%!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: paper.dvi %%Pages: 16 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentPaperSizes: Letter %%EndComments %DVIPSCommandLine: dvips -o jianglu.ps paper %DVIPSParameters: dpi=600, compressed, comments removed %DVIPSSource: TeX output 1999.06.20:0831 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B 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Fl(n;k)1127 4736 y Fn(=)17 b Fp(log)1319 4760 y Fm(2)1375 4736 y Fn(n)28 b Fi(!)f Fp(1)32 b Fh(almost)f(surely)-8 b(.)1872 5280 y Fk(2)p eop %%Page: 3 3 3 2 bop 0 83 a Fo(Theorem)38 b(2.)55 b Fh(In)33 b(a)f Fo(urrd)p Fh(,)h(for)f(\014xed)i(in)m(tegers)f Fn(r)i Fh(and)e Fn(m)f Fh(with)h Fp(1)27 b Fi(\024)h Fn(r)j Fi(\024)d Fn(m)p Fh(,)1208 301 y Fp(lim)1183 361 y Fl(n)p Fg(!1)1523 234 y Fn(M)1617 249 y Fl(n)p 1394 278 400 4 v 1394 370 a Fp(log)1520 393 y Fm(1+1)p Fl(=r)1735 370 y Fn(n)1831 301 y Fp(=)g(1)k Fh(almost)f(surely)-8 b(.)0 652 y Fo(Remark.)102 b Fp(If)34 b(w)m(e)h(de\014ne)g(a)f(random)g (graph)g(on)g Fn(n)g Fp(no)s(des)g(b)m(y)h(c)m(ho)s(osing)g Fn(n)23 b Fi(\000)g Fp(1)34 b(edges)h(at)f(random)0 785 y(from)c(the)h Fn(n)p Fp(\()p Fn(n)19 b Fi(\000)f Fp(1\))p Fn(=)p Fp(2)30 b(p)s(ossible)i(edges,)g(then)f(the)g(maximal)h(degree)f (is)g(in)g(probabilit)m(y)g(asymptotic)0 918 y(to)i(log)17 b Fn(n=)g Fp(log)g(log)h Fn(n)p Fp(.)44 b(This)34 b(follo)m(ws)g(from)f 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y(Theorem)34 b(1)e(via)h(the)g(follo)m(wing)g(device.)1872 5280 y Fk(3)p eop %%Page: 4 4 4 3 bop 0 83 a Fo(Lemma)36 b(1.)55 b Fh(Let)31 b Fn(a)735 98 y Fl(n)812 83 y Fh(b)s(e)g(a)f(sequence)j(of)d(p)s(ositiv)m(e)g(n)m (um)m(b)s(ers,)i(and)f(let)f Fn(A)2711 98 y Fl(ni)2812 83 y Fh(b)s(e)h(the)g(ev)m(en)m(t)h Fp([)p Fn(D)3472 98 y Fl(i)3528 83 y Fi(\025)c Fn(a)3684 98 y Fl(n)3732 83 y Fp(])p Fh(.)0 216 y(Then)921 263 y Fl(n)871 293 y Ff(X)885 503 y Fl(i)p Fm(=1)1031 387 y Fj(P)p Fi(f)p Fn(A)1231 402 y Fl(ni)1302 387 y Fi(g)f(\025)h Fj(P)17 b Fi(f)o([)1693 346 y Fl(n)1693 412 y(i)p Fm(=1)1812 387 y Fn(A)1885 402 y Fl(ni)1956 387 y Fi(g)28 b(\025)2233 245 y Ff(P)2338 271 y Fl(n)2338 348 y(i)p Fm(=1)2473 319 y Fj(P)p Fi(f)p Fn(A)2673 334 y Fl(ni)2744 319 y Fi(g)p 2149 364 730 4 v 2149 456 a Fp(1)22 b(+)2318 381 y Ff(P)2423 407 y Fl(n)2423 485 y(i)p Fm(=1)2558 456 y Fj(P)p Fi(f)p Fn(A)2758 471 y Fl(ni)2828 456 y Fi(g)2888 387 y Fn(:)0 623 y Fh(Also,)680 795 y Fp(lim)656 855 y Fl(n)p 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b(\025)1860 1507 y Fp(\()1898 1433 y Ff(P)2004 1459 y Fl(n)2004 1536 y(i)p Fm(=1)2138 1507 y Fj(P)17 b Fi(f)o Fn(A)2354 1522 y Fl(ni)2426 1507 y Fi(g)o Fp(\))2513 1471 y Fm(2)p 1495 1552 1424 4 v 1495 1569 a Ff(P)1600 1595 y Fl(n)1600 1673 y(i)p Fm(=1)1735 1644 y Fj(P)g Fi(f)o Fn(A)1951 1659 y Fl(ni)2022 1644 y Fi(g)22 b Fp(+)2192 1569 y Ff(P)2297 1673 y Fl(i)p Fg(6)p Fm(=)p Fl(j)2429 1644 y Fj(P)17 b Fi(f)o Fn(A)2645 1659 y Fl(ni)2716 1644 y Fn(A)2789 1659 y Fl(nj)2869 1644 y Fi(g)2928 1575 y Fn(:)0 1796 y Fp(As)33 b(w)m(e)h(will)f(sho)m (w)h(further)f(on,)1211 1974 y Fj(P)17 b Fi(f)o Fn(A)1427 1989 y Fl(ni)1498 1974 y Fn(A)1571 1989 y Fl(nj)1650 1974 y Fi(g)28 b(\024)g Fj(P)17 b Fi(f)o Fn(A)2049 1989 y Fl(ni)2120 1974 y Fi(g)f Fj(P)h Fi(f)o Fn(A)2402 1989 y Fl(nj)2482 1974 y Fi(g)f Fn(;)0 2152 y Fp(for)32 b(all)h Fn(i)27 b Fi(6)p Fp(=)h Fn(j)6 b Fp(,)33 b(so)f(that)462 2372 y Fj(P)17 b Fi(f)o([)671 2331 y Fl(n)671 2397 y(i)p Fm(=1)790 2372 y Fn(A)863 2387 y Fl(ni)934 2372 y Fi(g)28 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Fp(T)-8 b(o)48 b(sho)m(w)g(the)g (former)g(inequalit)m(y)-8 b(,)53 b(w)m(e)48 b(note)g(that)f(the)h(m)m (ultinomial)h(distribution)g(is)f(negativ)m(e)0 3000 y(orthan)m(t)26 b(dep)s(enden)m(t)i(\(see)g(Joag-Dev)d(and)i(Prosc)m (han,)h(1983,)f(for)f(de\014nitions)h(and)g(a)f(discussion\).)43 b(In)0 3133 y(particular,)29 b(if)e(\()p Fn(X)676 3148 y Fm(1)715 3133 y Fn(;)17 b(:)g(:)g(:)f(;)h(X)1015 3148 y Fl(k)1057 3133 y Fp(\))27 b(is)h(a)f(m)m(ultinomial)i(random)e(v)m (ector,)i(then)f(w)m(e)h(ha)m(v)m(e)f(for)f(all)g Fn(x)3402 3148 y Fm(1)3442 3133 y Fn(;)17 b(:)g(:)g(:)f(;)h(x)3716 3148 y Fl(k)3759 3133 y Fp(,)1087 3392 y Fj(P)1181 3311 y Ff(\010)1238 3392 y Fi(\\)1304 3351 y Fl(k)1304 3417 y(i)p Fm(=1)1423 3392 y Fp([)p Fn(X)1531 3407 y Fl(i)1586 3392 y Fi(\025)29 b Fn(x)1747 3407 y Fl(i)1775 3392 y Fp(])1802 3311 y Ff(\011)1888 3392 y Fi(\024)2038 3268 y Fl(k)1993 3297 y Ff(Y)2000 3507 y Fl(i)p Fm(=1)2137 3392 y Fj(P)17 b Fi(f)o Fn(X)2361 3407 y Fl(i)2417 3392 y Fi(\025)28 b Fn(x)2577 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4733 y Fp(])g Fi(\021)g([)3251 4697 y Fl(n)3251 4758 y(i)p Fm(=1)3370 4733 y Fn(A)3443 4748 y Fl(ni)3514 4733 y Fp(.)p 3584 4679 65 4 v 3584 4737 4 59 v 3646 4737 V 3584 4740 65 4 v 299 4947 a(W)-8 b(e)36 b(\014rst)g(sho)m(w)h(the)g(\014rst)f (statemen)m(t)h(of)f(Theorem)h(1)e(with)i(\\almost)f(surely")h (replaced)g(b)m(y)0 5080 y(\\in)d(probabilit)m(y".)49 b(Clearly)-8 b(,)36 b(w)m(e)f(need)g(only)g(v)m(erify)g(the)g(limits)g (of)e(the)i(sums)g(sho)m(wn)h(in)e(Lemma)h(1,)1872 5280 y Fk(4)p eop %%Page: 5 5 5 4 bop 0 83 a Fp(\014rst)36 b(for)e Fn(a)406 98 y Fl(n)486 83 y Fp(=)e Fi(d)p Fn(c)17 b Fp(log)h Fn(n)p Fi(e)35 b Fp(with)h Fn(c)c(>)g Fp(1)p Fn(=)17 b Fp(log)g(2,)36 b(and)f(later)h(with)g Fn(c)c(<)g Fp(1)p Fn(=)17 b Fp(log)g(2.)51 b(In)36 b(the)f(second)i(part,)0 216 y(w)m(e)d(obtain)e(strong)h(con)m (v)m(ergence.)46 b(In)33 b(the)g(third)g(part,)f(w)m(e)i(deal)f(with)g Fn(M)2761 231 y Fl(n;k)2866 216 y Fp(.)0 535 y Fo(Pr)n(oof)38 b(of)h(the)e(weak)h(conver)n(gence:)48 b(upper)36 b(bound.)99 b Fp(W)-8 b(e)33 b(will)g(b)s(egin)g(b)m(y)g(sho)m(wing)1367 790 y(lim)1342 849 y Fl(n)p Fg(!1)1593 665 y Fl(n)1543 695 y Ff(X)1558 905 y Fl(i)p Fm(=1)1703 790 y Fj(P)17 b Fi(f)o Fn(D)1927 805 y Fl(i)1983 790 y Fi(\025)28 b Fn(a)2139 805 y Fl(n)2187 790 y Fi(g)f Fp(=)h(0)p Fn(;)0 1062 y Fp(where)23 b Fn(a)322 1077 y Fl(n)397 1062 y Fp(=)28 b Fi(d)p Fn(c)17 b Fp(log)g Fn(n)p Fi(e)22 b Fp(with)h Fn(c)k(>)h Fp(1)p Fn(=)17 b Fp(log)g(2.)39 b(W)-8 b(e)23 b(\014rst)f(obtain)g(the)g(follo)m(wing)g(exp)s(onen)m (tial)i(inequalit)m(y:)1214 1258 y Fj(P)17 b Fi(f)o Fn(D)1438 1273 y Fl(i)1494 1258 y Fi(\025)28 b Fn(a)1650 1273 y Fl(n)1697 1258 y Fi(g)g(\024)g Fn(e)1925 1217 y Fl(c)11 b Fm(log)i Fl(n)p Fm(\(1)p Fg(\000)p Fl(u)p Fm(+log)q(\()p Fl(u)p Fm(\)\))2545 1258 y Fn(;)0 1455 y Fp(where)34 b Fn(u)27 b Fp(=)h(\()p Fn(H)588 1470 y Fl(n)p Fg(\000)p Fm(1)747 1455 y Fi(\000)22 b Fn(H)927 1470 y Fl(i)955 1455 y Fp(\))p Fn(=)p Fp(\()p Fn(c)17 b Fp(log)g Fn(n)p Fp(\).)44 b(Indeed,)34 b(for)e Fn(t)c(>)f Fp(0,)33 b(b)m(y)g(Mark)m(o)m (v's)i(inequalit)m(y)-8 b(,)450 1651 y Fj(P)17 b Fi(f)o Fn(D)674 1666 y Fl(i)729 1651 y Fi(\025)29 b Fn(a)886 1666 y Fl(n)933 1651 y Fi(g)e(\024)h Fj(E)1206 1570 y Ff(\010)1264 1651 y Fn(e)1309 1610 y Fl(tD)1392 1620 y Fe(i)1419 1610 y Fg(\000)p Fl(ta)1536 1618 y Fe(n)1583 1570 y Ff(\011)1010 1809 y Fi(\024)g Fj(E)1206 1728 y Ff(\010)1264 1809 y Fn(e)1309 1768 y Fl(tD)1392 1778 y Fe(i)1419 1768 y Fg(\000)p Fl(tc)11 b Fm(log)i Fl(n)1691 1728 y Ff(\011)1010 2021 y Fi(\024)28 b Fn(e)1160 1980 y Fg(\000)p Fl(tc)12 b Fm(log)h Fl(n)1450 1927 y Ff(Y)1458 2137 y Fl(j)t(>i)1594 1881 y Ff(\022)1667 2021 y Fp(1)22 b Fi(\000)1931 1954 y Fp(1)p 1848 1998 217 4 v 1848 2090 a Fn(j)28 b Fi(\000)22 b Fp(1)2096 2021 y(+)2275 1954 y Fn(e)2320 1918 y Fl(t)p 2204 1998 V 2204 2090 a Fn(j)28 b Fi(\000)23 b Fp(1)2431 1881 y Ff(\023)2553 2021 y Fp(\(b)m(y)33 b(indep)s(endence\))1010 2288 y Fi(\024)28 b Fn(e)1160 2247 y Fg(\000)p Fl(tc)12 b Fm(log)h Fl(n)p Fm(+)1484 2197 y Fd(P)1554 2268 y Fe(j)s(>i)1657 2247 y Fm(\()p Fl(e)1717 2223 y Fe(t)1745 2247 y Fg(\000)p Fm(1\))p Fl(=)p Fm(\()p Fl(j)t Fg(\000)p Fm(1\))1010 2446 y Fp(=)28 b Fn(e)1159 2405 y Fg(\000)p Fl(tc)12 b Fm(log)h Fl(n)p Fm(+\()p Fl(e)1543 2381 y Fe(t)1570 2405 y Fg(\000)p Fm(1\)\()p Fl(H)1772 2414 y Fe(n)p Fc(\000)p Fb(1)1894 2405 y Fg(\000)p Fl(H)2007 2415 y Fe(i)2034 2405 y Fm(\))2098 2446 y Fp(\()p Fn(i)28 b Fi(\025)g Fp(2\))p Fn(:)0 2642 y Fp(The)46 b(last)f(upp)s(er)h(b)s(ound)f(is)g(minimal)h(when)g Fn(t)g Fp(is)f(the)h(solution)f(of)g Fn(e)2706 2606 y Fl(t)2784 2642 y Fp(=)k Fn(c)17 b Fp(log)g Fn(n=)p Fp(\()p Fn(H)3336 2657 y Fl(n)p Fg(\000)p Fm(1)3504 2642 y Fi(\000)31 b Fn(H)3693 2657 y Fl(i)3721 2642 y Fp(\).)0 2775 y(Resubstitution)j (yields)g(the)f(b)s(ound.)44 b(Since)34 b Fn(u)e Fp(is)h(a)f (decreasing)i(function)f(of)f Fn(i)p Fp(,)h(w)m(e)h(note)f(that)931 2877 y Ff(X)927 3087 y Fl(in)932 4032 y Fb(1)p Fc(\000)p Fe(\017)1056 3928 y Fj(P)17 b Fi(f)o Fn(D)1280 3943 y Fl(i)1336 3928 y Fi(\025)28 b Fn(a)1492 3943 y Fl(n)1540 3928 y Fi(g)f(\024)h Fn(n)p Fj(P)1874 3848 y Ff(\010)1932 3928 y Fn(D)2013 3945 y Fg(d)p Fl(n)2087 3926 y Fb(1)p Fc(\000)p Fe(\017)2196 3945 y Fg(e)2259 3928 y Fi(\025)g Fn(a)2415 3943 y Fl(n)2462 3848 y Ff(\011)1617 4189 y Fi(\024)g Fn(ne)1825 4148 y Fl(c)12 b Fm(log)h Fl(n)f Fm(\(1+)p Fl(o)p Fm(\(1\)\)\(1)p Fg(\000)p Fl(\017=c)p Fm(+log)s(\()p Fl(\017=c)p Fm(\)\))1617 4347 y Fi(\024)28 b Fn(e)1767 4306 y Fg(\000)12 b Fm(log)h Fl(n)f Fm(\(1+)p Fl(o)p Fm(\(1\)\)\()p Fg(\000)p Fl(c)p Fm(+)p Fl(\017)p Fg(\000)p Fl(c)h Fm(log)q(\()p Fl(\017=c)p Fm(\))p Fg(\000)p Fm(1\))2949 4347 y Fn(;)0 4543 y Fp(whic)m(h)40 b(tends)f(to)e(0)h(pro)m(vided)i(that)e Fn(\017)f(<)g(c=e)1711 4507 y Fm(2)1750 4543 y Fp(|again,)i(this)g(is)f(satis\014ed)i(for)d (all)h Fn(\017)g Fp(small)h(enough.)0 4676 y(The)f(\014rst)f(part)g(of) g(the)g(theorem)h(follo)m(ws)g(if)f(w)m(e)h(can)f(sho)m(w)h(that)f(for) f(all)h Fn(\017)f(>)f Fp(0)i(small)g(enough)h(w)m(e)0 4809 y(ha)m(v)m(e)1259 4956 y(lim)1235 5015 y Fl(n)p Fg(!1)1543 4861 y Ff(X)1436 5078 y Fl(n)1479 5059 y Fe(\017)1509 5078 y Fg(\024)p Fl(i)p Fg(\024)p Fl(n)1686 5059 y Fb(1)p Fc(\000)p Fe(\017)1811 4956 y Fj(P)17 b Fi(f)o Fn(D)2035 4971 y Fl(i)2090 4956 y Fi(\025)29 b Fn(a)2247 4971 y Fl(n)2294 4956 y Fi(g)e Fp(=)h(0)p Fn(:)1872 5280 y Fk(5)p eop %%Page: 6 6 6 5 bop 0 83 a Fp(If)26 b(w)m(e)i(set)f Fn(i)h Fp(=)f Fn(n)597 47 y Fl(\014)645 83 y Fp(,)g(with)g Fn(\014)33 b Fi(2)28 b Fp([)p Fn(\017;)17 b Fp(1)9 b Fi(\000)g Fn(\017)p Fp(],)30 b(then)d Fn(u)g Fp(=)g(\(1)9 b Fi(\000)g Fn(\014)d Fp(\))p Fn(=c)j Fp(+)g Fn(o)p Fp(\(1\),)28 b(where)g(the)f Fn(o)p Fp(\(1\))e(term)i(do)s(es)g(not)0 216 y(dep)s(end)d(up)s(on)f Fn(\014)6 b Fp(.)40 b(Also,)25 b(for)e Fn(n)1131 180 y Fl(\017)1192 216 y Fi(\024)28 b Fn(y)t(;)17 b(z)31 b Fi(\024)d Fn(n)1632 180 y Fm(1)p Fg(\000)p Fl(\017)1755 216 y Fp(,)d Fi(j)p Fn(y)6 b Fi(\000)s Fn(z)t Fi(j)28 b(\024)g Fp(1,)d(w)m(e)f(ha)m(v)m(e)g Fi(j)p Fn(\036)p Fp(\()p Fn(u)p Fp(\()p Fn(y)t Fp(\)\))s Fi(\000)s Fn(\036)p Fp(\()p Fn(u)p Fp(\()p Fn(z)t Fp(\)\))p Fi(j)h(\024)j Fp(1)p Fn(=\017n)3726 180 y Fl(\017)3759 216 y Fp(,)0 361 y(where)34 b Fn(\036)p Fp(\()p Fn(s)p Fp(\))489 305 y Fm(d)o(ef)499 361 y Fp(=)k(1)22 b Fi(\000)g Fn(s)h Fp(+)f(log)q(\()p Fn(s)p Fp(\).)43 b(Th)m(us,)34 b(b)m(y)g(the)f (uniformit)m(y)g(of)f(this)h(estimate,)261 518 y Ff(X)154 735 y Fl(n)197 716 y Fe(\017)227 735 y Fg(\024)p Fl(i)p Fg(\024)p Fl(n)404 716 y Fb(1)p Fc(\000)p Fe(\017)529 613 y Fj(P)17 b Fi(f)o Fn(D)753 628 y Fl(i)809 613 y Fi(\025)28 b Fn(a)965 628 y Fl(n)1012 613 y Fi(g)g(\024)g Fn(o)p Fp(\(1\))22 b(+)g(\(1)g(+)g Fn(o)p Fp(\(1\)\))1921 477 y Ff(Z)2019 503 y Fl(n)2062 480 y Fb(1)p Fc(\000)p Fe(\017)1975 703 y Fl(n)2018 684 y Fe(\017)2191 613 y Fn(e)2236 566 y Fl(c)12 b Fm(log)h Fl(n)2425 574 y Fp(\()2463 566 y Fm(1)p Fg(\000)2563 539 y Fb(1)p 2563 551 31 3 v 2564 592 a Fe(c)2603 574 y Fp(\()2641 566 y Fm(1)p Fg(\000)2743 536 y Fb(log)f Fe(x)p 2741 551 129 3 v 2741 592 a Fb(log)g Fe(n)2879 574 y Fp(\))2917 566 y Fm(+log)3085 539 y Fb(1)p 3085 551 31 3 v 3086 592 a Fe(c)3125 574 y Fp(\()3163 566 y Fm(1)p Fg(\000)3265 536 y Fb(log)g Fe(x)p 3264 551 129 3 v 3264 592 a Fb(log)f Fe(n)3402 574 y Fp(\)\))3498 613 y Fn(dx:)0 893 y Fp(By)33 b(the)g(transform)g Fn(v)e Fp(=)d(1)22 b Fi(\000)g Fp(log)c Fn(x=)f Fp(log)g Fn(n)p Fp(,)33 b(w)m(e)h(can)f(rewrite)g(the)g(last)g(in)m(tegral)g(as) 202 996 y Ff(Z)301 1022 y Fm(1)p Fg(\000)p Fl(\017)257 1221 y(\017)441 1131 y Fn(e)486 1090 y Fl(c)11 b Fm(log)i Fl(n)p Fm(\(1)p Fg(\000)p Fl(v)r(=c)p Fm(+log)r(\()p Fl(v)r(=c)p Fm(\)\))1229 1131 y Fn(n)1287 1090 y Fm(1)p Fg(\000)p Fl(v)1435 1131 y Fp(log)18 b Fn(n)f(dv)31 b Fp(=)c Fn(n)1943 1090 y Fm(1+)p Fl(c)p Fg(\000)p Fl(c)11 b Fm(log)i Fl(c)2316 1131 y Fp(log)k Fn(n)2533 996 y Ff(Z)2633 1022 y Fm(1)p Fg(\000)p Fl(\017)2588 1221 y(\017)2772 1131 y Fn(n)2830 1090 y Fl(c)12 b Fm(log)h Fl(v)r Fg(\000)p Fm(2)p Fl(v)3160 1131 y Fn(dv)1782 1414 y Fp(=)1895 1346 y Fn(n)1953 1310 y Fm(1+)p Fl(c)p Fg(\000)p Fl(c)e Fm(log)i Fl(c)2326 1346 y Fp(log)k Fn(n)p 1895 1391 632 4 v 1898 1482 a Fp(\(2)g(log)h Fn(n)p Fp(\))2241 1453 y Fm(1+)p Fl(c)11 b Fm(log)i Fl(n)2553 1278 y Ff(Z)2653 1304 y Fm(2\(1)p Fg(\000)p Fl(\017)p Fm(\))f(log)h Fl(n)2608 1504 y Fm(2)p Fl(\017)f Fm(log)h Fl(n)3040 1414 y Fn(w)3113 1373 y Fl(c)e Fm(log)h Fl(n)3304 1414 y Fn(e)3349 1373 y Fg(\000)p Fl(w)3461 1414 y Fn(dw)1782 1699 y Fi(\024)1897 1632 y Fn(n)1955 1596 y Fm(1+)p Fl(c)p Fg(\000)p Fl(c)f Fm(log)i Fl(c)2327 1632 y Fp(log)18 b Fn(n)p 1897 1677 V 1900 1768 a Fp(\(2)f(log)g Fn(n)p Fp(\))2242 1739 y Fm(1+)p Fl(c)12 b Fm(log)h Fl(n)2538 1699 y Fp(\000\(1)22 b(+)g Fn(c)17 b Fp(log)g Fn(n)p Fp(\))1782 1924 y Fi(\030)28 b Fn(n)1945 1883 y Fm(1)p Fg(\000)p Fl(c)12 b Fm(log)h(2)2237 1834 y Ff(p)p 2336 1834 416 4 v 2336 1924 a Fn(\031)t(c)k Fp(log)g Fn(n=)p Fp(2)33 b Fn(;)0 2110 y Fp(where)39 b(w)m(e)f(used)h(the)f(gamma)f(in)m(tegral)h(and)f(Stirling's)i(form)m (ula.)58 b(The)38 b(last)g(expression)i(tends)e(to)0 2243 y(zero)33 b(with)g Fn(n)p Fp(.)0 2535 y Fo(Pr)n(oof)g(of)f(the)g (weak)g(conver)n(gence:)45 b(lo)n(wer)32 b(bound.)97 b Fp(Assume)30 b(next)e(that)f Fn(c)h(<)f Fp(1)p Fn(=)17 b Fp(log)g(2.)0 2668 y(De\014ne)29 b Fn(b)339 2683 y Fl(n)414 2668 y Fp(=)e Fn(b)i Fp(=)e Fi(b)p Fn(\016)21 b Fp(log)d Fn(n)p Fi(c)p Fp(,)29 b(where)h Fn(\016)h Fi(2)d Fp(\(0)p Fn(;)17 b Fp(1)p Fn(=)g Fp(log)g(2)c Fi(\000)g Fn(c)p Fp(\))29 b(is)f(an)g(arbitrary)h(small)g(p)s(ositiv)m (e)g(n)m(um)m(b)s(er.)0 2801 y(F)-8 b(urthermore,)33 b Fn(\017)c(>)f Fp(0)k(is)h(a)g(\014xed)h(small)f(n)m(um)m(b)s(er)h (suc)m(h)h(that)d(2)p Fn(\017)c(<)g(c)22 b Fp(+)h Fn(\016)32 b(<)27 b Fp(2\(1)22 b Fi(\000)h Fn(\017)p Fp(\).)44 b(W)-8 b(e)33 b(consider)0 2934 y(an)k(in)m(teger)h Fn(i)g Fp(with)g Fn(n)823 2897 y Fl(\017)892 2934 y Fi(\024)e Fn(i)g Fi(\024)g Fn(n)1245 2897 y Fm(1)p Fg(\000)p Fl(\017)1368 2934 y Fp(,)j(and)e(assume)i(that)e Fn(n)g Fp(is)h(so)g(large)f(that)g Fn(b)p Fp(\()p Fn(i)26 b Fi(\000)g Fp(1\))35 b Fn(>)h Fp(4)h(for)g(all)0 3066 y(suc)m(h)d Fn(i)p Fp(.)44 b(Let)33 b Fn(B)573 3081 y Fm(2)613 3066 y Fn(;)17 b(B)731 3081 y Fm(3)770 3066 y Fn(;)g(:)g(:)g(:)32 b Fp(b)s(e)h(indep)s(enden)m(t)i (Bernoulli)e(random)g(v)-5 b(ariables)33 b(with)h Fj(E)o Fn(B)3200 3081 y Fl(j)3265 3066 y Fp(=)28 b(1)p Fn(=)p Fp(\()p Fn(j)f Fi(\000)c Fp(1\),)0 3199 y(and)36 b(let)g Fn(P)400 3214 y Fm(2)439 3199 y Fn(;)17 b(P)546 3214 y Fm(3)585 3199 y Fn(;)g(:)g(:)g(:)f(;)35 b Fp(b)s(e)h(indep)s(enden)m (t)i(P)m(oisson)f(random)f(v)-5 b(ariables)36 b(with)g Fj(E)p Fn(P)2999 3214 y Fl(j)3068 3199 y Fp(=)d(1)p Fn(=)p Fp(\()p Fn(j)c Fi(\000)c Fp(1\).)52 b(Set)0 3332 y Fn(a)45 b Fp(=)f Fn(a)267 3347 y Fl(n)358 3332 y Fp(=)h Fi(d)p Fn(c)17 b Fp(log)g Fn(n)p Fi(e)p Fp(.)73 b(In)43 b(trying)g(to)f(deriv) m(e)i(a)e(lo)m(w)m(er)i(b)s(ound)e(for)g Fj(P)17 b Fi(f)o Fn(D)2833 3347 y Fl(i)2889 3332 y Fi(\025)28 b Fn(a)p Fi(g)p Fp(,)45 b(w)m(e)e(cannot)g(use)0 3465 y(the)d(standard)f(b)s (ounds)h(on)f(the)h(closeness)i(of)c(the)i(distribution)g(of)f(a)g(sum) h(of)f(Bernoulli)h(random)0 3598 y(v)-5 b(ariables)43 b(to)e(a)g(P)m(oisson)j(random)e(v)-5 b(ariable)42 b(\(see)g(Le)h(Cam)f (\(1960\);)j(see)e(Deheuv)m(els)h(and)e(Pfeifer)0 3731 y(\(1986\))33 b(for)h(the)h(most)f(recen)m(t)i(references\),)h(as)d (these)i(b)s(ounds)f(are)f(to)s(o)g(large.)48 b(Ho)m(w)m(ev)m(er,)37 b(in)e(a)f(ric)m(h)0 3863 y(enough)f(probabilit)m(y)h(space,)f(there)h (exists)g(an)e(em)m(b)s(edding)j(suc)m(h)f(that)1054 4050 y Fn(D)1135 4065 y Fl(i)1191 4050 y Fp(=)1294 3955 y Ff(X)1310 4165 y Fl(j)t(>i)1455 4050 y Fn(I)1498 4065 y Fm([)p Fl(R)1571 4075 y Fe(j)1603 4065 y Fm(=)p Fl(i)p Fm(])1733 4050 y Fp(=)1837 3955 y Ff(X)1853 4165 y Fl(j)t(>i)1997 4050 y Fn(B)2071 4065 y Fl(j)1733 4326 y Fi(\025)1838 4231 y Ff(X)1855 4441 y Fl(j)t(>i)1999 4326 y Fn(P)2062 4341 y Fl(j)2098 4326 y Fn(I)2141 4341 y Fm([)p 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Fp(and)j(th)m(us)0 216 y(that)41 b Fn(P)283 231 y Fl(j)361 216 y Fp(=)g(0.)68 b(If)41 b Fn(B)802 231 y Fl(j)881 216 y Fp(=)g(1,)i(w)m(e)f(trivially)g (ha)m(v)m(e)g Fn(B)1953 231 y Fl(j)2031 216 y Fi(\025)g Fn(P)2213 231 y Fl(j)2250 216 y Fn(I)2293 231 y Fm([)p Fl(P)2358 241 y Fe(j)2390 231 y Fg(\024)p Fm(1])2504 216 y Fp(.)68 b(Rep)s(eat)41 b(this)g(for)g(all)f Fn(j)6 b Fp(.)69 b(Note)0 371 y(that)38 b Fn(W)309 386 y Fl(i)375 371 y Fp(is)h(P)m(oisson)g(with)g(parameter)g Fn(\026)1632 316 y Fm(d)o(ef)1642 371 y Fp(=)1765 297 y Ff(P)1870 400 y Fl(j)t(>i)2002 371 y Fp(1)p Fn(=)p Fp(\()p Fn(j)32 b Fi(\000)26 b Fp(1\))37 b(=)g Fn(H)2631 386 y Fl(n)p Fg(\000)p Fm(1)2794 371 y Fi(\000)26 b Fn(H)2978 386 y Fl(i)3006 371 y Fp(.)60 b(F)-8 b(rom)38 b(the)g(trivial)0 504 y(inequalit)m(y)924 661 y Fj(P)17 b Fi(f)o Fn(D)1148 676 y Fl(i)1204 661 y Fi(\025)28 b Fn(a)p Fi(g)g(\025)g Fj(P)17 b Fi(f)o Fn(W)1778 676 y Fl(i)1834 661 y Fi(\025)28 b Fn(a)22 b Fp(+)g Fn(b)p Fi(g)h(\000)f Fj(P)17 b Fi(f)o Fn(Z)2533 676 y Fl(i)2589 661 y Fi(\025)28 b Fn(b)p Fi(g)50 b Fn(;)0 852 y Fp(w)m(e)34 b(note)e(that)h(it)f(su\016ces)j(to)d(sho)m (w)i(that)1155 1059 y(lim)1131 1119 y Fl(n)p Fg(!1)1438 964 y Ff(X)1331 1181 y Fl(n)1374 1162 y Fe(\017)1404 1181 y Fg(\024)p Fl(i)p Fg(\024)p Fl(n)1581 1162 y Fb(1)p Fc(\000)p Fe(\017)1706 1059 y Fj(P)17 b Fi(f)o Fn(W)1941 1074 y Fl(i)1997 1059 y Fi(\025)28 b Fn(a)22 b Fp(+)g Fn(b)p Fi(g)28 b Fp(=)g Fi(1)k Fn(;)1006 b Fp(\(1\))0 1354 y(and)1279 1511 y(lim)1254 1571 y Fl(n)p Fg(!1)1562 1416 y Ff(X)1455 1634 y Fl(n)1498 1615 y Fe(\017)1528 1634 y Fg(\024)p Fl(i)p Fg(\024)p Fl(n)1705 1615 y Fb(1)p Fc(\000)p Fe(\017)1830 1511 y Fj(P)17 b Fi(f)o Fn(Z)2040 1526 y Fl(i)2095 1511 y Fi(\025)29 b Fn(b)p Fi(g)f Fp(=)f(0)32 b Fn(;)1130 b Fp(\(2\))0 1795 y(to)32 b(conclude)i(b)m(y)g(Lemma)f(1)f (that)h(lim)17 b Fj(P)p Fi(f)p Fn(M)1670 1810 y Fl(n)1744 1795 y Fi(\025)28 b Fn(a)1900 1810 y Fl(n)1947 1795 y Fi(g)g Fp(=)f(1.)166 1936 y Fo(Verifica)-7 b(tion)37 b(of)f(\(1\).)44 b Fp(If)31 b Fn(V)1361 1951 y Fl(i)1419 1936 y Fp(is)h(P)m(oisson)g(with)g(parameter)f(log)q(\(\()p Fn(n)19 b Fi(\000)g Fp(1\))p Fn(=)p Fp(\()p Fn(i)f Fi(\000)h Fp(1\)\),)31 b(then)166 2069 y(uniformly)i(o)m(v)m(er)h(our)f Fn(i)p Fp(,)629 2276 y Fj(P)17 b Fi(f)o Fn(W)864 2291 y Fl(i)920 2276 y Fi(\025)28 b Fn(a)23 b Fp(+)f Fn(b)p Fi(g)28 b(\025)g Fj(P)17 b Fi(f)o Fn(V)1621 2291 y Fl(i)1677 2276 y Fi(\025)28 b Fn(a)22 b Fp(+)g Fn(b)p Fi(g)1316 2479 y(\025)1564 2411 y Fp(1)p 1431 2456 316 4 v 1431 2547 a(\()p Fn(a)g Fp(+)g Fn(b)p Fp(\)!)1773 2479 y(log)1899 2436 y Fl(a)p Fm(+)p Fl(b)2043 2338 y Ff(\022)2126 2411 y Fn(n)g Fi(\000)h Fp(1)p 2126 2456 229 4 v 2138 2547 a Fn(i)g Fi(\000)f Fp(1)2365 2338 y Ff(\023)2477 2411 y Fn(i)g Fi(\000)h Fp(1)p 2465 2456 V 2465 2547 a Fn(n)f Fi(\000)h Fp(1)1316 2771 y(=)1429 2703 y(log)1555 2660 y Fl(a)p Fm(+)p Fl(b)1682 2703 y Fp(\()p Fn(n)f Fi(\000)h Fp(1\))p 1429 2748 558 4 v 1550 2839 a(\()p Fn(a)f Fp(+)g Fn(b)p Fp(\)!)2019 2703 y Fn(i)g Fi(\000)h Fp(1)p 2007 2748 229 4 v 2007 2839 a Fn(n)f Fi(\000)h Fp(1)2262 2630 y Ff(\022)2335 2771 y Fp(1)f Fi(\000)2528 2703 y Fp(log)q(\()p Fn(i)h Fi(\000)f Fp(1\))p 2516 2748 431 4 v 2516 2839 a(log)q(\()p Fn(n)g Fi(\000)h Fp(1\))2956 2630 y Ff(\023)3030 2653 y Fl(a)p Fm(+)p Fl(b)1316 3063 y Fp(=)1429 2995 y(\(1)f Fi(\000)h Fn(o)p Fp(\(1\)\))17 b(log)1990 2953 y Fl(a)p Fm(+)p Fl(b)2117 2995 y Fp(\()p Fn(n)p Fp(\))p 1429 3040 822 4 v 1682 3131 a(\()p Fn(a)22 b Fp(+)g Fn(b)p Fp(\)!)2283 2995 y Fn(i)p 2271 3040 59 4 v 2271 3131 a(n)2355 2922 y Ff(\022)2429 3063 y Fp(1)g Fi(\000)2622 2995 y Fp(log)q(\()p Fn(i)p Fp(\))p 2609 3040 261 4 v 2609 3131 a(log)q(\()p Fn(n)p Fp(\))2879 2922 y Ff(\023)2953 2945 y Fl(a)p Fm(+)p Fl(b)3096 3063 y Fn(:)166 3309 y Fp(Th)m(us,)283 3481 y Ff(X)176 3698 y Fl(n)219 3679 y Fe(\017)249 3698 y Fg(\024)p Fl(i)p Fg(\024)p Fl(n)426 3679 y Fb(1)p Fc(\000)p Fe(\017)551 3576 y Fj(P)17 b Fi(f)o Fn(W)786 3591 y Fl(i)842 3576 y Fi(\025)28 b Fn(a)22 b Fp(+)g 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b(monotonicit)m(y)f(of)f(b)s(oth)g Fn(M)3145 98 y Fl(n)3227 83 y Fp(and)h(log)17 b Fn(n)36 b Fp(are)0 216 y(used)e(in)f(the)g(inequalities)769 447 y(inf)748 509 y Fl(N)7 b Fg(\024)p Fl(n)1011 379 y Fn(M)1105 394 y Fm(2)1140 375 y Fe(n)p 935 424 329 4 v 935 515 a Fp(log)18 b(2)1127 486 y Fl(n)p Fm(+1)1301 447 y Fi(\024)65 b Fp(inf)1406 516 y Fm(2)1441 498 y Fe(N)1499 516 y Fg(\024)p Fl(n)1653 379 y Fn(M)1747 394 y Fl(n)p 1623 424 201 4 v 1623 515 a Fp(log)18 b Fn(n)1862 447 y Fi(\024)50 b Fp(sup)1967 536 y Fm(2)2002 517 y Fe(N)2060 536 y Fg(\024)p Fl(n)2214 379 y Fn(M)2308 394 y Fl(n)p 2184 424 V 2184 515 a Fp(log)18 b Fn(n)2423 447 y Fi(\024)35 b Fp(sup)2528 528 y Fl(N)7 b Fg(\024)p Fl(n)2715 379 y Fn(M)2809 396 y Fm(2)2844 377 y Fe(n)p Fb(+1)p 2715 424 254 4 v 2722 515 a Fp(log)18 b(2)2914 486 y Fl(n)3011 447 y Fn(:)0 702 y Fp(T)-8 b(o)33 b(sho)m(w)g(that)1123 864 y(lim)17 b(sup)1180 943 y Fl(n)p Fg(!1)1478 797 y Fn(M)1572 812 y Fl(n)p 1448 841 201 4 v 1448 932 a Fp(log)h Fn(n)1687 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Fm(2)1156 2673 y Fe(n)p 982 2721 267 4 v 982 2812 a Fn(n)17 b Fp(log)h(2)1286 2744 y Fn(<)28 b(c)1432 2603 y Ff(\033)1534 2744 y Fp(=)f(1)22 b Fi(\000)h Fj(P)1902 2603 y Ff(\032)2031 2677 y Fn(M)2125 2692 y Fm(2)2160 2673 y Fe(n)p 1986 2721 V 1986 2812 a Fn(n)17 b Fp(log)g(2)2290 2744 y Fi(\025)28 b Fn(c)2437 2603 y Ff(\033)1534 3033 y Fi(\024)g Fp(1)22 b Fi(\000)1904 2890 y Ff(P)2009 2917 y Fm(2)2044 2893 y Fe(n)2009 2994 y Fl(i)p Fm(=1)2144 2965 y Fj(P)17 b Fi(f)o Fn(D)2368 2980 y Fl(i)2424 2965 y Fi(\025)28 b Fn(cn)17 b Fp(log)h(2)p Fi(g)p 1820 3010 1153 4 v 1820 3113 a Fp(1)j(+)1988 3039 y Ff(P)2094 3065 y Fm(2)2129 3046 y Fe(n)2094 3142 y Fl(i)p Fm(=1)2229 3113 y Fj(P)c Fi(f)o Fn(D)2453 3128 y Fl(i)2508 3113 y Fi(\025)29 b Fn(cn)17 b Fp(log)g(2)p Fi(g)1534 3307 y(\024)1773 3240 y Fp(1)p 1649 3284 298 4 v 1649 3376 a(2)1698 3347 y Fl("n)1799 3376 y Fp(+)22 b(1)1989 3307 y Fn(:)0 3528 y Fp(This)34 b(to)s(o)d(is)j(a)e(summable)i(series.)45 b(By)33 b(the)g(Borel-Can)m(telli)h(lemma,)1154 3759 y(lim)17 b(inf)1197 3819 y Fl(n)p Fg(!1)1480 3692 y Fn(M)1574 3707 y Fl(n)p 1450 3737 201 4 v 1450 3828 a Fp(log)h Fn(n)1689 3759 y Fi(\025)1875 3692 y Fp(1)p 1804 3737 192 4 v 1804 3828 a(log)f(2)2038 3759 y(almost)33 b(surely)q Fn(:)1872 5280 y Fk(9)p eop %%Page: 10 10 10 9 bop 0 83 a Fo(Pr)n(oof)48 b(of)g(the)e(last)h(p)-7 b(ar)g(t)47 b(of)h(Theorem)g(1.)124 b Fp(Here)43 b(w)m(e)f(use)g(a)f (simple)i(result)g(on)e(P\023)-49 b(oly)m(a)0 216 y(urns)36 b(that)g(ma)m(y)h(b)s(e)e(found)h(in)g(A)m(threy)m(a)h(and)f(Ney)h (\(1972\).)52 b(It)36 b(has)g(b)s(een)g(used)h(in)f(the)g(analysis)h (of)0 349 y(uniform)h(random)g(recursiv)m(e)i(trees)f(in)f(other)g(con) m(texts)i(b)m(y)f(Mahmoud)f(and)g(Sm)m(ythe)i(\(1991\).)58 b(As)0 482 y(w)m(e)34 b(gro)m(w)g(our)f Fo(urr)-7 b(t)34 b Fp(and)f(follo)m(w)h Fi(j)p Fn(T)1380 497 y Fl(k)1423 482 y Fi(j)p Fp(,)f(the)h(size)g(of)f(the)h Fo(urr)-7 b(t)p Fp(,)34 b(w)m(e)h(observ)m(e)g(that)e Fi(j)p Fn(T)3230 497 y Fl(k)3273 482 y Fi(j)p Fp(,)g(for)g Fn(n)c Fi(\025)h Fn(k)s Fp(,)0 614 y(follo)m(ws)36 b(a)f(P\023)-49 b(oly)m(a)37 b(urn)f(pro)s(cess:)50 b(at)35 b Fn(n)e Fp(=)g Fn(k)s Fp(,)j(the)g(urn)g(con)m(tains)g Fn(k)28 b Fi(\000)c Fp(1)36 b(blac)m(k)g(balls)g(and)g(one)g(white)0 747 y(ball,)29 b(the)g(white)h(ball)e(corresp)s(onding)h(to)f(no)s(de)h Fn(k)s Fp(.)42 b(Eac)m(h)29 b(subsequen)m(t)j(no)s(de)c(pic)m(ks)j(a)d (previous)i(no)s(de)0 880 y(uniformly)40 b(at)f(random.)64 b(Therefore,)42 b(w)m(e)f(ma)m(y)f(mo)s(del)f(this)h(b)m(y)g(pic)m (king)h(a)e(ball)h(from)f(the)h(urn)f(at)0 1013 y(random,)31 b(and)f(thro)m(wing)g(t)m(w)m(o)h(balls)g(bac)m(k)g(of)e(the)i(same)f (colour.)43 b(It)30 b(is)h(kno)m(wn)g(that)f(the)g(prop)s(ortion)0 1146 y(of)i(white)i(balls)f(tends)g(almost)g(surely)h(to)f(a)f(b)s(eta) g(\(1)p Fn(;)17 b(k)25 b Fi(\000)e Fp(1\))32 b(random)h(v)-5 b(ariable)32 b(when)i Fn(k)d(>)c Fp(1:)1295 1371 y(lim)1271 1431 y Fl(n)p Fg(!1)1481 1304 y Fi(j)p Fn(T)1566 1319 y Fl(k)1609 1304 y Fi(j)p 1481 1348 156 4 v 1530 1439 a Fn(n)1674 1371 y Fp(=)g Fn(Y)54 b Fp(almost)33 b(surely)h Fn(:)0 1574 y Fp(where)g Fn(Y)54 b Fp(is)33 b(b)s(eta)f(\(1)p Fn(;)17 b(k)25 b Fi(\000)d Fp(1\).)44 b(Since)33 b(0)28 b Fn(<)f(Y)49 b(<)28 b Fp(1)k(with)h(probabilit)m(y)h(one,)f(w)m(e)g (ha)m(v)m(e)1238 1799 y(lim)1214 1859 y Fl(n)p Fg(!1)1425 1732 y Fp(log)17 b Fi(j)p Fn(T)1652 1747 y Fl(k)1695 1732 y Fi(j)p 1425 1776 298 4 v 1473 1868 a Fp(log)h Fn(n)1760 1799 y Fp(=)27 b(1)33 b(almost)g(surely)h Fn(:)0 2020 y Fp(The)f(next)g(observ)-5 b(ation)33 b(is)g(that)f Fn(T)1303 2035 y Fl(k)1378 2020 y Fp(gro)m(ws)h(as)f(a)g Fo(urr)-7 b(t)32 b Fp(in)h(its)f(o)m(wn)h(righ)m(t.)44 b(Hence,)34 b(on)e([)p Fi(j)p Fn(T)3407 2035 y Fl(k)3449 2020 y Fi(j)c(!)f(1)p Fp(],)0 2153 y(w)m(e)34 b(ha)m(v)m(e)1280 2230 y Fn(M)1374 2245 y Fl(n;k)p 1231 2274 V 1231 2365 a Fp(log)18 b Fi(j)p Fn(T)1459 2380 y Fl(k)1501 2365 y Fi(j)1567 2297 y(!)1775 2230 y Fp(1)p 1704 2274 192 4 v 1704 2365 a(log)f(2)1938 2297 y(almost)33 b(surely)h Fn(;)0 2501 y Fp(and)f(th)m(us,)1205 2665 y(lim)1181 2724 y Fl(n)p Fg(!1)1392 2597 y Fn(M)1486 2612 y Fl(n;k)p 1392 2642 201 4 v 1392 2733 a Fp(log)17 b Fn(n)1630 2665 y Fp(=)1815 2597 y(1)p 1744 2642 192 4 v 1744 2733 a(log)g(2)1978 2665 y(almost)32 b(surely)j Fn(:)0 3113 y Fj(3.)50 b(Pro)s(of)37 b(of)h(Theorem)f(2.)299 3296 y Fp(F)-8 b(or)36 b Fn(i)e(>)h(j)6 b Fp(,)37 b(the)g(indicator)g(of)f(the)h(presence)i(of)d(a)h(directed)h (edge)f(b)s(et)m(w)m(een)i(\\c)m(hild")e Fn(i)g Fp(and)0 3429 y(\\paren)m(t")47 b Fn(j)52 b Fp(is)47 b(denoted)h(b)m(y)f Fn(R)1228 3444 y Fl(ij)1289 3429 y Fp(.)85 b(W)-8 b(e)47 b(de\014ne)h Fn(R)1953 3444 y Fl(j)2036 3429 y Fp(as)f(the)g(v)m(ector) h(\()p Fn(R)2771 3444 y Fl(j)t Fm(1)2843 3429 y Fn(;)17 b(:)g(:)g(:)f(;)h(R)3136 3444 y Fl(j;j)t Fg(\000)p Fm(1)3310 3429 y Fp(\).)86 b(Clearly)-8 b(,)0 3562 y Fi(f)p Fn(R)124 3577 y Fl(j)177 3562 y Fn(;)33 b(j)h(>)28 b(m)p Fi(g)k Fp(are)h(indep)s(enden)m(t.)0 3795 y Fo(Lemma)38 b(2.)55 b Fh(Let)33 b Fn(a)28 b(>)f Fp(0)33 b Fh(b)s(e)f(\014xed.)45 b(De\014ne)33 b(the)g(degree)h(of)e(the)h Fn(i)p Fh(-th)f(no)s(de,)1545 4035 y Fn(D)1626 4050 y Fl(ni)1725 4035 y Fp(=)1908 3911 y Fl(n)1857 3941 y Ff(X)1828 4151 y Fl(j)t Fm(=)p Fl(i)p Fm(+1)2046 4035 y Fn(R)2120 4050 y Fl(j)t(i)2214 4035 y Fn(;)0 4307 y Fh(and)h(the)g(ev)m(en)m(t)h Fn(A)688 4322 y Fl(ni)787 4307 y Fp(=)28 b([)p Fn(D)999 4322 y Fl(ni)1098 4307 y Fi(\025)g Fn(a)p Fp(])p Fh(,)33 b(where)h Fn(a)28 b(>)g Fp(0)k Fh(is)g(a)h(giv)m(en)f(n)m(um)m(b)s(er.)45 b(Then,)33 b(for)g Fn(k)d(>)e(i)g(>)g(j)34 b(>)27 b(m)p Fh(,)0 4439 y Fn(R)74 4454 y Fl(k)r(i)173 4439 y Fh(and)33 b Fn(R)437 4454 y Fl(k)r(j)545 4439 y Fh(are)g(negativ)m(ely)f(orthan)m (t)g(dep)s(enden)m(t,)j(i.e.,)d(for)g(an)m(y)i(increasing)d(functions)i Fn(f)43 b Fh(and)33 b Fn(g)t Fh(,)986 4622 y Fj(E)p Fi(f)p Fn(f)11 b Fp(\()p Fn(R)1281 4637 y Fl(k)r(i)1347 4622 y Fp(\))p Fn(g)t Fp(\()p Fn(R)1548 4637 y Fl(k)r(j)1622 4622 y Fp(\))p Fi(g)28 b(\024)g Fj(E)p Fi(f)p Fn(f)11 b Fp(\()p Fn(R)2138 4637 y Fl(k)r(i)2204 4622 y Fp(\))p Fi(g)p Fj(E)p Fi(f)p Fn(g)t Fp(\()p Fn(R)2579 4637 y Fl(k)r(j)2653 4622 y Fp(\))p Fi(g)32 b Fn(:)0 4805 y Fh(F)-8 b(urthermore,)28 b Fn(D)669 4820 y Fl(ni)769 4805 y Fh(and)g Fn(D)1035 4820 y Fl(nj)1142 4805 y Fh(are)g(negativ)m (ely)g(orthan)m(t)g(dep)s(enden)m(t.)44 b(In)29 b(particular,)e(for)h (an)m(y)h Fn(a)e(>)h Fp(0)p Fh(,)1236 4988 y Fj(P)p Fi(f)p Fn(A)1436 5003 y Fl(ni)1506 4988 y Fn(A)1579 5003 y Fl(nj)1659 4988 y Fi(g)f(\024)h Fj(P)p Fi(f)p Fn(A)2041 5003 y Fl(ni)2112 4988 y Fi(g)p Fj(P)p Fi(f)p Fn(A)2362 5003 y Fl(nj)2441 4988 y Fi(g)k Fn(:)1851 5280 y Fk(10)p eop %%Page: 11 11 11 10 bop 0 83 a Fo(Pr)n(oof.)56 b Fp(Clearly)-8 b(,)88 288 y Fj(E)o Fi(f)p Fn(f)11 b Fp(\()p Fn(R)382 303 y Fl(k)r(i)449 288 y Fp(\))p Fn(g)t Fp(\()p Fn(R)650 303 y Fl(k)r(j)724 288 y Fp(\))p Fi(g)189 492 y Fp(=)27 b Fn(f)11 b Fp(\(1\))p Fn(g)t Fp(\(1\))668 424 y Fn(r)s Fp(\()p Fn(r)25 b Fi(\000)d Fp(1\))p 662 469 355 4 v 662 560 a Fn(k)s Fp(\()p Fn(k)i Fi(\000)f Fp(1\))1047 492 y(+)f(\()p Fn(f)11 b Fp(\(1\))p Fn(g)t Fp(\(0\))21 b(+)h Fn(f)11 b Fp(\(0\))p Fn(g)t Fp(\(1\)\))2075 424 y Fn(r)s Fp(\()p Fn(k)23 b Fi(\000)f Fn(r)s Fp(\))p 2070 469 V 2070 560 a Fn(k)s Fp(\()p Fn(k)h Fi(\000)g Fp(1\))2455 492 y(+)f Fn(f)11 b Fp(\(0\))p Fn(g)t Fp(\(0\))2923 424 y(\()p Fn(k)23 b Fi(\000)g Fn(r)s Fp(\)\()p Fn(k)h Fi(\000)f Fn(r)i Fi(\000)d Fp(1\))p 2923 469 768 4 v 3127 560 a Fn(k)s Fp(\()p Fn(k)j Fi(\000)e Fp(1\))189 762 y Fi(\024)294 622 y Ff(\022)381 695 y Fn(r)p 377 740 55 4 v 377 831 a(k)441 762 y(f)11 b Fp(\(1\))22 b(+)755 695 y Fn(k)j Fi(\000)e Fn(r)p 755 740 223 4 v 839 831 a(k)987 762 y(f)11 b Fp(\(0\))1171 622 y Ff(\023)17 b(\022)1348 695 y Fn(r)p 1344 740 55 4 v 1344 831 a(k)1408 762 y(g)t Fp(\(1\))k(+)1713 695 y Fn(k)k Fi(\000)e Fn(r)p 1713 740 223 4 v 1797 831 a(k)1946 762 y(g)t Fp(\(0\))2122 622 y Ff(\023)189 965 y Fp(=)k Fj(E)p Fi(f)p Fn(f)11 b Fp(\()p Fn(R)587 980 y Fl(k)r(i)653 965 y Fp(\))p Fi(g)p Fj(E)p Fi(f)p Fn(g)t Fp(\()p Fn(R)1028 980 y Fl(k)r(j)1102 965 y Fp(\))p Fi(g)32 b Fn(;)0 1170 y Fp(b)s(ecause)38 b(0)d Fi(\024)g Fp(\()p Fn(k)28 b Fi(\000)e Fn(r)s Fp(\))p Fn(r)s Fp(\()p Fn(f)11 b Fp(\(1\))23 b Fi(\000)j Fn(f)11 b Fp(\(0\)\)\()p Fn(g)t Fp(\(1\))23 b Fi(\000)i Fn(g)t Fp(\(0\)\).)55 b(Also,)39 b(the)e(pairs)g(\()p Fn(R)2898 1185 y Fl(k)r(i)2965 1170 y Fn(;)17 b(R)3083 1185 y Fl(k)r(j)3158 1170 y Fp(\))37 b(for)f Fn(k)i(>)d(i)i Fp(are)0 1303 y(indep)s(enden)m(t.)51 b(Hence,)36 b(a)e(straigh)m(tforw)m(ard)h (argumen)m(t)g(sho)m(ws)h(that)2619 1228 y Ff(P)2724 1332 y Fl(k)r(>i)2862 1303 y Fn(R)2936 1318 y Fl(k)r(i)3037 1303 y Fp(and)3229 1228 y Ff(P)3334 1332 y Fl(k)r(>i)3472 1303 y Fn(R)3546 1318 y Fl(k)r(j)3656 1303 y Fp(are)0 1436 y(negativ)m(ely)24 b(orthan)m(t)f(dep)s(enden)m(t.)42 b(By)22 b(conditioning,)k(the)c(addition)h(of)f(an)g(indep)s(enden)m(t) j(comp)s(onen)m(t)0 1494 y Ff(P)105 1520 y Fl(i)105 1598 y(k)r Fm(=)p Fl(j)t Fm(+1)342 1569 y Fn(R)416 1584 y Fl(k)r(j)539 1569 y Fp(do)s(es)48 b(not)f(c)m(hange)i(matters,)j(and)c (w)m(e)g(note)g(that)f Fn(D)2606 1584 y Fl(ni)2731 1569 y Fp(=)2860 1494 y Ff(P)2965 1598 y Fl(k)r(>i)3103 1569 y Fn(R)3177 1584 y Fl(k)r(i)3292 1569 y Fp(and)g Fn(D)3577 1584 y Fl(nj)3710 1569 y Fp(=)0 1627 y Ff(P)105 1731 y Fl(k)r(>j)252 1702 y Fn(R)326 1717 y Fl(k)r(j)433 1702 y Fp(are)33 b(negativ)m(ely)h(orthan)m(t)f(dep)s(enden)m(t)h(as)f(w)m (ell.)p 2236 1648 65 4 v 2236 1706 4 59 v 2297 1706 V 2236 1709 65 4 v 299 1907 a(W)-8 b(e)42 b(note)h(at)f(this)g(p)s(oin)m (t)h(that)f(Lemma)g(1)g(remains)i(formally)e(v)-5 b(alid)42 b(if)g(w)m(e)h(de\014ne)h Fn(A)3595 1922 y Fl(ni)3710 1907 y Fp(=)0 2040 y([)p Fn(D)108 2055 y Fl(ni)219 2040 y Fi(\025)d Fn(a)388 2055 y Fl(n)435 2040 y Fp(].)66 b(The)41 b(pro)s(of)f(of)f(Theorem)i(2)f(pro)s(ceeds)h(along)f(the)h (lines)g(of)e(the)i(pro)s(of)e(of)g(Theorem)0 2173 y(1.)89 b(Strong)47 b(con)m(v)m(ergence)k(is)d(obtained)h(from)e(tail)h (estimates)h(that)f(w)m(ere)h(used)g(to)e(pro)m(v)m(e)i(w)m(eak)0 2306 y(con)m(v)m(ergence)30 b(b)m(y)f(applying)g(the)f(Borel-Can)m (telli)h(lemma)f(to)g(the)g(subsequence)k Fi(b)p Fp(\(1)12 b(+)g(1)p Fn(=r)s Fp(\))3390 2269 y Fl(n)3436 2306 y Fi(c)p Fp(,)30 b(where)0 2438 y Fn(n)46 b Fi(\025)f Fp(1,)h(and)d (emplo)m(ying)h(simple)h(monotonicit)m(y)f(argumen)m(ts.)75 b(Th)m(us,)48 b(w)m(e)c(only)f(consider)h(upp)s(er)0 2571 y(b)s(ounds)33 b(for)1537 2641 y Fl(n)1487 2671 y Ff(X)1501 2881 y Fl(i)p Fm(=1)1647 2765 y Fj(P)17 b Fi(f)o Fn(D)1871 2780 y Fl(ni)1970 2765 y Fi(\025)28 b Fn(a)2126 2780 y Fl(n)2173 2765 y Fi(g)49 b Fn(;)0 3029 y Fp(when)38 b Fn(a)310 3044 y Fl(n)393 3029 y Fp(=)e(\(1)25 b(+)g Fn(\017)p Fp(\))17 b(log)938 3053 y Fm(1+1)p Fl(=r)1153 3029 y Fn(n)p Fp(,)39 b(and)e(lo)m(w)m(er)h(b)s(ounds)g(for)f(the)h (same)g(expression)h(with)f Fn(a)3402 3044 y Fl(n)3485 3029 y Fp(=)d(\(1)25 b Fi(\000)0 3162 y Fn(\017)p Fp(\))17 b(log)220 3186 y Fm(1+1)p Fl(=r)435 3162 y Fn(n)p Fp(.)49 b(The)35 b(upp)s(er)g(b)s(ound)f(should)h(tend)g(to)f(zero,)h(and)f (the)h(lo)m(w)m(er)g(b)s(ound)f(should)h(tend)g(to)0 3295 y Fi(1)d Fp(in)h(order)g(to)f(b)s(e)h(able)g(to)f(deduce)i(that) 1384 3444 y Fn(M)1478 3459 y Fl(n)p 1255 3489 400 4 v 1255 3580 a Fp(log)1381 3603 y Fm(1+1)p Fl(=r)1597 3580 y Fn(n)1692 3511 y Fi(!)28 b Fp(1)k(in)h(probabilit)m(y)g Fn(:)1851 5280 y Fk(11)p eop %%Page: 12 12 12 11 bop 0 83 a Fo(An)36 b(upper)f(bound.)99 b Fp(W)-8 b(e)31 b(pro)s(ceed)h(as)g(in)f(the)h(pro)s(of)e(of)h(Theorem)h(1.)43 b(T)-8 b(ak)m(e)33 b Fn(a)3039 98 y Fl(n)3113 83 y Fp(=)28 b Fi(d)p Fn(c)17 b Fp(log)g Fn(n)p Fi(e)32 b Fp(with)0 216 y Fn(c)37 b(>)f Fp(1)p Fn(=)17 b Fp(log\(1)25 b(+)h(1)p Fn(=r)s Fp(\),)38 b(and)g(de\014ne)h Fn(u)d Fp(=)h Fn(r)s Fp(\()p Fn(H)1746 231 y Fl(n)p Fg(\000)p Fm(1)1908 216 y Fi(\000)26 b Fn(H)2092 231 y Fl(i)2120 216 y Fp(\))p Fn(=)p Fp(\()p Fn(c)17 b Fp(log)g Fn(n)p Fp(\).)59 b(Note)38 b(that)g Fn(D)3167 231 y Fl(ni)3274 216 y Fp(=)3387 141 y Ff(P)3492 245 y Fl(j)t(>i)3624 216 y Fn(R)3698 231 y Fl(j)t(i)3759 216 y Fp(,)0 349 y(where)c Fn(R)356 364 y Fl(j)t(i)449 349 y Fp(is)f(Bernoulli)h(\()p Fn(r)s(=)p Fp(\()p Fn(j)27 b Fi(\000)c Fp(1\)\).)43 b(Th)m(us,)34 b(for)e Fn(i)c Fi(\025)h Fn(m)p Fp(,)k(w)m(e)g(ha)m(v)m(e,)h(along)f (the)g(lines)g(of)g(the)g(pro)s(of)0 482 y(of)f(Theorem)i(1,)e(for)g Fn(t)c(>)g Fp(0,)428 713 y Fj(P)17 b Fi(f)o Fn(D)652 728 y Fl(ni)751 713 y Fi(\025)28 b Fn(a)907 728 y Fl(n)954 713 y Fi(g)g(\024)g Fj(E)1228 632 y Ff(\010)1285 713 y Fn(e)1330 672 y Fl(tD)1413 682 y Fe(ni)1478 672 y Fg(\000)p Fl(ta)1595 680 y Fe(n)1643 632 y Ff(\011)1750 713 y Fp(\(b)m(y)34 b(Mark)m(o)m(v's)g(inequalit)m(y\))1032 871 y Fi(\024)28 b Fj(E)1228 790 y Ff(\010)1285 871 y Fn(e)1330 829 y Fl(tD)1413 839 y Fe(ni)1478 829 y Fg(\000)p Fl(tc)12 b Fm(log)h Fl(n)1751 790 y Ff(\011)1032 1083 y Fi(\024)28 b Fn(e)1182 1042 y Fg(\000)p Fl(tc)12 b Fm(log)h Fl(n)1471 989 y Ff(Y)1479 1198 y Fl(j)t(>i)1615 943 y Ff(\022)1688 1083 y 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Fp(0)i(when)i Fn(\017)d(<)e(e)2635 5044 y Fg(\000)p Fm(\(1+1)p Fl(=r)r Fm(\))2944 5080 y Fp(.)51 b(The)36 b(sough)m(t)f(upp)s(er)1851 5280 y Fk(12)p eop %%Page: 13 13 13 12 bop 0 83 a Fp(b)s(ound)33 b(follo)m(ws)g(if)f(w)m(e)i(can)f(sho)m (w)h(that)e(for)g(all)h Fn(\017)28 b(>)f Fp(0)32 b(small)i(enough)f(w)m (e)g(ha)m(v)m(e)1238 261 y(lim)1214 321 y Fl(n)p Fg(!1)1521 167 y Ff(X)1414 384 y Fl(n)1457 365 y Fe(\017)1487 384 y Fg(\024)p Fl(i)p Fg(\024)p Fl(n)1664 365 y Fb(1)p Fc(\000)p Fe(\017)1789 261 y Fj(P)17 b Fi(f)o Fn(D)2013 276 y Fl(ni)2112 261 y Fi(\025)28 b Fn(a)2268 276 y Fl(n)2315 261 y Fi(g)g Fp(=)f(0)p Fn(:)0 541 y Fp(If)36 b(w)m(e)g(set)h Fn(i)c Fp(=)g Fn(n)636 505 y Fl(\014)719 541 y Fp(with)k Fn(\014)h Fi(2)c Fp([)p Fn(\017;)17 b Fp(1)24 b Fi(\000)h Fn(\017)p Fp(],)37 b(then)f(w)m(e)h(note)f(that)f Fn(u)e Fp(=)g Fn(r)s Fp(\(1)24 b Fi(\000)g Fn(\014)6 b Fp(\))p Fn(=c)24 b Fp(+)g Fn(o)p Fp(\(1\),)36 b(where)h(the)0 673 y Fn(o)p 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