen)o(ts)791 2156 y(I)-9 b(E)868 2104 y Fj(n)852 2116 y Fh(Y)870 2207 y Fi(1)928 2109 y Fh(\020)953 2156 y Fn(\()p Fl(\021)994 2163 y Fj(N)1027 2156 y Fn(\()p Fl(s)1066 2163 y Fj(i)1080 2156 y Fn(\))10 b Fk(\000)g Fn(I)-9 b(E)15 b Fl(\021)1229 2163 y Fj(N)1263 2156 y Fn(\()p Fl(s)1302 2163 y Fj(i)1316 2156 y Fn(\)\))7 b Fl(=)m(N)1421 2138 y Fi(1)p Fj(=)p Fi(2)1475 2109 y Fh(\021)249 2295 y Fn(to)15 b(the)g(momen)o(ts)f(of)h(Gaussian)g(random)g(pro)q(cess)46 b Fl(\030)r Fn(\()p Fl(s)p Fn(\))f(can)15 b(b)q(e)h(pro)o(v)o(en)f(in)h(the)f (same)g(w)o(a)o(y)l(.)249 2361 y(Since)h(the)e(con)o(v)o(ergence)g(of)f(all)i (momen)o(ts)e(to)h(the)g(gaussian)g(ones)g(implies)i(the)e(con)o(v)o(ergence) g(of)f(\014nite)249 2428 y(dimensional)k(distributions,)g(Theorem)e(1)g(is)g (pro)o(v)o(en.)p 1244 2432 21 30 v 1123 2578 a(26)p eop %%Page: 27 27 bop 249 403 a Fm(3)69 b(Pro)r(of)24 b(of)f(Theorem)d(1.2)k(and)g(Corollaries) 249 514 y Fn(W)l(e)15 b(start)f(with)i(the)f(pro)q(of)g(of)g(Theorem)g(1.2)f (.)20 b(Since)902 567 y Fh(\014)902 592 y(\014)902 617 y(\014)920 604 y Fn(~)916 616 y Fl(\030)936 623 y Fj(N)969 616 y Fn(\()p Fl(s)p Fn(\))10 b Fk(\000)g Fl(\030)1101 623 y Fj(N)1135 616 y Fn(\()p Fl(s)p Fn(\))1192 567 y Fh(\014)1192 592 y(\014)1192 617 y(\014)1218 616 y Fk(\024)j Fl(N)1308 597 y Fg(\000)p Fi(1)p Fj(=)p Fi(2)1962 616 y Fn(\(51\))249 724 y(the)i(\014nite-dimensional)k (distributions)e(of)1047 712 y(~)1042 724 y Fl(\030)1062 731 y Fj(N)1096 724 y Fn(\()p Fk(\001)p Fn(\))44 b(also)16 b(con)o(v)o(erge)e(to) h(those)g(of)g(limiting)i(Gaussian)249 790 y(pro)q(cess)33 b Fl(\030)445 797 y Fj(N)479 790 y Fn(\()p Fk(\001)p Fn(\))e(as)i Fl(N)17 b Fk(!)c(1)22 b Fl(;)11 b Fn(and)g(for)g(the)g(functional)h(con)o(v)o (ergence)f(of)g(probabilit)o(y)h(distributions)249 856 y(of)332 844 y(~)327 856 y Fl(\030)347 863 y Fj(N)381 856 y Fn(\()p Fk(\001)p Fn(\))40 b(on)i Fl(C)10 b Fn([0)p Fl(;)e Fk(1)p Fn(\))40 b(it)15 b(is)f(enough)g(to)g(pro)o(v)o(e)f(the)h(tigh)o(tness)g(\(relativ)o (e)g(compactness)g(\))f(of)h(an)o(y)249 923 y(sequence)i(of)f(distributions)i (of)842 911 y(~)837 923 y Fl(\030)857 930 y Fj(N)885 934 y Fd(n)908 923 y Fn(\()p Fk(\001)p Fn(\))45 b(on)g Fl(C)10 b Fn([0)p Fl(;)e(T)e Fn(])44 b(as)30 b Fl(N)1393 930 y Fj(n)1429 923 y Fk(!)13 b(1)30 b Fl(;)8 b(T)36 b Fn(is)16 b(arbitary)e(large)249 989 y(\()h([20)o(])g(\).)249 1056 y(Let)f(us)f(de\014ne)i(for)e(an)o(y)39 b Fl(f)18 b Fk(2)13 b Fl(C)e Fn([0)p Fl(;)d(T)e Fn(])38 b(and)i Fl(\016)15 b(>)e Fn(0)26 b Fl(;)13 b Fn(the)h(mo)q(dulus)g(of)f(con)o(tin)o (uit)o(y)h(of)f(the)g(function)249 1122 y Fl(f)51 b Fn(as)588 1188 y Fl(!)616 1195 y Fj(f)639 1188 y Fn(\()p Fl(\016)r Fn(\))12 b(=)20 b(sup)c Fk(f)f(j)p Fl(f)5 b Fn(\()p Fl(s)p Fn(\))k Fk(\000)i Fl(f)5 b Fn(\()p Fl(t)p Fn(\))p Fk(j)27 b Fn(:)g(0)13 b Fk(\024)g Fl(s;)8 b(t)k Fk(\024)h Fl(T)t(;)22 b Fk(j)p Fl(s)10 b Fk(\000)g Fl(t)p Fk(j)j Fl(<)g(\016)k Fk(g)249 1277 y Fn(The)22 b(classical)g (criterion)h(of)e(relativ)o(e)g(compactness)h(\()f([20)o(])f(\))h(tells)i (that)d(the)i(family)64 b Fk(fP)t(g)f Fn(of)249 1343 y(probabilit)o(y)17 b(measures)e(on)45 b Fl(C)s Fn([0)p Fl(;)8 b(T)e Fn(])43 b(is)16 b(relativ)o(ely)g(compact)f(i\013)249 1410 y(\(i\))61 b(for)14 b(eac)o(h)i(arbitary)e(small)i(p)q(ositiv)o(e)32 b Fl(\013)e Fn(there)15 b(exists)h(an)30 b Fl(A)p Fn(\()p Fl(\013)p Fn(\))g Fl(;)15 b Fn(suc)o(h)g(that)735 1511 y Fk(P)j(f)p Fl(f)g Fn(:)27 b Fk(j)p Fl(f)5 b Fn(\(0\))p Fk(j)11 b Fl(>)i(A)j Fk(g)c Fl(<)h(\013)30 b(;)38 b(f)5 b(or)31 b(any)i Fk(P)t Fl(:)404 b Fn(\(52\))249 1672 y(\(ii\))61 b(for)15 b(eac)o(h)45 b Fl(\013;)23 b(\014)15 b(>)e Fn(0)45 b(there)15 b(exists)h(some)45 b Fl(\016)r Fn(\()p Fl(\013;)22 b(\014)r Fn(\))45 b(suc)o(h)16 b(that)724 1774 y Fk(P)i(f)p Fl(f)g Fn(:)27 b Fl(!)905 1781 y Fj(f)928 1774 y Fn(\()p Fl(\016)r Fn(\))12 b Fl(>)h(\014)k Fk(g)c Fl(<)28 b(\013)i(;)38 b(f)5 b(or)31 b(any)i Fk(P)18 b Fl(:)394 b Fn(\(53\))249 1934 y(The)16 b(results)f(of)g(O.Costin)g(and)g(J.Leb)q(o)o(witz)h(\()f([6])f (\))h(tell)i(us)e(that)890 2024 y(~)885 2036 y Fl(\030)905 2043 y Fj(N)939 2036 y Fn(\(0\))c(=)i Fl(O)1101 1989 y Fh(\020)1126 2036 y Fn(\(log)8 b Fl(N)q(=)m(N)d Fn(\))1328 2017 y Fi(1)p Fj(=)p Fi(2)1382 1989 y Fh(\021)249 2138 y Fn(whic)o(h)16 b(giv)o(es)g(us)f (\(i\).)20 b(T)l(o)15 b(pro)o(v)o(e)f(\(ii\))i(w)o(e)f(need)249 2254 y Fc(Lemm)o(a)f(3.1)23 b Fb(Ther)n(e)16 b(exist)g(some)g(c)n(onstants)47 b Fl(c)1129 2261 y Fi(1)1148 2254 y Fl(;)8 b(c)1189 2261 y Fi(2)1257 2254 y Fb(dep)n(ending)15 b(on)49 b Fl(T)18 b(>)13 b Fn(0)33 b Fl(;)15 b Fb(such)i(that)327 2356 y Fk(P)359 2363 y Fj(N)511 2309 y Fh(n)555 2356 y Fk(j)573 2344 y Fn(~)568 2356 y Fl(\030)588 2363 y Fj(N)621 2356 y Fn(\()p Fl(t)p Fn(\))10 b Fk(\000)733 2344 y Fn(~)728 2356 y Fl(\030)748 2363 y Fj(N)782 2356 y Fn(\()p Fl(s)p Fn(\))p Fk(j)i Fl(<)20 b(c)939 2363 y Fi(1)959 2356 y Fn(\()o Fk(j)p Fl(t)11 b Fk(\000)f Fl(s)p Fk(j)1095 2337 y Fi(1)p Fj(=)p Fi(20)1178 2356 y Fn(+)g Fl(N)1265 2337 y Fg(\000)p Fi(1)p Fj(=)p Fi(4)1347 2356 y Fn(\))16 b Fl(;)24 b Fk(8)16 b Fl(t;)8 b(s)13 b Fk(2)g Fn([0)p Fl(;)8 b(T)e Fn(])k(:)i Fk(j)p Fl(t)f Fk(\000)f Fl(s)p Fk(j)j Fl(<)g(\016)1929 2309 y Fh(o)434 2442 y Fl(>)58 b Fn(1)10 b Fk(\000)18 b Fl(c)633 2449 y Fi(2)660 2442 y Fn(\()p Fl(\016)700 2423 y Fi(4)p Fj(=)p Fi(5)765 2442 y Fn(+)10 b Fl(N)852 2423 y Fg(\000)p Fi(1)p Fj(=)p Fi(20)959 2442 y Fn(log)f Fl(N)c Fn(\))876 b(\(54\))1123 2578 y(27)p eop %%Page: 28 28 bop 249 403 a Fn(Assuming)16 b(Lemma)f(3.1)f(is)i(pro)o(v)o(en)f(,)g(w)o(e)g (can)g(quic)o(kly)i(\014nish)f(the)f(pro)q(of)g(of)g(Theorem)g(1.2)f(:)249 469 y(Let)i(us)f(c)o(ho)q(ose)45 b Fl(N)600 476 y Fg(\003)620 469 y Fl(;)8 b(\016)661 476 y Fg(\003)725 469 y Fn(suc)o(h)16 b(that)882 576 y Fl(c)902 583 y Fi(1)921 576 y Fl(N)963 552 y Fg(\000)p Fi(1)p Fj(=)p Fi(4)958 583 y Fg(\003)1058 576 y Fl(<)d(\014)r(=)p Fn(2)56 b Fl(;)41 b(c)1310 583 y Fi(1)1330 576 y Fl(\016)1352 552 y Fi(1)p Fj(=)p Fi(20)1350 583 y Fg(\003)1437 576 y Fl(<)13 b(\014)r(=)p Fn(2)733 655 y Fl(c)753 662 y Fi(2)780 655 y Fn(log)8 b Fl(N)883 662 y Fg(\003)918 655 y Fl(N)960 631 y Fg(\000)p Fi(1)p Fj(=)p Fi(20)955 661 y Fg(\003)1072 655 y Fl(<)13 b(\013=)p Fn(2)41 b Fl(;)g(c)1310 662 y Fi(2)1330 655 y Fl(\016)1352 631 y Fi(4)p Fj(=)p Fi(5)1350 661 y Fg(\003)1419 655 y Fl(<)13 b(\013=)p Fn(2)249 821 y(F)l(or)h(an)o(y)g(\014xed)h (probabilit)o(y)h(distribution)45 b Fk(P)h Fn(on)e Fl(C)s Fn([0)p Fl(;)8 b(T)e Fn(])41 b(and)14 b(arbitary)43 b Fl(\013)15 b(;)8 b(\014)14 b(>)f Fn(0)43 b(one)14 b(can)249 887 y(\014nd)i(some)45 b Fl(\016)r Fn(\()p Fl(\013;)8 b(\014)r(;)g Fk(P)t Fn(\))i Fl(>)j Fn(0)45 b(suc)o(h)16 b(that)899 994 y Fk(P)j(f)p Fl(f)e Fn(:)28 b Fl(!)1081 1001 y Fj(f)1103 994 y Fn(\()p Fl(\016)r Fn(\))12 b Fl(>)h(\014)18 b Fk(g)12 b Fl(<)28 b(\013)249 1101 y Fn(Let)13 b(us)g(c)o(ho)q(ose)g(suc)o(h)g(a)38 b Fl(\016)i Fn(for)12 b(an)o(y)928 1089 y(~)923 1101 y Fl(\030)943 1108 y Fj(N)971 1113 y Fd(i)1012 1101 y Fl(;)c(N)1070 1108 y Fj(i)1095 1101 y Fl(<)13 b(N)1180 1108 y Fg(\003)1225 1101 y Fl(;)f Fn(and)h(de\014ne)h (the)f(\014nal)39 b Fl(\016)h Fn(as)12 b(the)h(minim)o(um)249 1168 y(of)i(suc)o(h)46 b Fl(\016)456 1151 y Fg(0)467 1168 y Fl(s)g Fn(and)g Fl(\016)673 1175 y Fg(\003)723 1168 y Fn(.)249 1234 y(Condition)15 b(\(ii\))f(is)h(satis\014ed)f(.The)g(family)h(of)e (probabilit)o(y)i(disributions)h(giv)o(en)e(b)o(y)1735 1222 y(~)1730 1234 y Fl(\030)1750 1241 y Fj(N)1778 1245 y Fd(n)1801 1234 y Fn(\()p Fk(\001)p Fn(\))41 b(is)14 b(tigh)o(t.)249 1301 y(No)o(w)h(w)o(e)g(will)h(pro)o(v)o(e)f(Lemma)g(3.1)f(.)249 1367 y(First,)i(let)h(us)f(note)g(that)g(if)33 b Fl(t;)8 b(s)49 b Fn(b)q(elong)17 b(to)f(some)g(in)o(terv)m(al)h(of)f(the)h(length)49 b Fl(c)1676 1374 y Fi(3)1696 1367 y Fl(N)1738 1351 y Fg(\000)p Fi(3)p Fj(=)p Fi(4)1836 1367 y Fl(;)8 b(c)1877 1374 y Fi(3)1910 1367 y Fk(\024)15 b Fn(1)47 b(:)249 1434 y Fl(s;)8 b(t)13 b Fk(2)g Fn([)p Fl(s)397 1417 y Fg(0)408 1434 y Fl(;)8 b(s)450 1417 y Fg(0)471 1434 y Fn(+)j Fl(c)537 1441 y Fi(3)564 1434 y Fl(N)606 1417 y Fg(\000)p Fi(3)p Fj(=)p Fi(4)688 1434 y Fn(])45 b(then)545 1541 y Fk(j)563 1529 y Fn(~)558 1541 y Fl(\030)578 1548 y Fj(N)611 1541 y Fn(\()p Fl(t)p Fn(\))10 b Fk(\000)723 1529 y Fn(~)718 1541 y Fl(\030)738 1548 y Fj(N)772 1541 y Fn(\()p Fl(s)p Fn(\))p Fk(j)i(\024)h(j)920 1529 y Fn(~)915 1541 y Fl(\030)935 1548 y Fj(N)968 1541 y Fn(\()p Fl(s)1007 1522 y Fg(0)1029 1541 y Fn(+)d Fl(c)1094 1548 y Fi(3)1121 1541 y Fl(N)1163 1522 y Fg(\000)p Fi(3)p Fj(=)p Fi(4)1245 1541 y Fn(\))g Fk(\000)1323 1529 y Fn(~)1318 1541 y Fl(\030)1338 1548 y Fj(N)1372 1541 y Fn(\()p Fl(s)1411 1522 y Fg(0)1422 1541 y Fn(\))p Fk(j)g Fn(+)g Fl(const)e(N)1664 1522 y Fg(\000)p Fi(1)p Fj(=)p Fi(4)1962 1541 y Fn(\(55\))249 1648 y(Indeed,)16 b(b)o(y)g(de\014nition,)g(for)45 b Fl(s)808 1631 y Fg(0)833 1648 y Fk(\024)13 b Fl(s)g Fk(\024)g Fl(t)g Fk(\024)g Fl(s)1061 1631 y Fg(0)1083 1648 y Fn(+)d Fl(c)1148 1655 y Fi(3)1175 1648 y Fl(N)1217 1631 y Fg(\000)p Fi(1)p Fj(=)p Fi(4)416 1821 y Fn(0)41 b Fk(\024)562 1809 y Fn(~)557 1821 y Fl(\030)577 1828 y Fj(N)611 1821 y Fn(\()p Fl(t)p Fn(\))9 b Fk(\000)723 1809 y Fn(~)718 1821 y Fl(\030)738 1828 y Fj(N)771 1821 y Fn(\()p Fl(s)p Fn(\))h(+)h Fl(F)913 1828 y Fj(N)946 1821 y Fn(\()p Fl(t)p Fn(\))d Fl(N)1048 1802 y Fi(1)p Fj(=)p Fi(2)1112 1821 y Fk(\000)j Fl(F)1187 1828 y Fj(N)1221 1821 y Fn(\()p Fl(s)p Fn(\))d Fl(N)1328 1802 y Fi(1)p Fj(=)p Fi(2)480 1900 y Fk(\024)562 1888 y Fn(~)557 1900 y Fl(\030)577 1907 y Fj(N)611 1900 y Fn(\()p Fl(s)650 1881 y Fg(0)671 1900 y Fn(+)j Fl(c)737 1907 y Fi(3)756 1900 y Fl(N)798 1881 y Fg(\000)p Fi(3)p Fj(=)p Fi(4)880 1900 y Fn(\))f Fk(\000)958 1888 y Fn(~)953 1900 y Fl(\030)973 1907 y Fj(N)1007 1900 y Fn(\()p Fl(s)1046 1881 y Fg(0)1057 1900 y Fn(\))g(+)h Fl(F)1160 1907 y Fj(N)1194 1900 y Fn(\()p Fl(s)1233 1881 y Fg(0)1254 1900 y Fn(+)g Fl(c)1320 1907 y Fi(3)1339 1900 y Fl(N)1381 1881 y Fg(\000)p Fi(3)p Fj(=)p Fi(4)1463 1900 y Fn(\))d Fl(N)1531 1881 y Fi(1)p Fj(=)p Fi(2)1595 1900 y Fk(\000)i Fl(F)1669 1907 y Fj(N)1703 1900 y Fn(\()p Fl(s)1742 1881 y Fg(0)1754 1900 y Fn(\))e Fl(N)1822 1881 y Fi(1)p Fj(=)p Fi(2)249 2066 y Fn(whic)o(h)16 b(implies)269 2173 y Fk(j)287 2161 y Fn(~)282 2173 y Fl(\030)302 2180 y Fj(N)335 2173 y Fn(\()p Fl(t)p Fn(\))10 b Fk(\000)447 2161 y Fn(~)442 2173 y Fl(\030)462 2180 y Fj(N)496 2173 y Fn(\()p Fl(s)p Fn(\))p Fk(j)i(\024)h(j)644 2161 y Fn(~)639 2173 y Fl(\030)659 2180 y Fj(N)692 2173 y Fn(\()p Fl(s)731 2154 y Fg(0)753 2173 y Fn(+)d Fl(c)818 2180 y Fi(3)845 2173 y Fl(N)887 2154 y Fg(\000)p Fi(3)p Fj(=)p Fi(4)969 2173 y Fn(\))g Fk(\000)1047 2161 y Fn(~)1042 2173 y Fl(\030)1062 2180 y Fj(N)1096 2173 y Fn(\()p Fl(s)1135 2154 y Fg(0)1146 2173 y Fn(\))p Fk(j)g Fn(+)g Fl(N)1274 2154 y Fi(1)p Fj(=)p Fi(2)1339 2173 y Fk(\001)g Fn(2)k Fl(V)c(ar)q(iation)1603 2186 y Fi([)p Fj(s)1629 2176 y Fp(0)1640 2186 y Fj(;s)1666 2176 y Fp(0)1678 2186 y Fi(+)p Fj(c)1720 2191 y Ff(3)1738 2186 y Fj(N)1770 2176 y Fp(\000)p Ff(3)p Fd(=)p Ff(4)1842 2186 y Fi(])1853 2126 y Fh(\020)1878 2173 y Fl(F)1907 2180 y Fj(N)1941 2173 y Fn(\()p Fl(s)p Fn(\))1998 2126 y Fh(\021)249 2280 y Fn(Since)42 b Fl(F)422 2287 y Fj(N)456 2280 y Fn(\()p Fl(s)p Fn(\))e(are)13 b(uniformly)i(con)o(tin)o(uously)f(di\013eren)o(tiable)i (functions)e(of)27 b Fl(s)13 b(;)h Fn(the)f(last)h(inequalit)o(y)249 2346 y(pro)o(v)o(es)30 b(\(55\))f(.)20 b(No)o(w)14 b(w)o(e)h(divide)i(the)f (segmen)o(t)45 b([0)p Fl(;)8 b(T)e Fn(])43 b(in)o(to)30 b(2)1415 2330 y Fj(k)1467 2346 y Fn(disjoin)o(t)16 b(subsegmen)o(ts)660 2453 y(\001)698 2429 y Fi(\()p Fj(k)q Fi(\))698 2467 y Fj(l)759 2453 y Fn(=)d([)p Fl(l)i(T)5 b(=)p Fn(2)927 2435 y Fj(k)948 2453 y Fl(;)j Fn(\()p Fl(l)h Fn(+)i(1\))p Fl(T)5 b(=)p Fn(2)1175 2435 y Fj(k)1194 2453 y Fn(])p Fl(;)68 b(l)13 b Fn(=)g(0)p Fl(;)8 b Fn(1)p Fl(;)g(:)g(:)f(:)t Fn(2)1532 2435 y Fj(k)1564 2453 y Fk(\000)j Fn(1)1123 2578 y(28)p eop %%Page: 29 29 bop 249 410 a Fn(with)29 b Fl(k)h Fn(ranging)15 b(from)f(1)g(to)783 363 y Fh(h)802 410 y Fn(\(log)8 b Fl(T)16 b Fn(+)980 392 y Fi(3)p 980 399 18 2 v 980 426 a(4)1010 410 y Fn(log)8 b Fl(N)d Fn(\))p Fl(=)j Fn(log)f(2)1255 363 y Fh(i)1283 410 y Fn(+)i(1)43 b(\()14 b(i.e.)49 b Fk(j)p Fn(\001)1583 386 y Fi(\()p Fj(k)q Fi(\))1583 424 y Fj(l)1631 410 y Fk(j)13 b(\025)1710 392 y Fi(1)p 1710 399 V 1710 426 a(2)1732 410 y Fl(N)1774 394 y Fg(\000)p Fi(3)p Fj(=)p Fi(4)1885 410 y Fn(\))p Fl(:)h Fn(Using)249 476 y(Cheb)o(yshev)i(inequalit)o(y)h(and)45 b(\(50\))g(:)580 606 y Fk(P)612 613 y Fj(N)653 559 y Fh(\020)678 606 y Fk(j)p Fl(\030)711 613 y Fj(N)744 606 y Fn(\()p Fl(t)p Fn(\))10 b Fk(\000)g Fl(\030)871 613 y Fj(N)905 606 y Fn(\()p Fl(s)p Fn(\))p Fk(j)i Fl(>)h Fk(j)p Fl(t)d Fk(\000)g Fl(s)p Fk(j)1153 588 y Fi(1)p Fj(=)p Fi(20)1226 559 y Fh(\021)1263 606 y Fk(\024)1316 576 y Fn(I)-9 b(E)q Fk(j)p Fl(\030)1388 583 y Fj(N)1421 576 y Fn(\()p Fl(t)p Fn(\))10 b Fk(\000)g Fl(\030)1548 583 y Fj(N)1581 576 y Fn(\()p Fl(s)p Fn(\))p Fk(j)1651 559 y Fi(4)p 1316 596 355 2 v 1407 639 a Fk(j)p Fl(t)g Fk(\000)g Fl(s)p Fk(j)1525 626 y Fi(1)p Fj(=)p Fi(5)503 740 y Fk(\024)585 709 y Fn(3)e(\()o Fl(b)653 716 y Fj(N)686 709 y Fn(\()p Fl(t;)g(t)p Fn(\))i Fk(\000)g Fl(b)850 716 y Fj(N)883 709 y Fn(\()p Fl(s;)e(t)p Fn(\))i Fk(\000)g Fl(b)1052 716 y Fj(N)1085 709 y Fn(\()p Fl(t;)e(s)p Fn(\))i Fk(\000)g Fl(b)1254 716 y Fj(N)1287 709 y Fn(\()p Fl(s;)e(s)p Fn(\)\))1403 688 y Fi(2)1433 709 y Fn(+)i Fl(const)p Fn(\()p Fl(T)t(;)e(")p Fn(\))p Fl(N)1735 693 y Fg(\000)p Fi(1+)p Fj(")p 585 729 1240 2 v 1118 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1431 y Fk(\024)441 1583 y(\024)42 b Fn(2)541 1564 y Fj(k)577 1583 y Fl(const)p Fn(\()p Fl(T)6 b Fn(\))772 1552 y(\()p Fl(T)f(=)p Fn(2)868 1536 y Fj(k)888 1552 y Fn(\))906 1536 y Fi(2)936 1552 y Fn(+)10 b Fl(N)1023 1536 y Fg(\000)p Fi(19)p Fj(=)p Fi(20)p 772 1573 369 2 v 862 1616 a Fn(\()p Fl(T)5 b(=)p Fn(2)958 1603 y Fj(k)978 1616 y Fn(\))996 1602 y Fi(1)p Fj(=)p Fi(5)1158 1583 y Fk(\024)13 b Fl(const)p Fn(\()p Fl(T)6 b Fn(\))1404 1536 y Fh(\020)1429 1583 y Fn(\(2)1470 1564 y Fg(\000)p Fj(k)1518 1583 y Fn(\))1536 1564 y Fi(4)p Fj(=)p Fi(5)1601 1583 y Fn(+)k Fl(N)1688 1564 y Fg(\000)p Fi(19)p Fj(=)p Fi(20)1805 1583 y Fn(\(2)1846 1564 y Fj(k)1867 1583 y Fn(\))1885 1564 y Fi(6)p Fj(=)p Fi(5)1940 1536 y Fh(\021)441 1692 y Fk(\024)42 b Fl(const)p Fn(\()p Fl(T)6 b Fn(\))716 1645 y Fh(\020)741 1692 y Fn(\(2)782 1674 y Fg(\000)p Fj(k)830 1692 y Fn(\))848 1674 y Fi(4)p Fj(=)p Fi(5)913 1692 y Fn(+)k Fl(N)1000 1674 y Fg(\000)p Fi(19)p Fj(=)p Fi(20)1117 1692 y Fl(N)1159 1674 y Fi(18)p Fj(=)p Fi(20)1249 1645 y Fh(\021)1286 1692 y Fk(\024)j Fl(const)p Fn(\()p Fl(T)6 b Fn(\))1532 1645 y Fh(\020)1557 1692 y Fn(\(2)1598 1674 y Fg(\000)p Fj(k)1646 1692 y Fn(\))1664 1674 y Fi(4)p Fj(=)p Fi(5)1729 1692 y Fn(+)k Fl(N)1816 1674 y Fg(\000)p Fi(1)p Fj(=)p Fi(20)1916 1645 y Fh(\021)249 1864 y Fn(and)328 2033 y Fk(P)360 2040 y Fj(N)511 1924 y Fh(0)511 1997 y(B)511 2022 y(B)511 2049 y(@)548 1966 y Fi([)563 1943 y Ff(log)c Fd(T)t Ff(+)658 1934 y(3)p 658 1938 16 2 v 658 1956 a(4)683 1943 y(log)h Fd(N)p 562 1959 193 2 v 629 1979 a Ff(log)f(2)760 1966 y Fi(]+1)658 1993 y Fh(G)640 2085 y Fj(k)q Fi(=)p Fj(k)704 2090 y Ff(0)853 1981 y Fi(2)871 1969 y Fd(k)889 1981 y Fg(\000)p Fi(1)870 1993 y Fh(G)866 2085 y Fj(l)p Fi(=0)972 1986 y Fh(n)1000 2033 y Fk(j)p Fl(\030)1033 2040 y Fj(N)1066 2033 y Fn(\(\()p Fl(l)k Fn(+)g(1\))p Fl(T)5 b(=)p Fn(2)1290 2015 y Fj(k)1310 2033 y Fn(\))10 b Fk(\000)h Fl(\030)1404 2040 y Fj(N)1437 2033 y Fn(\()p Fl(l)k(T)5 b(=)p Fn(2)1562 2015 y Fj(N)1595 2033 y Fn(\))p Fk(j)12 b Fl(>)h Fn(\()p Fl(T)5 b(=)p Fn(2)1782 2015 y Fj(k)1802 2033 y Fn(\))1820 2015 y Fi(1)p Fj(=)p Fi(20)1892 1986 y Fh(o)1920 1924 y(1)1920 1997 y(C)1920 2022 y(C)1920 2049 y(A)434 2187 y Fk(\024)42 b Fl(const)p Fn(\()p Fl(T)6 b Fn(\))686 2140 y Fh(\020)711 2187 y Fn(\(2)752 2168 y Fg(\000)p Fj(k)797 2173 y Ff(0)817 2187 y Fn(\))835 2168 y Fi(4)p Fj(=)p Fi(5)899 2187 y Fn(+)11 b Fl(N)987 2168 y Fg(\000)p Fi(1)p Fj(=)p Fi(20)1094 2187 y Fn(log)d Fl(N)1202 2140 y Fh(\021)1962 2187 y Fn(\(56\))249 2358 y(Cho)q(osing)30 b Fl(k)486 2365 y Fi(0)534 2358 y Fn(suc)o(h)15 b(that)43 b(2)786 2342 y Fg(\000)p Fj(k)831 2347 y Ff(0)849 2342 y Fi(+1)909 2358 y Fk(\024)13 b Fl(\016)h Fk(\024)f Fn(2)1062 2342 y Fg(\000)p Fj(k)1107 2347 y Ff(0)1125 2342 y Fi(+2)1216 2358 y Fn(and)i(com)o(bining)45 b(\(56\))13 b(with)30 b(\(55\))13 b(and)30 b(\(51\))249 2425 y(,)15 b(w)o(e)g(\014nish)h(the)g(pro)q(of.)p 705 2429 21 30 v 1123 2578 a(29)p eop %%Page: 30 30 bop 249 403 a Fn(Corollary)15 b(1.3)f(can)i(b)q(e)g(pro)o(v)o(en)e(b)o(y)i (using)f(the)h(same)f(mac)o(hinery)g(as)g(in)h(Theorem)f(1.1.)249 469 y(T)l(o)g(pro)o(v)o(e)g(Corollary)g(1.4)f(consider)i(the)f(con)o(tin)o (uous)h(functional)g(on)46 b Fl(C)s Fn([0)p Fl(;)8 b Fk(1)p Fn(\))13 b(:)942 577 y Fl(G)978 584 y Fj(T)1005 577 y Fn(\()p Fl(f)5 b Fn(\))12 b(=)35 b(sup)1128 617 y Fj(s)p Fg(2)p Fi([0)p Fj(;T)5 b Fi(])1241 577 y Fk(j)o Fl(f)g Fn(\()p Fl(s)p Fn(\))p Fk(j)249 706 y Fn(apply)16 b(Theorem)f(1.2)f(and)i(tak)o(e)e(in)o(to)h (accoun)o(t)854 814 y(lim)835 843 y Fj(N)t Fg(!1)982 814 y Fn(sup)960 854 y Fj(s)p Fg(2)p Fi([0)p Fj(;T)5 b Fi(])1072 814 y Fk(j)1089 802 y Fn(~)1085 814 y Fl(\030)1105 821 y Fj(N)1138 814 y Fn(\()p Fl(s)p Fn(\))10 b Fk(\000)g Fl(\030)1270 821 y Fj(N)1304 814 y Fn(\()p Fl(s)p Fn(\))p Fk(j)i Fn(=)h(0)249 944 y(Before)i(w)o(e)g(pro)q(ceed)h(with)g(the)f(pro)q(of)g(of)g(Corollary)g (1.5)f(,)h(w)o(e)g(w)o(an)o(t)f(to)g(obtain)i(an)f(estimate)g(on)837 1062 y(sup)828 1103 y Fi([0)p Fj(;)p Fg(1)p Fi(\))929 1062 y Fk(j)p Fn(I)-9 b(E)15 b Fl(\021)r Fn(\()p Fl(I)1058 1069 y Fj(N)1091 1062 y Fl(;)8 b(s)p Fn(\))h Fk(\000)i Fl(F)6 b Fn(\()p Fl(s)p Fn(\))1318 1032 y Fl(N)12 b Fk(j)p Fl(I)1400 1039 y Fj(N)1433 1032 y Fk(j)p 1318 1052 128 2 v 1357 1094 a Fn(2)p Fl(\031)1451 1062 y Fk(j)249 1192 y Fn(Consider)k(\014rst)f(the)g (case)940 1259 y Fl(s)e Fk(\024)g Fn(\(log)8 b Fl(N)d Fn(\))1166 1240 y Fi(1)p Fj(=)p Fi(2)1339 1259 y Fn(:)610 b(\(57\))402 1419 y Fk(j)p Fl(F)444 1426 y Fj(N)478 1419 y Fn(\()p Fl(s)p Fn(\))10 b Fk(\000)g Fl(F)c Fn(\()p Fl(s)p Fn(\))p Fk(j)41 b Fn(=)813 1367 y Fj(N)t Fg(\000)p Fi(1)822 1379 y Fh(X)818 1470 y Fj(n)p Fi(=0)902 1389 y Fn(\()p Fk(\000)p Fn(1\))996 1372 y Fj(n)p 902 1409 117 2 v 941 1451 a Fl(n)p Fn(!)1032 1362 y Fh(Z)1055 1456 y Fi([0)p Fj(;s)p Fi(])1119 1447 y Fd(n)1164 1419 y Fl(\032)1188 1395 y Fi(\()p Fj(N)t Fi(\))1188 1432 y Fj(n)p Fi(+1)1256 1419 y Fn(\(0)p Fl(;)10 b Fn(\026)-25 b Fl(x)o Fn(\))10 b Fk(\000)g Fl(\032)1440 1395 y Fi(\()p Fg(1)p Fi(\))1440 1432 y Fj(n)p Fi(+1)1508 1419 y Fn(\(0)p Fl(;)g Fn(\026)-25 b Fl(x)o Fn(\))15 b Fl(d)s Fn(\026)-26 b Fl(x)736 1562 y Fn(+)836 1509 y Fg(1)823 1521 y Fh(X)813 1613 y Fj(n)p Fi(=)p Fj(N)906 1531 y Fn(\()p Fk(\000)p Fn(1\))1000 1514 y Fj(n)p Fg(\000)p Fi(1)p 906 1551 163 2 v 967 1593 a Fl(n)p Fn(!)1081 1504 y Fh(Z)1104 1599 y Fi([0)p Fj(;s)p Fi(])1168 1589 y Fd(n)1213 1562 y Fl(\032)1237 1538 y Fi(\()p Fg(1)p Fi(\))1237 1574 y Fj(n)p Fi(+1)1305 1562 y Fn(\(0)p Fl(;)10 b Fn(\026)-25 b Fl(x)o Fn(\))15 b Fl(d)s Fn(\026)-26 b Fl(x)736 1714 y Fk(\024)813 1661 y Fj(N)t Fg(\000)p Fi(1)822 1673 y Fh(X)818 1764 y Fj(n)p Fi(=0)926 1683 y Fn(1)p 918 1703 40 2 v 918 1745 a Fl(n)p Fn(!)978 1714 y Fl(Ar)q(ea)22 b Fn(\([0)p Fl(;)8 b(s)p Fn(])1210 1695 y Fj(n)1232 1714 y Fn(\))52 b(sup)1275 1754 y Fi(\026)-21 b Fj(x)p Fg(2)p Fi([0)p Fj(;s)p Fi(])1380 1745 y Fd(n)1416 1714 y Fk(j)o Fl(\032)1452 1690 y Fi(\()p Fj(N)t Fi(\))1452 1726 y Fj(n)p Fi(+1)1520 1714 y Fn(\(0)p Fl(;)10 b Fn(\026)-25 b Fl(x)o Fn(\))10 b Fk(\000)g Fl(\032)1704 1690 y Fi(\()p Fg(1)p Fi(\))1704 1726 y Fj(n)p Fi(+1)1772 1714 y Fn(\(0)p Fl(;)g Fn(\026)-25 b Fl(x)o Fn(\))p Fk(j)736 1856 y Fn(+)836 1803 y Fg(1)823 1815 y Fh(X)813 1907 y Fj(n)p Fi(=)p Fj(N)930 1825 y Fn(1)p 921 1845 V 921 1887 a Fl(n)p Fn(!)981 1856 y Fl(Ar)q(ea)23 b Fn(\([0)p Fl(;)8 b(s)p Fn(])1214 1837 y Fj(n)1236 1856 y Fn(\))i Fk(\001)f Fn(1)249 2047 y(Using)16 b(the)f(inequalit)o(y)706 2162 y(sup)699 2203 y Fg(j)p Fj(x)p Fg(j\024)p Fj(s)797 2114 y Fh(\014)797 2138 y(\014)797 2163 y(\014)885 2132 y Fn(sin)8 b Fl(\031)r(x)p 816 2152 256 2 v 816 2193 a(N)k Fn(sin)q(\()p Fl(\031)r(x=)m(N)5 b Fn(\))1086 2162 y Fk(\000)1137 2132 y Fn(sin)j Fl(\031)r(x)p 1137 2152 117 2 v 1168 2193 a(\031)r(x)1259 2114 y Fh(\014)1259 2138 y(\014)1259 2163 y(\014)1285 2162 y Fl(<)13 b(const)j Fn(\()p Fl(s=)m(N)5 b Fn(\))1574 2144 y Fi(2)249 2309 y Fn(and)15 b(the)h(represen)o(tation)f(of)45 b Fl(\032)819 2285 y Fi(\()p Fj(N)t Fi(\))819 2322 y Fj(n)p Fi(+1)887 2309 y Fl(;)38 b(\032)962 2285 y Fi(\()p Fg(1)p Fi(\))962 2322 y Fj(n)p Fi(+1)1075 2309 y Fn(as)15 b(determinan)o(ts)46 b(\(6\))29 b(,)h(\(7\))44 b(w)o(e)15 b(ha)o(v)o(e)552 2417 y(sup)524 2457 y Fi(\026)-20 b Fj(x)p Fg(2)p Fi([0)p Fj(;s)p Fi(])630 2448 y Fd(n)666 2417 y Fk(j)o Fl(\032)702 2393 y Fi(\()p Fj(N)t Fi(\))702 2429 y Fj(n)p Fi(+1)770 2417 y Fn(\(0)p Fl(;)10 b Fn(\026)-25 b Fl(x)o Fn(\))10 b Fk(\000)g Fl(\032)954 2393 y Fi(\()p Fg(1)p Fi(\))954 2429 y Fj(n)p Fi(+1)1022 2417 y Fn(\(0)p Fl(;)g Fn(\026)-25 b Fl(x)o Fn(\))p Fk(j)12 b(\024)h Fl(const)i Fn(\()p Fl(n)c Fn(+)f(1\)!)e(\()p Fl(n)h Fn(+)h(1\))e(\()p Fl(s=)m(N)d Fn(\))1751 2398 y Fi(2)1123 2578 y Fn(30)p eop %%Page: 31 31 bop 249 410 a Fn(Since)47 b Fl(\032)422 386 y Fi(\()p Fj(N)t Fi(\))422 422 y Fj(n)p Fi(+1)490 410 y Fl(;)38 b(\032)565 386 y Fi(\()p Fg(1)p Fi(\))565 422 y Fj(n)p Fi(+1)648 410 y Fn(are)15 b(not)g(greater)f(than)h(1)g(b)o(y)g(absolute)g(v)m(alue,)i(w)o(e)d (\014nally)j(arriv)o(e)e(at)390 554 y Fk(j)p Fl(F)432 561 y Fj(N)466 554 y Fn(\()p Fl(s)p Fn(\))9 b Fk(\000)i Fl(F)6 b Fn(\()p Fl(s)p Fn(\))p Fk(j)41 b(\024)801 501 y Fj(N)t Fg(\000)p Fi(1)809 514 y Fh(X)806 605 y Fj(n)p Fi(=0)890 524 y Fl(s)911 507 y Fj(n)p 890 544 45 2 v 892 586 a Fl(n)p Fn(!)963 554 y(min)1046 507 y Fh(\020)1071 554 y Fn(2)p Fl(;)22 b(const)16 b Fn(\()p Fl(n)10 b Fn(+)g(1\)!)e(\()p Fl(n)h Fn(+)i(1\))d(\()p Fl(s=)m(N)d Fn(\))1681 536 y Fi(2)1699 507 y Fh(\021)1733 554 y Fn(+)1791 501 y Fg(1)1779 514 y Fh(X)1793 606 y Fj(N)1851 524 y Fl(s)1872 507 y Fj(n)p 1851 544 V 1853 586 a Fl(n)p Fn(!)724 668 y(=)42 b Fl(o)p Fn(\(\()p Fl(s=)m(N)5 b Fn(\))960 649 y Fi(1)p Fg(\000)p Fj(")1023 668 y Fn(\))k(+)i Fl(O)q Fn(\()p Fl(s)1171 649 y Fj(N)1204 668 y Fl(=)d(N)d Fn(!\))249 779 y(for)15 b(an)o(y)45 b Fl(")13 b(>)g Fn(0)45 b(.)20 b(Th)o(us)836 890 y(sup)769 934 y Fj(s)p Fg(\024)p Fi(\(log)6 b Fj(N)t Fi(\))924 925 y Ff(1)p Fd(=)p Ff(2)987 890 y Fk(j)p Fl(F)1029 897 y Fj(N)1062 890 y Fn(\()p Fl(s)p Fn(\))k Fk(\000)h Fl(F)6 b Fn(\()p Fl(s)p Fn(\))p Fk(j)12 b Fn(=)h Fl(o)p Fn(\()p Fl(N)1422 871 y Fg(\000)p Fi(1+)p Fj(")1512 890 y Fn(\))432 b(\(58\))249 1089 y(The)16 b(function)46 b Fl(F)6 b Fn(\()p Fl(s)p Fn(\))46 b(deca)o(ys)15 b(at)f(in\014nit)o(y)j(sup)q(erexp)q(onen)o(tially)h(:)881 1200 y(log)9 b Fl(F)d Fn(\()p Fl(s)p Fn(\))12 b(=)h Fk(\000)p Fl(\031)1163 1181 y Fi(2)1183 1200 y Fl(s)1204 1181 y Fi(2)1224 1200 y Fl(=)p Fn(8)d(+)g Fl(O)q Fn(\()p Fl(s)p Fn(\))544 b(\(59\))249 1311 y(\(see)46 b([13)o(],[12)n(],[14)n(])30 b(\),that)14 b(giv)o(es)h(us)890 1422 y Fl(F)6 b Fn(\(\(log)i Fl(N)d Fn(\))1087 1403 y Fi(1)p Fj(=)p Fi(2)1142 1422 y Fn(\))12 b(=)h Fl(o)p Fn(\()p Fl(N)1302 1403 y Fg(\000)p Fi(9)p Fj(=)p Fi(8)1384 1422 y Fn(\))560 b(\(60\))249 1533 y(Since)47 b(0)12 b Fk(\024)h Fl(F)510 1540 y Fj(N)544 1533 y Fn(\()p Fl(s)p Fn(\))g Fk(\024)g Fl(F)691 1540 y Fj(N)724 1533 y Fn(\(\(log)8 b Fl(N)d Fn(\))886 1516 y Fi(1)p Fj(=)p Fi(2)940 1533 y Fn(\))15 b Fl(;)22 b Fn(0)13 b Fk(\024)g Fl(F)6 b Fn(\()p Fl(s)p Fn(\))13 b Fk(\024)g Fl(F)6 b Fn(\(\(log)i Fl(N)d Fn(\))1442 1516 y Fi(1)p Fj(=)p Fi(2)1496 1533 y Fn(\))30 b Fl(f)5 b(or)16 b(any)32 b(s)13 b Fk(\025)g Fn(\(log)8 b Fl(N)d Fn(\))1961 1516 y Fi(1)p Fj(=)p Fi(2)249 1599 y Fn(\(58\))29 b(,)h(\(59\))44 b(establish)749 1710 y(sup)740 1751 y Fi([0)p Fj(;)p Fg(1)p Fi(\))841 1710 y Fk(j)p Fl(F)883 1717 y Fj(N)917 1710 y Fn(\()p Fl(s)p Fn(\))10 b Fk(\000)g Fl(F)c Fn(\()p Fl(s)p Fn(\))p Fk(j)13 b Fn(=)f Fl(o)p Fn(\()p Fl(N)1276 1691 y Fg(\000)p Fi(1+)p Fj(")1367 1710 y Fn(\))749 1845 y(sup)740 1885 y Fi([0)p Fj(;)p Fg(1)p Fi(\))841 1845 y Fk(j)p Fn(I)-9 b(E)15 b Fl(\021)r Fn(\()p Fl(I)970 1852 y Fj(N)1003 1845 y Fl(;)8 b(s)p Fn(\))h Fk(\000)i Fl(F)6 b Fn(\()p Fl(s)p Fn(\))1230 1814 y Fl(N)f Fk(j)p Fl(I)1305 1821 y Fj(N)1338 1814 y Fk(j)p 1230 1834 121 2 v 1265 1876 a Fn(2)p Fl(\031)1355 1845 y Fk(j)13 b Fn(=)f Fl(o)p Fn(\()p Fl(N)1510 1826 y Fj(")1533 1814 y Fk(j)p Fl(I)1566 1821 y Fj(N)1599 1814 y Fk(j)p 1533 1834 79 2 v 1547 1876 a Fn(2)p Fl(\031)1617 1845 y Fn(\))327 b(\(61\))249 1978 y(W)l(e)15 b(\014nish)i(this)e(section)h(with)g(the)f(pro)q(of)g(of)g(Corollary)g(1.5)f (.)249 2045 y(Since)j(the)e(tails)h(of)f(distribution)h(functions)47 b Fl(F)6 b Fn(\()p Fl(s)p Fn(\))15 b Fl(;)22 b(F)1242 2052 y Fj(N)1276 2045 y Fn(\()p Fl(s)p Fn(\))45 b(are)15 b(small)h(enough)g(\()f (see)30 b(\(58\))14 b(,)249 2111 y(\(60\))29 b(\),)15 b(w)o(e)g(ha)o(v)o(e)f (to)h(pro)o(v)o(e)g(only)g(that)663 2236 y(sup)573 2280 y Fi(0)p Fg(\024)p Fj(s)p Fg(\024)p Fi(\(log)6 b Fj(N)t Fi(\))773 2270 y Ff(1)p Fd(=)p Ff(2)836 2236 y Fk(j)p Fl(\021)r Fn(\()p Fl(I)912 2243 y Fj(N)944 2236 y Fl(;)i(s)p Fn(\))i Fk(\000)g Fn(I)-9 b(E)15 b Fl(\021)r Fn(\()p Fl(I)1175 2243 y Fj(N)1208 2236 y Fl(;)8 b(s)p Fn(\))p Fk(j)j Fn(=)i Fl(o)1370 2176 y Fh(\022)1400 2236 y Fn(\()p Fl(N)1472 2205 y Fk(j)p Fl(I)1505 2212 y Fj(N)1538 2205 y Fk(j)p 1472 2225 V 1486 2267 a Fn(2)p Fl(\031)1556 2236 y Fn(\))1574 2217 y Fi(1)p Fj(=)p Fi(2)h(+)p Fj(")1688 2176 y Fh(\023)249 2376 y Fn(T)l(o)19 b(do)h(this)g(w)o(e)g(will)h(estimate)e(the) h(momen)o(ts)f(of)39 b Fl(\021)r Fn(\()p Fl(I)1249 2383 y Fj(N)1281 2376 y Fl(;)8 b(s)p Fn(\))13 b Fk(\000)g Fn(I)-9 b(E)20 b Fl(\021)r Fn(\()p Fl(I)1523 2383 y Fj(N)1556 2376 y Fl(;)8 b(s)p Fn(\))38 b(,)20 b(when)g(s)g(is)g(sligh)o(tly)249 2442 y(gro)o(wing)15 b(up)g(,)45 b Fl(s)13 b Fk(\024)g Fn(\(log)8 b Fl(N)d Fn(\))769 2425 y Fi(1)p Fj(=)p Fi(2)854 2442 y Fl(:)1123 2578 y Fn(31)p eop %%Page: 32 32 bop 249 403 a Fc(Lemm)o(a)14 b(3.2)23 b Fb(F)m(or)14 b(any)f(even)h(inte)n (ger)27 b Fn(2)p Fl(k)42 b Fb(,)15 b(arbitary)f(smal)r(l)42 b Fl(")13 b(>)g Fn(0)41 b Fb(and)h Fn(0)12 b Fk(\024)h Fl(s)g Fk(\024)g Fn(\(log)8 b Fl(N)d Fn(\))1960 386 y Fi(1)p Fj(=)p Fi(2)249 469 y Fb(ther)n(e)16 b(exists)g(some)g(c)n(onstant)48 b Fl(c)p Fn(\()p Fl(";)8 b Fn(2)p Fl(k)q Fn(\))14 b Fl(;)32 b Fb(dep)n(ending)16 b(only)g(on)48 b Fl(")h Fb(and)g Fn(2)p Fl(k)33 b Fb(,)16 b(such)h(that)608 605 y(I)-9 b(E)23 b Fn(\()p Fl(\021)r Fn(\()p Fl(I)752 612 y Fj(N)784 605 y Fl(;)8 b(s)p Fn(\))i Fk(\000)g Fb(I)-9 b(E)15 b Fl(\021)r Fn(\()p Fl(I)1017 612 y Fj(N)1050 605 y Fl(;)8 b(s)p Fn(\)\))1127 584 y Fi(2)p Fj(k)1194 605 y Fk(\024)13 b Fl(c)p Fn(\()p Fl(";)8 b Fn(2)p Fl(k)q Fn(\))1411 546 y Fh(\022)1441 605 y Fl(N)1504 574 y Fk(j)p Fl(I)1537 581 y Fj(N)1570 574 y Fk(j)p 1504 595 79 2 v 1518 636 a Fn(2)p Fl(\031)1588 546 y Fh(\023)1618 553 y Fj(k)q Fi(+)p Fj(")249 747 y Fb(Pr)n(o)n(of:)249 813 y Fn(Again,)14 b(without)g(a)f(loss)h(in)h(generalit)o(y)l(,)f(w)o(e)f(can)h(assume)41 b Fl(I)1301 820 y Fj(N)1376 813 y Fn(to)13 b(b)q(e)h(a)g(unit)g(circle,)56 b Fl(I)1812 820 y Fj(N)1859 813 y Fn(=)12 b([)p Fk(\000)p Fl(\031)r(;)c(\031) r Fn(])249 880 y(.)249 946 y(Examinating)19 b(the)g(pro)q(of)f(of)g(Theorem)h (1.1)f(w)o(e)g(realize)i(that)e(all)i(what)e(w)o(e)g(need)i(is)f(the)g(follo) o(wing)249 1012 y(generalization)e(of)d(estimates)46 b(\(42\))e(from)14 b(Lemma)i(2.2)e(:)385 1131 y(sup)296 1175 y Fi(0)p Fg(\024)p Fj(s)p Fg(\024)p Fi(\(log)6 b Fj(N)t Fi(\))496 1166 y Ff(1)p Fd(=)p Ff(2)558 1082 y Fh(\014)558 1107 y(\014)558 1132 y(\014)703 1074 y(Z)669 1202 y Fi([0)p Fj(;N)t Fi(])749 1190 y Fd(l)568 1240 y Fi(min)627 1246 y Fd(i)p Fp(6)p Ff(=)p Fd(j)683 1240 y Fg(j)p Fj(x)713 1245 y Fd(i)726 1240 y Fg(\000)p Fj(x)773 1245 y Fd(j)790 1240 y Fg(j)800 1245 y Ff(1)817 1240 y Fj(>s)928 1131 y Fl(r)949 1138 y Fj(l)961 1131 y Fn(\()p Fl(x)1005 1138 y Fi(1)1025 1131 y Fl(;)i(:)g(:)g(:)d(x)1132 1138 y Fj(l)1145 1131 y Fl(;)j(s)p Fn(\))p Fl(dx)1255 1138 y Fi(1)1281 1131 y Fl(:)g(:)g(:)d(dx)1391 1138 y Fj(l)1404 1082 y Fh(\014)1404 1107 y(\014)1404 1132 y(\014)1430 1131 y Fk(\024)13 b Fl(const)p Fn(\()p Fl(";)8 b(l)q Fn(\))14 b Fl(N)1733 1112 y Fj(l=)p Fi(2)g(+)p Fj(")1886 1131 y Fl(;)8 b(l)k Fk(\025)h Fn(2)1962 1293 y(\(62\))249 1419 y(Going)j(along)g(the)g(lines)h(of)f(calculations)h(from)e(Theorem)h (1.1)f(,w)o(e)g(will)i(clarify)g(the)f(dep)q(endence)i(on)249 1485 y Fl(s)31 b Fn(of)14 b(the)i(constan)o(ts)e(app)q(earing)i(there.)k(W)l (e)15 b(remind)h(that)f(w)o(e)g(de\014ned)47 b Fl(r)1583 1492 y Fj(l)1641 1485 y Fn(in)31 b(\(41\))e(as)428 1626 y Fl(r)449 1633 y Fj(l)461 1626 y Fn(\()p Fl(x)505 1633 y Fi(1)524 1626 y Fl(;)8 b(:)g(:)g(:)d(x)631 1633 y Fj(l)644 1626 y Fl(;)j(s)p Fn(\))k(=)782 1573 y Fj(N)t Fg(\000)p Fj(l)787 1586 y Fh(X)779 1676 y Fj(m)p Fi(=0)868 1595 y Fn(\()p Fk(\000)p Fn(1\))962 1579 y Fj(m)p 868 1616 127 2 v 905 1657 a Fl(m)p Fn(!)1007 1569 y Fh(Z)1076 1626 y Fn(~)-27 b Fl(\032)1096 1602 y Fi(\()p Fj(N)t Fi(\))1096 1640 y Fj(l)p Fi(+)p Fj(m;l)1187 1626 y Fn(\()p Fl(x)1231 1633 y Fi(1)1251 1626 y Fl(;)8 b(:)g(:)g(:)d(x)1358 1633 y Fj(l)1371 1626 y Fn(;)j Fl(:)g(:)g(:)t(x)1477 1633 y Fj(l)p Fi(+)p Fj(m)1549 1626 y Fn(\))p Fl(dx)1617 1633 y Fj(l)p Fi(+1)1682 1626 y Fl(:)g(:)g(:)e(dx)1793 1633 y Fj(l)p Fi(+)p Fj(m)1962 1626 y Fn(\(63\))249 1780 y(\()30 b(the)15 b(in)o(tegration)h(is)f (o)o(v)o(er)780 1733 y Fh(\020)820 1748 y(F)854 1761 y Fj(l)854 1792 y Fi(1)889 1780 y Fn([)p Fl(x)928 1787 y Fj(i)942 1780 y Fl(;)8 b(x)989 1787 y Fj(i)1012 1780 y Fn(+)j Fl(s)p Fn(])1107 1733 y Fh(\021)1132 1745 y Fj(m)1210 1780 y Fn(\).)249 1847 y(F)l(or)k(simplicit)o(y)i(w)o(e)e(will)i(consider)f(cases)45 b Fl(l)13 b Fn(=)g(2)46 b(and)f Fl(l)13 b(>)g Fn(2)46 b(separately)l(.)249 1913 y(Let)16 b(us)f(\014rst)30 b Fl(l)h Fn(equal)16 b(2.)j(Rewrite)46 b(\(63\))14 b(:)389 2054 y Fl(r)410 2061 y Fj(l)423 2054 y Fn(\()p Fl(x)467 2061 y Fi(1)486 2054 y Fl(;)8 b(x)533 2061 y Fi(2)552 2054 y Fl(;)g(s)p Fn(\))k(=)687 2001 y Fj(N)t Fg(\000)p Fi(2)695 2014 y Fh(X)687 2104 y Fj(m)p Fi(=0)776 2023 y Fn(\()p Fk(\000)p Fn(1\))870 2007 y Fj(m)p 776 2043 V 813 2085 a Fl(m)p Fn(!)1013 1996 y Fh(Z)906 2123 y Fi(\()p Fg(t)944 2112 y Ff(2)944 2134 y(1)961 2123 y Fi([)p Fj(x)991 2128 y Fd(i)1004 2123 y Fj(;x)1034 2128 y Fd(i)1047 2123 y Fi(+)p Fj(s)p Fi(]\))1114 2114 y Fd(m)1175 2054 y Fl(\032)1199 2030 y Fi(\()p Fj(N)t Fi(\))1199 2066 y(2+)p Fj(m;)p Fi(2)1304 2054 y Fn(\()p Fl(x)1348 2061 y Fi(1)1368 2054 y Fl(;)c(x)1415 2061 y Fi(2)1434 2054 y Fl(;)g(:)g(:)g(:)d(x)1541 2061 y Fi(2+)p Fj(m)1619 2054 y Fn(\))p Fl(dx)1687 2061 y Fi(3)1713 2054 y Fl(:)j(:)g(:)e(dx)1824 2061 y Fi(2+)p Fj(m)249 2222 y Fn(The)16 b(estimates)45 b(\(21\))29 b(,)h(\(22\))44 b(giv)o(e)16 b(us)532 2357 y Fk(j)p Fl(\032)569 2364 y Fi(2+)p Fj(m;)p Fi(2)674 2357 y Fn(\()s(\026)-26 b Fl(x)p Fn(\))p Fk(j)12 b(\024)h Fn(min)892 2285 y Fh( )925 2357 y Fn(2)p Fl(;)22 b Fn(\()p Fl(m)10 b Fn(+)g(2\)!)1434 2326 y(2)p 1169 2347 553 2 v 1169 2394 a(1)g(+)g(max)e(\()o Fk(j)p Fl(x)1396 2401 y Fi(1)1426 2394 y Fk(\000)i Fl(x)1497 2401 y Fi(2)1517 2394 y Fk(j)1530 2401 y Fi(1)1559 2394 y Fk(\000)h Fl(s;)22 b Fn(0\))1702 2373 y Fi(2)1727 2285 y Fh(!)1123 2578 y Fn(32)p eop %%Page: 33 33 bop 249 403 a Fn(Using)16 b(the)f(inequalit)o(y)714 508 y(1)p Fl(=)768 461 y Fh(\020)792 508 y Fn(1)10 b(+)g(max\()p Fl(x)f Fk(\000)i Fl(s;)d Fn(0\))1137 490 y Fi(2)1155 461 y Fh(\021)1193 508 y Fk(\024)13 b Fn(\(2)c(+)i Fl(s)1358 490 y Fi(2)1378 508 y Fn(\))p Fl(=)p Fn(\(1)e(+)h Fl(x)1540 490 y Fi(2)1560 508 y Fn(\))384 b(\(64\))249 614 y(w)o(e)15 b(obtain)444 731 y Fk(j)p Fl(\032)481 738 y Fi(2+)p Fj(m;)p Fi(2)601 731 y Fn(\()s(\026)-26 b Fl(x)p Fn(\))p Fk(j)40 b(\024)i Fn(min)877 659 y Fh( )909 731 y Fn(2;)23 b(\()p Fl(m)9 b Fn(+)i(2\)!)1193 700 y(2)k(\(2)9 b(+)i Fl(s)1348 684 y Fi(2)1368 700 y Fn(\))p 1154 720 271 2 v 1154 762 a(1)f(+)g Fk(j)p Fl(x)1271 769 y Fi(1)1301 762 y Fk(\000)g Fl(x)1372 769 y Fi(2)1392 762 y Fk(j)1405 747 y Fi(2)1405 775 y(1)1429 659 y Fh(!)716 863 y Fk(\024)42 b Fn(2)15 b(\(2)10 b(+)g(log)e Fl(N)d Fn(\))22 b(min)1159 804 y Fh(\022)1189 863 y Fn(1;)1304 833 y(\()p Fl(m)10 b Fn(+)g(2\)!)p 1252 853 V 1252 895 a(1)g(+)h Fk(j)p Fl(x)1370 902 y Fi(1)1399 895 y Fk(\000)f Fl(x)1470 902 y Fi(2)1490 895 y Fk(j)1503 879 y Fi(2)1503 907 y(1)1527 804 y Fh(\023)462 1003 y Fk(j)p Fl(r)496 1010 y Fi(2)515 1003 y Fn(\()p Fl(x)559 1010 y Fi(1)579 1003 y Fl(;)e(x)626 1010 y Fi(2)645 1003 y Fn(\))p Fk(j)40 b(\024)793 951 y Fj(N)t Fg(\000)p Fi(2)802 963 y Fh(X)793 1054 y Fj(m)p Fi(=0)913 973 y Fn(1)p 898 993 53 2 v 898 1035 a Fl(m)p Fn(!)955 1003 y(\(2)p Fl(s)p Fn(\))1035 985 y Fj(m)1083 1003 y Fn(2)15 b(\(2)9 b(+)i(log)d Fl(N)d Fn(\))22 b(min)1448 944 y Fh(\022)1479 1003 y Fn(1;)1594 973 y(\()p Fl(m)10 b Fn(+)g(2\)!)p 1542 993 271 2 v 1542 1035 a(1)g(+)g Fk(j)p Fl(x)1659 1042 y Fi(1)1689 1035 y Fk(\000)g Fl(x)1760 1042 y Fi(2)1780 1035 y Fk(j)1793 1019 y Fi(2)1793 1047 y(1)1817 944 y Fh(\023)249 1202 y Fn(Fix)16 b(some)45 b Fl(")13 b(>)g Fn(0)30 b(.)19 b(If)46 b Fk(j)p Fl(x)757 1209 y Fi(1)787 1202 y Fk(\000)10 b Fl(x)858 1209 y Fi(2)878 1202 y Fk(j)891 1209 y Fi(1)923 1202 y Fl(<)j(N)1013 1185 y Fj("=)p Fi(2)1096 1202 y Fn(,)i(w)o(e)g(arriv)o(e)g(at)460 1307 y Fk(j)o Fl(r)493 1314 y Fi(2)513 1307 y Fn(\()p Fl(x)557 1314 y Fi(1)576 1307 y Fl(;)8 b(x)623 1314 y Fi(2)642 1307 y Fn(\))p Fk(j)k(\024)h Fn(2)i(\(2)9 b(+)i(log)d Fl(N)d Fn(\))22 b(exp\(2)p Fl(s)p 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Fn(\))p Fk(j)k(\024)h Fn(2)i(\(2)9 b(+)i(log)d Fl(N)d Fn(\))14 b Fl(const)p Fn(\()p Fl(")p Fn(\))1267 1642 y Fh(\022)1427 1670 y Fn(1)p 1303 1691 V 1303 1733 a(1)c(+)g Fk(j)p Fl(x)1420 1740 y Fi(1)1449 1733 y Fk(\000)h Fl(x)1521 1740 y Fi(2)1541 1733 y Fk(j)1554 1717 y Fi(2)1554 1745 y(1)1578 1642 y Fh(\023)1609 1653 y Fi(1)p Fj(=)p Fi(2)p Fg(\000)p Fj("=)p Fi(4)1962 1701 y Fn(\(66\))249 1866 y(In)o(tegrating)k(the)g(r.h.s.)k(of)45 b(\(65\))30 b(,)g(\(66\))44 b(o)o(v)o(er)760 1972 y Fk(f)p Fl(x)809 1979 y Fi(1)828 1972 y Fl(;)8 b(x)875 1979 y Fi(2)907 1972 y Fk(2)13 b Fn([0)p Fl(;)8 b(N)d Fn(])p Fl(;)35 b Fk(j)p Fl(x)1149 1979 y Fi(1)1178 1972 y Fk(\000)11 b Fl(x)1250 1979 y Fi(2)1270 1972 y Fk(j)h Fl(<)h Fn(\()p Fk(\025)p Fn(\))p Fl(N)1456 1953 y Fj("=)p Fi(2)1509 1972 y Fk(g)249 2077 y Fn(w)o(e)i(obtain)h(the)g(desired)h(result.)k(The)16 b(case)46 b Fl(l)14 b Fk(\025)g Fn(2)46 b(will)17 b(b)q(e)f(treated)f(in)i(a) e(similar)i(fashion.)k(Again)249 2144 y(the)d(inequalities)57 b(\(43\))35 b(,)i(\(45\))53 b(are)17 b(crucial)j(in)f(our)f(calculations.)30 b(W)l(e)18 b(k)o(eep)h(the)f(former)f(and)249 2210 y(sligh)o(tly)c(re\014ne)f (the)g(latter.)18 b(Namely)l(,)13 b(using)f(the)g(pro)q(cedure)h(describ)q (ed)g(on)f(the)g(pages)f(17-18)g(,)h(section)249 2277 y(2)j(and)g(Remark)h (from)e(the)h(page)g(19)g(,)g(w)o(e)g(can)g(estimate)g(eac)o(h)g(term)g(in)51 b(~)-28 b Fl(\032)1577 2284 y Fj(l)p Fi(+)p Fj(m;l)1715 2277 y Fn(as)482 2340 y Fh(\014)482 2365 y(\014)482 2390 y(\014)482 2415 y(\014)482 2440 y(\014)496 2361 y Fj(l)p Fi(+)p Fj(m)504 2374 y Fh(Y)522 2464 y Fi(1)663 2379 y Fn(sin)8 b Fl(\031)r Fn(\()p Fl(x)798 2386 y Fj(i)821 2379 y Fk(\000)j Fl(x)893 2388 y Fj(\033)q Fi(\()p Fj(i)p Fi(\))956 2379 y Fn(\))p 593 2403 450 2 v 593 2445 a Fl(N)h Fn(sin)q(\()p Fl(\031)r Fn(\()p Fl(x)788 2452 y Fj(i)811 2445 y Fk(\000)f Fl(x)883 2454 y Fj(\033)q Fi(\()p Fj(i)p Fi(\))946 2445 y Fn(\))p Fl(=)m(N)5 b Fn(\))1048 2340 y Fh(\014)1048 2365 y(\014)1048 2390 y(\014)1048 2415 y(\014)1048 2440 y(\014)1074 2414 y Fk(\024)1138 2361 y Fj(K)1128 2374 y Fh(Y)1122 2466 y Fj(k)q Fi(=1)1498 2383 y Fn(2)p 1214 2403 591 2 v 1214 2445 a(1)10 b(+)g(max)1385 2411 y Fh(\000)1404 2445 y Fk(j)p Fl(x)1443 2452 y Fj(j)1457 2458 y Fd(k)q Ff(+1)1526 2445 y Fk(\000)h Fl(x)1598 2452 y Fj(j)1612 2458 y Fd(k)1633 2445 y Fk(j)1646 2452 y Fi(1)1676 2445 y Fk(\000)f Fl(s;)e Fn(0)1786 2411 y Fh(\001)1123 2578 y Fn(33)p eop %%Page: 34 34 bop 249 403 a Fn(where)46 b Fl(\033)29 b Fk(2)13 b Fl(S)537 410 y Fj(l)p Fi(+)p Fj(m)654 403 y Fn(is)j(suc)o(h,)f(that)30 b Fk(J)8 b Fn(\()p Fl(\033)r Fn(\))44 b(is)16 b(connected,)g(and)932 515 y Fl(j)951 522 y Fi(1)983 515 y Fk(!)d Fl(j)1060 522 y Fi(2)1087 515 y Fl(:)8 b(:)g(:)j Fk(!)i Fl(j)1230 522 y Fj(k)1263 515 y Fk(!)g Fl(j)1340 522 y Fi(1)249 627 y Fn(is)i(some)f(closed)h(path)e (on)43 b Fk(J)8 b Fn(\()p Fl(\033)r Fn(\))41 b(with)14 b(p)q(ossible)i(in)o (tersections,)f(visiting)h(all)f(v)o(ertices)42 b Fk(f)p Fn(1)p Fl(;)8 b(:)g(:)g(:)t(l)q Fk(g)28 b Fn(,)249 693 y(with)17 b(the)f(n)o(um)o(b) q(er)h(of)f(steps)48 b Fl(K)k Fn(not)16 b(greater)f(than)49 b Fl(l)q Fn(\()p Fl(l)10 b Fn(+)i(1\))p Fl(=)p Fn(2)26 b Fk(\000)11 b Fn(1)48 b(.)23 b(Using)17 b(the)g(inequalit)o(y)249 760 y(\(64\))44 b(and)16 b(eliminating)h(p)q(ossible)g(sel\014n)o(tersections)f(as)f (explained)j(in)e(section)f(2,w)o(e)g(arriv)o(e)g(at)361 852 y Fh(\014)361 877 y(\014)361 902 y(\014)375 848 y Fj(l)p Fi(+)p Fj(m)384 861 y Fh(Y)382 952 y Fj(i)p Fi(=1)527 866 y Fn(sin)8 b Fl(\031)r Fn(\()p Fl(x)662 873 y Fj(i)685 866 y Fk(\000)j Fl(x)757 875 y Fj(\033)q Fi(\()p Fj(i)p Fi(\))820 866 y Fn(\))p 457 890 450 2 v 457 932 a Fl(N)h Fn(sin)q(\()p Fl(\031)r Fn(\()p Fl(x)652 939 y Fj(i)675 932 y Fk(\000)f Fl(x)747 941 y Fj(\033)q Fi(\()p Fj(i)p Fi(\))810 932 y Fn(\))p Fl(=)m(N)5 b Fn(\))912 852 y Fh(\014)912 877 y(\014)912 902 y(\014)938 901 y Fk(\024)13 b Fn(\(2)i(\(2)9 b(+)i Fl(s)1159 882 y Fi(2)1179 901 y Fn(\)\))1214 879 y Fg(\000)p Fi(1+\()p Fj(l)p Fi(+1\))p Fj(l=)p Fi(2)1466 848 y Fj(l)1445 861 y Fh(Y)1441 952 y Fj(j)r Fi(=1)1676 870 y Fn(2)p 1530 890 315 2 v 1530 932 a(1)f(+)g Fk(j)p Fl(x)1647 939 y Fj(j)1675 932 y Fk(\000)h Fl(x)1747 941 y Fj(\034)t Fi(\()p Fj(j)r Fi(\))1812 932 y Fk(j)1825 939 y Fi(1)1962 901 y Fn(\(67\))249 1044 y(where)46 b Fl(\034)k Fn(is)16 b(some)f(cyclic)i(p)q(erm)o(utation)e (of)g(in)o(tegers)46 b(1)p Fl(;)8 b(:)g(:)g(:)t(l)q(:)249 1110 y Fn(The)16 b(estimates)45 b(\(43\))29 b(,)h(\(67\))44 b(imply)249 1257 y Fk(j)t Fn(~)-27 b Fl(\032)286 1264 y Fj(l)p Fi(+)p Fj(m;l)378 1257 y Fk(j)12 b(\024)459 1217 y Fh(X)451 1309 y Fj(\034)t Fg(2)p Fj(S)516 1315 y Fd(l)545 1257 y Fn(min)639 1173 y Fh(0)639 1248 y(@)690 1257 y Fl(const)p Fn(\()p Fl(l)q Fn(\))j Fl(;)22 b Fn(\()p Fl(m)10 b Fn(+)g(2\)!)k(\(2)h(\(2)9 b(+)i(log)d Fl(N)d Fn(\)\))1373 1235 y Fg(\000)p Fi(1+\()p Fj(l)p Fi(+1\))p Fj(l=)p Fi(2)1625 1204 y Fj(l)1604 1217 y Fh(Y)1600 1308 y Fj(j)r Fi(=1)1835 1227 y Fn(2)p 1689 1247 V 1689 1289 a(1)10 b(+)g Fk(j)p Fl(x)1806 1296 y Fj(j)1834 1289 y Fk(\000)h Fl(x)1906 1298 y Fj(\034)t Fi(\()p Fj(j)r Fi(\))1971 1289 y Fk(j)1984 1296 y Fi(1)2023 1173 y Fh(1)2023 1248 y(A)249 1402 y Fn(and)582 1514 y Fk(j)p Fl(r)616 1521 y Fj(l)629 1514 y Fn(\()p Fl(x)673 1521 y Fi(1)692 1514 y Fl(;)d(:)g(:)g(:)d(x)799 1521 y Fj(l)812 1514 y Fl(;)j(s)p Fn(\))p Fk(j)j(\024)28 b Fl(const)p Fn(\()p Fl(l)q Fn(\))15 b(\(2)f(\(2)c(+)g(log)f Fl(N)c Fn(\)\))1426 1492 y Fg(\000)p Fi(1+\()p Fj(l)p Fi(+1\))p Fj(l=)p Fi(2)1640 1514 y Fk(\001)528 1622 y(\001)606 1582 y Fh(X)598 1674 y Fj(\034)t Fg(2)p Fj(S)663 1680 y Fd(l)717 1569 y Fg(1)705 1582 y Fh(X)697 1672 y Fj(m)p Fi(=0)801 1591 y Fn(1)p 786 1612 53 2 v 786 1653 a Fl(m)p Fn(!)858 1622 y(\()p Fl(l)16 b(s)p Fn(\))945 1603 y Fj(m)1001 1622 y Fn(min)8 b(\()15 b(1;)22 b(\()p Fl(m)10 b Fn(+)g(2\)!)1389 1569 y Fj(l)1368 1582 y Fh(Y)1364 1673 y Fj(j)r Fi(=1)1598 1591 y Fn(2)p 1453 1612 315 2 v 1453 1653 a(1)g(+)g Fk(j)p Fl(x)1570 1660 y Fj(j)1598 1653 y Fk(\000)g Fl(x)1669 1663 y Fj(\034)t Fi(\()p Fj(j)r Fi(\))1735 1653 y Fk(j)1748 1660 y Fi(1)1787 1622 y Fn(\))157 b(\(68\))249 1824 y(Fix)46 b Fl(\034)17 b Fk(2)c Fl(S)469 1831 y Fj(l)513 1824 y Fn(.)19 b(If)857 1863 y Fj(l)836 1875 y Fh(Y)832 1966 y Fj(j)r Fi(=1)908 1916 y Fn(\()c(1)10 b(+)g Fk(j)p Fl(x)1058 1923 y Fj(j)1086 1916 y Fk(\000)h Fl(x)1158 1925 y Fj(\034)t Fi(\()p Fj(j)r Fi(\))1223 1916 y Fk(j)1236 1923 y Fi(1)1255 1916 y Fn(\))i Fl(<)g(N)1376 1897 y Fj("=)p Fi(\(2)p Fj(l)p Fi(\))1962 1916 y Fn(\(69\))249 2042 y(the)i(corresp)q (onding)i(summand)e(in)46 b(\(68\))e(will)17 b(b)q(e)f(estimated)g(from)e(ab) q(o)o(v)o(e)h(b)o(y)668 2154 y Fl(const)p Fn(\()p Fl(l)q Fn(\))g(\(4)9 b(+)i(2)d(log)f Fl(N)e Fn(\))1092 2136 y Fi(\()p Fj(l)p Fi(+1\))p Fj(l=)p Fi(2)p Fg(\000)p Fi(1)1291 2154 y Fn(exp\()p Fl(l)16 b Fn(\(log)8 b Fl(N)d Fn(\))1552 2136 y Fi(1)p Fj(=)p Fi(2)1606 2154 y Fn(\))338 b(\(70\))249 2266 y(If)857 2298 y Fj(l)836 2310 y Fh(Y)832 2401 y Fj(j)r Fi(=1)908 2351 y Fn(\()15 b(1)10 b(+)g Fk(j)p Fl(x)1058 2358 y Fj(j)1086 2351 y Fk(\000)h Fl(x)1158 2360 y Fj(\034)t Fi(\()p Fj(j)r Fi(\))1223 2351 y Fk(j)1236 2358 y Fi(1)1255 2351 y Fn(\))i Fk(\025)g Fl(N)1376 2332 y Fj("=)p Fi(\(2)p Fj(l)p Fi(\))1962 2351 y Fn(\(71\))1123 2578 y(34)p eop %%Page: 35 35 bop 249 403 a Fn(then)16 b(for)e(an)o(y)45 b Fl(")559 410 y Fi(1)592 403 y Fl(>)13 b Fn(0)352 540 y(\()p Fl(l)i(s)p Fn(\))438 521 y Fj(m)484 540 y Fn(=)e Fl(o)p Fn(\()612 487 y Fj(l)591 499 y Fh(Y)587 590 y Fj(j)r Fi(=1)663 540 y Fn(\(1)d(+)g Fk(j)p Fl(x)798 547 y Fj(j)826 540 y Fk(\000)h Fl(x)898 549 y Fj(\034)t Fi(\()p Fj(j)r Fi(\))963 540 y Fk(j)976 547 y Fi(1)995 540 y Fn(\))1013 518 y Fi(1)p Fj(=)p Fi(2+)p Fj(")1110 523 y Ff(1)1129 540 y Fn(\))30 b Fl(;)8 b(if)34 b Fn(\()p Fl(m)10 b Fn(+)g Fl(l)q Fn(\)!)i Fk(\030)1529 487 y Fj(l)1508 499 y Fh(Y)1504 590 y Fj(j)r Fi(=1)1580 540 y Fn(\()1749 509 y(2)p 1603 529 315 2 v 1603 571 a(1)e(+)g Fk(j)p Fl(x)1720 578 y Fj(j)1748 571 y Fk(\000)h Fl(x)1820 580 y Fj(\034)t Fi(\()p Fj(j)r Fi(\))1885 571 y Fk(j)1898 578 y Fi(1)1922 540 y Fn(\))249 681 y(and)k(the)h(corresp)q (onding)g(summand)f(is)h(not)f(greater)f(than)444 818 y Fl(const)p Fn(\()p Fl(l)q Fn(\))h(\(4)9 b(+)i(2)d(log)f Fl(N)e Fn(\))868 799 y Fi(\()p Fj(l)p Fi(+1\))p Fj(l=)p Fi(2)p Fg(\000)p Fi(1)1059 818 y Fl(const)p Fn(\()p Fl(")1204 825 y Fi(1)1225 818 y Fn(\))15 b(\()1316 765 y Fj(l)1294 777 y Fh(Y)1290 869 y Fj(j)r Fi(=1)1525 787 y Fn(1)p 1379 807 V 1379 849 a(1)10 b(+)h Fk(j)p Fl(x)1497 856 y Fj(j)1525 849 y Fk(\000)f Fl(x)1596 858 y Fj(\034)t Fi(\()p Fj(j)r Fi(\))1661 849 y Fk(j)1674 856 y Fi(1)1699 818 y Fn(\))1716 796 y Fi(1)p Fj(=)p Fi(2)k Fg(\000)p Fj(")1827 801 y Ff(1)1962 818 y Fn(\(72\))249 961 y(for)h(an)o(y)47 b Fl(")458 968 y Fi(1)491 961 y Fl(>)14 b Fn(0)30 b(.)22 b(In)o(tegrating)46 b(\(70\))30 b(,)h(\(72\))46 b(o)o(v)o(er)14 b(the)i(domains)47 b(\(69\))30 b(,)h(\(71\))46 b(w)o(e)15 b(obtain)249 1028 y(the)g(desired)i (result.)j(Lemma)15 b(3.2)g(is)g(pro)o(v)o(en)g(.)p 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b Fn(=)c([\(log)8 b Fl(N)d Fn(\))1419 1365 y Fi(1)p Fj(=)p Fi(2)1473 1381 y Fl(N)1515 1365 y Fi(3)p Fj(=)p Fi(4)1570 1381 y Fn(])45 b(segmen)o(ts)649 1492 y([)p Fl(s)683 1499 y Fj(i)697 1492 y Fl(;)8 b(s)739 1499 y Fj(i)p Fi(+1)797 1492 y Fn(])46 b Fl(;)22 b(s)912 1499 y Fj(i)939 1492 y Fn(=)13 b(\(log)8 b Fl(N)d Fn(\))1131 1473 y Fi(1)p Fj(=)p Fi(2)1200 1492 y Fl(i=)m(M)20 b Fn(;)j Fl(i)12 b Fn(=)h(0)p Fl(;)8 b(:)g(:)g(:)t(M)16 b Fk(\000)10 b Fn(1)249 1602 y(w)o(e)15 b(estimate)g(the)g(probabilit)o(y)694 1713 y Fk(P)726 1720 y Fj(N)774 1713 y Fk(f)g Fn(sup)831 1746 y Fj(s)847 1751 y Fd(i)888 1713 y Fk(j)p Fl(\021)924 1720 y Fj(N)957 1713 y Fn(\()p Fl(s)996 1720 y Fj(i)1010 1713 y Fn(\))9 b Fk(\000)i Fn(I)-9 b(E)15 b Fl(\021)1159 1720 y Fj(N)1193 1713 y Fn(\()p Fl(s)1232 1720 y Fj(i)1246 1713 y Fn(\))p Fk(j)d Fl(>)h Fn(1)p Fl(=)p Fn(2)h Fl(N)1462 1694 y Fi(1)p Fj(=)p Fi(2)f(+)p Fj(")1575 1713 y Fk(g)249 1835 y Fn(b)o(y)i(the)h(sum)f(of)f(probabilities)k(:)309 1946 y Fk(P)341 1953 y Fj(N)390 1946 y Fk(f)d Fn(sup)447 1980 y Fj(s)463 1985 y Fd(i)504 1946 y Fk(j)p Fl(\021)540 1953 y Fj(N)572 1946 y Fn(\()p Fl(s)611 1953 y Fj(i)625 1946 y Fn(\))10 b Fk(\000)h Fn(I)-9 b(E)15 b Fl(\021)775 1953 y Fj(N)808 1946 y Fn(\()p Fl(s)847 1953 y Fj(i)861 1946 y Fn(\))p Fk(j)d Fl(>)h Fn(1)p Fl(=)p Fn(2)i Fl(N)1078 1927 y Fi(1)p Fj(=)p Fi(2)e(+)p Fj(")1191 1946 y Fk(g)f Fl(<)h(const)p Fn(\()p Fl(";)8 b(n)p Fn(\))15 b(\(log)8 b Fl(N)d Fn(\))1644 1927 y Fi(1)p Fj(=)p Fi(2)1713 1946 y Fl(N)1755 1927 y Fg(\000)p Fj(n)16 b Fi(+3)p Fj(=)p Fi(4)1962 1946 y Fn(\(74\))249 2130 y(T)l(o)h(\014nish)h(the)f(pro)q (of)g(w)o(e)f(claim)i(that)f(v)m(ariations)g(of)51 b Fl(\021)1255 2137 y Fj(N)1288 2130 y Fn(\()p Fl(s)p Fn(\))f(on)17 b(the)g(segmen)o(ts)51 b([)p Fl(s)1804 2137 y Fj(i)1818 2130 y Fl(;)8 b(s)1860 2137 y Fj(i)p Fi(+1)1918 2130 y Fn(])51 b(are)249 2196 y(negligibly)18 b(small,as)d(w)o(ell)h(as)f(the)g(tails)h(of)45 b Fl(\021)1057 2203 y Fj(N)1090 2196 y Fn(\()p Fl(s)p Fn(\))p Fl(;)22 b Fn(I)-9 b(E)16 b Fl(\021)1259 2203 y Fj(N)1292 2196 y Fn(\()p Fl(s)p Fn(\))f Fl(;)8 b(w)q(hen)30 b(s)13 b Fk(\025)g Fn(\(log)8 b Fl(N)d Fn(\))1749 2180 y Fi(1)p Fj(=)p Fi(2)1848 2196 y Fn(.)249 2263 y(Namely)l(,similar)17 b(to)45 b(\(55\))452 2373 y Fl(V)9 b(ar)q(iation)655 2383 y Fi([)p Fj(s)681 2388 y Fd(i)694 2383 y Fj(;s)720 2388 y Fd(i)p Ff(+1)772 2383 y Fi(])799 2373 y Fl(\021)822 2380 y Fj(N)855 2373 y Fn(\()p Fl(s)p Fn(\))k Fk(\024)g(j)o Fl(\021)1008 2380 y Fj(N)1042 2373 y Fn(\()p Fl(s)1081 2380 y Fj(i)p Fi(+1)1140 2373 y Fn(\))c Fk(\000)i Fl(\021)1236 2380 y Fj(N)1269 2373 y Fn(\()p Fl(s)1308 2380 y Fj(i)1322 2373 y Fn(\))p Fk(j)375 2452 y(\024)42 b(j)o Fl(\021)487 2459 y Fj(N)520 2452 y Fn(\()p Fl(s)559 2459 y Fj(i)p Fi(+1)619 2452 y Fn(\))9 b Fk(\000)i Fn(I)-9 b(E)15 b Fl(\021)768 2459 y Fj(N)801 2452 y Fn(\()p Fl(s)840 2459 y Fj(i)p Fi(+1)900 2452 y Fn(\))p Fk(j)9 b Fn(+)h Fk(j)p Fl(\021)1021 2459 y Fj(N)1054 2452 y Fn(\()p Fl(s)1093 2459 y Fj(i)1107 2452 y Fn(\))g Fk(\000)h Fn(I)-9 b(E)15 b Fl(\021)1257 2459 y Fj(N)1290 2452 y Fn(\()p Fl(s)1329 2459 y Fj(i)1343 2452 y Fn(\))p Fk(j)10 b Fn(+)25 b Fl(V)10 b(ar)q(iation)1648 2461 y Fi([)p Fj(s)1674 2466 y Fd(i)1687 2461 y Fj(;s)1713 2466 y Fd(i)p Ff(+1)1765 2461 y Fi(])1792 2452 y Fn(I)-9 b(E)15 b Fl(\021)1868 2459 y Fj(N)1902 2452 y Fn(\()p Fl(s)p Fn(\))s(\(75\))1123 2578 y(35)p eop %%Page: 36 36 bop 249 403 a Fn(and)15 b(b)o(y)46 b(\(58\))29 b(,)h(\(59\))44 b(and)16 b(smo)q(othness)f(of)45 b Fl(F)6 b Fn(\()p Fl(s)p Fn(\))264 501 y Fl(V)k(ar)q(iation)468 510 y Fi([)p Fj(s)494 515 y Fd(i)507 510 y Fj(;s)533 515 y Fd(i)p Ff(+1)585 510 y Fi(])612 501 y Fn(I)-9 b(E)15 b Fl(\021)688 508 y Fj(N)721 501 y Fn(\()p Fl(s)p Fn(\))42 b Fk(\024)110 b Fn(sup)896 545 y Fi([0)p Fj(;)p Fi(\(log)5 b Fj(N)t Fi(\))1045 535 y Ff(1)p Fd(=)p Ff(2)1093 545 y Fi(])1118 501 y Fk(j)14 b Fn(I)-9 b(E)16 b Fl(\021)1222 508 y Fj(N)1255 501 y Fn(\()p Fl(s)p Fn(\))10 b Fk(\000)18 b Fl(N)h(F)6 b Fn(\()p Fl(s)p Fn(\))p Fk(j)k Fn(+)26 b Fl(N)20 b(V)9 b(ar)q(iation)1867 510 y Fi([)p Fj(s)1893 515 y Fd(i)1906 510 y Fj(;s)1932 515 y Fd(i)p Ff(+1)1984 510 y Fi(])2011 501 y Fl(F)d Fn(\()p Fl(s)p Fn(\))820 618 y(=)41 b Fl(o)p Fn(\()p Fl(N)978 599 y Fj(")996 618 y Fn(\))10 b(+)g Fl(O)q Fn(\()p Fl(N)1165 599 y Fi(1)p Fj(=)p Fi(4)1219 618 y Fn(\))786 b(\(76\))249 775 y(Estimates)45 b(\(74\))29 b(,)h(\(75\))f(,)h (\(76\))44 b(imply)327 872 y Fk(P)359 879 y Fj(N)408 872 y Fk(f)104 b Fn(sup)445 917 y Fi(0)p Fg(\024)p Fj(s)p Fg(\024)p Fi(\(log)6 b Fj(N)t Fi(\))645 907 y Ff(1)p Fd(=)p Ff(2)708 872 y Fk(j)p Fl(\021)744 879 y Fj(N)777 872 y Fn(\()p Fl(s)p Fn(\))k Fk(\000)g Fn(I)-9 b(E)16 b Fl(\021)966 879 y Fj(N)999 872 y Fn(\()p Fl(s)p Fn(\))p Fk(j)c Fl(>)h(N)1171 854 y Fi(1)p Fj(=)p Fi(2)g(+)p Fj(")1284 872 y Fk(g)g Fl(<)g(const)p Fn(\()p Fl(";)8 b(n)p Fn(\))14 b Fl(N)1635 854 y Fg(\000)p Fj(n)p Fi(+3)p Fj(=)p Fi(4)1766 872 y Fn(\(log)8 b Fl(N)d Fn(\))1910 854 y Fi(1)p Fj(=)p Fi(2)249 999 y Fn(Cho)q(osing)46 b Fl(n)13 b(>)g Fn(7)p Fl(=)p Fn(4)44 b(w)o(e)15 b(ha)o(v)o(e)602 1064 y Fg(1)590 1077 y Fh(X)581 1169 y Fj(N)t Fi(=1)681 1117 y Fk(P)j(f)84 b Fn(sup)769 1161 y Fi([0)p Fj(;)p Fi(\(log)5 b Fj(N)t Fi(\))918 1152 y Ff(1)p Fd(=)p Ff(2)965 1161 y Fi(])990 1117 y Fk(j)p Fl(\021)1026 1124 y Fj(N)1059 1117 y Fn(\()p Fl(s)p Fn(\))10 b Fk(\000)g Fn(I)-9 b(E)16 b Fl(\021)1248 1124 y Fj(N)1281 1117 y Fn(\()p Fl(s)p Fn(\))p Fk(j)c Fl(>)h(N)1453 1098 y Fi(1)p Fj(=)p Fi(2)g(+)p Fj(")1566 1117 y Fk(g)f Fl(<)29 b Fk(1)249 1241 y Fn(and)12 b(applying)i(Borel-Can)o(telli)g(lemma)f(w)o(e)e(\014nd)i (that)f(with)g(probabilit)o(y)i(1)e(there)g(exists)g(some)g(in)o(teger)249 1307 y Fl(N)286 1314 y Fi(0)336 1307 y Fn(,)i(suc)o(h)i(that)e(for)h(an)o(y) 45 b Fl(N)17 b(>)c(N)889 1314 y Fi(0)812 1405 y Fn(sup)743 1450 y Fi([0)p Fj(;)p Fi(\(log)5 b Fj(N)t Fi(\))892 1440 y Ff(1)p Fd(=)p Ff(2)940 1450 y Fi(])965 1405 y Fk(j)o Fl(\021)1000 1412 y Fj(N)1033 1405 y Fn(\()p Fl(s)p Fn(\))10 b Fk(\000)h Fn(I)-9 b(E)15 b Fl(\021)1222 1412 y Fj(N)1255 1405 y Fn(\()p Fl(s)p Fn(\))p Fk(j)27 b(\024)13 b Fl(N)1442 1387 y Fi(1)p Fj(=)p Fi(2)h(+)p Fj(")1962 1405 y Fn(\(77\))249 1539 y(Since)j(for)30 b Fl(s)12 b(>)h Fn(\(log)8 b Fl(N)d Fn(\))678 1523 y Fi(1)p Fj(=)p Fi(2)763 1539 y Fn(w)o(e)15 b(ha)o(v)o(e)739 1637 y Fl(\021)762 1644 y Fj(N)795 1637 y Fn(\()p Fl(s)p Fn(\))d Fk(\024)h Fl(\021)r Fn(\()o(\(log)8 b Fl(N)d Fn(\))1098 1618 y Fi(1)p Fj(=)p Fi(2)1152 1637 y Fn(\))30 b Fl(;)739 1716 y Fn(I)-9 b(E)15 b Fl(\021)815 1723 y Fj(N)848 1716 y Fn(\()p Fl(s)p Fn(\))28 b(=)13 b Fl(N)19 b(F)1066 1723 y Fj(N)1100 1716 y Fn(\()p Fl(s)p Fn(\))27 b Fk(\024)13 b Fl(N)20 b(F)1318 1723 y Fj(N)1352 1716 y Fn(\(\(log)8 b Fl(N)d Fn(\))1514 1697 y Fi(1)p Fj(=)p Fi(2)1568 1716 y Fn(\))30 b Fl(;)739 1795 y(N)19 b(F)6 b Fn(\()p Fl(s)p Fn(\))28 b Fk(\024)13 b Fl(N)20 b(F)6 b Fn(\(\(log)i Fl(N)d Fn(\))1217 1776 y Fi(1)p Fj(=)p Fi(2)1271 1795 y Fn(\))249 1893 y(and)735 1990 y Fl(N)19 b(F)6 b Fn(\()q(\(log)i Fl(N)d Fn(\))989 1972 y Fi(1)p Fj(=)p Fi(2)1043 1990 y Fn(\))12 b(=)h Fl(o)p Fn(\()p Fl(N)1203 1972 y Fg(\000)p Fi(1)p Fj(=)p Fi(8)1285 1990 y Fn(\))h Fl(;)735 2069 y(N)777 2022 y Fh(\020)801 2069 y Fl(F)6 b Fn(\(\(log)i Fl(N)d Fn(\))998 2051 y Fi(1)p Fj(=)p Fi(2)1052 2069 y Fn(\))10 b Fk(\000)h Fl(F)1155 2076 y Fj(N)1188 2069 y Fn(\(\(log)d Fl(N)d Fn(\))1350 2051 y Fi(1)p Fj(=)p Fi(2)1404 2069 y Fn(\))1422 2022 y Fh(\021)1459 2069 y Fn(=)28 b Fl(o)p Fn(\()p Fl(N)1604 2051 y Fj(")1622 2069 y Fn(\))249 2226 y(\()15 b(see)46 b(\(60\))29 b(,)h(\(61\))f(\))45 b(,w)o(e)15 b(can)g(extend)h(the)45 b(sup)h(in)g(\(77\))f(to)14 b(the)i(whole)f(real)h(axis)f(:)719 2324 y(sup)710 2365 y Fi([0)p Fj(;)p Fg(1)p Fi(\))812 2324 y Fk(j)o Fl(\021)847 2331 y Fj(N)881 2324 y Fn(\()p Fl(s)p Fn(\))9 b Fk(\000)i Fn(I)-9 b(E)15 b Fl(\021)1069 2331 y Fj(N)1102 2324 y Fn(\()p Fl(s)p Fn(\))p Fk(j)27 b(\024)13 b Fl(const)p Fn(\()p Fl(")p Fn(\))j Fl(N)1468 2305 y Fi(1)p Fj(=)p Fi(2)d(+)p Fj(")249 2444 y Fn(Corollary)i(1.5)f(is)i (pro)o(v)o(en.)p 759 2448 21 30 v 1123 2578 a(36)p eop %%Page: 37 37 bop 249 403 a Fm(4)69 b(Orthogonal)16 b(and)g(Symplec)o(ti)o(c)c(Groups.)29 b(Conclusiv)n(e)14 b(Rem)o(arks)296 514 y Fn(The)i(results)g(of)g(our)f(pap)q (er)i(are)e(v)m(alid)i(for)f(the)f(other)h(Classical)h(Compact)e(Groups)g(as) h(w)o(ell.)22 b(The)249 581 y(k)o(ey)17 b(factor)e(here)j(is)f(the)g(V)l (andermonde)g(determinan)o(t)g(nature)g(of)f(the)h(densit)o(y)g(of)g(the)g (distribution)249 647 y(function)f(of)e(eigen)o(v)m(alues.)21 b(F)l(orm)o(ulas)14 b(for)g(the)h(disribution)i(of)d(the)h(eigen)o(v)m(alues) h(with)f(resp)q(ect)g(to)f(the)249 713 y(normalized)j(Haar)d(measure)h(are)g (classical\()i(see)e(\()g([5)o(])g(\))30 b(\).)249 780 y(Ho)o(w)o(ev)o(er)16 b(it)i(has)f(b)q(een)i(noted)e(only)h(recen)o(tly)g(b)o(y)f(N.Katz)g(and)h(P) l(.Sarnak)f(\()f([11])g(\))h(that)g(the)g(corre-)249 846 y(sp)q(onding)i(n-p) q(oin)o(t)f(correlation)g(functions)g(ha)o(v)o(e)f(the)h(form)e(of)h 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Fn(exp)q(\()p Fk(\000)p Fl(i\022)1607 1572 y Fj(N)1640 1565 y Fn(\))716 1644 y(0)13 b Fk(\024)g Fl(\022)821 1651 y Fi(1)853 1644 y Fk(\024)g Fl(\022)922 1651 y Fi(2)955 1644 y Fk(\024)g Fl(:)8 b(:)g(:)e(\022)1085 1651 y Fj(N)1131 1644 y Fk(\024)13 b Fl(\031)757 b Fn(\(78\))249 1737 y(The)16 b(probabilit)o(y)g(distribution)h(of)e(eigen)o(v)m(alues)i(is)e (de\014ned)i(b)o(y)e(its)g(densit)o(y)h(:)595 1847 y Fl(P)624 1854 y Fj(N)658 1847 y Fn(\()p Fl(\022)697 1854 y Fi(1)716 1847 y Fl(;)8 b(:)g(:)g(:)d(\022)818 1854 y Fj(N)852 1847 y Fn(\))13 b(=)g(2)976 1787 y Fh(\022)1025 1816 y Fn(1)p 1012 1836 51 2 v 1012 1878 a(2)p Fl(\031)1067 1787 y Fh(\023)1097 1799 y Fj(N)1207 1806 y Fh(Y)1154 1898 y Fi(1)p Fg(\024)p Fj(i)g Fn(0)30 b(.)19 b(Similary)e(to)e(the)g(calculations)i(in)f (the)f(section)h(3)f(one)g(can)g(sho)o(w)g(that)711 739 y(sup)702 780 y Fi([0)p Fj(;)p Fg(1)p Fi(\))804 739 y Fk(j)o Fn(I)-9 b(E)16 b Fl(\021)893 746 y Fj(N)926 739 y Fn(\([0)p Fl(;)8 b(\031)r Fn(])p Fl(;)g(s)p Fn(\))f 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