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3462 y(cesses)37 b(of)c(higher)h(order,)g(treated)h(in)e(the)h(con)m(text)i(of)d(random) g(matrices)g(for)h(instance)236 3583 y(b)m(y)g(Kesten)f([24].)236 3703 y(|)50 b(Ev)m(en)44 b(more)e(general)g(is)g(a)h(treatmen)m(t)f(in) g(the)i(framew)m(ork)e(of)h(top)s(ological)c(semi-)236 3824 y(groups)31 b(as)g(b)s(egun)f(b)m(y)i(Mukherjea)g(and)e(Tserp)s (es)j([31])d(and)g(con)m(tin)m(ued)i(recen)m(tly)f(in)f(the)236 3944 y(sp)s(ecial)i(case)h(of)f(nonnegativ)m(e)h(matrices)f(b)m(y)i (Mukherjea)g([30].)236 4064 y(|)50 b(There)38 b(are,)h(moreo)m(v)m(er,) g(attempts)e(to)g(w)m(eak)m(en)j(the)d(asumption)g(on)g(the)h(sequence) 236 4185 y(\()p Fp(Y)331 4200 y Fo(n)378 4185 y Fp(;)17 b(Z)489 4200 y Fo(n)535 4185 y Fr(\))573 4200 y Fo(n)p Fn(2)p Fk(N)769 4185 y Fr(to)38 b(stationarit)m(y)e(and)j(ergo)s(dicit) m(y;)h(here)f(Boro)m(vk)m(o)m(v)h([4])e(and)g(Brandt)g([5])236 4305 y(ha)m(v)m(e)c(to)e(b)s(e)h(men)m(tioned.)236 4425 y(|)50 b(Finally)-8 b(,)35 b(it)h(is)h(only)f(natural)g(to)h(ask)h(for) e(con)m(tin)m(uous{time)h(analogues;)h(for)f(these)236 4546 y(W)-8 b(olfe)32 b([40])g(and)h(de)g(Haan)f(and)h(Karandik)-5 b(ar)31 b([19])h(ma)m(y)h(b)s(e)f(consulted.)236 4899 y Fq(8.)50 b(P)m(ositiv)m(e)35 b(recurrence)j(and)g(n)m(ull)e (recurrence)236 5135 y Fr(A)d(further)g(classi\014cation)e(in)h(the)h (recurren)m(t)h(case)f(is)f(suggested)i(b)m(y)g(\(5.6\):)236 5314 y Fq(\(8.1\))58 b(De\014nition.)48 b Fi(L)-5 b(et)48 b Fp(\027)53 b Fi(b)-5 b(e)46 b(r)-5 b(e)g(curr)g(ent)48 b(with)e(invariant)h(me)-5 b(asur)g(e)46 b Fp(\026)p Fi(.)80 b(Then)46 b(the)236 5434 y(distribution)35 b Fp(\027)41 b Fr(\()p Fi(or)35 b(the)g(kernel)f Fp(P)48 b Fi(or)35 b(the)g(pr)-5 b(o)g(c)g(ess)34 b Fr(\()p Fp(X)2311 5449 y Fo(n)2358 5434 y Fr(\))2396 5449 y Fo(n)p Fn(\025)p Fm(0)2533 5434 y Fr(\))h Fi(is)f(c)-5 b(al)5 b(le)-5 b(d)1865 5753 y Fr(2)p eop %%Page: 3 4 3 3 bop 236 171 a Fr(\(a\))174 b(\\)p Fi(p)-5 b(ositive)34 b(r)-5 b(e)g(curr)g(ent")100 b(if)f Fp(\026)p Fr(\()p Fq(R)1807 186 y Fm(+)1866 171 y Fr(\))27 b Fp(<)h Fl(1)p Fi(,)236 349 y Fr(\(b\))175 b(\\)p Fi(nul)5 b(l)34 b(r)-5 b(e)g(curr)g(ent")253 b(if)99 b Fp(\026)p Fr(\()p Fq(R)1811 364 y Fm(+)1869 349 y Fr(\))28 b(=)g Fl(1)p Fi(.)383 528 y Fr(The)23 b(simplest)f(example)h(for)f(p)s(ositiv)m(e)g (recurrence)j(is)e(pro)m(vided)g(b)m(y)h(the)f(case)p 3230 475 56 4 v 24 w Fp(x)28 b(<)g Fl(1)p Fr(,)236 648 y(in)k(whic)m(h)i(the)f(supp)s(ort)g(of)f Fp(\026)h Fr(is)f(b)s (ounded.)45 b(A)33 b(\014rst)g(general)f(criterion,)g(deriv)m(ed)h (under)236 768 y(momen)m(t)47 b(conditions)f(but)i(not)f(restricted)h (to)f(the)h(case)g Fp(Y)5 b(;)17 b(Z)60 b Fl(\025)53 b Fr(0,)e(w)m(as)e(giv)m(en)e(b)m(y)236 889 y(V)-8 b(erv)j(aat)33 b([39].)43 b(In)33 b(the)g(presen)m(t)h(framew)m(ork)f(it)e(is)h (easily)g(established:)236 1067 y Fq(\(8.2\))58 b(Theorem.)49 b Fp(\027)42 b Fi(is)34 b(p)-5 b(ositive)35 b(r)-5 b(e)g(curr)g(ent)35 b(if)f(and)h(only)f(if)535 1264 y Fp(W)41 b Fr(:=)799 1197 y Fj(P)903 1279 y Fo(n)p Fn(2)p Fk(N)1077 1264 y Fp(Y)1134 1279 y Fm(1)1190 1264 y Fp(:)17 b(:)g(:)f(Y)1378 1279 y Fo(n)p Fn(\000)p Fm(1)1515 1264 y Fp(Z)1582 1279 y Fo(n)1656 1264 y Fp(<)28 b Fl(1)49 b Fi(a.s.)236 1460 y(In)34 b(this)h(c)-5 b(ase)34 b(the)h(stationary)g(distribution)g(is)g (given)f(by)h Fl(L)p Fr(\()p Fp(W)14 b Fr(\))p Fi(.)236 1638 y(Pr)-5 b(o)g(of.)50 b Fr(1.)55 b(If)37 b(the)g(in)m(v)-5 b(arian)m(t)35 b(measure)i Fp(\026)f Fr(is)g(\014nite,)i Fp(\026)c Fl(2)h(M)2538 1653 y Fm(1)2577 1638 y Fr(\()p Fq(R)2699 1653 y Fm(+)2758 1638 y Fr(\))h(ma)m(y)h(b)s(e)g(assumed)236 1759 y(as)e(w)m(ell.)49 b(Under)36 b(the)f(initial)c(la)m(w)j Fp(\026)h Fr(the)g(sequence)i(\()p Fp(X)2319 1774 y Fo(n)2366 1759 y Fr(\))2404 1774 y Fo(n)p Fn(\025)p Fm(0)2576 1759 y Fr(is)d(stationary)-8 b(,)35 b(and)g(with)236 1879 y(the)d(dual)f(sequence)k(\()p Fp(W)1153 1894 y Fo(n)1200 1879 y Fr(\))1238 1894 y Fo(n)p Fn(\025)p Fm(0)1375 1879 y Fr(,)d(de\014ned)h(in)e(Section)h(0,)g(this)f(implies)e(b)m(y)k (monotonicit)m(y)535 2076 y Fp(\026)p Fr(\([0)p Fp(;)17 b(t)p Fr(]\))83 b(=)h(\(lim)17 b(inf)1420 2091 y Fo(n)p Fn(!1)1608 2076 y Fr(\))1663 2005 y Fj(R)1734 2076 y Fq(P)p Fr(\()p Fp(X)1938 2034 y Fo(x)1930 2100 y(n)2009 2076 y Fl(\024)28 b Fp(t)p Fr(\))17 b Fp(\026)p Fr(\()p Fp(dx)p Fr(\))935 2279 y Fl(\024)83 b Fr(lim)17 b(inf)1382 2294 y Fo(n)p Fn(!1)1587 2279 y Fq(P)p Fr(\()p Fp(X)1791 2238 y Fm(0)1783 2304 y Fo(n)1857 2279 y Fl(\024)28 b Fp(t)p Fr(\))935 2482 y(=)84 b(lim)17 b(inf)1382 2497 y Fo(n)p Fn(!1)1587 2482 y Fq(P)p Fr(\()p Fp(W)1794 2497 y Fo(n)1868 2482 y Fl(\024)28 b Fp(t)p Fr(\))935 2686 y(=)84 b Fq(P)p Fr(\()p Fp(W)41 b Fl(\024)28 b Fp(t)p Fr(\))935 2889 y Fl(\024)83 b Fq(P)p Fr(\()p Fp(W)41 b(<)27 b Fl(1)p Fr(\))p Fp(;)236 3086 y Fr(whic)m(h)33 b(for)f Fp(t)c Fl(!)g(1)k Fr(yields)g(the)h(assertion.)383 3206 y(2.)56 b(T)-8 b(o)37 b(pro)m(v)m(e)h(the)f(con)m(v)m(erse,)k(let) 36 b(\()p Fp(Y)1763 3221 y Fm(0)1802 3206 y Fp(;)17 b(Z)1913 3221 y Fm(0)1952 3206 y Fr(\))37 b(b)s(e)g(indep)s(enden)m(t)h(of)f(\() p Fp(Y)2932 3221 y Fo(n)2978 3206 y Fp(;)17 b(Z)3089 3221 y Fo(n)3136 3206 y Fr(\))p Fp(;)33 b(n)i Fl(2)h Fq(N)p Fp(;)236 3326 y Fr(with)28 b(distribution)f Fp(\027)6 b Fr(.)43 b(Then)30 b Fp(Z)1424 3341 y Fm(0)1477 3326 y Fr(+)14 b Fp(Y)1624 3341 y Fm(0)1663 3326 y Fp(W)42 b Fr(is)28 b(distributed)h(as)f Fp(W)14 b Fr(,)30 b(hence)g Fp(\026)d Fr(=)h Fl(L)p Fr(\()p Fp(W)14 b Fr(\))27 b(is)i(a)236 3447 y(stationary)34 b(distribution)f(for)h Fp(P)14 b Fr(.)48 b(Moreo)m(v)m(er,)37 b(the)e(recurrence)h(of)e Fp(\027)41 b Fr(is)34 b(a)g(consequence)236 3567 y(of)27 b(\(2.2b\),)h(b)s(ecause)g(the)g(p)s(oten)m(tial)d(k)m(ernel)j Fp(G)f Fr(:=)2093 3501 y Fj(P)2197 3582 y Fo(n)p Fn(\025)p Fm(0)2351 3567 y Fp(P)2428 3531 y Fo(n)2501 3567 y Fr(satis\014es)h(b)m (y)g(monotonicit)m(y)535 3763 y Fp(G)p Fr(\(0;)17 b([0)p Fp(;)g(t)p Fr(]\))82 b Fl(\025)1206 3692 y Fj(R)1278 3763 y Fp(G)p Fr(\()p Fp(x)p Fr(;)17 b([0)p Fp(;)g(t)p Fr(]\))g Fp(\026)p Fr(\()p Fp(dx)p Fr(\))1046 3967 y(=)1206 3900 y Fj(P)1310 3982 y Fo(n)p Fn(\025)p Fm(0)1464 3967 y Fr(\()p Fp(\026P)1638 3926 y Fo(n)1684 3967 y Fr(\)\([0)p Fp(;)g(t)p Fr(]\))1046 4170 y(=)1206 4104 y Fj(P)1310 4185 y Fo(n)p Fn(\025)p Fm(0)1464 4170 y Fp(\026)p Fr(\([0)p Fp(;)g(t)p Fr(]\))174 b(for)32 b(all)47 b Fp(t)28 b Fl(\025)g Fr(0)p Fp(:)100 b Fh(2)383 4367 y Fr(T)-8 b(o)28 b(deriv)m(e)h(the)g (stationary)f(distribution)e Fp(\026)i Fr(asso)s(ciated)h(with)f(a)g(p) s(ositiv)m(e)g(recurren)m(t)236 4487 y(distribution)j Fp(\027)6 b Fr(,)33 b(\(8.2\))f(is)g(not)g(of)g(m)m(uc)m(h)h(use.)45 b(F)-8 b(or)32 b(some)g(explicit)f(examples,)i(obtained)236 4607 y(b)m(y)h(an)e(ad)h(ho)s(c)f(principle)f(in)h(the)h(framew)m(ork)g (of)f(exp)s(onen)m(tial)g(families,)d(see)34 b([7].)383 4728 y(The)f(next)h(result)e(strengthens)i(\(2.3\):)236 4906 y Fq(\(8.3\))58 b(Prop)s(osition.)48 b Fp(\027)41 b Fi(is)35 b(p)-5 b(ositive)34 b(r)-5 b(e)g(curr)g(ent)35 b(whenever)f Fq(P)p Fr(\()p Fp(Y)48 b Fr(=)28 b(0\))f Fp(>)h Fr(0)p Fi(.)236 5085 y(Pr)-5 b(o)g(of.)50 b Fr(Since)31 b Fp(Y)853 5100 y Fo(n)927 5085 y Fr(=)d(0)j(for)g(some)g Fp(n)d Fl(2)g Fq(N)j Fr(with)g(probabilit)m(y)e(1,)j(the)f(series)h(in) f(\(8.2\))g(is)g(a)236 5205 y(\014nite)i(sum)f(almost)f(surely)p Fp(:)100 b Fh(2)383 5384 y Fr(In)32 b(this)h(con)m(text)h(it)d(is)h(w)m (orth)m(while)h(to)f(p)s(oin)m(t)g(out)g(the)h(follo)m(wing:)236 5504 y(|)50 b(The)33 b(stationary)e(distribution)g Fp(\026)h Fr(can)g(b)s(e)h(computed,)f(at)g(least)g(in)g(principle,)e(from)1865 5753 y(3)p eop %%Page: 4 5 4 4 bop 236 171 a Fr(the)31 b(probabilities)c Fp(p)h Fr(:=)g Fq(P)p Fr(\()p Fp(Y)48 b Fr(=)28 b(0\))i(and)g Fp(q)h Fr(:=)d(1)17 b Fl(\000)h Fp(p)30 b Fr(and)h(the)f(distributions) f Fp(\027)3208 186 y Fm(0)3278 171 y Fr(and)i Fp(\027)3520 134 y Fn(0)236 291 y Fr(of)i(\()p Fp(Y)5 b(;)17 b(Z)7 b Fr(\))33 b(under)i(the)f(condition)e Fp(Y)50 b Fr(=)29 b(0)34 b(and)f Fp(Y)51 b Fl(6)p Fr(=)29 b(0,)k(resp)s(ectiv)m(ely)i (\(the)f(assumption)236 411 y Fp(q)48 b Fl(6)p Fr(=)43 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b(its)g(su\016ciency)j(w)m(as)e(sho)m(wn)h(b)m(y)f(V)-8 b(erv)j(aat)236 5231 y([39])31 b(along)e(the)i(same)g(lines)f(but)h (under)g(a)g(w)m(eak)m(er)i(momen)m(t)c(assumption.)43 b(Emplo)m(ying)236 5351 y(Kesten's)27 b(extension)e(of)g(the)g(strong)g (la)m(w)f(of)g(large)g(n)m(um)m(b)s(ers,)j(this)d(assumption)h (actually)236 5472 y(can)33 b(b)s(e)g(dropp)s(ed)g(completely:)1865 5753 y(4)p eop %%Page: 5 6 5 5 bop 236 171 a Fq(\(8.5\))58 b(Theorem.)49 b Fi(If)35 b Fp(S)1206 186 y Fo(n)1280 171 y Fl(!)28 b(\0001)p Fi(,)34 b(then)h(the)g(c)-5 b(ondition)535 374 y Fq(E)p Fr(\(log)773 398 y Fm(+)832 374 y Fp(Z)7 b Fr(\))27 b Fp(<)h Fl(1)236 577 y Fr(\(a\))174 b Fi(is)35 b(always)f(su\016cient)h(for)g(p)-5 b(ositive)34 b(r)-5 b(e)g(curr)g(enc)g(e,)236 756 y Fr(\(b\))175 b Fi(is)34 b(ne)-5 b(c)g(essary)34 b(for)h(p)-5 b(ositive)34 b(r)-5 b(e)g(curr)g(enc)g(e)35 b(whenever)e Fq(E)p Fr(\()p Fl(j)17 b Fr(log)f Fp(Y)21 b Fl(j)p Fr(\))27 b Fp(<)h Fl(1)p Fi(.)236 934 y(Pr)-5 b(o)g(of.)50 b Fr(Applying)31 b(Borel{Can)m(telli)f(it)i(follo)m(ws)f(easily)h(that)535 1195 y(lim)17 b(sup)851 1210 y Fo(n)p Fn(!1)1070 1127 y Fr(1)p 1065 1171 59 4 v 1065 1263 a Fp(n)1167 1195 y Fr(log)1293 1218 y Fm(+)1352 1195 y Fp(Z)1419 1210 y Fo(n)1504 1048 y Fj(\()1603 1127 y Fr(=)27 b(0)175 b(a.s.)100 b(if)e Fq(E)p Fr(\(log)2565 1151 y Fm(+)2624 1127 y Fp(Z)7 b Fr(\))27 b Fp(<)h Fl(1)p Fp(;)1598 1263 y Fr(=)f(+)p Fl(1)50 b Fr(a.s.)100 b(if)e Fq(E)p Fr(\(log)2562 1286 y Fm(+)2621 1263 y Fp(Z)7 b Fr(\))27 b(=)h Fl(1)p Fp(:)5491 1048 y Fj(\))236 1460 y Fr(Applying)k([23],)g(moreo)m(v)m (er,)h(it)f(follo)m(ws)f(from)h(the)h(h)m(yp)s(othesis)g Fp(S)2674 1475 y Fo(n)2749 1460 y Fl(!)27 b(\0001)33 b Fr(that)535 1726 y(lim)687 1741 y Fo(n)p Fn(!1)907 1658 y Fr(1)p 902 1703 V 902 1794 a Fp(n)987 1659 y Fj(P)1091 1741 y Fm(1)p Fn(\024)p Fo(m)27 b(z)2167 5290 y Fo(n)2214 5275 y Fr(\))p Fp(;)236 5503 y Fr(\(2\))174 b(1)28 b(=)f Fp(y)763 5518 y Fm(0)830 5503 y Fp(>)g(y)981 5518 y Fm(1)1048 5503 y Fp(>)h(:)17 b(:)g(:)27 b Fl(!)g Fr(0)174 b(and)1976 5437 y Fj(P)2080 5518 y Fo(n)p Fn(2)p Fk(N)2260 5503 y Fp(y)2308 5518 y Fo(n)2354 5503 y Fp(z)2399 5518 y Fo(n)2474 5503 y Fp(<)28 b Fl(1)p Fp(;)1865 5753 y Fr(5)p eop %%Page: 6 7 6 6 bop 236 171 a Fr(\(3\))174 b(0)28 b(=)f Fp(p)764 186 y Fm(0)831 171 y Fp(<)h(p)984 186 y Fm(1)1051 171 y Fp(<)f(:)17 b(:)g(:)28 b Fl(!)f Fr(1)174 b(and)1979 104 y Fj(P)2083 186 y Fo(n)p Fn(2)p Fk(N)2262 171 y Fp(p)2311 129 y Fo(n)p Fn(\000)p Fm(1)2311 195 y Fo(n)2476 171 y Fp(<)28 b Fl(1)236 399 y Fr(\(e.g.)44 b Fp(p)513 414 y Fo(n)587 399 y Fr(=)28 b Fp(n)749 363 y Fn(\000)p Fm(2)p Fo(=)p Fm(\()p Fo(n)p Fn(\000)p Fm(1\))1098 399 y Fr(for)j Fp(n)d(>)g Fr(1\).)43 b(Indep)s(enden)m(tly)33 b(of)e Fp(Z)2408 414 y Fo(n)2455 399 y Fp(;)17 b(n)11 b Fl(2)g Fq(N)p Fp(;)32 b Fr(let)f(no)m(w)i Fp(Y)3192 414 y Fo(n)3238 399 y Fp(;)17 b(n)11 b Fl(2)g Fq(N)p Fp(;)236 519 y Fr(b)s(e)33 b(indep)s(enden)m(t)h(with)e(a)g(distribution)f Fp(\027)1805 534 y Fo(y)1879 519 y Fr(suc)m(h)j(that)535 723 y Fq(P)p Fr(\()p Fp(Y)49 b(>)27 b(y)907 738 y Fo(n)954 723 y Fr(\))g(=)h Fp(p)1172 738 y Fo(n)1393 723 y Fr(for)49 b Fp(n)28 b Fl(\025)g Fr(0)p Fp(:)236 926 y Fr(Then)34 b(0)27 b Fp(<)h(Y)49 b Fl(\024)28 b Fr(1)k(and)h(th)m(us)535 1129 y Fq(P)p Fr(\()p Fp(Y)707 1144 y Fm(1)762 1129 y Fp(:)17 b(:)g(:)f(Y)950 1144 y Fo(n)p Fn(\000)p Fm(1)1087 1129 y Fp(Z)1154 1144 y Fo(n)1228 1129 y Fp(>)28 b(y)1380 1144 y Fo(n)1427 1129 y Fp(z)1472 1144 y Fo(n)1519 1129 y Fr(\))83 b Fl(\024)g Fq(P)p Fr(\()p Fp(Y)1972 1144 y Fm(1)2011 1129 y Fp(;)17 b(:)g(:)g(:)f(;)h(Y)2287 1144 y Fo(n)p Fn(\000)p Fm(1)2451 1129 y Fp(>)27 b(y)2602 1144 y Fo(n)2649 1129 y Fr(\))39 b(+)f Fq(P)p Fr(\()p Fp(Z)3022 1144 y Fo(n)3096 1129 y Fp(>)28 b(z)3245 1144 y Fo(n)3292 1129 y Fr(\))1641 1333 y(=)83 b Fp(p)1849 1292 y Fo(n)p Fn(\000)p Fm(1)1849 1358 y Fo(n)2009 1333 y Fr(+)22 b Fp(q)2150 1348 y Fo(n)2371 1333 y Fr(for)49 b Fp(n)28 b Fl(2)g Fq(N)p Fp(:)236 1536 y Fr(Therefore,)34 b(b)m(y)f(the)g(summabilit)m(y)d(in)i(\(1\))g(and)h (\(3\),)f(with)g(probabilit)m(y)f(1)535 1740 y Fp(Y)592 1755 y Fm(1)648 1740 y Fp(:)17 b(:)g(:)f(Y)836 1755 y Fo(n)p Fn(\000)p Fm(1)973 1740 y Fp(Z)1040 1755 y Fo(n)1114 1740 y Fl(\024)28 b Fp(y)1267 1755 y Fo(n)1314 1740 y Fp(z)1359 1755 y Fo(n)1580 1740 y Fr(for)k(almost)f(all)48 b Fp(n)28 b Fl(2)g Fq(N)p Fp(;)236 1943 y Fr(hence,)34 b(b)m(y)g(the)f(summabilit)m(y)d(in)h(\(2\),)i(the)g(assertion)f(follo) m(ws)f(from)h(\(8.2\))p Fp(:)99 b Fh(2)383 2122 y Fr(No)m(w)44 b(the)h(close)f(relationship)e(b)s(et)m(w)m(een)47 b(the)d(b)s(eha)m (viour)g(of)g(an)g(a\016ne)h(recursion)236 2242 y(\()p Fp(X)355 2257 y Fo(n)402 2242 y Fr(\))440 2257 y Fo(n)p Fn(\025)p Fm(0)611 2242 y Fr(and)35 b(the)f(asso)s(ciated)g(random)f(w) m(alk)i(\()p Fp(S)2125 2257 y Fo(n)2171 2242 y Fr(\))2209 2257 y Fo(n)p Fn(\025)p Fm(0)2381 2242 y Fr(can)f(b)s(e)g(summarized.) 47 b(Under)236 2362 y(a)27 b(w)m(eak)i(b)s(oundedness)h(condition)25 b(on)j Fp(Z)7 b Fr(,)28 b(satis\014ed)g(for)f(instance)g(in)g(the)h(m)m (ultiplicativ)m(e)236 2483 y(mo)s(del,)j(there)j(is)e(in)f(fact)i(a)f (bijection:)236 2661 y Fq(\(8.7\))58 b(Theorem.)49 b Fi(With)30 b(arbitr)-5 b(ary)29 b Fp(\013)f(>)g Fr(0)h Fi(let)g(one)f(of)h(the)g(fol)5 b(lowing)27 b(two)i(c)-5 b(onditions)236 2782 y(b)g(e)35 b(satis\014e)-5 b(d:)236 2985 y Fr(\(a\))174 b Fq(E)p Fr(\()p Fp(Z)721 2944 y Fo(\013)798 2985 y Fl(j)27 b Fp(Y)21 b Fr(\))28 b Fl(\024)g Fp(\015)105 b Fi(for)34 b(some)50 b Fp(\015)32 b(<)c Fl(1)p Fp(;)236 3262 y Fr(\(b\))175 b Fq(P)p Fr(\()p Fp(Y)48 b Fr(=)28 b(0\))f(=)h(0)174 b Fp(and)g Fq(E)1714 3165 y Fj(\020)o(\020)1825 3194 y Fp(Z)p 1823 3238 79 4 v 1823 3330 a(Y)1911 3165 y Fj(\021)1961 3188 y Fo(\013)2010 3165 y Fj(\021)2087 3262 y Fp(<)28 b Fl(1)p Fp(:)236 3515 y Fi(Then)34 b(the)h(fol)5 b(lowing)34 b(trichotomy)h(holds:)535 3719 y Fp(S)595 3734 y Fo(n)670 3719 y Fl(!)27 b Fr(+)p Fl(1)55 b Fr(=)-17 b Fl(\))55 b Fp(\027)41 b(tr)s(ansient;)535 3947 y(S)595 3962 y 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(the)f(asymptotic)g(b)s(eha)m(viour)h(of)236 411 y(the)k(asso)s(ciated) g(random)e(w)m(alk.)44 b(The)33 b(recurren)m(t)h(case)g(follo)m(ws)d (from)h(the)h(inequalities)535 610 y Fp(Z)602 625 y Fm(1)641 610 y Fp(Y)698 625 y Fm(2)754 610 y Fp(:)17 b(:)g(:)f(Y)942 625 y Fo(n)1011 610 y Fr(+)22 b Fp(:)17 b(:)g(:)k Fr(+)h Fp(Z)1410 625 y Fo(n)1485 610 y Fl(\024)28 b Fr(\()p Fp(Z)1702 569 y Fn(0)1695 635 y Fm(1)1734 610 y Fp(Y)1812 569 y Fn(0)1791 635 y Fm(2)1852 610 y Fp(:)17 b(:)g(:)f(Y)2062 569 y Fn(0)2040 635 y Fo(n)2109 610 y Fr(+)22 b Fp(:)17 b(:)g(:)22 b Fr(+)g Fp(Z)2516 569 y Fn(0)2509 635 y Fo(n)2556 610 y Fr(\))2609 542 y Ff(1)p 2603 554 41 4 v 2603 595 a Fg(\013)2658 610 y Fp(;)535 846 y(Z)602 861 y Fm(1)663 846 y Fr(+)g Fp(:)17 b(:)g(:)22 b Fr(+)g Fp(Y)1053 861 y Fm(1)1109 846 y Fp(:)17 b(:)g(:)f(Y)1297 861 y Fo(n)p Fn(\000)p Fm(1)1434 846 y Fp(Z)1501 861 y Fo(n)1575 846 y Fl(\024)28 b Fr(\()p Fp(Z)1792 804 y Fn(0)1785 870 y Fm(1)1846 846 y Fr(+)22 b Fp(:)17 b(:)g(:)22 b Fr(+)g Fp(Y)2257 804 y Fn(0)2236 870 y Fm(1)2297 846 y Fp(:)17 b(:)g(:)f(Y)2507 804 y Fn(0)2485 870 y Fo(n)p Fn(\000)p Fm(1)2622 846 y Fp(Z)2696 804 y Fn(0)2689 870 y Fo(n)2736 846 y Fr(\))2789 777 y Ff(1)p 2784 789 V 2784 830 a Fg(\013)2838 846 y Fp(:)236 1068 y Fr(Indeed,)34 b(if)e Fp(x)p 670 1081 56 4 v -36 x Fn(0)781 1068 y Fr(and)h Fp(W)1077 1032 y Fn(0)1132 1068 y Fr(denote)g(the)g(analogues)f(of)g Fp(x)p 2170 1081 V 33 w Fr(and)h Fp(W)14 b Fr(,)32 b(this)h(means)535 1266 y Fp(x)p 535 1279 V 28 w Fl(\024)28 b Fr(\()p Fp(x)p 761 1279 V 817 1225 a Fn(0)840 1266 y Fr(\))893 1198 y Ff(1)p 888 1210 41 4 v 888 1251 a Fg(\013)1116 1266 y Fr(and)175 b Fp(W)41 b Fl(\024)28 b Fr(\()p Fp(W)1830 1225 y Fn(0)1853 1266 y Fr(\))1906 1198 y Ff(1)p 1901 1210 V 1901 1251 a Fg(\013)1955 1266 y Fp(:)100 b Fh(2)383 1465 y Fr(It)33 b(should)h(b)s(e)g(men)m(tioned)f(that)h(conditions)e (\(a\))i(and)g(\(b\))f(according)g(to)h(\(3.4\))f(can)236 1586 y(b)s(e)g(simpli\014ed)e(to)535 1784 y Fq(E)p Fr(\()p Fp(Z)721 1743 y Fo(\013)770 1784 y Fr(\))d Fp(<)f Fl(1)174 b Fr(for)32 b(some)50 b Fp(\013)28 b(>)g Fr(0)p Fp(;)236 1983 y Fr(pro)m(vided)33 b Fp(Y)54 b Fr(and)33 b Fp(Z)39 b Fr(are)33 b(indep)s(enden)m(t)h(or)e Fp(Y)54 b Fr(is)32 b(b)s(ounded)h(a)m(w)m(a)m(y)h(from)d(0.)383 2104 y(This)g(section)h (will)d(b)s(e)j(concluded)h(b)m(y)f(a)f(stabilit)m(y)f(result.)44 b(Related,)31 b(but)h(not)f(com-)236 2224 y(parable,)40 b(conditions)d(for)h(con)m(tin)m(uous)i(dep)s(endence)h(of)d Fp(\026)g Fr(on)g Fp(\027)45 b Fr(can)39 b(b)s(e)g(found)g(in)e([5])236 2344 y(\(see)d(also)e([6],)g(c)m(hapter)i(9.1\).)43 b(The)33 b(follo)m(wing)d(result)j(strengthens)h(\(6.5\):)236 2523 y Fq(\(8.8\))58 b(Prop)s(osition.)48 b Fi(L)-5 b(et)40 b Fl(N)51 b(3)38 b Fp(\027)1632 2538 y Fo(k)1735 2470 y Fm(w)1711 2523 y Fl(!)f Fp(\027)46 b Fi(and)39 b Fp(\027)2184 2538 y Fo(k)2267 2523 y Fi(b)-5 b(e)39 b(p)-5 b(ositive)40 b(r)-5 b(e)g(curr)g(ent)40 b(with)f(sta-)236 2643 y(tionary)c (distributions)g Fp(\026)1197 2658 y Fo(k)1274 2643 y Fi(such)g(that)535 2842 y Fp(\015)e Fr(:=)27 b(sup)913 2857 y Fo(k)r Fn(2)p Fk(N)1083 2771 y Fj(R)1155 2842 y Fp(y)1207 2801 y Fo(\013)1272 2842 y Fp(d\027)1371 2857 y Fo(k)1441 2842 y Fp(<)h Fr(1)174 b Fi(and)f Fp(\016)32 b Fr(:=)c(sup)2465 2857 y Fo(k)r Fn(2)p Fk(N)2635 2771 y Fj(R)2707 2842 y Fp(z)2756 2801 y Fo(\013)2823 2842 y Fp(d\027)2922 2857 y Fo(k)2992 2842 y Fp(<)f Fl(1)236 3041 y Fi(for)35 b(some)f Fp(\013)28 b(>)g Fr(0)p Fi(.)44 b(Then)236 3240 y Fr(\(a\))174 b Fp(\027)42 b(is)35 b(positiv)t(e)g(r)s (ecur)s(r)s(ent;)236 3462 y Fr(\(b\))175 b Fp(\026)600 3477 y Fo(k)694 3409 y Fm(w)670 3462 y Fl(!)27 b Fp(\026;)236 3684 y Fi(wher)-5 b(e)34 b Fp(\026)h Fi(is)g(the)g(stationary)g (distribution)f(asso)-5 b(ciate)g(d)34 b(with)h Fp(\027)6 b Fi(.)236 3862 y(Pr)-5 b(o)g(of.)50 b Fr(\(a\))32 b(In)h(view)g(of)f (the)h(w)m(eak)h(con)m(v)m(ergence)535 3990 y Fj(R)607 4061 y Fp(y)659 4020 y Fo(\013)724 4061 y Fp(d\027)g Fl(\024)28 b Fp(\015)180 b Fr(and)1524 3990 y Fj(R)1596 4061 y Fp(z)1645 4020 y Fo(\013)1712 4061 y Fp(d\027)34 b Fl(\024)28 b Fp(\016)n(;)236 4260 y Fr(whic)m(h)33 b(b)m(y)h(the)f(h)m(yp)s(otheses)i(on)d Fp(\015)38 b Fr(and)32 b Fp(\016)37 b Fr(\(and)c(Jensen's)h(inequalit)m(y\))e (yields)535 4459 y Fl(\0001)c(\024)845 4388 y Fj(R)934 4459 y Fr(log)16 b Fp(y)k(d\027)34 b(<)27 b Fr(0)174 b(and)1935 4388 y Fj(R)2024 4459 y Fr(log)2150 4482 y Fm(+)2209 4459 y Fp(z)21 b(d\027)34 b(<)27 b Fl(1)p Fp(:)236 4658 y Fr(Th)m(us)34 b Fp(\027)39 b Fr(is)32 b(p)s(ositiv)m(e)g (recurren)m(t)i(according)e(to)h(\(8.5a\).)383 4778 y(\(b\))45 b(Since)g(the)h Fi(L)1067 4793 y Fe(p)1112 4778 y Fr({norms)f(increase) g(with)g(the)h(order,)j(clearly)44 b Fp(\013)50 b Fl(\024)g Fr(1)45 b(ma)m(y)g(b)s(e)236 4898 y(assumed.)f(In)33 b(this)g(case)535 5026 y Fj(R)607 5097 y Fp(x)662 5056 y Fo(\013)729 5097 y Fp(d\026)839 5112 y Fo(k)964 5097 y Fr(=)1124 5026 y Fj(R)g(R)1268 5097 y Fr(\()p Fp(xy)26 b Fr(+)c Fp(z)t Fr(\))1620 5056 y Fo(\013)1686 5097 y Fp(\026)1745 5112 y Fo(k)1788 5097 y Fr(\()p Fp(dx)p Fr(\))g Fp(\027)2040 5112 y Fo(k)2083 5097 y Fr(\()p Fp(dy)t(;)17 b(dz)t Fr(\))964 5301 y Fl(\024)1124 5230 y Fj(R)33 b(R)1268 5301 y Fr(\(\()p Fp(xy)t Fr(\))1489 5259 y Fo(\013)1560 5301 y Fr(+)22 b Fp(z)1707 5259 y Fo(\013)1757 5301 y Fr(\))g Fp(\026)1876 5316 y Fo(k)1919 5301 y Fr(\()p Fp(dx)p Fr(\))g Fp(\027)2171 5316 y Fo(k)2214 5301 y Fr(\()p Fp(dy)t(;)17 b(dz)t Fr(\))964 5504 y Fl(\024)83 b Fr(\()1162 5433 y Fj(R)1234 5504 y Fp(x)1289 5463 y Fo(\013)1356 5504 y Fp(d\026)1466 5519 y Fo(k)1508 5504 y Fr(\))17 b Fp(\015)26 b Fr(+)c Fp(\016)n(;)1865 5753 y Fr(7)p eop %%Page: 8 9 8 8 bop 236 171 a Fr(whic)m(h)33 b(implies)535 415 y(sup)699 430 y Fo(k)r Fn(2)p Fk(N)868 344 y Fj(R)940 415 y Fp(x)995 374 y Fo(\013)1062 415 y Fp(d\026)1172 430 y Fo(k)1241 415 y Fl(\024)1446 348 y Fp(\016)p 1357 392 227 4 v 1357 484 a Fr(1)22 b Fl(\000)g Fp(\015)1621 415 y(<)27 b Fl(1)p Fp(:)236 666 y Fr(In)e(view)g(of)f Fp(\013)k(>)g 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1447 y Fr(refers)i(to)f Fp(\027)3072 1462 y Fo(k)3115 1447 y Fr(.)236 1687 y(\(1\))174 b Fq(P)612 1702 y Fo(k)654 1687 y Fr(\()p Fp(Z)35 b Fr(=)927 1620 y Fp(k)p 907 1664 94 4 v 907 1755 a(y)959 1726 y Fo(k)1011 1687 y Fr(\))28 b(=)1235 1620 y(1)p 1190 1664 138 4 v 1190 1682 a Fl(p)p 1273 1682 55 4 v 84 x Fp(k)1337 1687 y(;)67 b Fq(P)1508 1702 y Fo(k)1550 1687 y Fr(\()p Fp(Z)35 b Fr(=)27 b Fp(z)t Fr(\))i(=)e(1)22 b Fl(\000)2237 1620 y Fr(1)p 2193 1664 138 4 v 2193 1682 a Fl(p)p 2276 1682 55 4 v 84 x Fp(k)236 1941 y Fr(with)32 b Fp(z)h(>)27 b Fr(0)33 b(and)f Fp(Y)49 b Fr(=)28 b Fp(y)j Fl(2)17 b Fr(]0)p Fp(;)g Fr(1[)32 b(yield)f(p)s(ositiv)m(e)h(recurren)m(t)i Fp(\027)2574 1956 y Fo(k)2617 1941 y Fp(;)g(k)c Fl(2)e Fq(N)p Fp(;)33 b Fr(suc)m(h)h(that)535 2141 y Fp(\027)583 2156 y Fo(k)677 2088 y Fm(w)654 2141 y Fl(!)27 b Fp(")827 2157 y Fm(\()p Fo(y)r(;z)s Fm(\))1152 2141 y Fr(and)175 b Fp(\026)1543 2156 y Fo(k)1637 2088 y Fm(w)1613 2141 y Fl(!)27 b Fp(")1786 2156 y Fn(1)1861 2141 y Fp(:)236 2342 y Fr(The)34 b(\014rst)f(con)m(v)m(ergence)i(is)d(clear)g(and)h (the)g(second)h(one)f(follo)m(ws)e(from)535 2543 y Fq(P)612 2558 y Fo(k)654 2543 y Fr(\()p Fp(W)42 b Fl(\024)28 b Fp(k)s Fr(\))83 b Fl(\024)g Fq(P)1343 2558 y Fo(k)1386 2543 y Fr(\()p Fp(Z)1491 2558 y Fo(i)1546 2543 y Fr(=)28 b Fp(z)54 b Fr(for)c(1)27 b Fl(\024)h Fp(i)g Fl(\024)g Fp(k)s Fr(\))1107 2791 y(=)83 b(\(1)22 b Fl(\000)1529 2723 y Fr(1)p 1485 2767 138 4 v 1485 2785 a Fl(p)p 1568 2785 55 4 v 85 x Fp(k)1632 2791 y Fr(\))1670 2750 y Fo(k)1740 2791 y Fl(!)27 b Fr(0)p Fp(:)236 3101 y Fr(\(2\))174 b Fq(P)612 3116 y Fo(k)654 3101 y Fr(\()p Fp(Y)49 b Fr(=)28 b(0\))f(=)1152 3034 y(1)p 1130 3078 94 4 v 1130 3170 a Fp(k)1184 3141 y Fm(3)1233 3101 y Fp(;)67 b Fq(P)1404 3116 y Fo(k)1446 3101 y Fr(\()p Fp(Y)49 b Fr(=)1706 3034 y(1)p 1703 3078 55 4 v 1703 3170 a Fp(k)1767 3101 y Fr(\))28 b(=)g(1)21 b Fl(\000)2120 3034 y Fr(1)p 2117 3078 V 2117 3170 a Fp(k)2181 3101 y(;)67 b Fq(P)2352 3116 y Fo(k)2394 3101 y Fr(\()p Fp(Y)49 b Fr(=)27 b Fp(k)2695 3060 y Fm(\()p Fo(k)2761 3037 y Ff(2)2796 3060 y Fm(\))2828 3101 y Fr(\))g(=)3009 3034 y(1)p 3007 3078 V 3007 3170 a Fp(k)3093 3101 y Fl(\000)3225 3034 y Fr(1)p 3202 3078 94 4 v 3202 3170 a Fp(k)3256 3141 y Fm(3)236 3328 y Fr(and)33 b Fp(Z)h Fr(=)28 b(1)k(yield)g(p)s (ositiv)m(e)g(recurren)m(t)i Fp(\027)1777 3343 y Fo(k)1820 3328 y Fp(;)f(k)e Fl(2)d Fq(N)p Fp(;)k Fr(suc)m(h)i(that)535 3528 y Fp(\027)583 3543 y Fo(k)677 3475 y Fm(w)654 3528 y Fl(!)27 b Fp(")827 3544 y Fm(\(0)p Fo(;)p Fm(1\))1150 3528 y Fr(and)175 b Fp(\026)1541 3543 y Fo(k)1635 3475 y Fm(w)1611 3528 y Fl(!)27 b Fp(")1784 3543 y Fn(1)1858 3528 y Fp(:)236 3729 y Fr(Again)32 b(the)h(\014rst)g(con)m(v)m(ergence) i(is)d(clear,)g(while)g(the)h(second)h(one)f(follo)m(ws)e(from)535 3930 y Fq(P)612 3945 y Fo(k)654 3930 y Fr(\()p Fp(W)42 b(<)27 b(k)s Fr(\))83 b Fl(\024)h Fq(P)1342 3945 y Fo(k)1384 3930 y Fr(\()p Fp(Y)1479 3945 y Fm(1)1534 3930 y Fp(:)17 b(:)g(:)g(Y)1723 3947 y Fo(k)1762 3928 y Ff(2)1827 3930 y Fp(<)28 b(k)s Fr(\))1104 4178 y Fl(\024)84 b Fq(P)1342 4193 y Fo(k)1384 4178 y Fr(\()1422 4111 y Fj(S)1508 4194 y Fm(1)p Fn(\024)p Fo(i)p Fn(\024)p Fo(k)1716 4175 y Ff(2)1754 4178 y Fl(f)p Fp(Y)1861 4193 y Fo(i)1916 4178 y Fr(=)28 b(0)p Fl(g)p Fr(\))21 b(+)h Fq(P)2353 4193 y Fo(k)2396 4178 y Fr(\()2434 4111 y Fj(T)2519 4194 y Fm(1)p Fn(\024)p Fo(i)p Fn(\024)p Fo(k)2727 4175 y Ff(2)2766 4178 y Fl(f)p Fp(Y)2873 4193 y Fo(i)2928 4178 y Fr(=)3044 4110 y(1)p 3042 4154 55 4 v 3042 4246 a Fp(k)3106 4178 y Fl(g)p Fr(\))1105 4469 y(=)84 b(1)22 b Fl(\000)g Fr(\(1)g Fl(\000)1676 4402 y Fr(1)p 1654 4446 94 4 v 1654 4537 a Fp(k)1708 4508 y Fm(3)1757 4469 y Fr(\))1795 4428 y Fo(k)1834 4404 y Ff(2)1894 4469 y Fr(+)g(\(1)g Fl(\000)2214 4402 y Fr(1)p 2211 4446 55 4 v 2211 4537 a Fp(k)2275 4469 y Fr(\))2313 4428 y Fo(k)2352 4404 y Ff(2)2418 4469 y Fl(!)27 b Fr(0)p Fp(:)236 4906 y Fq(9.)50 b(F)-9 b(urther)37 b(ergo)s(dic)g(theorems)236 5143 y Fr(The)f(mean)d(ergo)s(dic)h (theorem)g(\(7.1\))g(can)h(b)s(e)f(strengthened)i(to)e(w)m(eak)i(con)m (v)m(ergence)h(in)236 5263 y(the)28 b(p)s(ositiv)m(e)f(recurren)m(t)i (case,)g(holding)d(in)g(an)i(extended)h(sense)g(also)e(in)f(b)s(oth)h (the)h(other)236 5384 y(cases.)53 b(This)35 b(is)g(a)g(consequence)j (of)d(more)g(general)f(results)i(in)e([18],)i(but)g(for)e(the)i(state) 236 5504 y(space)j Fq(R)586 5519 y Fm(+)683 5504 y Fr(a)f(direct)g(pro) s(of)f(is)h(to)s(o)f(simple)g(to)h(refer)g(to)g(other)g(w)m(ork.)61 b(In)39 b(the)g(presen)m(t)1865 5753 y(8)p eop %%Page: 9 10 9 9 bop 236 171 a Fr(framew)m(ork)33 b(the)g(follo)m(wing)d(holds,)i (indep)s(enden)m(tly)h(of)g(the)g(initial)28 b(la)m(w:)236 349 y Fq(\(9.1\))58 b(Theorem.)49 b Fi(If)35 b Fp(\026)1205 364 y Fo(n)1279 349 y Fr(:=)28 b Fl(L)p Fr(\()p Fp(X)1598 364 y Fo(n)1644 349 y Fr(\))35 b Fi(for)g Fp(n)28 b Fl(\025)g Fr(0)p Fi(,)34 b(then)236 552 y Fr(\(a\))174 b Fp(\026)594 567 y Fo(n)692 499 y Fm(w)669 552 y Fl(!)27 b Fp(\026;)236 756 y Fi(whenever)34 b Fp(\027)41 b Fi(is)35 b(p)-5 b(ositive)34 b(r)-5 b(e)g(curr)g(ent)35 b(with)g(stationary)g(distribution)g Fp(\026)p Fi(,)236 959 y Fr(\(b\))175 b Fp(\026)600 974 y Fo(n)698 906 y Fm(w)674 959 y Fl(!)27 b Fp(")847 974 y Fn(1)922 959 y Fp(;)66 b Fi(i.e.)49 b Fp(\026)1258 974 y Fo(n)1305 959 y Fr(\([0)p Fp(;)17 b(t)p Fr(]\))28 b Fl(!)f Fr(0)49 b Fi(for)35 b(al)5 b(l)50 b Fp(t)27 b(<)h Fl(1)p Fp(;)236 1163 y Fi(whenever)34 b Fp(\027)41 b Fi(is)35 b(nul)5 b(l)35 b(r)-5 b(e)g(curr)g(ent)35 b(or)g(tr)-5 b(ansient.)236 1341 y(Pr)g(o)g(of.)50 b Fr(\(a\))32 b(By)h(b)s(ounded)g(con)m(v)m(ergence)i(\(2.6\))e(implies) 535 1545 y Fp(\026)594 1560 y Fo(n)641 1545 y Fp(f)g Fl(\000)22 b Fp(\026f)38 b Fr(=)28 b Fq(E)p Fr(\()1182 1474 y Fj(R)1253 1545 y Fr(\()p Fp(f)11 b Fr(\()p Fp(X)1469 1560 y Fo(n)1516 1545 y Fr(\))22 b Fl(\000)h Fp(f)11 b Fr(\()p Fp(X)1862 1503 y Fo(x)1854 1569 y(n)1905 1545 y Fr(\)\))17 b Fp(\026)p Fr(\()p Fp(dx)p Fr(\)\))27 b Fl(!)g Fr(0)174 b(for)32 b(all)48 b Fp(f)38 b Fl(2)28 b(K)q Fr(\()p Fq(R)3335 1560 y Fm(+)3394 1545 y Fr(\))p Fp(:)383 1748 y Fr(\(b\))k(It)h(follo)m(ws)e(from)g(the)i(represen)m (tation)535 1951 y Fp(W)41 b Fr(=)28 b Fp(Z)839 1966 y Fm(1)900 1951 y Fr(+)22 b Fp(:)17 b(:)g(:)22 b Fr(+)g Fp(Y)1290 1966 y Fm(1)1346 1951 y Fp(:)17 b(:)g(:)f(Y)1534 1966 y Fo(n)p Fn(\000)p Fm(1)1670 1951 y Fr(\()p Fp(Z)1775 1966 y Fo(n)1844 1951 y Fr(+)22 b Fp(Y)1999 1966 y Fo(n)2046 1951 y Fp(Z)2113 1966 y Fo(n)p Fm(+1)2272 1951 y Fr(+)g Fp(:)17 b(:)g(:)o Fr(\))175 b(for)49 b Fp(n)28 b Fl(2)g Fq(N)236 2155 y Fr(that)35 b(the)h(ev)m(en)m(t)h Fl(f)p Fp(W)45 b Fr(=)32 b Fl(1g)j Fr(is)g(con)m(tained)g(in)f(\(the)i (completion)d(of)7 b(\))34 b(the)i(tail{\014eld)d(of)236 2275 y(\()p Fp(Y)331 2290 y Fo(n)378 2275 y Fp(;)17 b(Z)489 2290 y Fo(n)535 2275 y Fr(\))573 2290 y Fo(n)p Fn(2)p Fk(N)731 2275 y Fr(,)34 b(b)s(ecause)h Fq(P)p Fr(\()p Fp(Y)50 b Fr(=)30 b(0\))f(=)h(0)j(b)m(y)i(\(8.3\).)47 b(Therefore)35 b Fp(W)43 b Fr(=)29 b Fl(1)50 b Fr(a.s.)34 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b(p)s(oin)m(t)m(wise)f(ergo)s(dic)f(theorem)h(for)g(the)g (p)s(ositiv)m(e)g(recurren)m(t)h(case)h(can)e(b)s(e)g(deriv)m(ed)236 3691 y(from)45 b(\(7.4\).)81 b(T)-8 b(o)46 b(obtain)f(it)f(for)h(as)h (man)m(y)f(functions)h(as)g(p)s(ossible,)i(it)c(is,)49 b(ho)m(w)m(ev)m(er,)236 3811 y(preferable)35 b(to)f(establish)g (\014rst)h(ergo)s(dicit)m(y)f(under)i(stationarit)m(y)-8 b(.)48 b(Actually)34 b(a)g(stronger)236 3931 y(result)f(holds:)236 4110 y Fq(\(9.2\))58 b(Prop)s(osition.)48 b Fi(If)34 b(the)h(pr)-5 b(o)g(c)g(ess)34 b Fr(\()p Fp(X)1887 4125 y Fo(n)1934 4110 y Fr(\))1972 4125 y Fo(n)p Fn(\025)p Fm(0)2144 4110 y Fi(is)g(stationary,)h(it)h(is)e(mixing.)236 4288 y(Pr)-5 b(o)g(of.)50 b Fr(1.)41 b(Extending)28 b(\()p Fp(Y)1220 4303 y Fo(n)1266 4288 y Fp(;)17 b(Z)1377 4303 y Fo(n)1424 4288 y Fr(\))p Fp(;)33 b(n)28 b Fl(2)g Fq(N)p Fp(;)f Fr(let)f(\()p Fp(Y)2074 4303 y Fo(n)2121 4288 y Fp(;)17 b(Z)2232 4303 y Fo(n)2278 4288 y Fr(\))p Fp(;)33 b(n)28 b Fl(2)g Fq(Z)p Fp(;)g 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b(stationary)-8 b(,)46 b(if)c(the)i(initial)39 b(la)m(w)k(is)g(the)h (uniform)e(distribution)f(on)i([0,2].)236 2412 y(Since)33 b Fp(X)572 2427 y Fo(n)p Fn(\000)p Fm(1)742 2412 y Fr(can)f(b)s(e)h (reconstructed)i(from)c Fp(X)1972 2427 y Fo(n)2051 2412 y Fr(via)535 2616 y Fp(X)616 2631 y Fo(n)p Fn(\000)p Fm(1)781 2616 y Fr(=)c(2)17 b(\()p Fp(X)1069 2631 y Fo(n)1138 2616 y Fl(\000)22 b Fr(1)1286 2631 y Fn(f)p Fo(X)1379 2639 y Fg(n)1422 2631 y Fo(>)p Fm(1)p Fn(g)1552 2616 y Fr(\))49 b(a.s.)q Fp(;)236 2819 y Fr(the)26 b(tail{\014eld)c(of)i(\() p Fp(X)1012 2834 y Fo(n)1059 2819 y Fr(\))1097 2834 y Fo(n)p Fn(\025)p Fm(0)1259 2819 y Fr(coincides)h(in)f(this)h(case)h (with)e(the)i(full)d Fp(\033)t Fr({algebra)g(generated)236 2939 y(b)m(y)34 b(\()p Fp(X)491 2954 y Fo(n)538 2939 y Fr(\))576 2954 y Fo(n)p Fn(\025)p Fm(0)745 2939 y Fr(\(mo)s(dulo)d(n) m(ull)g(sets\).)383 3060 y(No)m(w)23 b(the)h(p)s(oin)m(t)m(wise)f(ergo) s(dic)f(theorem)h(is)g(a)g(simple)f(consequence.)43 b(Related)23 b(results)236 3180 y(in)32 b(the)h(con)m(text)i(of)d(Lipsc)m(hitz)h (maps)f(can)h(b)s(e)g(found)g(in)f([3],)h([10],)f([11].)44 b(In)33 b(the)h(presen)m(t)236 3301 y(framew)m(ork)f(the)g(follo)m (wing)d(is)i(true,)h(again)e(indep)s(enden)m(tly)i(of)f(the)h(initial)c (la)m(w:)236 3479 y Fq(\(9.3\))58 b(Theorem.)49 b Fi(L)-5 b(et)48 b Fp(\027)54 b Fi(b)-5 b(e)47 b(p)-5 b(ositive)47 b(r)-5 b(e)g(curr)g(ent)48 b(with)f(stationary)g(distribution)h Fp(\026)236 3599 y Fi(and)35 b Fp(f)45 b Fi(b)-5 b(e)35 b Fp(\026)p Fi({inte)-5 b(gr)g(able.)43 b(Then)34 b(the)h(c)-5 b(onver)g(genc)g(e)550 3774 y Fr(1)p 545 3818 59 4 v 545 3910 a Fp(n)630 3775 y Fj(P)734 3857 y Fm(0)p Fn(\024)p Fo(m)g Fr(0)i(the)i (conditions)e(\(a\))g(and)h(\(b\))g(can)h(b)s(e)f(sho)m(wn)236 3257 y(to)k(b)s(e)g(sup)s(er\015uous,)j(in)d(the)g(general)g(case)h (\(9.3\))e(can)h(hardly)g(b)s(e)g(impro)m(v)m(ed.)63 b(Indeed)236 3378 y(con)m(v)m(ergence)38 b(can)e(fail,)e(if)h Fp(f)46 b Fr(is)35 b(assumed)h(to)f(b)s(e)h Fp(\026)p 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Fr(:=)27 b([2)886 5463 y Fo(k)929 5504 y Fp(;)17 b Fr(2)1022 5463 y Fo(k)r Fm(+1)1154 5504 y Fr([)1841 5753 y(11)p eop %%Page: 12 13 12 12 bop 236 171 a Fr(and)30 b(the)h(stationary)e(distribution)f Fp(\026)i Fr(is)g(nonatomic)e(b)m(y)j(\(6.4\),)f(a)g(con)m(tin)m(uous)h (function)236 291 y Fp(f)39 b Fr(:)27 b Fq(R)461 306 y Fm(+)548 291 y Fl(!)g Fq(R)759 306 y Fm(+)850 291 y Fr(can)33 b(b)s(e)g(constructed)h(suc)m(h)g(that)236 488 y(\(2\))174 b Fp(f)11 b Fr(\()p Fp(x)p Fr(\))28 b Fl(\025)g Fp(k)20 b(n)987 503 y Fo(k)1204 488 y Fr(for)49 b Fp(x)28 b Fl(2)h Fp(A)1621 503 y Fo(k)1663 488 y Fp(;)236 708 y Fr(\(3\))535 637 y Fj(R)591 723 y Fo(B)644 735 y Fg(k)703 708 y Fp(f)e(d\026)g Fl(\024)h Fr(2)1069 667 y Fn(\000)p Fo(k)1341 708 y Fr(for)49 b Fp(k)31 b Fl(2)d Fq(N)p Fp(:)236 927 y Fr(Then)34 b(b)m(y)f(\(2\))g(the)g(sequence)535 1169 y Fp(V)592 1184 y Fo(n)667 1169 y Fr(:=)812 1101 y(1)p 807 1146 59 4 v 807 1237 a Fp(n)892 1102 y Fj(P)996 1184 y Fm(1)p Fn(\024)p Fo(m)p Fn(\024)p Fo(n)1267 1169 y Fp(f)11 b Fr(\()p Fp(X)1445 1184 y Fo(m)1511 1169 y Fr(\))175 b(for)49 b Fp(n)28 b Fl(2)g Fq(N)236 1391 y Fr(satis\014es)535 1588 y Fq(P)p Fr(\(lim)17 b(sup)965 1603 y Fo(n)p Fn(!1)1170 1588 y Fp(V)1227 1603 y Fo(n)1301 1588 y Fr(=)28 b Fl(1)p Fr(\))83 b Fl(\025)g Fq(P)p Fr(\(lim)17 b(sup)2216 1603 y Fo(k)r Fn(!1)2416 1588 y Fl(f)p Fp(V)2523 1603 y Fo(T)2564 1615 y Fg(k)2634 1588 y Fl(\025)28 b Fp(k)s Fl(g)p Fr(\))1626 1792 y Fl(\025)83 b Fq(P)p Fr(\(lim)17 b(sup)2216 1807 y Fo(k)r Fn(!1)2416 1792 y Fl(f)p Fp(T)2523 1807 y Fo(k)2594 1792 y Fl(\024)28 b Fp(n)2757 1807 y Fo(k)2800 1792 y Fl(g)p Fr(\))p Fp(;)236 1989 y Fr(where)34 b(the)f(last)f(inequalit)m(y)f(follo)m(ws)h(from)535 2225 y Fp(V)592 2240 y Fo(T)633 2252 y Fg(k)703 2225 y Fl(\025)844 2157 y Fr(1)p 818 2201 100 4 v 818 2293 a Fp(T)875 2308 y Fo(k)928 2225 y Fp(f)11 b Fr(\()p Fp(X)1106 2240 y Fo(T)1147 2252 y Fg(k)1189 2225 y Fr(\))27 b Fl(\025)i Fp(k)177 b Fr(for)49 b Fp(T)1811 2240 y Fo(k)1882 2225 y 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b(:)g(:)g(:)g(;)g(X)2025 4669 y Fm(0)2017 4734 y Fo(k)r(n)2130 4710 y Fp(<)27 b(x)p Fl(g)1192 4913 y(\032)1352 4847 y Fj(T)1455 4928 y Fm(0)p Fn(\024)p Fo(i)27 b(k)s(n)p Fr(\))h Fl(\024)g Fr(\()p Fq(P)p Fr(\()p Fp(X)1536 5266 y Fm(0)1528 5332 y Fo(n)1603 5307 y Fp(<)f(x)p Fr(\)\))1837 5266 y Fo(k)1908 5307 y Fr(=)g(\(1)22 b Fl(\000)h Fp(#)p Fr(\))2315 5266 y Fo(k)2532 5307 y Fr(for)50 b Fp(k)30 b Fl(\025)f Fr(0)p Fp(:)236 5504 y Fr(Therefore)34 b(the)f(assertion)f(holds)h(for)f(an)m (y)h(initial)c(distribution)p Fp(:)98 b Fh(2)1841 5753 y Fr(12)p eop %%Page: 13 14 13 13 bop 383 171 a Fr(The)45 b(follo)m(wing)d(coun)m(terpart)j(of)f (\(9.4\))g(distinguishes)g(p)s(ositiv)m(e)g(and)h(n)m(ull)e(recur-)236 291 y(rence:)236 469 y Fq(\(9.5\))58 b(Prop)s(osition.)48 b Fi(L)-5 b(et)36 b Fp(T)1400 484 y Fo(B)1497 469 y Fi(b)-5 b(e)36 b(de\014ne)-5 b(d)35 b(as)h(in)g Fr(\(9.4\))f Fi(and)h Fp(\027)43 b Fi(b)-5 b(e)35 b(r)-5 b(e)g(curr)g(ent.)49 b(Then,)236 590 y(for)35 b(arbitr)-5 b(ary)35 b 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Fp(:)1841 5753 y Fr(14)p eop %%Page: 15 16 15 15 bop 236 171 a Fr(Therefore)34 b(b)m(y)f(\(1\))535 427 y Fq(E)609 386 y Fo(x)653 427 y Fr(\()p Fp(T)748 442 y Fo(G)807 427 y Fr(\))27 b Fl(\024)i Fq(E)1052 386 y Fo(x)1092 363 y Fd(0)1117 427 y Fr(\()p Fp(T)1212 442 y Fo(G)1267 423 y Fd(0)1294 427 y Fr(\))f Fp(<)f Fl(1)174 b Fr(for)1950 360 y(2)p Fp(")2045 324 y Fn(0)2068 360 y Fp(x)p 1913 404 247 4 v 1913 496 a(x)23 b Fl(\000)g Fp(")2137 467 y Fn(0)2197 427 y Fl(\024)28 b Fp(";)236 668 y Fr(where)34 b(the)f(last)f(condition)f(is)h(satis\014ed)h(for)f (su\016cien)m(tly)h(small)e Fp(")2711 632 y Fn(0)2734 668 y Fr(.)383 789 y(2.)43 b(T)-8 b(o)32 b(pro)m(v)m(e)i(the)f(con)m(v) m(erse)i(supp)s(ose)f(only)535 992 y Fq(E)609 951 y Fo(x)653 992 y Fr(\()p Fp(T)748 1008 y Fm([0)p Fo(;t)p Fm(])871 992 y Fr(\))28 b Fp(<)g Fl(1)174 b Fr(for)32 b(all)47 b Fp(t)28 b(>)g(x:)236 1195 y Fr(Assuming)k(no)m(w)i Fp(\027)39 b Fr(not)32 b(to)g(b)s(e)h(p)s(ositiv)m(e)f(recurren)m(t)i (implies)236 1399 y(\(2\))174 b Fq(P)p Fr(\()p Fp(Y)21 b(x)i Fr(+)f Fp(Z)34 b(>)28 b(t)p Fr(\))g(=)f(0)174 b(for)32 b(all)48 b Fp(t)28 b(>)f(x:)236 1602 y Fr(Indeed,)34 b(with)f Fp(\026)862 1617 y Fm(1)928 1602 y Fr(:=)28 b Fl(L)p Fr(\()p Fp(X)1255 1566 y Fo(x)1247 1627 y Fm(1)1298 1602 y Fr(\))33 b(monotonicit)m(y)d(yields)535 1806 y Fq(E)609 1765 y Fo(x)653 1806 y Fr(\()p Fp(T)748 1821 y Fm([0)p Fo(;t)p Fm(])871 1806 y Fr(\))83 b Fl(\025)1153 1735 y Fj(R)1225 1821 y Fo(s>t)1364 1806 y Fq(E)1438 1765 y Fo(s)1474 1806 y Fr(\(1)22 b(+)g Fp(T)1738 1821 y Fm([0)p Fo(;t)p Fm(])1862 1806 y Fr(\))g Fp(\026)1981 1821 y Fm(1)2020 1806 y Fr(\()p Fp(ds)p Fr(\))992 2009 y Fl(\025)84 b Fq(P)p Fr(\()p Fp(Y)21 b(x)h Fr(+)g Fp(Z)35 b(>)27 b(t)p Fr(\))c Fq(E)1896 1968 y Fo(t)1925 2009 y Fr(\()p Fp(T)2020 2025 y Fm([0)p Fo(;t)p Fm(])2144 2009 y Fr(\))p Fp(;)236 2212 y Fr(where)34 b Fq(E)592 2176 y Fo(t)621 2212 y Fr(\()p Fp(T)716 2228 y Fm([0)p Fo(;t)p Fm(])840 2212 y Fr(\))28 b(=)f Fl(1)32 b Fr(b)m(y)i(\(9.5b\).) 43 b(Letting)32 b Fp(t)c Fl(#)f Fp(x)33 b Fr(in)f(\(2\))g(leads)h(to) 535 2416 y Fq(P)p Fr(\()p Fp(Y)21 b(x)h Fr(+)g Fp(Z)35 b Fl(\024)28 b Fp(x)p Fr(\))g(=)g(1)p Fp(;)236 2619 y Fr(hence)40 b(b)m(y)f(\(1.2b\))f(to)p 1071 2566 56 4 v 37 w Fp(x)g Fl(\024)f Fp(x)h(<)f Fl(1)p Fr(.)60 b(This)38 b(implies)e(p)s(ositiv)m(e)h(recurrence)j(and)e(th)m(us)i(a)236 2740 y(con)m(tradiction)32 b(to)g(the)h(assumption)p Fp(:)99 b Fh(2)383 2918 y Fr(T)-8 b(ogether,)56 b(\(9.1\))51 b(and)g(\(9.6\))f(sho)m(w)i(that)f(the)h(t)m(w)m(o)g(main)d(c)m (haracterizations)i(of)236 3039 y(p)s(ositiv)m(e/n)m(ull)39 b(recurrence)j(from)e(classical)f(Mark)m(o)m(v)i(c)m(hain)g(theory)g (in)f(essence)j(carry)236 3159 y(o)m(v)m(er)34 b(to)e(a\016ne)h (recursions.)236 3512 y Fq(10.)50 b(The)38 b(con)m(tractiv)m(e)e(case) 236 3748 y Fr(As)e(outlined)e(in)g(Section)h(5)g(it)f(is)h(in)f (general)h(imp)s(ossible)e(to)h(determine)h(the)h(in)m(v)-5 b(arian)m(t)236 3869 y(measure)36 b Fp(\026)f Fr(of)f(a)h(recurren)m(t) i(distribution)c Fp(\027)42 b Fr(explicitly)-8 b(.)49 b(In)36 b(spite)f(of)g(\(8.2\))g(this)g(holds)236 3989 y(as)43 b(w)m(ell)e(for)h(the)h(stationary)f(distribution)e(in)i(the)h (p)s(ositiv)m(e)e(recurren)m(t)j(case)g(\(for)d(an)236 4110 y(exception)f(see)g(\(10.3\)\).)61 b(This)39 b(is)g(comp)s (ensated)g(to)g(some)f(exten)m(t)j(b)m(y)e(the)h(fact)e(that,)236 4230 y(due)f(to)g(the)g(algebraic)d(form)i(of)g(the)h(underlying)f (recursion,)i(the)f(momen)m(ts)f(of)g Fp(\026)g Fr(can)236 4350 y(b)s(e)c(easily)e(computed)i(from)e(those)i(of)f Fp(\027)6 b Fr(,)32 b(as)g(far)f(as)g(they)h(exist.)44 b(The)32 b(follo)m(wing)d(simple)236 4471 y(criterion)i(for)h(their)g (existence)j(extends)f(a)e(result)h(b)m(y)g(V)-8 b(erv)j(aat)33 b([39]:)236 4649 y Fq(\(10.1\))58 b(Prop)s(osition.)48 b Fi(L)-5 b(et)35 b Fr(\()p Fp(X)1517 4664 y Fo(n)1564 4649 y Fr(\))1602 4664 y Fo(n)p Fn(\025)p Fm(0)1774 4649 y Fi(b)-5 b(e)34 b(stationary)h(and)57 b Fr(0)27 b Fp(<)h(\013)g(<)g Fl(1)p Fi(.)44 b(Then)535 4853 y Fq(E)p Fr(\()p Fp(X)736 4811 y Fo(\013)728 4877 y(n)785 4853 y Fr(\))27 b Fp(<)h Fl(1)236 5056 y Fi(if)35 b(and)f(only)h(if)535 5259 y Fq(E)p Fr(\()p Fp(Y)725 5218 y Fo(\013)774 5259 y Fr(\))28 b Fp(<)g Fr(1)174 b Fi(and)f Fq(E)p Fr(\()p Fp(Z)1681 5218 y Fo(\013)1730 5259 y Fr(\))28 b Fp(<)f Fl(1)p Fp(:)1841 5753 y Fr(15)p eop %%Page: 16 17 16 16 bop 236 171 a Fi(Pr)-5 b(o)g(of.)50 b Fr(With)32 b Fp(W)46 b Fr(as)33 b(de\014ned)h(in)d(\(8.2\))h(it)g(follo)m(ws)f (from)535 369 y Fq(E)p Fr(\()p Fp(W)753 328 y Fo(\013)802 369 y Fr(\))c(=)h(sup)1134 384 y Fo(n)p Fn(2)p Fk(N)1308 369 y Fq(E)p Fr(\(\()p Fp(Z)1525 384 y Fm(1)1586 369 y Fr(+)22 b Fp(:)17 b(:)g(:)22 b Fr(+)g Fp(Y)1976 384 y Fm(1)2031 369 y Fp(:)17 b(:)g(:)g(Y)2220 384 y Fo(n)p Fn(\000)p Fm(1)2356 369 y Fp(Z)2423 384 y Fo(n)2470 369 y Fr(\))2508 328 y Fo(\013)2557 369 y Fr(\))236 567 y(b)m(y)34 b(elemen)m(tary)e(inequalities)f(that)535 804 y Fq(E)p Fr(\()p Fp(W)753 763 y Fo(\013)802 804 y Fr(\))862 707 y Fj(n)944 736 y 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Fp(Z)7 b Fl(k)3238 3506 y Fn(1)3340 3491 y Fp(<)28 b Fl(1)236 3611 y Fr(ob)m(viously)36 b(imply)p 949 3559 56 4 v 35 w Fp(x)e(<)g Fl(1)p Fr(,)j(con)m(v)m(ersely)h(this)e (condition)f(yields)g(only)h Fl(k)p Fp(Y)21 b Fl(k)3081 3626 y Fn(1)3190 3611 y Fl(\024)34 b Fr(1)i(and)236 3732 y Fl(k)p Fp(Z)7 b Fl(k)410 3747 y Fn(1)518 3732 y Fp(<)33 b Fl(1)i Fr(\(consider)h(for)g(instance)g(the)g(case)h Fp(N)44 b Fr(=)33 b Fl(f)p Fr(\(1)p Fp(;)17 b Fr(0\))p Fp(;)g Fr(\(0)p Fp(;)g Fr(1\))p Fl(g)p Fr(,)34 b(where)j Fp(\026)c Fr(=)g Fp(")3504 3747 y Fm(1)236 3852 y Fr(according)f(to)g (\(1.4\)\).)383 3973 y(In)38 b(view)h(of)f(\(10.1\))f(it)g(is)h(of)g (in)m(terest,)i(under)f(whic)m(h)g(conditions)e Fp(\026)h Fr(has)h(an)f(exp)s(o-)236 4093 y(nen)m(tial)29 b(momen)m(t)f(and)i(is) f(th)m(us)i(determined)e(b)m(y)i(its)e(momen)m(ts)g(of)g(natural)g (order.)43 b(Here)236 4213 y(con)m(tractivit)m(y)33 b(en)m(ters:)236 4392 y Fq(\(10.2\))58 b(Prop)s(osition.)48 b Fi(L)-5 b(et)35 b Fr(\()p Fp(X)1517 4407 y 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b(for)32 b(all)48 b Fp(u)27 b Fl(\025)i Fr(0)p Fp(:)1841 5753 y Fr(17)p eop %%Page: 18 19 18 18 bop 236 171 a Fr(F)-8 b(or)32 b Fp(u)467 186 y Fm(0)534 171 y Fp(>)27 b Fr(0)32 b(with)h Fp( )1004 186 y Fm(1)1043 171 y Fr(\()p Fp(u)1137 186 y Fm(0)1176 171 y Fr(\))28 b Fp(<)g Fr(1)k(and)h Fp( )1680 186 y Fm(2)1719 171 y Fr(\()p Fp(u)1813 186 y Fm(0)1852 171 y Fr(\))28 b Fp(<)f Fl(1)33 b Fr(this)f(yields)236 368 y(\(3\))174 b Fq(E)p Fr(\()p Fp(e)692 327 y Fo(u)733 336 y Ff(0)767 327 y Fo(Z)820 304 y Fd(0)846 368 y Fr(\))28 b Fl(\024)g Fp( )1080 383 y Fm(2)1120 368 y Fr(\()p Fp(u)1214 383 y Fm(0)1253 368 y Fr(\))17 b Fp(=)g Fr(\(1)k Fl(\000)h Fp( )1644 383 y Fm(1)1684 368 y Fr(\()p Fp(u)1778 383 y Fm(0)1817 368 y Fr(\)\))28 b Fp(<)f Fl(1)p Fp(:)236 566 y Fr(By)33 b(\(1\))f({)h(\(3\))f(the)h(reduction)f(to)h(the)g(sp)s (ecial)e(case)j(is)e(settled)p Fp(:)100 b Fh(2)383 744 y Fr(The)42 b(simplest)e(example)h(for)f(the)i(situation)e(of)g (\(10.2\))h(is)g(pro)m(vided)g(b)m(y)i(the)e(case)p 236 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y(additiv)m(e)32 b(mo)s(del)535 814 y Fp(Y)49 b Fr(=)842 746 y(1)p 755 791 224 4 v 755 882 a Fp(k)25 b Fr(+)d(1)1162 814 y(and)174 b Fq(P)p Fr(\()p Fp(Z)34 b Fr(=)1918 746 y Fp(i)p 1823 791 V 1823 882 a(k)25 b Fr(+)d(1)2056 814 y(\))28 b(=)2322 746 y(1)p 2235 791 V 2235 882 a Fp(k)d Fr(+)d(1)2642 814 y(for)50 b(0)27 b Fl(\024)h Fp(i)g Fl(\024)g Fp(k)s(:)236 1056 y Fr(Since)k(\(10.3\))f (extends)j(easily)d(to)h(coun)m(table)g(partitions,)e(there)j(are)f (similar)c(examples)236 1176 y(in)k(the)h(m)m(ultiplicativ)m(e)c(mo)s (del:)535 1380 y Fq(P)p Fr(\()p Fp(Y)49 b Fr(=)27 b(2)908 1339 y Fn(\000)p Fo(k)1005 1380 y Fr(\))h(=)g(2)1224 1339 y Fn(\000)p Fo(k)1371 1380 y Fr(for)49 b Fp(k)31 b Fl(2)d Fq(N)174 b Fr(and)g Fp(Z)35 b Fr(=)28 b(1)236 1583 y(for)k(instance)h(yields)f(the)h(uniform)e(distribution)g(on)h ([1,2].)383 1703 y(It)h(is)g(ob)m(vious)h(that)g(the)g(h)m(yp)s (othesis)h(in)d(\(10.3\))h(implies)2535 1637 y Fj(P)2640 1718 y Fm(0)p Fn(\024)p Fo(i)p Fn(\024)p Fo(k)2868 1703 y 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Fp(u)p 1796 370 74 4 v 1796 462 a(l)1827 433 y Fo(k)1880 394 y Fr(\)])1945 352 y Fn(\000)2010 325 y Ff(1)p 2009 337 31 4 v 2009 378 a(2)2228 394 y Fr(with)50 b Fp(\015)33 b Fr(:=)2692 326 y(2)p Fp(q)p 2692 370 96 4 v 2696 462 a(p)2745 433 y Fm(2)236 642 y Fr(it)f(su\016ces)i(to)f(sho)m(w)236 845 y(\(1\))174 b(lim)17 b(inf)822 860 y Fo(n)p Fn(!1)1027 779 y Fj(Q)1122 860 y Fo(k)r Fn(\025)p Fm(0)1271 845 y Fr([)p Fp(:)g(:)g(:)p Fr(])27 b Fp(<)h Fl(1)p Fp(;)236 1049 y Fr(b)s(ecause)f(then)g Fp(\026)e Fr(cannot)h(b)s(e)g(absolutely) f(con)m(tin)m(uous)i(with)e(resp)s(ect)i(to)e(Leb)s(esgue)i(mea-)236 1169 y(sure)g(b)m(y)f(the)g(Riemann{Leb)s(esgue)f(lemma)e(and)j (\(6.2\))e(applies.)41 b(T)-8 b(o)25 b(v)m(erify)h(\(1\))f(consider)236 1289 y(the)33 b(v)-5 b(alues)535 1493 y Fp(u)591 1508 y Fo(m)685 1493 y Fr(=)27 b(2)p Fp(\031)f(l)949 1452 y Fo(m)1044 1493 y Fl(!)h(1)99 b Fr(for)g Fp(m)28 b Fl(!)g(1)p Fp(;)236 1696 y Fr(whic)m(h)33 b(b)m(y)h(shifting)d(the)i(index)g Fp(k)i Fr(to)e Fp(k)25 b Fl(\000)d Fp(m)33 b Fr(yield)f(the)h(constan)m (t)g(v)-5 b(alue)535 1872 y Fj(Q)630 1953 y Fo(k)r Fn(\025)p Fm(0)780 1938 y Fr([)p Fp(:)17 b(:)g(:)o Fr(])28 b(=)1079 1872 y Fj(Q)1174 1953 y Fo(k)r Fn(2)p Fk(N)1344 1938 y Fr(\(1)22 b(+)g Fp(\015)g Fr(\(1)g Fl(\000)g Fr(cos)1989 1871 y(2)p Fp(\031)p 1989 1915 108 4 v 2006 2007 a(l)2037 1978 y Fo(k)2107 1938 y Fr(\)\))p Fp(:)236 2180 y Fr(Since)47 b(1)32 b Fl(\000)g Fr(cos)q(\(2)p Fp(\031)t(=l)1052 2144 y Fo(k)1094 2180 y Fr(\))47 b(is)f(b)s(ounded)i(b)m(y)1864 2141 y Fm(1)p 1864 2157 36 4 v 1864 2215 a(2)1909 2180 y Fr(\(2)p Fp(\031)t(=l)2135 2144 y Fo(k)2177 2180 y Fr(\))2215 2144 y Fm(2)2302 2180 y Fr(for)e(almost)f(all)g Fp(k)50 b Fr(and)d(th)m(us)h(is)236 2301 y(summable,)32 b(the)h(latter)e(pro)s(duct)i(con)m(v)m(erges)i(and)d(\(1\))h(is)f (established)p Fp(:)99 b Fh(2)383 2479 y Fr(The)25 b(\014nal)f(result,) j(whic)m(h)e(originally)c(motiv)-5 b(ated)24 b(the)h(presen)m(t)h(w)m (ork,)i(requires)d(some)236 2600 y(preparation:)236 2778 y Fq(\(10.5\))58 b(Lemma.)49 b Fi(With)36 b Fr(0)27 b Fl(\024)h Fp(\015)33 b(<)28 b Fr(1)34 b Fi(and)h Fr(0)27 b Fp(<)h(p)f(<)h Fr(1)p Fp(;)38 b(q)32 b Fr(=)27 b(1)22 b Fl(\000)h Fp(p)35 b Fi(assume)535 2982 y Fp(Y)49 b Fr(=)28 b Fp(\015)179 b(and)174 b Fq(P)p Fr(\()p Fp(Z)35 b Fr(=)27 b Fp(k)s Fr(\))h(=)f Fp(p)22 b(q)1970 2940 y Fo(k)r Fn(\000)p Fm(1)2203 2982 y Fp(f)11 b(or)37 b(k)31 b Fl(2)d Fq(N)p Fp(:)236 3185 y Fi(Then)535 3388 y Fr(lim)17 b(sup)851 3403 y Fo(n)p Fn(!1)1055 3388 y Fp(X)1136 3403 y Fo(n)1183 3388 y Fp(=)g Fr(log)f Fp(n)28 b Fr(=)g Fl(\000)p Fr(1)p Fp(=)17 b Fr(log)f Fp(q)54 b Fi(a.s.)236 3592 y(Pr)-5 b(o)g(of.)50 b Fr(1.)43 b(It)33 b(is)f(less)h(cum)m(b)s(ersome) f(to)h(pro)m(v)m(e)g(instead)535 3795 y(lim)17 b(sup)851 3810 y Fo(n)p Fn(!1)1055 3795 y Fp(X)1144 3754 y Fn(0)1136 3820 y Fo(n)1183 3795 y Fp(=)g Fr(log)f Fp(n)28 b Fr(=)g(1)p Fp(=\016)53 b Fr(a.s.)236 3999 y(for)31 b(a)g(sequence)j(\()p Fp(X)994 3962 y Fn(0)986 4023 y Fo(n)1033 3999 y Fr(\))1071 4014 y Fo(n)p Fn(\025)p Fm(0)1239 3999 y Fr(where)e Fp(Y)1598 3962 y Fn(0)1649 3999 y Fr(=)27 b Fp(\015)5 b Fr(,)32 b Fp(Z)1941 3962 y Fn(0)1995 3999 y Fr(has)g(an)f(exp)s(onen)m(tial)f (distribution)g(with)236 4119 y(parameter)i Fp(\016)37 b Fr(and)32 b(in)g(addition)f Fp(X)1559 4083 y Fn(0)1551 4144 y Fm(0)1618 4119 y Fr(=)d(0.)43 b(Then)535 4322 y Fp(Z)35 b Fr(=)740 4256 y Fj(P)844 4337 y Fo(k)r Fn(2)p Fk(N)1020 4322 y Fp(k)25 b Fr(1)1145 4338 y Fn(f)p Fo(k)r Fn(\000)p Fm(1)p Fn(\024)p Fo(Z)1417 4319 y Fd(0)1438 4338 y Fo()p Fm(1)793 374 y Fq(P)p Fr(\()p Fp(Z)975 389 y Fo(n)1021 374 y Fp(=)17 b Fr(log)f Fp(n)28 b(>)g Fr(1\))f(=)1637 308 y Fj(P)1741 389 y Fo(n>)p Fm(1)1895 374 y Fq(P)p Fr(\()p Fp(e)2055 333 y Fo(Z)2103 341 y Fg(n)2177 374 y Fp(>)g(n)p Fr(\))h(=)f Fl(1)p Fp(;)236 577 y Fr(whic)m(h)33 b(b)m(y)h(Borel{Can)m(telli)c(and)i Fp(X)1562 592 y Fo(n)1637 577 y Fl(\025)c Fp(Z)1809 592 y Fo(n)1888 577 y Fr(pro)m(v)m(es)35 b(one)d(half)g(of)g(the)h (equation.)383 698 y(3.)43 b(T)-8 b(o)32 b(pro)m(v)m(e)i(the)f(other)g (half,)e(c)m(ho)s(ose)j(an)e(arbitrary)g Fp(t)c(>)f Fr(1.)44 b(Then)535 901 y Fq(P)p Fr(\()p Fp(X)731 916 y Fo(n)778 901 y Fp(=)17 b Fr(log)f Fp(n)27 b(>)h(t)p Fr(\))84 b(=)g Fq(P)p Fr(\()p Fp(Z)1674 916 y Fo(n)1748 901 y Fp(>)27 b(t)17 b Fr(log)g Fp(n)22 b Fl(\000)h Fr(\()p Fp(\015)2320 860 y Fo(n)p Fn(\000)p Fm(1)2457 901 y Fp(Z)2524 916 y Fm(1)2585 901 y Fr(+)f Fp(:)17 b(:)g(:)22 b Fr(+)g Fp(\015)2974 860 y Fm(1)3013 901 y Fp(Z)3080 916 y Fo(n)p Fn(\000)p Fm(1)3217 901 y Fr(\)\))1331 1105 y Fl(\024)84 b Fq(E)p Fr(\(exp\()p Fp(\015)1846 1063 y Fo(n)p Fn(\000)p Fm(1)1983 1105 y Fp(Z)2050 1120 y Fm(1)2112 1105 y Fr(+)22 b Fp(:)17 b(:)g(:)k Fr(+)h Fp(\015)2500 1063 y Fm(1)2540 1105 y Fp(Z)2607 1120 y Fo(n)p Fn(\000)p Fm(1)2766 1105 y Fl(\000)g Fp(t)17 b Fr(log)g Fp(n)p Fr(\)\))236 1308 y(for)33 b Fp(n)c(>)f Fr(1,)33 b(b)s(ecause)i Fq(P)p Fr(\()p Fp(Z)g(>)28 b(z)t Fr(\))h Fl(\024)g Fp(e)1636 1272 y Fn(\000)p Fo(z)1764 1308 y Fr(holds)k(for)g Fp(z)g Fl(\024)c Fr(0)k(as)g(w)m(ell)g(and)g(F)-8 b(ubini)31 b(applies)236 1428 y(due)i(to)g(indep)s(endence.)45 b(By)33 b(\(1\))f(therefore)535 1676 y Fq(P)p Fr(\()p Fp(X)731 1691 y Fo(n)778 1676 y Fp(=)17 b Fr(log)f Fp(n)27 b(>)h(t)p Fr(\))83 b Fl(\024)h Fp(n)1550 1635 y Fn(\000)p Fo(t)1651 1610 y Fj(Q)1746 1691 y Fm(1)p Fn(\024)p Fo(m)f Fr(1)p Fp(;)236 2141 y Fr(where)34 b(the)f(in\014nite)f (pro)s(duct)g(is)h(strictly)e(p)s(ositiv)m(e)h(b)m(y)2311 2074 y Fj(P)2415 2156 y Fo(m)p Fn(2)p Fk(N)2609 2141 y Fp(\015)2665 2104 y Fo(m)2759 2141 y Fp(<)27 b Fl(1)p Fr(.)44 b(Th)m(us)535 2278 y Fj(P)639 2359 y Fo(n>)p Fm(1)793 2344 y Fq(P)p Fr(\()p Fp(X)989 2359 y Fo(n)1036 2344 y Fp(=)17 b Fr(log)e Fp(n)28 b(>)g(t)p Fr(\))g Fp(<)f Fl(1)174 b Fr(for)32 b(all)48 b Fp(t)28 b(>)g Fr(1)p Fp(;)236 2547 y Fr(i.e.)43 b(1)33 b(is)f(also)f(an)i(upp)s(er)g(b)s (ound)p Fp(:)100 b Fh(2)383 2726 y Fr(In)34 b(conclusion)f(the)h(order) g(in)f(whic)m(h)h(an)f(upp)s(er)i(limit)p 2450 2673 56 4 v 30 w Fp(x)30 b Fr(=)f Fl(1)34 b Fr(is)f(approac)m(hed)i(will)236 2846 y(b)s(e)30 b(studied)f(for)g(the)h(w)m(eakly)g(con)m(tractiv)m(e)g (m)m(ultiplicativ)m(e)c(mo)s(del,)j(b)s(eing)f(of)h(particular)236 2967 y(in)m(terest)k(in)e(applications.)42 b(While)31 b(the)h(result)h(pro)m(vides)g(just)f(an)g(upp)s(er)h(b)s(ound)f(in)g (the)236 3087 y(case)h Fq(P)p Fr(\()p Fp(Y)49 b Fr(=)27 b(1\))h(=)f(0,)32 b(it)g(is)g(exact)h(otherwise)f(and,)h(somewhat)f (surprisingly)-8 b(,)32 b(dep)s(ends)236 3207 y(only)g(on)h(this)f (probabilit)m(y:)236 3386 y Fq(\(10.6\))58 b(Theorem.)49 b Fi(If)35 b Fp(Y)49 b Fl(\024)28 b Fr(1)34 b Fi(and)h Fp(Z)f Fr(=)28 b(1)p Fi(,)34 b(then)535 3589 y Fr(lim)17 b(sup)851 3604 y Fo(n)p Fn(!1)1055 3589 y Fp(X)1136 3604 y Fo(n)1183 3589 y Fp(=)g Fr(log)f Fp(n)28 b Fr(=)g Fl(\000)p Fr(1)17 b Fp(=)g Fr(log)f Fq(P)p Fr(\()p Fp(Y)48 b Fr(=)28 b(1\))49 b Fi(a.s.)236 3793 y(Pr)-5 b(o)g(of.)50 b Fr(1.)81 b(Clearly)-8 b(,)48 b Fp(X)1155 3808 y Fm(0)1244 3793 y Fr(=)h(1)c(can)h(b)s(e)f(supp)s(osed)i(in)e(the)g(sequel.)83 b(In)46 b(pro)m(ving)f(the)236 3913 y(righ)m(t{hand)d(side)h(to)f(b)s (e)g(a)h(lo)m(w)m(er)f(b)s(ound)h(for)f(the)h(upp)s(er)g(limit,)e (moreo)m(v)m(er,)46 b Fp(Y)63 b Fr(ma)m(y)236 4034 y(b)s(e)46 b(decreased)i(to)e(1)1017 4049 y Fn(f)p Fo(Y)16 b Fm(=1)p Fn(g)1238 4034 y Fp(Y)21 b Fr(.)84 b(Therefore)47 b(in)e(this)h(part)g Fp(Y)67 b Fr(will)44 b(b)s(e)i(assumed)g(to)g(b)s(e)236 4154 y(0,)16 b(1{v)-5 b(alued,)32 b(where)535 4357 y Fp(p)c Fr(:=)f Fq(P)p Fr(\()p Fp(Y)49 b Fr(=)27 b(0\))h Fp(>)f Fr(0)p Fp(:)236 4561 y Fr(Then)45 b(the)g(random)e(times)g(0)j (=)h Fp(T)1598 4576 y Fm(0)1685 4561 y Fp(<)g(T)1865 4576 y Fm(1)1951 4561 y Fp(<)g(:)17 b(:)g(:)44 b Fr(with)f Fp(Y)2523 4576 y Fo(n)2617 4561 y Fr(=)k(0)c(are)h(de\014ned)i(with)236 4681 y(probabilit)m(y)31 b(1)h(and)h(ha)m(v)m(e)h(indep)s(enden)m(t)f (and)g(geometrically)d(distributed)i(incremen)m(ts)535 4885 y Fp(U)601 4900 y Fo(k)672 4885 y Fr(:=)27 b Fp(T)859 4900 y Fo(k)924 4885 y Fl(\000)c Fp(T)1081 4900 y Fo(k)r Fn(\000)p Fm(1)1388 4885 y Fr(for)50 b Fp(k)30 b Fl(2)e Fq(N)p Fp(:)236 5088 y Fr(Th)m(us)34 b(b)m(y)g(the)f(strong)f(la)m(w)h (of)f(large)f(n)m(um)m(b)s(ers)548 5263 y(1)p 545 5307 55 4 v 545 5399 a Fp(k)626 5330 y(T)683 5345 y Fo(k)753 5330 y Fr(=)869 5263 y(1)p 867 5307 V 867 5399 a Fp(k)947 5330 y Fr(\()p Fp(U)1051 5345 y Fm(1)1113 5330 y Fr(+)22 b Fp(:)17 b(:)g(:)22 b Fr(+)g Fp(U)1512 5345 y Fo(k)1555 5330 y Fr(\))27 b Fl(!)h Fq(E)p Fr(\()p Fp(T)1917 5345 y Fm(1)1956 5330 y Fr(\))f(=)2135 5263 y(1)p 2135 5307 49 4 v 2135 5399 a Fp(p)2244 5330 y Fr(a.s.)p Fp(;)1841 5753 y Fr(21)p eop %%Page: 22 23 22 22 bop 236 171 a Fr(whic)m(h)33 b(implies)236 374 y(\(1\))191 b(log)16 b Fp(T)751 389 y Fo(k)794 374 y Fp(=)h Fr(log)f Fp(k)31 b Fl(!)c Fr(1)50 b(a.s.)236 577 y(Therefore)34 b(the)f(equation)535 781 y Fp(X)616 796 y Fo(T)657 808 y Fg(k)707 796 y Fn(\000)p Fm(1)829 781 y Fr(=)28 b Fp(U)999 796 y Fo(k)1216 781 y Fr(for)49 b Fp(k)31 b Fl(2)d Fq(N)236 984 y Fr(yields)k(the)h(estimate)535 1188 y(lim)17 b(sup)851 1203 y Fo(n)p Fn(!1)1055 1188 y Fp(X)1136 1203 y Fo(n)1183 1188 y Fp(=)g Fr(log)f Fp(n)84 b Fl(\025)f Fr(lim)17 b(sup)2009 1203 y Fo(k)r Fn(!1)2209 1188 y Fp(X)2290 1203 y Fo(T)2331 1215 y Fg(k)2381 1203 y Fn(\000)p Fm(1)2492 1188 y Fp(=)33 b Fr(log\()p Fp(T)2795 1203 y Fo(k)2860 1188 y Fl(\000)22 b Fr(1\))1533 1391 y(=)84 b(lim)17 b(sup)2009 1406 y Fo(k)r Fn(!1)2209 1391 y Fp(U)2275 1406 y Fo(k)2335 1391 y Fp(=)33 b Fr(log)16 b Fp(k)s(:)236 1594 y Fr(No)m(w)33 b(\(10.5\))f(applies)g(with)g Fp(\015)h Fr(=)27 b(0,)33 b(in)f(whic)m(h)h(case)g Fp(X)2264 1609 y Fo(n)2344 1594 y Fr(and)f Fp(Z)2600 1609 y Fo(n)2679 1594 y Fr(agree.)383 1715 y(2.)49 b(In)34 b(pro)m(ving)h(the)g(in)m(v)m (erse)h(inequalit)m(y)d Fp(Y)56 b Fr(ma)m(y)34 b(b)s(e)h(increased)g (to)f Fp(\015)40 b Fr(on)34 b Fl(f)p Fp(Y)52 b Fl(\024)31 b Fp(\015)5 b Fl(g)236 1835 y Fr(and)44 b(to)g(1)f(on)h Fl(f)p Fp(Y)68 b(>)46 b(\015)5 b Fl(g)44 b Fr(for)f(an)m(y)h Fp(\015)52 b(<)47 b Fr(1)c(satisfying)g Fq(P)p Fr(\()p Fp(Y)68 b Fl(\024)47 b Fp(\015)5 b Fr(\))46 b Fp(>)h Fr(0,)f(b)s(ecause)g(a)236 1956 y(subsequen)m(t)38 b(limiting)30 b(pro)s(cedure)35 b Fp(\015)h Fl(!)30 b Fr(1)k(yields)h(the)f(desired)h (upp)s(er)g(b)s(ound.)49 b(There-)236 2076 y(fore)44 b(restrict)f Fp(Y)65 b Fr(to)44 b(the)g(v)-5 b(alues)44 b Fp(\015)k Fr(and)c(1)g(and,)i(with)e(0)f(replaced)h(b)m(y)h Fp(\015)5 b Fr(,)46 b(in)m(tro)s(duce)236 2196 y(the)e(random)f(v)-5 b(ariables)43 b Fp(T)1257 2211 y Fo(k)1343 2196 y Fr(and)h Fp(U)1610 2211 y Fo(k)1696 2196 y Fr(as)g(in)f(the)h(\014rst)h(part)e (of)g(the)h(pro)s(of.)76 b(No)m(w)45 b(for)236 2317 y Fp(T)293 2332 y Fo(k)r Fn(\000)p Fm(1)454 2317 y Fl(\024)28 b Fp(n)g(<)g(T)806 2332 y Fo(k)881 2317 y Fr(clearly)535 2520 y Fp(X)616 2535 y Fo(n)663 2520 y Fp(=)17 b Fr(log)f Fp(n)28 b Fl(\024)g Fp(X)1143 2535 y Fo(T)1184 2547 y Fg(k)1234 2535 y Fn(\000)p Fm(1)1345 2520 y Fp(=)33 b Fr(log)17 b Fp(T)1627 2535 y Fo(k)r Fn(\000)p Fm(1)1760 2520 y Fp(;)236 2724 y Fr(where)34 b(b)m(y)g(\(1\))535 2927 y(log)17 b Fp(T)735 2942 y Fo(k)r Fn(\000)p Fm(1)884 2927 y Fp(=)33 b Fr(log)17 b(\()p Fp(T)1204 2942 y Fo(k)1269 2927 y Fl(\000)22 b Fr(1\))28 b Fl(!)f Fr(1)50 b(a.s.)236 3130 y(Again)32 b(b)m(y)h(\(1\))f(this)h(implies)535 3334 y(lim)17 b(sup)851 3349 y Fo(n)p Fn(!1)1055 3334 y Fp(X)1136 3349 y Fo(n)1183 3334 y Fp(=)g Fr(log)f Fp(n)84 b Fl(\024)f Fr(lim)17 b(sup)2009 3349 y Fo(k)r Fn(!1)2209 3334 y Fp(X)2290 3349 y Fo(T)2331 3361 y Fg(k)2381 3349 y Fn(\000)p Fm(1)2492 3334 y Fp(=)33 b Fr(log\()p Fp(T)2795 3349 y Fo(k)2860 3334 y Fl(\000)22 b Fr(1\))1533 3537 y(=)84 b(lim)17 b(sup)2009 3552 y Fo(k)r Fn(!1)2209 3537 y Fp(X)2290 3552 y Fo(T)2331 3564 y Fg(k)2381 3552 y Fn(\000)p Fm(1)2492 3537 y Fp(=)33 b Fr(log)16 b Fp(k)s(;)236 3741 y Fr(where)535 3944 y Fp(X)616 3959 y Fo(T)657 3971 y Fg(k)707 3959 y Fn(\000)p Fm(1)829 3944 y Fr(=)933 3878 y Fj(P)1037 3959 y Fm(1)p Fn(\024)p Fo(i)p Fn(\024)p Fo(k)1265 3944 y Fp(\015)1321 3903 y Fo(k)r Fn(\000)p Fo(i)1459 3944 y Fp(U)1525 3959 y Fo(i)1728 3944 y Fr(for)49 b Fp(k)31 b Fl(2)d Fq(N)p Fp(:)236 4147 y Fr(Th)m(us)34 b(\(10.5\))e(applies)g(again,)f(with)h Fp(Z)1667 4162 y Fo(n)1747 4147 y Fr(replaced)g(b)m(y)i Fp(U)2333 4162 y Fo(n)2380 4147 y Fp(:)100 b Fh(2)383 4326 y Fr(Finally)-8 b(,)30 b(it)j(should)g(b)s(e)h(men)m(tioned)f(that)g(with)g(probabilit) m(y)e(1)i(the)h(limit)c(p)s(oin)m(ts)j(of)236 4446 y(the)g(normalized)e (sequence)k(\()p Fp(X)1426 4461 y Fo(n)1473 4446 y Fp(=)17 b Fr(log)f Fp(n)p Fr(\))1777 4461 y Fo(n>)p Fm(1)1947 4446 y Fr(exhaust)34 b(the)f(in)m(terv)-5 b(al)535 4650 y Fp(I)36 b Fr(:=)27 b([0)p Fp(;)17 b Fl(\000)p Fr(1)p Fp(=)g Fr(log)f Fq(P)p Fr(\()p Fp(Y)49 b Fr(=)27 b(1\)])p Fp(:)236 4853 y Fr(Indeed,)57 b(giv)m(en)51 b Fp(l)44 b Fl(2)e Fq(N)50 b Fr(and)h Fp(")58 b(>)g Fr(0,)d(b)m(y)d Fp(x)p 1926 4866 56 4 v 58 w(<)59 b Fl(1)50 b Fr(there)h(exists)h(a)e (random)g(time)236 4973 y Fp(T)42 b Fl(\025)28 b Fp(l)41 b Fl(_)e Fp(e)660 4937 y Fm(1)p Fo(=")800 4973 y Fr(suc)m(h)34 b(that)e Fp(X)1312 4988 y Fo(T)1367 4973 y Fp(=)17 b Fr(log)f Fp(T)42 b(<)27 b(")p Fr(,)33 b(hence)h(in)d(view)i(of)535 5177 y Fp(X)616 5192 y Fo(n)p Fm(+1)753 5177 y Fp(=)17 b Fr(log)22 b(\()p Fp(n)g Fr(+)g(1\))27 b Fl(\024)i Fr(\()p Fp(X)1522 5192 y Fo(n)1591 5177 y Fr(+)22 b(1\))p Fp(=)17 b Fr(log)e Fp(n)28 b Fl(\024)g Fp(X)2255 5192 y Fo(n)2302 5177 y Fp(=)17 b Fr(log)f Fp(n)39 b Fr(+)22 b Fp(")174 b Fr(for)50 b Fp(n)27 b Fl(\025)i Fp(T)236 5380 y Fr(the)k(v)-5 b(alues)33 b Fp(X)776 5395 y Fo(n)823 5380 y Fp(=)17 b Fr(log)f Fp(n;)33 b(n)28 b Fl(\025)g Fp(l)r(;)33 b Fr(are)g Fp(")p Fr({dense)g(in)f Fp(I)8 b Fr(.)1841 5753 y(22)p eop %%Page: 23 24 23 23 bop 236 171 a Fq(References)283 391 y Fc(1.)49 b(Alpuim,)37 b(M.:)56 b(An)37 b(extremal)h(Mark)m(o)m(vian)g(sequence.) 63 b(J.)38 b(Appl.)60 b(Prob.)i Fb(26)p Fc(,)40 b(219{232)402 504 y(\(1989\))283 622 y(3.)49 b(Barnsley)-8 b(,)31 b(M.,)h(Elton,)f (J.,)g(Hardin,)f(D.:)42 b(Recurren)m(t)31 b(iterated)g(function)f (systems.)42 b(Con-)402 735 y(str.)f(Appro)m(x.)f Fb(5)p Fc(,)31 b(3{31)h(\(1989\))283 853 y(4.)49 b(Boro)m(vk)m(o)m(v,)29 b(A.:)38 b(On)23 b(the)i(ergo)s(dicit)m(y)e(and)h(stabilit)m(y)f(of)h (the)h(sequence)f Fa(w)2895 867 y Fo(n)p Fm(+1)3058 853 y Fc(=)h Fa(f)10 b Fc(\()p Fa(w)3309 867 y Fo(n)3356 853 y Fa(;)15 b(\030)3436 867 y Fo(n)3483 853 y Fc(\).)402 966 y(Theory)30 b(Prob.)40 b(Appl.)f Fb(33)p Fc(,)31 b(595{611)j(\(1989\))283 1084 y(5.)49 b(Brandt,)31 b(A.:)43 b(The)30 b(sto)s(c)m(hastic)i(equation)f Fa(Y)1934 1098 y Fo(n)p Fm(+1)2097 1084 y Fc(=)26 b Fa(A)2262 1098 y Fo(n)2309 1084 y Fa(Y)2362 1098 y Fo(n)2429 1084 y Fc(+)21 b Fa(B)2590 1098 y Fo(n)2668 1084 y Fc(with)29 b(stationary)i(co)s (e\016-)402 1197 y(cien)m(ts.)41 b(Adv.)f(Appl.)g(Prob.)g Fb(18)p Fc(,)31 b(211{220)i(\(1986\))283 1315 y(6.)49 b(Brandt,)38 b(A.,)g(F)-8 b(rank)m(en,)38 b(P)-8 b(.,)39 b(Lisek,)e(B.:)52 b(Stationary)36 b(sto)s(c)m(hastic)h(mo)s(dels.)57 b(Chic)m(hester:)402 1428 y(Wiley)30 b(1990)283 1545 y(7.)49 b(Chama)m(y)m(ou,)36 b(J.,)f(Letac,)h(G.:)48 b(Explicit)32 b(stationary)i(distributions)c(for)k(comp)s(ositions)e (of)402 1658 y(random)i(functions)f(and)h(pro)s(ducts)f(of)i(random)e (matrices.)53 b(J.)35 b(Theor.)52 b(Prob.)h Fb(4)p Fc(,)35 b(3{36)402 1771 y(\(1991\))237 1889 y(10.)50 b(Elton,)36 b(J.:)50 b(An)35 b(ergo)s(dic)g(theorem)h(for)e(iterated)i(maps.)55 b(Ergo)s(dic)34 b(Theory)g(Dyn.)55 b(Syst.)402 2002 y Fb(7)p Fc(,)31 b(481{488)j(\(1987\))237 2120 y(11.)50 b(Elton,)28 b(J.:)40 b(A)28 b(m)m(ultiplicativ)m(e)e(ergo)s(dic)h (theorem)i(for)e(Lipsc)m(hitz)g(maps.)40 b(Sto)s(c)m(hastic)28 b(Pro-)402 2233 y(cesses)j(Appl.)39 b Fb(34)p Fc(,)32 b(39{47)g(\(1990\))237 2351 y(12.)50 b(Elton,)29 b(J.,)g(Y)-8 b(an,)30 b(Z.:)40 b(Appro)m(ximation)27 b(of)i(measures)f(b)m(y)h(Mark) m(o)m(v)h(pro)s(cesses)f(and)f(homo-)402 2464 y(geneous)j(a\016ne)g (iterated)f(function)f(systems.)41 b(Constr.)f(Appro)m(x.)g Fb(5)p Fc(,)31 b(69{87)i(\(1989\))237 2582 y(13.)50 b(Erd\177)-45 b(os,)29 b(P)-8 b(.:)40 b(On)28 b(a)i(family)d(of)i(symmetric)f (Bernoulli)f(con)m(v)m(olutions.)40 b(Am.)g(J.)29 b(Math.)41 b Fb(61)p Fc(,)402 2694 y(974{976)34 b(\(1939\))237 2812 y(15.)50 b(Garsia,)30 b(A.:)41 b(En)m(trop)m(y)30 b(and)f(singularit)m (y)f(of)h(in\014nite)f(con)m(v)m(olutions.)40 b(P)m(ac.)i(J.)29 b(Math.)42 b Fb(13)p Fc(,)402 2925 y(1159{1169)35 b(\(1963\))237 3043 y(16.)50 b(Goldie,)32 b(C.:)43 b(Implicit)29 b(renew)m(al)j (theory)g(and)f(tails)f(of)i(solutions)e(of)i(random)f(equations.)402 3156 y(Ann.)40 b(Appl.)f(Prob.)h Fb(1)p Fc(,)31 b(126{166)j(\(1991\)) 237 3274 y(18.)50 b(Grincevi)m(\024)-43 b(cius,)32 b(A.:)46 b(A)33 b(random)f(di\013erence)g(equation.)48 b(Lith.)e(Math.)i(T)-8 b(rans.)47 b Fb(21)p Fc(,)34 b(302{)402 3387 y(306)e(\(1982\))237 3505 y(19.)50 b(de)28 b(Haan,)g(L.,)h(Karandik)-5 b(ar,)26 b(R.:)40 b(Em)m(b)s(edding)25 b(a)j(sto)s(c)m(hastic)g(di\013erence)f (equation)g(in)m(to)h(a)402 3618 y(con)m(tin)m(uous{time)j(pro)s(cess.) 40 b(Sto)s(c)m(hastic)31 b(Pro)s(cesses)f(Appl.)39 b Fb(32)p Fc(,)31 b(225{235)j(\(1989\))237 3736 y(20.)50 b(Hutc)m(hinson,)35 b(J.:)48 b(F)-8 b(ractals)35 b(and)f(self)f (similarit)m(y)-8 b(.)50 b(Indiana)33 b(Univ.)51 b(Math.)i(J.)34 b Fb(30)p Fc(,)h(713{)402 3848 y(747)d(\(1981\))237 3966 y(21.)50 b(Kac,)44 b(M.:)61 b(On)39 b(the)h(notion)g(of)h(recurrence)f (in)f(discrete)g(sto)s(c)m(hastic)j(pro)s(cesses.)70 b(Bull.)402 4079 y(Am.)41 b(Math.)g(So)s(c.)g Fb(53)p Fc(,)31 b(1002{1010)j(\(1947\))237 4197 y(23.)50 b(Kesten,)34 b(H.:)45 b(The)33 b(limit)d(p)s(oin)m(ts)i(of)g(a)h(normalized)f (random)f(w)m(alk.)48 b(Ann.)e(Math.)i(Stat.)402 4310 y Fb(41)p Fc(,)31 b(1173{1205)k(\(1970\))237 4428 y(24.)50 b(Kesten,)41 b(H.:)58 b(Random)38 b(di\013erence)g(equations)g(and)g (renew)m(al)g(theory)h(for)f(pro)s(ducts)f(of)402 4541 y(random)30 b(matrices.)41 b(Acta)31 b(Math.)42 b Fb(131)p Fc(,)31 b(207{248)i(\(1973\))237 4659 y(27.)50 b(Lev,)c(G.:)65 b(Semi{Mark)m(o)m(v)44 b(pro)s(cesses)e(of)h(m)m(ultiplication)c(with)i (drift.)76 b(Theory)42 b(Prob.)402 4772 y(Appl.)d Fb(17)p Fc(,)31 b(159{164)j(\(1972\))237 4890 y(29.)50 b(Masimo)m(v,)28 b(V.:)38 b(A)26 b(generalized)g(Bernoulli)d(sc)m(heme)j(and)g(its)f (limit)e(distribution.)36 b(Theory)402 5002 y(Prob.)k(Appl.)f Fb(18)p Fc(,)32 b(521{530)h(\(1973\))237 5120 y(30.)50 b(Mukherjea,)34 b(A.:)47 b(Recurren)m(t)33 b(random)f(w)m(alk)h(in)f (nonnegativ)m(e)i(matrices:)46 b(attractors)35 b(of)402 5233 y(certain)k(iterated)h(function)d(systems.)67 b(Prob.)g(Theory)38 b(Related)h(Fields)f Fb(91)p Fc(,)k(297{306)402 5346 y(\(1992\))1841 5753 y Fr(23)p eop %%Page: 24 25 24 24 bop 237 171 a Fc(31.)50 b(Mukherjea,)36 b(A.,)g(Tserp)s(es,)e (N.:)50 b(Measures)34 b(on)h(top)s(ological)f(semigroups:)47 b(con)m(v)m(olution)402 283 y(pro)s(ducts)33 b(and)g(random)h(w)m (alks.)51 b(Lect.)i(Notes)35 b(Math.)52 b Fb(547)p Fc(,)36 b(Berlin{Heidelb)s(erg{New)402 396 y(Y)-8 b(ork:)42 b(Springer)28 b(1976)237 514 y(32.)50 b(Norman,)33 b(M.:)45 b(Limiting)30 b(distributions)f(for)j(some)h(random)e(w)m(alks)h(arising)f(in)g (learning)402 627 y(mo)s(dels.)40 b(Ann.)g(Math.)h(Stat.)g Fb(37)p Fc(,)31 b(393{405)j(\(1966\))237 745 y(34.)50 b(Rac)m(hev,)30 b(S.,)f(Samoro)s(dnitsky)-8 b(,)28 b(G.:)40 b(Limit)26 b(la)m(ws)i(for)h(a)f(sto)s(c)m(hastic)h(pro)s(cess)f(and)g (random)402 858 y(recursion)h(arising)g(in)g(probabilit)m(y)f(mo)s (delling.)38 b(Preprin)m(t)29 b(\(1992\))237 976 y(39.)50 b(V)-8 b(erv)j(aat,)35 b(W.:)46 b(On)32 b(a)h(sto)s(c)m(hastic)g (di\013erence)f(equation)h(and)f(a)h(represen)m(tation)g(of)f(non{)402 1089 y(negativ)m(e)k(in\014nitely)31 b(divisible)g(random)i(v)-5 b(ariables.)52 b(Adv.)g(Appl.)f(Prob.)h Fb(11)p Fc(,)36 b(750{783)402 1202 y(\(1979\))237 1320 y(40.)50 b(W)-8 b(olfe,)57 b(S.:)81 b(On)50 b(a)i(con)m(tin)m(uous)e(analogue)h(of)g (the)g(sto)s(c)m(hastic)h(di\013erence)e(equation)402 1433 y Fa(X)477 1447 y Fo(n)550 1433 y Fc(=)25 b Fa(\045X)768 1447 y Fo(n)p Fn(\000)p Fm(1)925 1433 y Fc(+)20 b Fa(B)1085 1447 y Fo(n)1132 1433 y Fc(.)41 b(Sto)s(c)m(hastic)31 b(Pro)s(cesses)f(Appl.)39 b Fb(12)p Fc(,)32 b(301{312)h(\(1982\))1841 5753 y Fr(24)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF