%!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: I.dvi %%Pages: 21 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips I %DVIPSParameters: dpi=300, compressed, comments removed %DVIPSSource: TeX output 1999.04.16:1726 %%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 /FMat[1 0 0 -1 0 0]N /FBB[0 0 0 0]N /nn 0 N /IE 0 N 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b(distributed)h Fp(R)645 1847 y Fl(2)665 1865 y Fq(-v)m(alued)h(random)g(v)m(ariables.) 191 1925 y(These)g(a\016ne)f(recursions)h(include)f(as)h(degenerate)f (cases)h(random)f(w)o(alks)h(\()p Fo(Y)1639 1932 y Fn(n)1677 1925 y Fq(=)d(1\))118 1985 y(and)g(in\014nite)d(pro)q(ducts)j(\()p Fo(Z)619 1992 y Fn(n)657 1985 y Fq(=)h(0\).)20 b(Of)11 b(particular)h(in)o(terest)f(are)g(the)h(\\additiv)o(e)f(mo)q(del",)118 2045 y(where)268 2147 y Fo(X)308 2154 y Fn(n)345 2147 y Fq(=)j Fo(y)r(X)463 2154 y Fn(n)p Fm(\000)p Fl(1)543 2147 y Fq(+)d Fo(Z)625 2154 y Fn(n)736 2147 y Fq(with)16 b(constan)o(t)25 b Fo(y)16 b Fk(2)e Fp(R)118 2248 y Fq(\(the)d (simplest)e(case)h(of)i(an)f(autoregressiv)o(e)g(pro)q(cess\),)h(and)f (the)g(\\m)o(ultiplic)o(ativ)o(e)c(mo)q(del")-8 b(,)118 2308 y(where)268 2410 y Fo(X)308 2417 y Fn(n)345 2410 y Fq(=)14 b Fo(Y)425 2417 y Fn(n)449 2410 y Fo(X)489 2417 y Fn(n)p Fm(\000)p Fl(1)569 2410 y Fq(+)d Fo(z)89 b Fq(with)16 b(constan)o(t)26 b Fo(z)15 b Fk(2)f Fp(R)p Fo(:)118 2511 y Fq(Both)k(mo)q(dels)f(can)h(b)q(e)g(subsumed)f(under)g (the)h(case)g(where)f Fo(Y)1293 2518 y Fn(n)1335 2511 y Fq(and)h Fo(Z)1464 2518 y Fn(n)1506 2511 y Fq(are)g(indep)q(en-)118 2571 y(den)o(t.)191 2632 y(Due)11 b(to)h(the)f(particular)f(role)h(of)h (0)f(with)g(resp)q(ect)g(to)g(m)o(ultiplicati)o(on,)f(the)h(situation)g (is)118 2692 y(esp)q(ecially)i(simple)e(for)k Fp(P)p Fq(\()p Fo(Y)25 b Fq(=)14 b(0\))g Fo(>)g Fq(0)p Fo(:)g Fq(This)g(\\regenerativ)o(e")f(case)h(can)g(b)q(e)g(studied)g(in)118 2752 y(a)h(more)f(general)g(con)o(text)g(\(see)g(e.g.)21 b(Numm)o(eli)o(n)12 b([33]\).)21 b(Another)14 b(particular)g(situation) 933 2877 y(1)p eop %%Page: 2 3 2 2 bop 118 77 a Fq(is)16 b(pro)o(vided)g(b)o(y)g(the)g(\\con)o (tractiv)o(e")f(case,)h(where)g(in)g(the)g(w)o(eak)g(v)o(ersion)g Fk(j)p Fo(Y)1548 84 y Fn(n)1572 77 y Fk(j)e(\024)f Fq(1)k(and)118 137 y(in)f(the)g(strong)h(v)o(ersion)f Fk(j)p Fo(Y)616 144 y Fn(n)640 137 y Fk(j)d(\024)h Fo(#)f(<)h Fq(1.)191 197 y(T)l(o)i(conclude)e(the)h(historical)f(remarks,)f(t)o(w)o(o)i(cen) o(tral)f(results)h(concerning)g(the)f(gen-)118 258 y(eral)i(case)g(ha)o (v)o(e)g(to)g(b)q(e)h(men)o(tioned:)118 318 y(|)25 b(If)12 b(an)i(a\016ne)f(recursion)g(has)g(a)h(stationary)g(distribution,)e(it) h(is)g(unique)f(and)i(the)f(la)o(ws)118 378 y Fk(L)p Fq(\()p Fo(X)211 385 y Fn(n)235 378 y Fq(\))k(con)o(v)o(erge)e(w)o (eakly)g(to)i(it,)e(indep)q(enden)o(tly)g(of)h(the)g(initial)f(la)o(w)h Fk(L)p Fq(\()p Fo(X)1530 385 y Fl(0)1551 378 y Fq(\).)118 438 y(|)25 b(Conditions)16 b(on)g(the)f(existence)f(of)i(a)g (stationary)h(distribution)e(can)h(b)q(e)f(form)o(ulated)118 498 y(via)h(logarithmic)e(momen)o(ts)g(of)j Fo(Y)747 505 y Fn(n)787 498 y Fq(and)f Fo(Z)914 505 y Fn(n)938 498 y Fq(.)191 559 y(The)k(presen)o(t)f(w)o(ork)g(originates)h(in)f (the)g(observ)m(ation)h(that)g(most)f(applications)g(in)118 619 y(economics,)c(biology)l(,)i(ph)o(ysics)f(etc.)23 b(\(see)16 b(the)h(long)h(list)e(of)h(references)f(in)g([39]\))h(in)g (fact)118 679 y(w)o(ork)i(in)g(the)h(state)f(space)h Fp(R)686 686 y Fl(+)715 679 y Fq(,)g(and)f(th)o(us)h(the)f(case)g Fo(X)1190 686 y Fl(0)1230 679 y Fk(\025)f Fq(0)i(and)g Fo(Y)1457 686 y Fn(n)1481 679 y Fo(;)8 b(Z)1536 686 y Fn(n)1579 679 y Fk(\025)18 b Fq(0)i(is)f(of)118 739 y(sp)q(ecial)h(imp) q(ortance.)31 b(F)l(rom)19 b(the)g(mathematical)e(p)q(oin)o(t)j(of)h (view)e(this)h(restriction)f(is)118 799 y(supp)q(orted)c(b)o(y)e(the)g (fact)h(that)g(a)g(state)g(space)g Fp(R)1013 806 y Fl(+)1056 799 y Fq(allo)o(ws)f(only)h(one)f(kind)h(of)f(div)o(ergence)118 859 y(to)25 b(in\014nit)o(y)d(and)j(the)f(assumption)f Fo(Y)849 866 y Fn(n)873 859 y Fo(;)8 b(Z)928 866 y Fn(n)979 859 y Fk(\025)27 b Fq(0)d(en)o(tails)f(additional)i(monotonicit)o(y)118 920 y(prop)q(erties)16 b(of)h(the)f(asso)q(ciated)h(transition)g(k)o (ernel.)191 980 y(In)g(the)g(existing)f(literature)g(these)g(asp)q (ects)i(do)f(not)h(\014nd)f(m)o(uc)o(h)e(atten)o(tion.)23 b(Apart)118 1040 y(from)g(an)h(approac)o(h)g(b)o(y)f(Lamp)q(erti)g ([25],)i([26],)g(to)q(o)g(general)e(for)h(a\016ne)f(recursions,)118 1100 y(there)f(are)g(only)g(t)o(w)o(o)h(exceptions:)32 b(a)23 b(recen)o(t)e(pap)q(er)i(b)o(y)f(Mukherjea)f([30],)j(coupling) 118 1160 y(in)17 b(the)g(con)o(text)f(of)h(nonnegativ)o(e)g(matrices)e (the)i(sequence)f(\()p Fo(X)1306 1167 y Fn(n)1330 1160 y Fq(\))1349 1167 y Fn(n)p Fm(\025)p Fl(0)1434 1160 y Fq(with)h(the)g(partial)118 1221 y(pro)q(ducts)h(of)f(\()p Fo(Y)424 1228 y Fn(n)448 1221 y Fq(\))467 1229 y Fn(n)p Fm(2)p Fj(N)549 1221 y Fq(,)f(and)i(a)f(preprin)o(t)f(b)o(y)h(Rac)o (hev)f([34],)h(concen)o(trating)f(on)i(cen)o(tral)118 1281 y(limit)c(theorems)h(for)h(suitably)g(normalized)e(v)m(ariables)i Fo(X)1195 1288 y Fn(n)1236 1281 y Fq(in)g(the)g(div)o(ergen)o(t)e (case.)191 1341 y(T)l(o)e(summarize)c(the)j(main)f(feature)h(of)g(the)g (presen)o(t)g(w)o(ork)g(b)q(efore)g(going)h(in)o(to)f(details:)118 1401 y(a\016ne)23 b(recursions)g(on)g Fp(R)606 1408 y Fl(+)659 1401 y Fq(seem)e(to)i(pro)o(vide)f(one)h(of)h(the)f(b)q(est)g (suited)f(mo)q(dels)g(for)118 1461 y(extending)16 b(classical)f(Mark)o (o)o(v)h(c)o(hain)g(theory)g(to)h(an)f(uncoun)o(table)g(state)h(space.) 22 b(Since)118 1522 y(the)d(Harris)f(theory)g(\(see)h(e.g.)28 b([35]\))18 b(is)h(easily)f(seen)g(not)h(to)g(b)q(e)g(adequate,)g(the)g (study)118 1582 y(has)d(to)f(b)q(e)f(based)i(on)f(the)f(top)q(ological) i(structure.)k(This,)15 b(in)f(general,)g(leads)h(to)g(v)m(arious)118 1642 y(notions)k(of)g(irreducibilit)o(y)c(and)j(ap)q(erio)q(dicit)o(y)l (,)g(of)g(\(p)q(ositiv)o(e)g(or)g(n)o(ull\))f(recurrence)g(and)118 1702 y(transience)c(\(see)f(the)h(pap)q(ers)h(b)o(y)f(Rosen)o(blatt)g ([37])g(and)h(Tw)o(eedie)e([38]\).)20 b(In)12 b(the)h(presen)o(t)118 1762 y(case,)j(ho)o(w)o(ev)o(er,)e(these)i(notions)h(merge)d(in)o(to)i (v)o(ery)f(natural)i(de\014nitions)f(satisfying)g(the)118 1823 y(classical)g(criteria.)191 1883 y(This)h(allo)o(ws)f(for)g(a)h (rather)f(complete)e(theory)l(,)i(dev)o(elop)q(ed)f(in)h(the)g (sections:)142 1984 y(1.)22 b(Lo)o(w)o(er)16 b(and)h(upp)q(er)g(limit) 142 2045 y(2.)22 b(Recurrence)15 b(and)i(transience)142 2105 y(3.)22 b(Recurrence)15 b(criteria)142 2165 y(4.)22 b(Excessiv)o(e)15 b(and)i(in)o(v)m(arian)o(t)e(measures)142 2225 y(5.)22 b(Existence)15 b(and)i(uniqueness)f(of)g(in)o(v)m(arian)o (t)g(measures)142 2285 y(6.)22 b(Main)16 b(prop)q(erties)g(of)h(the)f (in)o(v)m(arian)o(t)f(measure)142 2346 y(7.)22 b(Ratio)17 b(ergo)q(dic)f(theorems)142 2406 y(8.)22 b(P)o(ositiv)o(e)15 b(and)i(n)o(ull)e(recurrence)142 2466 y(9.)22 b(F)l(urther)16 b(ergo)q(dic)g(theorems)118 2526 y(10.)22 b(The)16 b(con)o(tractiv)o(e) f(case)191 2628 y(Th)o(us)30 b(the)f(pap)q(er)h(divides)e(in)o(to)h (three)f(parts:)48 b(Sections)29 b(1{3)h(classify)f(a\016ne)118 2688 y(recursions)20 b(on)g Fp(R)463 2695 y Fl(+)512 2688 y Fq(according)f(to)h(recurrence)e(and)i(transience,)g(Sections)f (4{7)i(treat)118 2748 y(existence)e(and)i(uniqueness)f(of)h(in)o(v)m (arian)o(t)e(measures)h(as)h(w)o(ell)e(as)i(ergo)q(dic)f(theorems,)933 2877 y(2)p eop %%Page: 3 4 3 3 bop 118 77 a Fq(Sections)21 b(8{10)h(con)o(tin)o(ue)e(to)h (classify)f(the)g(recurren)o(t)g(case)g(b)o(y)h(in)o(tro)q(ducing)f (the)h(no-)118 137 y(tions)16 b(of)g(p)q(ositiv)o(e)f(recurrence)f(and) i(n)o(ull)f(recurrence.)k(P)o(art)d(I)q(I)f(and)i(I)q(I)q(I)e(will)f (app)q(ear)j(in)118 197 y Fo(:)8 b(:)g(:)32 b Fq(and)17 b Fo(:)8 b(:)g(:)15 b Fq(;)h(the)g(con)o(ten)o(ts)g(of)g(P)o(art)h(I)f (are)g(summarized)d(b)q(elo)o(w.)191 287 y Fp(Section)21 b(1.)30 b Fq(Since)18 b(con)o(v)o(ergence)g(of)h(an)h(a\016ne)e (recursion,)h(ev)o(en)f(in)h(probabilit)o(y)l(,)118 347 y(o)q(ccurs)g(only)e(in)h(an)h(exceptional)e(case)h(\(1.4\),)g(the)g (lo)o(w)o(er)f(and)i(upp)q(er)f(limit)d(are)j(of)h(in-)118 407 y(terest.)34 b(It)20 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Fq(Clearly)l(,)11 b(\()p Fo(X)680 1265 y Fn(n)704 1258 y Fq(\))723 1265 y Fn(n)p Fm(\025)p Fl(0)804 1258 y Fq(has)h(to)g(b)q(e)f(called)g(\\transien)o (t")h(in)f(the)g(case)h Fo(x)p 1615 1265 V 14 w Fq(=)h Fk(1)p Fq(.)118 1319 y(It)k(is)f(less)h(clear)f({)i(and)f(one)g(of)g (the)g(cen)o(tral)f(questions)h(in)g(the)f(sequel)g({,)h(whether)g(the) 118 1379 y(sequence)i(ma)o(y)f(b)q(e)h(called)g(\\recurren)o(t")g(in)h (the)f(case)h Fo(x)p 1161 1386 V 19 w(<)f Fk(1)p Fq(.)32 b(A)19 b(\014rst)h(justi\014cation)118 1439 y(is)c(supplied)f(b)o(y)h (an)g(equiv)m(alen)o(t)f(c)o(haracterization)g(through)i(the)f(asso)q (ciated)h(p)q(oten)o(tial)118 1499 y(k)o(ernel:)h(the)12 b(mean)g(time)e(sp)q(en)o(t)j(in)g(a)g(b)q(ounded)g(in)o(terv)m(al)f ([0)p Fo(;)c(t)p Fq(])k(is)g(\014nite)g(in)h(the)f(transien)o(t)118 1559 y(case)20 b(and)g(in\014nite)f(in)h(the)f(recurren)o(t)g(case,)h (pro)o(vided)f Fo(t)h(>)f(x)p 1274 1566 V 40 w Fq(\(2.2\).)32 b(Examples)18 b(for)118 1620 y(b)q(oth)13 b(situations)g(are)g(easily)e (established:)19 b(the)12 b(a\016ne)h(recursion)f(is)g(certainly)f (recurren)o(t)118 1680 y(in)21 b(the)g(regenerativ)o(e)f(case)h Fp(P)p Fq(\()p Fo(Y)741 1687 y Fn(n)788 1680 y Fq(=)h(0\))h Fo(>)f Fq(0)g(\(2.3\),)g(and)g(it)f(is)g(transien)o(t)g(whenev)o(er)118 1740 y(the)e(asso)q(ciated)h(random)f(w)o(alk)f(\()p Fo(S)787 1747 y Fn(n)811 1740 y Fq(\))830 1747 y Fn(n)p Fm(\025)p Fl(0)917 1740 y Fq(with)h(incremen)o(ts)d(log)k Fo(Y)1389 1747 y Fn(n)1431 1740 y Fq(div)o(erges)f(to)g(+)p Fk(1)118 1800 y Fq(\(2.4\).)j(Though)17 b(b)q(oth)g(conditions)f(dep)q (end)g(on)g(the)g(\\primary")f(v)m(ariable)h Fo(Y)1548 1807 y Fn(n)1588 1800 y Fq(only)l(,)f(the)118 1860 y(\\secondary")h(v)m (ariable)e Fo(Z)606 1867 y Fn(n)644 1860 y Fq(ma)o(y)f(b)q(e)i(essen)o (tial)e(as)i(w)o(ell.)20 b(In)14 b(fact,)g(ho)o(w)o(ev)o(er)f(small)g Fo(Y)1687 1867 y Fn(n)1725 1860 y Fq(is,)118 1921 y(a)k(su\016cien)o (tly)d(large)i Fo(Z)555 1928 y Fn(n)595 1921 y Fq(yields)f(transience)h (\(2.5\).)191 2010 y Fp(Section)23 b(3.)34 b Fq(In)20 b(the)g(additiv)o(e)g(mo)q(del)f(a)i(nearly)f(complete)e(c)o (haracterization)h(of)118 2070 y(recurrence)i(or)h(transience)f(b)o(y)h (the)g(asymptotic)e(b)q(eha)o(viour)i(of)h Fo(t)8 b Fp(P)p Fq(\(log)h Fo(Z)1562 2077 y Fn(n)1601 2070 y Fo(>)16 b(t)p Fq(\))21 b(for)118 2130 y Fo(t)16 b Fk(!)g(1)i Fq(can)g(b)q(e)g(deriv)o(ed,)e(whic)o(h)h(extends)g(to)h(the)g(case)g (where)f Fo(Y)1373 2137 y Fn(n)1415 2130 y Fq(is)g(b)q(ounded)i(a)o(w)o (a)o(y)118 2190 y(from)f(1)i(or)f(0)h(\(3.1\).)31 b(The)19 b(situation)g(is)g(less)g(clear)g(in)g(the)g(m)o(ultipli)o(cativ)n(e)d (mo)q(del.)29 b(It)118 2251 y(is)20 b(not)g(surprising)f(that)h(in)g (example)d(\(E\))j(ab)q(o)o(v)o(e)f Fo(p)1109 2258 y Fm(\000)1159 2251 y Fo(<)h(p)1241 2258 y Fl(+)1290 2251 y Fq(implies)d(transience)i(and)118 2311 y Fo(p)142 2318 y Fm(\000)186 2311 y Fo(>)14 b(p)262 2318 y Fl(+)308 2311 y Fq(implies)f(recurrence.)20 b(It)c(is)f(a)i(non)o(trivial)e (problem,)f(ho)o(w)o(ev)o(er,)h(to)h(decide)f(the)118 2371 y(balanced)f(case)h Fo(p)445 2378 y Fm(\000)488 2371 y Fq(=)f Fo(p)564 2378 y Fl(+)594 2371 y Fq(.)21 b(Clearly)13 b(the)h(related)g(m)o(ultipli)o(cativ)n(e)d(random)j(w)o (alk,)g(where)268 2471 y Fo(X)308 2478 y Fl(0)342 2471 y Fq(=)f(1)88 b(and)g Fo(X)711 2478 y Fn(n)748 2471 y Fq(=)14 b Fo(Y)828 2478 y Fn(n)852 2471 y Fo(X)892 2478 y Fn(n)p Fm(\000)p Fl(1)986 2471 y Fq(for)25 b Fo(n)14 b Fk(2)g Fp(N)p Fo(;)118 2571 y Fq(oscillates)g(through)j(the)d(v)m (alues)h(2)757 2553 y Fn(k)779 2571 y Fo(;)8 b(k)16 b Fk(2)e Fp(Z)p Fq(.)21 b(But)15 b(it)f(requires)g(Spitzer's)g(com)o (binatorial)118 2632 y(iden)o(tit)o(y)i(to)j(pro)o(v)o(e)e(the)h(drift) g(term)e(+1)j(not)f(to)h(c)o(hange)f(recurrence)f(in)o(to)h (transience.)118 2692 y(The)h(relev)m(an)o(t)f(recurrence)g(criterion)g (settles)g(the)h(m)o(ultiplicativ)n(e)d(mo)q(del)i(completely)118 2752 y(and)k(extends)e(in)g(fact)h(to)g(the)g(case)g(where)f Fo(Z)1001 2759 y Fn(n)1046 2752 y Fq(is)g(only)h(b)q(ounded)h(\(3.3\).) 35 b(Assuming)933 2877 y(3)p eop %%Page: 4 5 4 4 bop 118 77 a Fo(Y)146 84 y Fn(n)186 77 y Fq(and)16 b Fo(Z)313 84 y Fn(n)352 77 y Fq(to)g(b)q(e)g(indep)q(enden)o(t,)f(ev)o (en)f(\014niteness)i(of)f(the)h(exp)q(ectation)f(of)h Fo(Z)1577 84 y Fn(n)1617 77 y Fq(ensures)118 137 y(recurrence)f (\(3.4\).)118 314 y Fp(0.)25 b(Preliminarie)o(s)118 432 y Fq(Throughout)18 b(the)e(pap)q(er)g(\()p Fo(X)668 439 y Fn(n)692 432 y Fq(\))711 439 y Fn(n)p Fm(\025)p Fl(0)796 432 y Fq(is)f(a)i(\014xed)e(a\016ne)h(recursion)g(on)g Fp(R)1450 439 y Fl(+)1480 432 y Fq(,)f(giv)o(en)g(b)o(y)h(the)118 492 y(sto)q(c)o(hastic)f(di\013erence)e(equation)h(\()p Fk(\003)p Fq(\))g(of)h(the)f(in)o(tro)q(duction.)20 b(Th)o(us)15 b(the)f(distribution)f(of)118 552 y(\()p Fo(X)177 559 y Fn(n)201 552 y Fq(\))220 559 y Fn(n)p Fm(\025)p Fl(0)307 552 y Fq(is)18 b(completely)d(determined)h(b)o(y)i(the)g(la)o(ws)g Fo(\026)1156 559 y Fl(0)1193 552 y Fq(=)g Fk(L)p Fq(\()p Fo(X)1342 559 y Fl(0)1362 552 y Fq(\))g(and)h Fo(\027)i Fq(=)c Fk(L)p Fq(\()p Fo(Y)s(;)8 b(Z)t Fq(\).)118 612 y(Here,)k Fo(Y)24 b Fq(and)13 b Fo(Z)k Fq(is)12 b(brie\015y)g(written)g (instead)h(of)g Fo(Y)1041 619 y Fn(n)1078 612 y Fq(and)g Fo(Z)1202 619 y Fn(n)1226 612 y Fq(,)g(as)g(will)f(b)q(e)h(done)g (whenev)o(er)118 673 y Fo(n)h Fk(2)g Fp(N)i Fq(pla)o(ys)g(no)h(role.) 191 733 y(The)k(initial)e(la)o(w)i Fo(\026)563 740 y Fl(0)583 733 y Fq(,)h(as)f(usual,)h(is)e(largely)g(of)h(only)g (secondary)g(signi\014cance.)34 b(If)118 793 y(in)19 b(particular)g Fo(\026)435 800 y Fl(0)475 793 y Fq(is)g(a)h(unit)f (measure)g Fo(")894 800 y Fn(x)915 793 y Fq(,)h(this)g(will)e(b)q(e)i (expressed)f(b)o(y)g(the)g(notation)118 853 y(\()p Fo(X)181 835 y Fn(x)177 866 y(n)203 853 y Fq(\))222 860 y Fn(n)p Fm(\025)p Fl(0)291 853 y Fq(,)d(i.e.)268 951 y Fo(X)312 930 y Fn(x)308 963 y(n)348 951 y Fq(:=)d Fo(xY)469 958 y Fl(1)489 951 y Fo(Y)517 958 y Fl(2)545 951 y Fo(:)8 b(:)g(:)g(Y)639 958 y Fn(n)674 951 y Fq(+)j Fo(Z)756 958 y Fl(1)776 951 y Fo(Y)804 958 y Fl(2)832 951 y Fo(:)d(:)g(:)f(Y)925 958 y Fn(n)960 951 y Fq(+)k Fo(:)d(:)g(:)j Fq(+)g Fo(Z)1160 958 y Fn(n)1271 951 y Fq(for)25 b Fo(x)13 b Fk(2)h Fp(R)1484 958 y Fl(+)1539 951 y Fq(and)25 b Fo(n)14 b Fk(\025)g Fq(0)p Fo(:)118 1048 y Fq(Th)o(us)j(conditional)f(probabilit)o(y)f(and) i(exp)q(ectation)f(are)g(simply)e(giv)o(en)h(b)o(y)268 1145 y Fp(P)306 1125 y Fn(x)328 1145 y Fq(\(\()p Fo(X)406 1152 y Fn(n)430 1145 y Fo(;)8 b(n)13 b Fk(\025)h Fq(0\))g Fk(2)g Fo(B)s Fq(\))g(=)f Fp(P)p Fq(\(\()p Fo(X)895 1125 y Fn(x)891 1157 y(n)918 1145 y Fo(;)8 b(n)14 b Fk(\025)f Fq(0\))i Fk(2)f Fo(B)s Fq(\))p Fo(;)268 1253 y Fp(E)305 1233 y Fn(x)326 1253 y Fq(\()p Fo(g)r Fq(\()p Fo(X)429 1260 y Fn(n)453 1253 y Fo(;)8 b(n)14 b Fk(\025)g Fq(0\)\))g(=)g Fp(E)p Fq(\()p Fo(g)r Fq(\()p Fo(X)843 1233 y Fn(x)839 1266 y(n)865 1253 y Fo(;)8 b(n)14 b Fk(\025)f Fq(0\)\))p Fo(:)191 1362 y Fq(Roughly)j(sp)q(eaking,)f(what)h(follo)o(ws)g(is)f(a) g(theory)g(of)h(distributions)f Fo(\027)k Fq(on)d Fp(R)1609 1344 y Fl(2)1609 1374 y(+)1638 1362 y Fq(.)21 b(Here)118 1422 y(an)15 b(essen)o(tial)f(role)g(will)g(b)q(e)g(pla)o(y)o(ed)g(b)o (y)g(their)g(supp)q(ort,)h(for)g(whic)o(h)f(the)g(notation)i Fo(N)k Fq(will)118 1482 y(b)q(e)d(\014xed.)24 b(Since)16 b(nothing)i(new)f(can)g(b)q(e)h(exp)q(ected)e(in)h(the)g(sp)q(ecial)f (cases)i Fo(Y)26 b Fq(=)16 b(1)h(resp.)118 1542 y Fo(Z)h Fq(=)c(0,)i(for)g(simpli\014cation)268 1640 y Fo(N)g Fk(\\)11 b(f)p Fq(\()p Fo(y)r(;)d(z)r Fq(\))13 b(:)h Fo(y)h Fk(6)p Fq(=)f(1)p Fk(g)g(6)p Fq(=)g Fk(;)f(6)p Fq(=)h Fo(N)i Fk(\\)c(f)p Fq(\()p Fo(y)r(;)c(z)r Fq(\))13 b(:)g Fo(z)j Fk(6)p Fq(=)e(0)p Fk(g)118 1737 y Fq(is)i(alw)o(a)o(ys)g (assumed.)21 b(The)c(sym)o(b)q(ol)e Fk(N)23 b Fq(will)16 b(throughout)h(refer)f(to)h(the)f(class)g(of)h(distri-)118 1797 y(butions)g Fo(\027)i Fq(on)e Fp(R)445 1779 y Fl(2)445 1809 y(+)491 1797 y Fq(that)f(are)h(admissible)d(in)i(this)g(sense.)191 1857 y(As)g(is)f(clear)g(from)f(the)i(in)o(tro)q(duction,)f(the)g(ergo) q(dic)h(b)q(eha)o(viour)f(of)h(\()p Fo(X)1508 1864 y Fn(n)1532 1857 y Fq(\))1551 1864 y Fn(n)p Fm(\025)p Fl(0)1635 1857 y Fq(is)g(in)o(ti-)118 1918 y(mately)e(related)i(to)h(the)f (random)f(w)o(alk)268 2015 y Fo(S)298 2022 y Fn(n)335 2015 y Fq(:=)400 1982 y Fi(P)452 2022 y Fl(1)p Fm(\024)p Fn(m)p Fm(\024)p Fn(n)588 2015 y Fq(log)9 b Fo(Y)687 2022 y Fn(m)808 2015 y Fq(for)25 b Fo(n)14 b Fk(\025)g Fq(0)p Fo(;)118 2112 y Fq(generalized)g(in)i(the)f(sense)h(that)g(it)f (ma)o(y)f(attain)i(the)f(\(absorbing\))i(v)m(alue)e Fk(\0001)p Fq(.)21 b(Due)15 b(to)118 2172 y Fp(P)p Fq(\()p Fo(Y)28 b Fq(=)16 b(1\))h Fo(<)f Fq(1)i(there)f(are)h(only)f(three)g(p)q (ossibilities)g(for)h(the)f(asymptotic)g(b)q(eha)o(viour)118 2233 y(of)g(this)f(random)g(w)o(alk:)118 2330 y(\(1\))88 b Fo(S)298 2337 y Fn(n)335 2330 y Fk(!)14 b Fq(+)p Fk(1)49 b Fq(a.s.)p Fo(;)118 2438 y Fq(\(2\))88 b Fo(S)298 2445 y Fn(n)335 2438 y Fk(!)14 b(\0061)49 b Fq(a.s.)p Fo(;)118 2546 y Fq(\(3\))88 b Fo(S)298 2553 y Fn(n)335 2546 y Fk(!)14 b(\0001)49 b Fq(a.s.)p Fo(;)118 2655 y Fq(where)16 b(the)g(sym)o(b)q(ol)f(in)h(\(2\))g(serv)o(es)g(as)h(a)f(short)h (notation)g(for)268 2752 y Fp(P)p Fq(\(lim)8 b(inf)468 2759 y Fn(n)p Fm(!1)571 2752 y Fo(S)601 2759 y Fn(n)638 2752 y Fq(=)14 b Fk(\0001)p Fo(;)h Fq(lim)8 b(sup)966 2759 y Fn(n)p Fm(!1)1068 2752 y Fo(S)1098 2759 y Fn(n)1136 2752 y Fq(=)14 b(+)p Fk(1)p Fq(\))f(=)h(1)p Fo(:)933 2877 y Fq(4)p eop %%Page: 5 6 5 5 bop 191 77 a Fq(Another)16 b(pro)q(cess)h(closely)e(related)h(to)g (\()p Fo(X)989 84 y Fn(n)1013 77 y Fq(\))1032 84 y Fn(n)p Fm(\025)p Fl(0)1117 77 y Fq(arises)g(if,)f(in)h(the)g(notation)h(at)g (the)118 137 y(b)q(eginning)12 b(of)g(the)f(in)o(tro)q(duction,)h(the)f (random)g(v)m(ariables)h Fo(g)1230 144 y Fn(n)1255 137 y Fk(\016)q Fo(:)c(:)g(:)q Fk(\016)q Fo(g)1389 144 y Fl(1)1409 137 y Fq(\()p Fo(X)1468 144 y Fl(0)1488 137 y Fq(\))k(are)g(replaced)118 197 y(b)o(y)k Fo(g)209 204 y Fl(1)240 197 y Fk(\016)11 b Fo(:)d(:)g(:)i Fk(\016)h Fo(g)403 204 y Fn(n)427 197 y Fq(\()p Fo(X)486 204 y Fl(0)506 197 y Fq(\).)21 b(Esp)q(ecially)16 b(for)g Fo(X)906 204 y Fl(0)940 197 y Fq(=)e(0)j(this)f(yields)268 295 y Fo(W)314 302 y Fn(n)351 295 y Fq(:=)d Fo(Z)449 302 y Fl(1)480 295 y Fq(+)e Fo(Y)557 302 y Fl(1)577 295 y Fo(Z)610 302 y Fl(2)642 295 y Fq(+)g Fo(:)d(:)g(:)i Fq(+)h Fo(Y)836 302 y Fl(1)864 295 y Fo(:)d(:)g(:)g(Y)958 302 y Fn(n)p Fm(\000)p Fl(1)1027 295 y Fo(Z)1060 302 y Fn(n)1171 295 y Fq(for)25 b Fo(n)14 b Fk(\025)f Fq(0)p Fo(:)118 392 y Fq(The)22 b(sequence)f(\()p Fo(W)497 399 y Fn(n)521 392 y Fq(\))540 399 y Fn(n)p Fm(\025)p Fl(0)631 392 y Fq(is)h(no)g(longer)h(a)f(Mark)o(o)o(v)g(pro)q(cess,)h(ho)o(w)o(ev)o (er,)f(due)g(to)h(the)118 452 y(exc)o(hangeabilit)o(y)14 b(of)j(\()p Fo(Y)565 459 y Fl(1)585 452 y Fo(;)8 b(Z)640 459 y Fl(1)660 452 y Fq(\))p Fo(;)g(:)g(:)g(:)f(;)h Fq(\()p Fo(Y)835 459 y Fn(n)859 452 y Fo(;)g(Z)914 459 y Fn(n)937 452 y Fq(\),)16 b(satis\014es)268 550 y Fk(L)p Fq(\()p Fo(W)367 557 y Fn(n)390 550 y Fq(\))e(=)g Fk(L)p Fq(\()p Fo(X)572 529 y Fl(0)568 562 y Fn(n)592 550 y Fq(\))88 b(for)25 b Fo(n)14 b Fk(\025)f Fq(0)p Fo(;)118 647 y Fq(an)k(equation,)f(whic)o(h)f(will)g(b)q(e)i(imp)q(ortan)o(t)e(later)h (on.)191 707 y(The)d(transition)f(k)o(ernel)f(of)i(the)f(Mark)o(o)o(v)g (pro)q(cess)h(\()p Fo(X)1177 714 y Fn(n)1201 707 y Fq(\))1220 714 y Fn(n)p Fm(\025)p Fl(0)1301 707 y Fq(will)e(alw)o(a)o(ys)h(b)q(e)h (denoted)118 767 y(b)o(y)p Fo(P)7 b Fq(.)20 b(Th)o(us)15 b(the)e(k)o(ernel)g Fo(P)21 b Fq(transforms)13 b(a)i(nonnegativ)o(e)f (function)f Fo(f)20 b Fq(in)o(to)13 b(the)h(function)118 828 y Fo(P)7 b(f)22 b Fq(giv)o(en)15 b(b)o(y)268 925 y(\()p Fo(P)7 b(f)e Fq(\)\()p Fo(x)p Fq(\))14 b(=)504 890 y Fi(R)540 925 y Fo(f)5 b Fq(\()p Fo(y)r(x)11 b Fq(+)g Fo(z)r Fq(\))d Fo(\027)s Fq(\()p Fo(dy)r(;)g(dz)r Fq(\))118 1022 y(and)16 b(a)f(measure)f Fo(\026)h Fq(on)g(the)g(Borel)f Fo(\033)r Fq({algebra)h Fk(B)r Fq(\()p Fp(R)1080 1029 y Fl(+)1109 1022 y Fq(\))g(in)o(to)f(the)h(measure)f Fo(\026P)22 b Fq(giv)o(en)14 b(b)o(y)268 1120 y(\()p Fo(\026P)7 b Fq(\)\()p Fo(B)s Fq(\))13 b(=)516 1084 y Fi(R)552 1120 y Fo(\027)s Fq(\()p Fk(f)p Fq(\()p Fo(y)r(;)8 b(z)r Fq(\))13 b(:)h Fo(y)r(x)c Fq(+)h Fo(z)16 b Fk(2)e Fo(B)s Fk(g)p Fq(\))8 b Fo(\026)p Fq(\()p Fo(dx)p Fq(\))p Fo(:)118 1217 y Fq(If)16 b Fo(\026)g Fq(is)h Fo(\033)r Fq({\014nite,)e(the)h(last)g(equation)g(amoun)o(ts)g(to)268 1315 y(\()p Fo(\026P)7 b Fq(\)\()p Fo(B)s Fq(\))13 b(=)h(\()p Fo(\026)e Fk(\012)e Fo(\027)s Fq(\)\()p Fk(f)p Fq(\()p Fo(x)p Fq(;)e Fo(y)r(;)g(z)r Fq(\))13 b(:)g Fo(y)r(x)e Fq(+)g Fo(z)16 b Fk(2)e Fo(B)s Fk(g)p Fq(\))p Fo(:)118 1412 y Fq(In)19 b(accordance)g(with)h(the)f(notations)h Fo(P)7 b(f)25 b Fq(and)20 b Fo(\026P)27 b Fq(the)19 b Fo(\026)p Fq({in)o(tegral)g(of)h(a)f(function)g Fo(f)118 1472 y Fq(sometimes)13 b(is)j(simply)f(denoted)h(b)o(y)g Fo(\026f)5 b Fq(.)191 1532 y(The)19 b(k)o(ernel)e Fo(P)26 b Fq(enjo)o(ys)18 b(t)o(w)o(o)h(imp)q(ortan)o(t)e(prop)q(erties.)29 b(First)18 b(it)g(is)g(clearly)f(a)i(F)l(eller)118 1592 y(k)o(ernel,)14 b(transforming)h(b)q(ounded)h(con)o(tin)o(uous)g (functions)f(in)o(to)g(the)h(same)e(t)o(yp)q(e.)21 b(More-)118 1653 y(o)o(v)o(er,)e(due)g(to)h Fo(Y)31 b Fk(\025)19 b Fq(0,)h(it)f(is)g(monotone)h(in)f(the)g(sense)g(that)h(it)f (transforms)h(b)q(ounded)118 1713 y(increasing)c(functions)g(in)o(to)g (the)g(same)f(t)o(yp)q(e,)h(to)q(o.)191 1773 y(If)i Fo(E)k Fq(is)d(a)g(lo)q(cally)f(compact)f(space)i(with)g(a)g(coun)o(table)f (base,)h(the)g(follo)o(wing)f(con-)118 1833 y(cepts)e(from)f(top)q (ological)i(measure)e(theory)h(will)f(b)q(e)i(used:)118 1893 y(|)25 b Fk(C)s Fq(\()p Fo(E)s Fq(\))18 b(denotes)h(the)f(space)g (of)h(b)q(ounded)g(con)o(tin)o(uous)g(functions)f Fo(f)23 b Fq(:)17 b Fo(E)k Fk(!)c Fp(R)h Fq(and)118 1954 y Fk(K)q Fq(\()p Fo(E)s Fq(\))e(the)g(subspace)h(consisting)f(of)h(functions)f Fo(f)j Fk(2)14 b(C)s Fq(\()p Fo(E)s Fq(\))j(with)f(compact)f(supp)q (ort.)118 2014 y(|)25 b(If)20 b Fo(\026)i Fq(is)e(an)o(y)h(measure)f (on)h Fk(B)r Fq(\()p Fo(E)s Fq(\),)g(then)g Fk(C)1002 2021 y Fn(\026)1026 2014 y Fq(\()p Fo(E)s Fq(\))g(denotes)g(the)f (space)h(of)h(b)q(ounded)118 2074 y(Borel{measurable)15 b(functions)i Fo(f)j Fq(:)14 b Fo(E)k Fk(!)c Fp(R)j Fq(that)g(are)g Fo(\026)p Fq({almost)f(ev)o(erywhere)f(con)o(tin-)118 2134 y(uous)i(and)g Fk(K)365 2141 y Fn(\026)388 2134 y Fq(\()p Fo(E)s Fq(\))g(the)f(corresp)q(onding)h(subspace.)118 2194 y(|)25 b(The)15 b(class)g Fk(M)p Fq(\()p Fo(E)s Fq(\))f(of)i(lo)q(cally)e(\014nite)g(measures)g(on)h Fo(E)j Fq(is)d(endo)o(w)o(ed)g(with)f(the)h(v)m(ague)118 2255 y(\(w)o(eak*\))h(top)q(ology)l(,)h(i.e.)j(the)c(initial)f(top)q (ology)j(with)e(resp)q(ect)g(to)g(the)g(mappings)268 2340 y Fo(\026)e Fk(!)f Fo(\026f)s(;)19 b(f)g Fk(2)c(K)q Fq(\()p Fo(E)s Fq(\))p Fo(:)118 2437 y Fq(In)h(this)g(top)q(ology)i (con)o(v)o(ergence)c(will)h(b)q(e)i(denoted)f(b)o(y)1170 2410 y Fl(v)1155 2437 y Fk(!)p Fq(.)118 2497 y(|)25 b(The)17 b(class)g Fk(M)468 2504 y Fl(1)487 2497 y Fq(\()p Fo(E)s Fq(\))g(of)g(probabilit)o(y)f(measures)g(on)h Fo(E)j Fq(is)d(endo)o(w)o(ed)g(with)f(the)h(w)o(eak)118 2557 y(\(narro)o(w\))g(top)q(ology)l(,)g(i.e.)j(the)c(initial)f(top)q(ology) i(with)g(resp)q(ect)e(to)i(the)f(mappings)268 2655 y Fo(\026)e Fk(!)f Fo(\026f)s(;)19 b(f)g Fk(2)c(C)s Fq(\()p Fo(E)s Fq(\))p Fo(:)118 2752 y Fq(In)h(this)g(top)q(ology)i(con)o(v)o (ergence)c(will)h(b)q(e)i(denoted)f(b)o(y)1167 2725 y Fl(w)1155 2752 y Fk(!)p Fq(.)933 2877 y(5)p eop %%Page: 6 7 6 6 bop 191 77 a Fq(Finally)17 b(it)h(has)i(to)e(b)q(e)h(emphasized)e (that)h(statemen)o(ts)f(concerning)h(random)g(v)m(ari-)118 137 y(ables)h(in)g(case)h(of)f(doubt)h(are)g(alw)o(a)o(ys)f(understo)q (o)q(d)i(mo)q(dulo)e Fp(P)p Fq({n)o(ull)g(sets.)30 b(Th)o(us)20 b(the)118 197 y(supplemen)o(t)13 b(\\almost)h(surely",)g(as)i(in)e(the) h(tric)o(hotom)o(y)d(concerning)j(\()p Fo(S)1470 204 y Fn(n)1493 197 y Fq(\))1512 204 y Fn(n)p Fm(\025)p Fl(0)1581 197 y Fq(,)f(will)g(fre-)118 258 y(quen)o(tly)h(b)q(e)h(deleted.)118 434 y Fp(1.)25 b(Lo)n(w)n(er)19 b(and)g(upp)r(er)f(limit)118 552 y Fq(As)e(outlined)g(in)g(the)g(in)o(tro)q(duction,)g(the)g(follo)o (wing)g(observ)m(ation)h(is)f(of)h(cen)o(tral)e(imp)q(or-)118 612 y(tance)h(throughout)i(the)e(pap)q(er:)118 702 y Fp(\(1.1\))28 b(Theorem.)23 b Fh(The)17 b(r)n(andom)g(variables)268 803 y Fo(X)p 268 810 45 2 v 18 w Fq(:=)c(lim)8 b(inf)534 810 y Fn(n)p Fm(!1)637 803 y Fo(X)677 810 y Fn(n)788 803 y Fo(and)p 955 764 V 87 w(X)18 b Fq(:=)c(lim)8 b(sup)1237 810 y Fn(n)p Fm(!1)1339 803 y Fo(X)1379 810 y Fn(n)118 905 y Fh(ar)n(e)18 b(almost)i(sur)n(ely)e(c)n(onstant)i(and)f(indep)n (endent)i(of)e(the)h(initial)f(value)i Fo(X)1519 912 y Fl(0)1539 905 y Fh(,)e(i.e.)28 b(ther)n(e)118 965 y(exist)18 b(c)n(onstants)h Fq(0)14 b Fk(\024)f Fo(x)p 536 972 28 2 v 14 w Fk(\024)p 631 939 V 14 w Fo(x)g Fk(\024)h(1)p Fh(,)j(dep)n(ending)i(only)e(on)h Fo(\027)s Fh(,)g(such)g(that)268 1067 y Fp(P)p Fq(\()p Fo(X)p 325 1074 45 2 v 18 w Fq(=)c Fo(x)p 435 1074 28 2 v Fq(\))f(=)h(1)75 b Fh(and)g Fp(P)p Fq(\()p 855 1028 45 2 v Fo(X)19 b Fq(=)p 965 1041 28 2 v 13 w Fo(x)p Fq(\))14 b(=)g(1)p Fo(:)118 1169 y Fh(Pr)n(o)n(of.)23 b Fq(1.)f(The)16 b(follo)o(wing)g(fact)g(will)f(b)q(e)i(needed:)118 1270 y(\(1\))88 b(sup)349 1277 y Fn(n)p Fm(\025)p Fl(0)426 1270 y Fo(S)456 1277 y Fn(n)494 1270 y Fq(=)13 b(+)p Fk(1)50 b Fq(implies)d(sup)964 1277 y Fn(n)p Fm(\025)p Fl(0)1041 1270 y Fo(X)1081 1277 y Fn(n)1118 1270 y Fq(=)14 b Fk(1)50 b Fq(a.s.)118 1372 y(Indeed,)15 b(since)268 1474 y(sup)349 1482 y Fn(m)p Fm(2)p Fj(N)449 1474 y Fo(Z)482 1481 y Fn(m)529 1474 y Fo(>)f Fq(0)26 b(a.s.)49 b(and)h(sup)961 1481 y Fn(n)p Fm(\025)p Fn(m)1052 1474 y Fo(e)1075 1453 y Fn(S)1096 1457 y Fg(n)1117 1453 y Fm(\000)p Fn(S)1165 1457 y Fg(m)1211 1474 y Fq(=)14 b Fk(1)25 b Fq(a.s.)o Fo(;)118 1576 y Fq(the)16 b(assertion)h(is)f(a)h(consequence)e(of)268 1677 y(sup)349 1684 y Fn(n)p Fm(\025)p Fl(0)426 1677 y Fo(X)466 1684 y Fn(n)504 1677 y Fk(\025)22 b Fq(sup)647 1686 y Fn(m)p Fm(2)p Fj(N)746 1677 y Fq(\()p Fo(Z)798 1684 y Fn(m)840 1677 y Fq(sup)922 1684 y Fn(n)p Fm(\025)p Fn(m)1012 1677 y Fo(e)1035 1657 y Fn(S)1056 1661 y Fg(n)1078 1657 y Fm(\000)p Fn(S)1126 1661 y Fg(m)1158 1677 y Fq(\))p Fo(:)191 1779 y Fq(2.)g(Let)16 b Fo(X)p 338 1786 45 2 v 383 1758 a Fn(x)421 1779 y Fq(and)p 516 1740 V 17 w Fo(X)560 1750 y Fn(x)598 1779 y Fq(b)q(e)h(de\014ned)f(in)g(analogy)h (to)f Fo(X)1173 1761 y Fn(x)1169 1791 y(n)1196 1779 y Fq(.)21 b(Then)268 1881 y Fo(X)p 268 1888 V 312 1860 a Fl(0)346 1881 y Fk(\025)13 b Fq(lim)8 b(inf)541 1888 y Fn(n)p Fm(!1)644 1881 y Fq(\()p Fo(Z)696 1888 y Fn(m)729 1881 y Fo(Y)757 1888 y Fn(m)p Fl(+1)844 1881 y Fo(:)g(:)g(:)g(Y)938 1888 y Fn(n)973 1881 y Fq(+)j Fo(:)d(:)g(:)i Fq(+)h Fo(Z)1172 1888 y Fn(m)p Fl(+)p Fn(n)1255 1881 y Fq(\))i(=:)h Fo(X)p 1353 1888 V 1397 1860 a Fl(0)1397 1893 y Fn(m)1430 1881 y Fo(;)118 1982 y Fq(where)i Fk(L)p Fq(\()p Fo(X)p 312 1989 V 357 1961 a Fl(0)376 1982 y Fq(\))h(and)g Fk(L)p Fq(\()p Fo(X)p 560 1989 V 604 1961 a Fl(0)604 1995 y Fn(m)638 1982 y Fq(\))f(are)g(iden)o(tical.)j(This)e(implies)268 2084 y Fo(X)p 268 2091 V 312 2063 a Fl(0)346 2084 y Fq(=)c Fo(X)p 397 2091 V 442 2063 a Fl(0)442 2096 y Fn(m)500 2084 y Fq(a.s.)87 b(for)16 b(all)25 b Fo(m)13 b Fk(2)h Fp(N)p Fo(;)118 2186 y Fq(i.e.)40 b Fo(X)p 222 2193 V 267 2165 a Fl(0)309 2186 y Fq(is)23 b(measurable)f(with)g(resp)q(ect)h (to)g(\(the)g(completion)e(of)s(\))i(the)g(tail{\014eld)f(of)118 2253 y(\()p Fo(Y)165 2260 y Fn(n)189 2253 y Fo(;)8 b(Z)244 2260 y Fn(n)268 2253 y Fq(\))287 2261 y Fn(n)p Fm(2)p Fj(N)368 2253 y Fq(.)20 b(Th)o(us)14 b Fo(X)p 523 2260 V 568 2232 a Fl(0)601 2253 y Fq({)g(and)g(similarly)p 927 2214 V 10 w Fo(X)972 2224 y Fl(0)1005 2253 y Fq({)g(are)g(almost)e (surely)h(constan)o(t.)21 b(There-)118 2313 y(fore)f(it)f(su\016ces)g (to)h(sho)o(w)g(that)g Fo(X)p 735 2320 V 779 2292 a Fn(x)820 2313 y Fq(and)p 919 2274 V 21 w Fo(X)963 2284 y Fn(x)1004 2313 y Fq(are)g(in)f(fact)h(indep)q(enden)o(t)e(of)i Fo(x)f Fk(2)h Fp(R)1729 2320 y Fl(+)1758 2313 y Fq(.)118 2373 y(Here)15 b(sup)315 2380 y Fn(n)p Fm(\025)p Fl(0)392 2373 y Fo(S)422 2380 y Fn(n)459 2373 y Fq(=)f(+)p Fk(1)i Fq(ma)o(y)f(b)q(e)h(assumed,)f(b)q(ecause)i(otherwise)268 2475 y Fo(X)312 2454 y Fn(x)308 2487 y(n)345 2475 y Fk(\000)11 b Fo(X)439 2454 y Fl(0)435 2487 y Fn(n)473 2475 y Fq(=)i Fo(x)8 b(e)583 2454 y Fn(S)604 2458 y Fg(n)642 2475 y Fk(!)13 b Fq(0)26 b(a.s.)191 2577 y(3.)c(T)l(o)17 b(treat)f(the)g(lo)o (w)o(er)f(limit)f(\014rst,)i(consider)g(the)g(random)g(time)268 2678 y Fo(T)k Fq(:=)13 b(inf)s Fk(f)p Fo(n)h Fk(\025)g Fq(0)g(:)f Fo(X)671 2658 y Fl(0)667 2691 y Fn(n)705 2678 y Fk(\025)h Fo(x)p Fk(g)p Fo(;)933 2877 y Fq(6)p eop %%Page: 7 8 7 7 bop 118 77 a Fq(whic)o(h)16 b(is)g(\014nite)f(almost)h(surely)f(b)o (y)h(\(1\).)22 b(Then)268 178 y Fo(X)p 268 185 45 2 v 312 157 a Fn(x)382 178 y Fk(\025)47 b Fo(X)p 468 185 V 513 157 a Fl(0)382 280 y Fk(\025)g Fq(lim)8 b(inf)612 287 y Fn(n)p Fm(!1)714 280 y Fq(\()p Fo(X)777 259 y Fl(0)773 292 y Fn(T)809 280 y Fo(Y)837 287 y Fn(T)d Fl(+1)919 280 y Fo(:)j(:)g(:)f(Y)1012 287 y Fn(T)e Fl(+)p Fn(n)1100 280 y Fq(+)11 b Fo(:)d(:)g(:)i Fq(+)h Fo(Z)1299 287 y Fn(T)5 b Fl(+)p Fn(n)1376 280 y Fq(\))382 382 y Fk(\025)47 b Fq(lim)8 b(inf)612 389 y Fn(n)p Fm(!1)714 382 y Fq(\()p Fo(x)g(Y)797 389 y Fn(T)d Fl(+1)878 382 y Fo(:)j(:)g(:)g(Y)972 389 y Fn(T)d Fl(+)p Fn(n)1060 382 y Fq(+)11 b Fo(:)d(:)g(:)i Fq(+)h Fo(Z)1259 389 y Fn(T)5 b Fl(+)p Fn(n)1336 382 y Fq(\))375 483 y(=:)41 b Fo(X)p 468 490 V 513 463 a Fn(x)513 496 y(T)540 483 y Fo(;)118 585 y Fq(where)23 b Fk(L)p Fq(\()p Fo(X)p 319 592 V 364 564 a Fn(x)386 585 y Fq(\))g(and)h Fk(L)p Fq(\()p Fo(X)p 583 592 V 627 564 a Fn(x)627 597 y(T)655 585 y Fq(\))f(are)g(iden)o(tical,)g(b)q (ecause)g Fo(T)30 b Fq(is)23 b(a)h(stopping)g(time)d(with)118 645 y(resp)q(ect)16 b(to)h(\()p Fo(Y)392 652 y Fn(n)415 645 y Fo(;)8 b(Z)470 652 y Fn(n)494 645 y Fq(\))513 653 y Fn(n)p Fm(2)p Fj(N)595 645 y Fq(.)21 b(Th)o(us,)16 b(indeed)268 746 y Fo(X)p 268 753 V 312 725 a Fn(x)348 746 y Fq(=)d Fo(X)p 399 753 V 444 725 a Fl(0)489 746 y Fq(a.s.)86 b(for)17 b(all)24 b Fo(x)14 b Fk(2)g Fp(R)928 753 y Fl(+)957 746 y Fo(:)118 847 y Fq(The)21 b(corresp)q(onding)i (result)d(for)i(the)f(upp)q(er)g(limit)e(is)i(another)h(consequence)e (of)i(\(1\),)118 908 y(whic)o(h)16 b(yields)p 268 974 V 268 1013 a Fo(X)312 984 y Fn(x)348 1013 y Fk(\025)p 400 974 V 13 w Fo(X)445 984 y Fl(0)478 1013 y Fq(=)e Fk(1)25 b Fq(a.s.)87 b(for)16 b(all)25 b Fo(x)13 b Fk(2)h Fp(R)1044 1020 y Fl(+)1073 1013 y Fo(:)50 b Ff(2)191 1115 y Fq(The)16 b(notation)h Fo(x)p 486 1122 28 2 v 16 w Fq(and)p 625 1088 V 17 w Fo(x)f Fq(will)f(b)q(e)h(used)h(in)e(the) h(sequel)f(without)i(further)f(reference.)118 1175 y(The)g(in)o(terv)m (al)f(de\014ned)g(b)o(y)h(these)f(limits)f(attracts)i(the)g(sequence)e (\()p Fo(X)1414 1182 y Fn(n)1438 1175 y Fq(\))1457 1182 y Fn(n)p Fm(\025)p Fl(0)1542 1175 y Fq(in)h(a)i(strong)118 1235 y(sense:)118 1324 y Fp(\(1.2\))28 b(Prop)r(osition.)c Fh(Whenever)d(\014nite,)g(the)g(c)n(onstants)f Fo(x)p 1261 1331 V 20 w Fh(and)p 1406 1298 V 20 w Fo(x)f Fh(ar)n(e)g (determine)n(d)118 1384 y(by)f(the)g(e)n(quivalenc)n(es)118 1486 y Fq(\(a\))88 b Fo(x)13 b Fk(\024)h Fo(x)p 362 1493 V 87 w(if)22 b(and)c(onl)q(y)h(if)92 b Fp(P)p Fq(\()p Fo(Y)11 b(x)g Fq(+)g Fo(Z)18 b Fk(\025)13 b Fo(x)p Fq(\))h(=)g(1)p Fo(;)118 1599 y Fq(\(b\))87 b Fo(x)14 b Fk(\025)p 364 1573 V 13 w Fo(x)87 b(if)23 b(and)17 b(onl)q(y)i(if)93 b Fp(P)p Fq(\()p Fo(Y)11 b(x)g Fq(+)g Fo(Z)18 b Fk(\024)13 b Fo(x)p Fq(\))h(=)f(1;)118 1713 y Fh(mor)n(e)n(over)k(the)h Fq(\\)p Fh(if{p)n(art")e(holds)i(without)g(the)g(\014niteness)h (assumption.)118 1802 y(Pr)n(o)n(of.)k Fq(1.)31 b(The)19 b(implication)d(from)j(righ)o(t)f(to)i(left)e(is)h(an)h(imme)o(diate)c (consequence)i(of)118 1862 y(\(1.1\),)e(b)q(ecause)h(e.g.)j(the)c (assumption)268 1964 y Fo(Y)11 b(x)f Fq(+)h Fo(Z)18 b Fk(\025)c Fo(x)24 b Fq(a.s.)118 2065 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Fq(\()p Fo(y)r(X)450 2567 y Fn(n)485 2560 y Fq(+)j Fo(z)r(;)d(y)r(X)15 b Fq(+)c Fo(z)r Fq(\)\))i Fk(!)h Fq(0)88 b(for)16 b(all)24 b(\()p Fo(y)r(;)8 b(z)r Fq(\))13 b Fk(2)i Fp(R)1327 2539 y Fl(2)1327 2572 y(+)1356 2560 y Fo(:)118 2656 y Fq(Since)g Fo(X)285 2663 y Fn(n)326 2656 y Fq(and)i(\()p Fo(Y)468 2663 y Fn(n)p Fl(+1)536 2656 y Fo(;)8 b(Z)591 2663 y Fn(n)p Fl(+1)660 2656 y Fq(\))16 b(are)h(indep)q(enden)o(t,)e(moreo)o(v)o(er)268 2717 y Fi(R)312 2752 y Fp(E)8 b Fq(\()p Fo(d)g Fq(\()p Fo(y)r(X)494 2759 y Fn(n)529 2752 y Fq(+)j Fo(z)r(;)d(X)665 2759 y Fn(n)689 2752 y Fq(\)\))g Fo(d\027)17 b Fq(=)d Fp(E)8 b Fq(\()p Fo(d)g Fq(\()p Fo(X)1009 2759 y Fn(n)p Fl(+1)1079 2752 y Fo(;)g(X)1141 2759 y Fn(n)1165 2752 y Fq(\)\))14 b Fk(!)f Fq(0)p Fo(:)933 2877 y Fq(9)p eop %%Page: 10 11 10 10 bop 118 77 a Fq(Th)o(us)17 b(there)e(is)h(a)h(subsequence)f(\()p Fo(X)791 84 y Fn(n)812 90 y Fg(k)834 77 y Fq(\))853 84 y Fn(k)q Fm(\025)p Fl(0)935 77 y Fq(suc)o(h)g(that)118 174 y(\(2\))88 b Fp(E)8 b Fq(\()p 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Fl(2)1159 634 y(+)1189 622 y Fq(;)118 719 y(therefore,)h(b)o(y)h (F)l(ubini,)268 816 y Fo(y)r(x)10 b Fq(+)h Fo(z)429 790 y Fn(\027)420 816 y Fq(=)i Fo(x)87 b Fq(for)25 b Fo(\026)p Fq({almost)17 b(all)e Fo(x)f Fk(2)g Fp(R)1079 823 y Fl(+)1108 816 y Fo(:)118 913 y Fq(Th)o(us)j(there)e(exists)h(indeed)f Fo(\015)i Fk(2)d Fp(R)786 920 y Fl(+)832 913 y Fq(suc)o(h)i(that)268 1010 y Fo(Z)h Fq(=)d Fo(\015)s Fq(\(1)e Fk(\000)e Fo(Y)i Fq(\))p Fo(;)118 1107 y Fq(where)k Fo(\015)h Fk(6)p Fq(=)d(0)i(\(since) g(otherwise)g Fo(Z)i Fq(=)13 b(0\))k(and)g Fo(Y)25 b Fk(\024)13 b Fq(1)k(\(b)q(ecause)f(of)h Fo(Z)h Fk(\025)13 b Fq(0\).)191 1167 y(3.)22 b(The)16 b(implication)e(\(c\))i Fk(\))g Fq(\(a\))g(is)g(immediate)d(from)i(\(1.2\))p Fo(:)50 b Ff(2)191 1256 y Fq(Clearly)l(,)20 b(the)g(constan)o(t)h Fo(\015)j Fq(is)c(a)h(common)d(\014xed)i(p)q(oin)o(t)g(of)h(the)f (underlying)g(a\016ne)118 1317 y(maps)c(almost)f(surely)l(.)118 1493 y Fp(2.)25 b(Recurrence)17 b(and)i(transience)118 1611 y Fq(In)d(view)g(of)g Fo(X)387 1618 y Fn(n)425 1611 y Fk(\025)d Fq(0)k(the)f(\014rst)h(classi\014cation)f(is)g(v)o(ery)f (natural:)118 1701 y Fp(\(2.1\))28 b(De\014nition.)c Fh(The)12 b(distribution)h Fo(\027)27 b Fq(\()p Fh(or)12 b(the)h(kernel)h Fo(P)19 b Fh(or)12 b(the)h(pr)n(o)n(c)n(ess)e Fq(\()p Fo(X)1643 1708 y Fn(n)1667 1701 y Fq(\))1686 1708 y Fn(n)p Fm(\025)p Fl(0)1755 1701 y Fq(\))118 1761 y Fh(is)17 b(c)n(al)r(le)n(d)118 1850 y Fq(\(a\))88 b(\\)p Fh(r)n(e)n(curr)n(ent")49 b(if)h Fo(x)p 638 1857 28 2 v 13 w(<)14 b Fk(1)p Fh(,)118 1939 y Fq(\(b\))87 b(\\)p Fh(tr)n(ansient")51 b(if)f Fo(x)p 637 1946 V 13 w Fq(=)14 b Fk(1)p Fh(.)191 2028 y Fq(Both)j(cases)h(can)f(b)q(e)g(distinguished) g(as)g(w)o(ell)f(b)o(y)h(the)g(asso)q(ciated)h(p)q(oten)o(tial)e(k)o (ernel)118 2089 y Fo(G)e Fq(:=)236 2055 y Fi(P)288 2096 y Fn(n)p Fm(\025)p Fl(0)365 2089 y Fo(P)403 2071 y Fn(n)426 2089 y Fq(,)i(indep)q(enden)o(tly)f(of)h(the)g(initial)f(la)o(w:)118 2178 y Fp(\(2.2\))28 b(Theorem.)23 b Fh(The)17 b(fol)r(lowing)j (dichotomy)d(holds:)118 2267 y Fq(\(a\))88 b Fh(if)17 b Fo(\027)k Fh(is)c(r)n(e)n(curr)n(ent,)f(then)268 2331 y Fi(P)320 2371 y Fn(n)p Fm(\025)p Fl(0)397 2364 y Fp(P)p Fq(\()p Fo(X)494 2371 y Fn(n)532 2364 y Fk(\024)d Fo(t)p Fq(\))h(=)f Fk(1)87 b Fh(for)25 b Fo(t)13 b(>)h(x)p 992 2371 V Fq(;)118 2461 y(\(b\))87 b Fh(if)18 b Fo(\027)i Fh(is)e(tr)n(ansient,)f(then)268 2525 y Fi(P)320 2565 y Fn(n)p Fm(\025)p Fl(0)397 2558 y Fp(P)p Fq(\()p Fo(X)494 2565 y Fn(n)532 2558 y Fk(\024)c Fo(t)p Fq(\))h Fo(<)f Fk(1)87 b Fh(for)25 b Fo(t)13 b(<)h Fk(1)p Fo(:)118 2655 y Fh(Pr)n(o)n(of.)23 b Fq(\(a\))17 b(This)f(is)g(an)h(immedi)o(ate)d (consequence)h(of)268 2752 y Fp(P)p Fq(\()p Fo(X)365 2759 y Fn(n)403 2752 y Fk(\024)e Fo(t)25 b Fq(in\014nitely)14 b(often\))g(=)g(1)p Fo(:)920 2877 y Fq(10)p eop %%Page: 11 12 11 11 bop 191 77 a Fq(\(b\))17 b(De\014ne)f(recursiv)o(ely)268 178 y Fo(T)297 185 y Fl(0)330 178 y Fq(:=)d(0)50 b(and)h Fo(T)627 185 y Fn(k)661 178 y Fq(:=)14 b(inf)s Fk(f)p Fo(n)g(>)f(T)934 185 y Fn(k)q Fm(\000)p Fl(1)1014 178 y Fq(:)h Fo(X)1082 185 y Fn(n)1119 178 y Fk(\024)g Fo(t)p Fk(g)24 b Fq(\()p Fk(\024)14 b(1)p Fq(\))p Fo(:)118 280 y Fq(Then)i(in)g(particular)268 381 y(0)e(=)g Fp(P)396 361 y Fl(0)416 381 y Fq(\()p Fo(X)475 388 y Fn(n)512 381 y Fk(\024)g Fo(t)24 b Fq(in\014nitely)15 b(often\))f(=)g(lim)1080 388 y Fn(k)q Fm(!1)1180 381 y Fp(P)1218 361 y Fl(0)1238 381 y Fq(\()p Fo(T)1286 388 y Fn(k)1321 381 y Fo(<)f Fk(1)p Fq(\))p Fo(;)118 483 y Fq(hence)j(there)f(exists)h Fo(l)e Fk(2)g Fp(N)i Fq(suc)o(h)h(that)268 584 y Fo(#)c Fq(:=)g Fp(P)413 564 y Fl(0)433 584 y Fq(\()p Fo(T)481 591 y Fn(l)508 584 y Fo(<)h Fk(1)p Fq(\))f Fo(<)h Fq(1)p Fo(:)118 686 y Fq(With)i(the)g(decreasing)g(function)268 787 y Fo(g)r Fq(\()p Fo(x)p Fq(\))d(:=)h Fp(P)476 767 y Fn(x)498 787 y Fq(\()p Fo(T)546 794 y Fn(l)572 787 y Fo(<)g Fk(1)p Fq(\))118 889 y(this)i(implies)268 990 y Fp(P)306 969 y Fl(0)326 990 y Fq(\()p Fo(T)374 998 y Fl(\()p Fn(k)q Fl(+1\))5 b Fn(l)498 990 y Fo(<)13 b Fk(1)p Fq(\))42 b(=)740 955 y Fi(R)776 998 y Fm(f)p Fn(T)815 1004 y Fg(k)q(l)843 998 y Fn(<)p Fm(1g)934 990 y Fo(g)r Fq(\()p Fo(X)1018 997 y Fn(T)1039 1003 y Fg(k)q(l)1070 990 y Fq(\))8 b Fo(d)p Fp(P)1160 969 y Fl(0)660 1092 y Fk(\024)740 1056 y Fi(R)776 1100 y Fm(f)p Fn(T)815 1106 y Fg(k)q(l)843 1100 y Fn(<)p Fm(1g)934 1092 y Fo(g)r Fq(\(0\))g Fo(d)p Fp(P)1092 1071 y Fl(0)660 1193 y Fq(=)42 b Fo(#)8 b Fp(P)815 1173 y Fl(0)835 1193 y Fq(\()p Fo(T)883 1200 y Fn(k)q(l)929 1193 y Fo(<)13 b Fk(1)p Fq(\))p Fo(:)118 1295 y Fq(Therefore)268 1396 y Fp(P)306 1376 y Fl(0)326 1396 y Fq(\()p Fo(T)374 1403 y Fn(k)q(l)419 1396 y Fo(<)h Fk(1)p Fq(\))g Fk(\024)f Fo(#)635 1376 y Fn(k)743 1396 y Fq(for)k(all)24 b Fo(k)16 b Fk(\025)e Fq(0)p Fo(;)118 1498 y Fq(and)j(again)g(b)o(y)f(monotonicit)o(y)e(this)i(yields)268 1566 y Fi(P)320 1606 y Fn(n)p Fm(\025)p Fl(0)397 1599 y Fp(P)p Fq(\()p Fo(X)494 1606 y Fn(n)532 1599 y Fk(\024)d Fo(t)p Fq(\))41 b Fk(\024)743 1566 y Fi(P)795 1606 y Fn(n)p Fm(\025)p Fl(0)872 1599 y Fp(P)910 1579 y Fl(0)930 1599 y Fq(\()p Fo(X)989 1606 y Fn(n)1026 1599 y Fk(\024)14 b Fo(t)p Fq(\))663 1701 y(=)42 b Fp(E)780 1680 y Fl(0)799 1701 y Fq(\()p Fk(jf)p Fo(n)14 b Fk(\025)f Fq(0)h(:)g Fo(X)1058 1708 y Fn(n)1096 1701 y Fk(\024)f Fo(t)p Fk(gj)p Fq(\))663 1803 y(=)743 1769 y Fi(P)795 1810 y Fn(i)p Fm(\025)p Fl(0)862 1803 y Fp(P)900 1782 y Fl(0)920 1803 y Fq(\()p Fo(T)968 1810 y Fn(i)996 1803 y Fo(<)g Fk(1)p Fq(\))662 1904 y Fk(\024)42 b Fo(l)772 1871 y Fi(P)824 1911 y Fn(k)q Fm(\025)p Fl(0)899 1904 y Fp(P)937 1884 y Fl(0)957 1904 y Fq(\()p Fo(T)1005 1911 y Fn(k)q(l)1050 1904 y Fo(<)14 b Fk(1)p Fq(\))662 2006 y Fk(\024)42 b Fo(l)772 1973 y Fi(P)824 2013 y Fn(k)q Fm(\025)p Fl(0)899 2006 y Fo(#)928 1986 y Fn(k)963 2006 y Fo(<)14 b Fk(1)p Fo(:)49 b Ff(2)191 2108 y Fq(The)25 b(next)f(result)g(is)g(a)h(simple)d (consequence)h(of)i(\(2.2\))g(\(and)g(needed)f(b)q(efore)g(a)118 2168 y(stronger)17 b(v)o(ersion)f(will)f(b)q(e)h(a)o(v)m(ailable\):)118 2257 y Fp(\(2.3\))28 b(Prop)r(osition.)c Fo(\027)d Fh(is)c(r)n(e)n (curr)n(ent)f(whenever)k Fp(P)p Fq(\()p Fo(Y)25 b Fq(=)14 b(0\))g Fo(>)g Fq(0)p Fh(.)118 2346 y(Pr)n(o)n(of.)23 b Fq(F)l(or)17 b(an)o(y)f Fo(t)d(<)h Fk(1)i Fq(satisfying)268 2448 y Fp(P)p Fq(\()p Fo(Y)25 b Fq(=)14 b(0)p Fo(;)8 b(Z)18 b Fk(\024)13 b Fo(t)p Fq(\))h Fo(>)f Fq(0)118 2549 y(the)j(assertion)h(follo)o(ws)f(from)f(\(2.2a\))i(in)f(view)f(of) 268 2617 y Fi(P)320 2658 y Fn(n)p Fm(\025)p Fl(0)397 2651 y Fp(P)p Fq(\()p Fo(X)494 2658 y Fn(n)532 2651 y Fk(\024)e Fo(t)p Fq(\))h Fk(\025)687 2617 y Fi(P)739 2659 y Fn(n)p Fm(2)p Fj(N)829 2651 y Fp(P)p Fq(\()p Fo(Y)914 2658 y Fn(n)952 2651 y Fq(=)g(0)p Fo(;)8 b(Z)1083 2658 y Fn(n)1121 2651 y Fk(\024)13 b Fo(t)p Fq(\))p Fo(:)49 b Ff(2)191 2752 y Fq(Equally)16 b(simple)e(is)i(the)g(follo)o(wing)g (coun)o(terpart:)920 2877 y(11)p eop %%Page: 12 13 12 12 bop 118 77 a Fp(\(2.4\))28 b(Prop)r(osition.)c Fo(\027)d Fh(is)c(tr)n(ansient)h(whenever)h Fo(S)1131 84 y Fn(n)1168 77 y Fk(!)14 b Fq(+)p Fk(1)p Fo(:)118 166 y Fh(Pr)n(o)n(of.)23 b Fq(Since)268 265 y Fo(T)d Fq(:=)13 b(inf)s Fk(f)p Fo(n)h Fk(2)g Fp(N)g Fq(:)f Fo(Z)674 272 y Fn(n)712 265 y Fo(>)h Fq(0)p Fk(g)118 363 y Fq(de\014nes)g(an)h (\(almost)e(surely)g(\014nite\))h(stopping)h(time)d(with)i(resp)q(ect)f (to)i(\()p Fo(Z)1486 370 y Fn(n)1510 363 y Fq(\))1529 372 y Fn(n)p Fm(2)p Fj(N)1610 363 y Fq(,)f(b)o(y)g(the)118 424 y(h)o(yp)q(othesis)268 522 y Fo(S)298 529 y Fn(n)332 522 y Fk(\000)d Fo(S)412 529 y Fn(T)453 522 y Fk(!)j(1)25 b Fq(a.s.)118 621 y(Th)o(us)17 b(the)f(assertion)g(is)g(a)h (consequence)e(of)268 720 y Fo(X)308 727 y Fn(n)345 720 y Fk(\025)f Fo(Z)431 727 y Fn(T)467 720 y Fo(e)490 699 y Fn(S)511 703 y Fg(n)532 699 y Fm(\000)p Fn(S)580 705 y Fg(T)695 720 y Fq(for)25 b Fo(n)14 b Fk(\025)g Fo(T)t(:)49 b Ff(2)191 818 y Fq(That)14 b(the)g(con)o(v)o(erse)e(of)i(\(2.4\))g(do) q(es)g(not)g(hold)g(in)f(general)g(can)h(b)q(e)f(demonstrated)g(b)o(y) 118 878 y(a)k(somewhat)g(surprising)f(result.)23 b(Ho)o(w)o(ev)o(er)15 b(small)g(the)h(primary)f(v)m(ariable)i Fo(Y)11 b Fq(,)16 b(in)g(view)118 939 y(of)i(\(2.3\))f(only)h(supp)q(osed)g(to)g(b)q(e)g (strictly)e(p)q(ositiv)o(e,)g(ma)o(y)g(b)q(e,)h(the)g(secondary)h(v)m (ariable)118 999 y Fo(Z)k Fq(can)c(b)q(e)f(made)g(large)h(enough)g(for) g(transience,)f(ev)o(en)f(if)i(in)f(addition)h(indep)q(endence)118 1059 y(of)f Fo(Y)27 b Fq(and)17 b Fo(Z)j Fq(is)c(p)q(ostulated.)22 b(More)16 b(precisely)l(,)e(in)i(terms)f(of)h(distributions:)118 1148 y Fp(\(2.5\))28 b(Prop)r(osition.)c Fh(F)l(or)13 b(any)g Fo(\027)787 1155 y Fn(y)822 1148 y Fk(2)h(M)929 1155 y Fl(1)949 1148 y Fq(\()p Fp(R)1010 1155 y Fl(+)1039 1148 y Fq(\))g Fh(with)f Fo(\027)1197 1155 y Fn(y)1218 1148 y Fq(\()p Fk(f)p Fq(0)p Fk(g)p Fq(\))h(=)g(0)g Fh(and)g Fo(\027)1549 1155 y Fn(y)1570 1148 y Fq(\()p Fk(f)p Fq(1)p Fk(g)p Fq(\))f Fk(6)p Fq(=)h(1)118 1208 y Fh(ther)n(e)j(exists)h Fo(\027)396 1215 y Fn(z)430 1208 y Fk(2)c(M)537 1215 y Fl(1)556 1208 y Fq(\()p Fp(R)617 1215 y Fl(+)646 1208 y Fq(\))j Fh(with)h Fo(\027)812 1215 y Fn(z)832 1208 y Fq(\()p Fk(f)p Fq(0)p Fk(g)p Fq(\))c Fk(6)p Fq(=)f(1)18 b Fh(such)f(that)g Fo(\027)g Fq(=)d Fo(\027)1377 1215 y Fn(y)1408 1208 y Fk(\012)c Fo(\027)1481 1215 y Fn(z)1517 1208 y Fh(is)17 b(tr)n(ansient.)118 1298 y(Pr)n(o)n(of.)23 b Fq(Let)12 b Fo(Y)382 1305 y Fn(n)406 1298 y Fo(;)k(n)e Fk(2)g Fp(N)p Fq(,)e(b)q(e)f(indep)q(enden)o(t)g(with)g(distribution)g Fo(\027)1320 1305 y Fn(y)1353 1298 y Fq(and)h(assume)e(without)118 1358 y(loss)17 b(of)f(generalit)o(y)f Fo(Y)i(<)d Fq(1.)21 b(Then)c(c)o(ho)q(ose)f(a)h(sequence)e(\()p Fo(c)1212 1365 y Fn(n)1236 1358 y Fq(\))1255 1365 y Fn(n)p Fm(\025)p Fl(0)1339 1358 y Fq(satisfying)118 1456 y(\(1\))88 b(0)14 b Fo(<)g(c)379 1463 y Fl(0)412 1456 y Fo(<)g(c)485 1463 y Fl(1)518 1456 y Fo(<)g(:)8 b(:)g(:)13 b Fk(!)h Fq(1)p Fo(;)118 1567 y Fq(\(2\))268 1533 y Fi(P)320 1574 y Fn(n)p Fm(\025)p Fl(0)402 1567 y Fo(c)423 1574 y Fl(1)451 1567 y Fo(:)8 b(:)g(:)g(c)538 1574 y Fn(n)575 1567 y Fo(<)14 b Fk(1)118 1677 y Fq(\(e.g.)21 b Fo(c)253 1684 y Fn(n)290 1677 y Fq(=)14 b(\()388 1657 y Fn(n)p 366 1665 67 2 v 366 1694 a(n)p Fl(+1)437 1677 y Fq(\))456 1658 y Fl(2)492 1677 y Fq(for)j Fo(n)d Fk(2)g Fp(N)p Fq(\))i(and)h(a)f(sequence)g(\()p Fo(")1116 1684 y Fn(n)1139 1677 y Fq(\))1158 1685 y Fn(n)p Fm(2)p Fj(N)1256 1677 y Fq(satisfying)118 1775 y(\(3\))88 b(1)14 b Fo(>)g(")381 1782 y Fl(1)414 1775 y Fo(>)g(")489 1782 y Fl(2)522 1775 y Fo(>)g(:)8 b(:)g(:)13 b Fk(!)h Fq(0)p Fo(;)118 1885 y Fq(\(4\))88 b Fo(b)289 1892 y Fn(n)326 1885 y Fq(:=)13 b Fp(P)p Fq(\()p Fo(Y)476 1892 y Fl(1)505 1885 y Fo(:)8 b(:)g(:)f(Y)598 1892 y Fn(n)p Fm(\000)p Fl(1)681 1885 y Fk(\024)13 b Fo(")756 1892 y Fn(n)780 1885 y Fq(\))25 b Fk(\024)f Fq(2)911 1865 y Fm(\000)p Fn(n)971 1885 y Fo(c)992 1892 y Fl(1)1020 1885 y Fo(:)8 b(:)g(:)f(c)1106 1892 y Fn(n)1217 1885 y Fq(for)25 b Fo(n)14 b Fk(2)g Fp(N)p Fo(;)118 1995 y Fq(where)j(the)g(assumption)g Fp(P)p Fq(\()p Fo(Y)27 b Fq(=)16 b(0\))g(=)g(0)h(en)o(ters.)24 b(Indep)q(enden)o(tly)16 b(of)i Fo(Y)1495 2002 y Fn(n)1519 1995 y Fo(;)e(n)g Fk(2)f Fp(N)p Fq(,)i(let)118 2055 y Fo(Z)151 2062 y Fn(n)175 2055 y Fo(;)f(n)e Fk(2)g Fp(N)p Fq(,)i(b)q(e)g(indep)q(enden)o(t)g (with)g(a)g(distribution)g Fo(\027)1153 2062 y Fn(z)1189 2055 y Fq(suc)o(h)g(that)268 2154 y Fp(P)p Fq(\()p Fo(Z)i Fq(=)13 b(0\))i(=)e Fo(c)557 2161 y Fl(0)664 2154 y Fq(and)88 b Fp(P)p Fq(\()p Fo(Z)18 b Fq(=)c(1)p Fo(=")1061 2161 y Fn(n)1085 2154 y Fq(\))g(=)f Fo(c)1190 2161 y Fn(n)1225 2154 y Fk(\000)e Fo(c)1296 2161 y Fn(n)p Fm(\000)p Fl(1)1414 2154 y Fq(for)25 b Fo(n)14 b Fk(2)g Fp(N)p Fo(;)118 2253 y Fq(hence)i(in)g(particular)268 2351 y Fp(P)p Fq(\()p Fo(Z)i Fk(\024)13 b Fq(1)p Fo(=")499 2358 y Fn(n)523 2351 y Fq(\))h(=)g Fo(c)629 2358 y Fn(n)739 2351 y Fq(for)26 b Fo(n)14 b Fk(2)g Fp(N)p Fo(:)118 2450 y Fq(With)g(the)f(dual)h (sequence)e(\()p Fo(W)693 2457 y Fn(n)717 2450 y Fq(\))736 2457 y Fn(n)p Fm(\025)p Fl(0)804 2450 y Fq(,)i(de\014ned)g(in)f (Section)g(0,)h(consider)f(no)o(w)i(the)e(ev)o(en)o(ts)268 2549 y Fo(A)305 2556 y Fn(n)370 2549 y Fq(:=)41 b Fk(f)p Fo(W)534 2556 y Fn(n)571 2549 y Fk(\024)13 b Fq(1)p Fk(g)p Fo(;)268 2650 y(B)305 2657 y Fn(n)370 2650 y Fq(:=)41 b Fk(f)p Fo(Y)516 2657 y Fl(1)544 2650 y Fo(:)8 b(:)g(:)f(Y)637 2657 y Fn(n)p Fm(\000)p Fl(1)720 2650 y Fk(\024)14 b Fo(")796 2657 y Fn(n)819 2650 y Fk(g)p Fo(;)270 2752 y(C)305 2759 y Fn(n)370 2752 y Fq(:=)41 b Fk(f)p Fo(Z)521 2759 y Fn(n)558 2752 y Fk(\024)14 b Fq(1)p Fo(=")682 2759 y Fn(n)706 2752 y Fk(g)p Fo(;)920 2877 y Fq(12)p eop %%Page: 13 14 13 13 bop 118 77 a Fq(whic)o(h)16 b(ob)o(viously)f(satisfy)268 179 y Fo(A)305 186 y Fn(n)341 179 y Fk(\032)f Fo(A)431 186 y Fn(n)p Fm(\000)p Fl(1)510 179 y Fk(\\)d Fq(\()p Fo(B)610 186 y Fn(n)645 179 y Fk([)g Fo(C)724 186 y Fn(n)748 179 y Fq(\))j Fk(\032)f Fq(\()p Fo(A)889 186 y Fn(n)p Fm(\000)p Fl(1)968 179 y Fk(\\)f Fo(C)1048 186 y Fn(n)1071 179 y Fq(\))f Fk([)g Fo(B)1182 186 y Fn(n)1293 179 y Fq(for)25 b Fo(n)14 b Fk(2)g Fp(N)p Fo(:)118 280 y Fq(With)i(the)g (notation)h Fo(a)549 287 y Fn(n)586 280 y Fq(:=)c Fp(P)p Fq(\()p Fo(A)745 287 y Fn(n)769 280 y Fq(\))j(this)g(yields)f(b)o(y)h (indep)q(endence)f(the)h(inequalit)o(y)268 382 y Fo(a)294 389 y Fn(n)330 382 y Fk(\024)e Fo(a)409 389 y Fn(n)p Fm(\000)p Fl(1)486 382 y Fo(c)507 389 y Fn(n)541 382 y Fq(+)d Fo(b)611 389 y Fn(n)721 382 y Fq(for)26 b Fo(n)14 b Fk(2)g Fp(N)p Fo(;)118 484 y Fq(whic)o(h)i(b)o(y)f(induction,)h (using)g(\(4\),)g(leads)h(to)268 586 y Fo(a)294 593 y Fn(n)330 586 y Fk(\024)d Fq(\(1)d(+)g Fo(:)d(:)g(:)j Fq(+)g(2)628 565 y Fm(\000)p Fn(n)679 586 y Fq(\))g Fo(c)730 593 y Fl(1)758 586 y Fo(:)d(:)g(:)g(c)845 593 y Fn(n)955 586 y Fq(for)26 b Fo(n)13 b Fk(\025)h Fq(0)p Fo(:)118 687 y Fq(By)i(\(2\),)g(therefore)268 756 y Fi(P)320 796 y Fn(n)p Fm(\025)p Fl(0)397 789 y Fp(P)p Fq(\()p Fo(X)498 768 y Fl(0)494 801 y Fn(n)532 789 y Fk(\024)d Fq(1\))43 b(=)749 756 y Fi(P)802 796 y Fn(n)p Fm(\025)p Fl(0)878 789 y Fp(P)p Fq(\()p Fo(W)981 796 y Fn(n)1019 789 y Fk(\024)14 b Fq(1\))669 891 y Fk(\024)41 b Fq(2)785 857 y Fi(P)837 898 y Fn(n)p Fm(\025)p Fl(0)917 891 y Fo(c)938 898 y Fl(1)966 891 y Fo(:)8 b(:)g(:)f(c)1052 898 y Fn(n)1090 891 y Fo(<)13 b Fk(1)p Fo(:)118 992 y Fq(According)18 b(to)g(\(1.3a\))h(the)e(lo)o(w)o(er)g(limit)f Fo(x)p 889 999 28 2 v -1 w Fq(,)i(due)g(to)h Fp(P)p Fq(\()p Fo(Y)h(<)c Fq(1)p Fo(;)8 b(Z)22 b Fq(=)16 b(0\))i Fo(>)e Fq(0,)j(can)f(only)118 1053 y(tak)o(e)h(the)h(v)m(alues)f(0)h(or)g Fk(1)p Fq(.)31 b(But)20 b(the)f(\014rst)h(p)q(ossibilit)o(y)f(is)g (ruled)g(out)h(b)o(y)f(\(2.2a\),)i(and)118 1113 y(th)o(us)16 b Fo(\027)h Fq(=)d Fo(\027)342 1120 y Fn(y)374 1113 y Fk(\012)d Fo(\027)448 1120 y Fn(z)484 1113 y Fq(is)16 b(transien)o(t)p Fo(:)49 b Ff(2)191 1202 y Fq(The)20 b(\014nal)g(result)g(of)g(this)g(section)f(is)h(a)g(consequence)f(of)h (the)g(monotonicit)o(y)e(and)118 1262 y(will)d(b)q(e)i(crucial)e(in)h (Sections)g(5)g(and)h(7:)118 1351 y Fp(\(2.6\))28 b(Lemma.)22 b Fh(If)17 b Fo(f)i Fk(2)14 b(K)q Fq(\()p Fp(R)724 1358 y Fl(+)753 1351 y Fq(\))p Fh(,)j(then)268 1453 y Fo(f)5 b Fq(\()p Fo(X)356 1460 y Fn(n)380 1453 y Fq(\))11 b Fk(\000)g Fo(f)5 b Fq(\()p Fo(X)552 1433 y Fn(x)548 1465 y(n)574 1453 y Fq(\))14 b Fk(!)g Fq(0)25 b Fh(a.s.)87 b(for)17 b(al)r(l)26 b Fo(x)14 b Fk(2)g Fp(R)1168 1460 y Fl(+)1197 1453 y Fo(:)118 1555 y Fh(Pr)n(o)n(of.)23 b Fq(1.)e(It)11 b(su\016ces)h(to)h(pro)o(v)o(e)e(the)h(assertion)g (under)h(the)e(h)o(yp)q(othesis)i Fo(X)1493 1562 y Fl(0)1527 1555 y Fq(=)g Fo(x)1606 1562 y Fl(0)1640 1555 y Fk(2)h Fp(R)1729 1562 y Fl(+)1758 1555 y Fq(,)118 1615 y(b)q(ecause)21 b(its)g(general)g(v)m(alidit)o(y)f(then)h(follo)o(ws)g(b)o(y)f(in)o (tegration.)36 b(Comparing)21 b Fo(f)5 b Fq(\()p Fo(X)1713 1597 y Fn(x)1733 1602 y Fe(0)1709 1627 y Fn(n)1753 1615 y Fq(\))118 1675 y(and)23 b Fo(f)5 b Fq(\()p Fo(X)311 1657 y Fn(x)307 1687 y(n)333 1675 y Fq(\))22 b(with)g Fo(f)5 b Fq(\()p Fo(X)583 1657 y Fl(0)579 1687 y Fn(n)604 1675 y Fq(\))22 b(sho)o(ws)g(that)h(in)e(fact)h Fo(x)1094 1682 y Fl(0)1137 1675 y Fq(=)i(0)e(ma)o(y)f(b)q(e)h(assumed.)38 b(Since)118 1735 y(the)17 b(assertion)h(is)g(ob)o(viously)f(true)g(in)g (the)g(transien)o(t)g(case,)h(moreo)o(v)o(er)d(recurrence)h(will)118 1796 y(b)q(e)j(assumed)f(in)g(the)g(sequel.)27 b(According)18 b(to)g(\(2.4\))h(the)f(random)g(w)o(alk)g(\()p Fo(S)1547 1803 y Fn(n)1571 1796 y Fq(\))1590 1803 y Fn(n)p Fm(\025)p Fl(0)1677 1796 y Fq(then)118 1856 y(hits)f(the)g(in)o(terv)m(al)e([)p Fk(\0001)p Fo(;)8 b Fk(\000)p Fo(\015)s Fq(])15 b(in\014nitely)h(often) h(with)f(probabilit)o(y)g(1,)h(where)g Fo(\015)j Fq(will)c(b)q(e)118 1916 y(c)o(hosen)f(at)h(the)f(end)g(of)g(the)g(pro)q(of.)22 b(If)15 b Fo(T)866 1923 y Fl(1)899 1916 y Fo(<)f(T)980 1923 y Fl(2)1013 1916 y Fo(<)g(:)8 b(:)g(:)30 b Fq(are)15 b(the)g(corresp)q(onding)h(hitting)118 1976 y(times,)e(the)i(random)g (v)m(ariables)268 2078 y(\()p Fo(Y)326 2057 y Fm(0)315 2090 y Fn(k)337 2078 y Fo(;)8 b(Z)396 2057 y Fm(0)392 2090 y Fn(k)414 2078 y Fq(\))14 b(:=)f(\()p Fo(Y)559 2085 y Fn(T)580 2091 y Fg(k)605 2085 y Fl(+1)652 2078 y Fo(;)8 b(Z)707 2085 y Fn(T)728 2091 y Fg(k)753 2085 y Fl(+1)800 2078 y Fq(\))87 b(for)25 b Fo(k)16 b Fk(2)e Fp(N)118 2180 y Fq(are)i(again)h(indep)q(enden)o(t)f(and)h(distributed) e(according)i(to)f Fo(\027)s Fq(.)191 2240 y(2.)22 b(Next,)15 b(in)h(view)f(of)268 2341 y(0)f Fo(<)g Fp(P)p Fq(\()p Fo(Z)k(>)13 b Fq(0\))i(=)e(lim)702 2348 y Fn(m)p Fm(!1)815 2341 y Fp(P)p Fq(\()p Fo(Y)25 b(<)14 b(mZ)t Fq(\))118 2443 y(there)i(exists)f Fo(l)g Fk(2)f Fp(N)i Fq(suc)o(h)g(that)268 2545 y Fo(#)d Fq(:=)g Fp(P)p Fq(\()p Fo(Y)26 b(<)13 b(l)q(Z)t Fq(\))h Fo(>)g Fq(0)p Fo(:)118 2647 y Fq(Therefore,)i(b)o(y)f(part)i(1) g(of)f(the)g(pro)q(of,)h(the)f(random)g(time)268 2748 y Fo(T)k Fq(:=)13 b(inf)s Fk(f)p Fo(T)495 2755 y Fn(k)530 2748 y Fq(:)g Fo(Y)585 2755 y Fn(T)606 2761 y Fg(k)631 2755 y Fl(+1)692 2748 y Fo(<)h(l)9 b(Z)801 2755 y Fn(T)822 2761 y Fg(k)846 2755 y Fl(+1)894 2748 y Fk(g)920 2877 y Fq(13)p eop %%Page: 14 15 14 14 bop 118 77 a Fq(ma)o(y)15 b(b)q(e)h(assumed)g(to)g(b)q(e)h (\014nite)e(for)i(all)f Fo(!)g Fk(2)e Fq(\012)p Fo(:)191 137 y Fq(3.)22 b(No)o(w)16 b(for)g(\014xed)g Fo(!)g Fk(2)e Fq(\012)j(and)g Fo(n)d(>)f(T)7 b Fq(\()p Fo(!)r Fq(\))16 b(ob)o(viously)268 239 y Fo(X)312 218 y Fn(x)308 251 y(n)334 239 y Fq(\()p Fo(!)r Fq(\))11 b Fk(\000)g Fo(X)509 218 y Fl(0)505 251 y Fn(n)529 239 y Fq(\()p Fo(!)r Fq(\))j(=)g Fo(x)706 205 y Fi(Q)754 246 y Fl(0)p Fn()f Fq(0,)k(c)o(ho)q(ose)f Fo(\016)g(>)d Fq(0)k(suc)o(h)f(that) 268 1177 y Fk(j)p Fo(f)5 b Fq(\()p Fo(t)348 1184 y Fl(1)367 1177 y Fq(\))11 b Fk(\000)g Fo(f)5 b Fq(\()p Fo(t)513 1184 y Fl(2)533 1177 y Fq(\))p Fk(j)13 b Fo(<)h(")87 b Fq(for)25 b Fk(j)p Fo(t)856 1184 y Fl(1)886 1177 y Fk(\000)11 b Fo(t)954 1184 y Fl(2)974 1177 y Fk(j)i Fo(<)h(\016)o(:)118 1278 y Fq(This)i(yields)268 1380 y Fo(f)5 b Fq(\()p Fo(X)360 1359 y Fn(x)356 1392 y(n)382 1380 y Fq(\()p Fo(!)r Fq(\)\))11 b Fk(\000)g Fo(f)5 b Fq(\()p Fo(X)624 1359 y Fl(0)620 1392 y Fn(n)645 1380 y Fq(\()p Fo(!)r Fq(\)\))13 b(=)h(0)88 b(for)25 b Fo(X)1038 1359 y Fl(0)1034 1392 y Fn(n)1058 1380 y Fq(\()p Fo(!)r Fq(\))14 b Fo(>)g(t;)118 1481 y Fq(in)i(view)f(of)i Fo(X)387 1463 y Fn(x)383 1493 y(n)409 1481 y Fq(\()p Fo(!)r Fq(\))d Fk(\025)g Fo(X)590 1463 y Fl(0)586 1493 y Fn(n)610 1481 y Fq(\()p Fo(!)r Fq(\),)i(and)268 1583 y Fk(j)p Fo(f)5 b Fq(\()p Fo(X)374 1562 y Fn(x)370 1595 y(n)396 1583 y Fq(\()p Fo(!)r Fq(\)\))11 b Fk(\000)g Fo(f)5 b Fq(\()p Fo(X)638 1562 y Fl(0)634 1595 y Fn(n)658 1583 y Fq(\()p Fo(!)r Fq(\)\))p Fk(j)14 b Fo(<)g(")87 b Fq(for)25 b Fo(X)1064 1562 y Fl(0)1060 1595 y Fn(n)1084 1583 y Fq(\()p Fo(!)r Fq(\))14 b Fk(\024)g Fo(t;)118 1684 y Fq(if)i(in)g(addition)268 1786 y Fk(j)p Fo(X)326 1765 y Fn(x)322 1798 y(n)348 1786 y Fq(\()p Fo(!)r Fq(\))11 b Fk(\000)g Fo(X)523 1765 y Fl(0)519 1798 y Fn(n)543 1786 y Fq(\()p Fo(!)r Fq(\))p Fk(j)j Fo(<)f(\016)o(:)118 1887 y Fq(By)i(\(1\))h(this)g(condition)g(is)f(satis\014ed)i(for)f Fo(x)8 b(e)950 1869 y Fm(\000)p Fn(\015)1007 1887 y Fo(l)h(t)14 b(<)f(\016)r Fq(,)i(i.e.)20 b(if)c Fo(\015)i Fq(is)e(c)o(hosen)g (su\016cien)o(tly)118 1947 y(large)p Fo(:)49 b Ff(2)118 2124 y Fp(3.)25 b(Recurrence)17 b(criteria)118 2242 y Fq(The)12 b(su\016cien)o(t)f(conditions)h(for)h(recurrence)e(and)h (transience,)g(giv)o(en)g(in)g(\(2.3\))g(and)h(\(2.4\),)118 2302 y(apply)i(only)f(to)h(extreme)d(cases.)21 b(T)l(o)15 b(deal)f(with)h(concrete)f(situations)h(stronger)g(criteria)118 2362 y(are)f(necessary)l(.)20 b(The)13 b(\014rst)h(relev)m(an)o(t)f (result)g(concerns)h(essen)o(tially)e(the)h(additiv)o(e)f(mo)q(del.)118 2423 y(A)k(sligh)o(t)g(generalization)f(yields:)118 2512 y Fp(\(3.1\))28 b(Theorem.)23 b Fh(F)l(or)17 b Fq(0)d Fo(<)g(\015)i(<)e Fq(1)k Fh(the)g(fol)r(lowing)i(dichotomy)d(holds:)118 2601 y Fq(\(a\))88 b Fo(\027)20 b Fh(is)e(tr)n(ansient,)f(if)268 2725 y Fo(Y)25 b Fk(\025)13 b Fo(\015)90 b(and)e Fq(lim)8 b(inf)799 2732 y Fn(t)p Fm(!1)895 2725 y Fo(t)g Fp(P)p Fq(\(log)i Fo(Z)18 b(>)13 b(t)p Fq(\))h Fo(>)g Fq(log)1333 2691 y(1)p 1331 2713 29 2 v 1331 2759 a Fo(\015)1364 2725 y(;)920 2877 y Fq(14)p eop %%Page: 15 16 15 15 bop 118 77 a Fq(\(b\))87 b Fo(\027)21 b Fh(is)c(r)n(e)n(curr)n (ent,)g(if)268 194 y Fo(Y)25 b Fk(\024)13 b Fo(\015)90 b(and)e Fq(lim)8 b(sup)813 201 y Fn(t)p Fm(!1)910 194 y Fo(t)g Fp(P)p Fq(\(log)h Fo(Z)18 b(>)c(t)p Fq(\))f Fo(<)h Fq(log)1347 160 y(1)p 1345 182 29 2 v 1345 228 a Fo(\015)1378 194 y(:)118 314 y Fh(Pr)n(o)n(of.)23 b Fq(\(a\))17 b(Clearly)l(,)e Fo(Y)25 b Fq(=)14 b Fo(\015)19 b Fq(ma)o(y)14 b(b)q(e)j(assumed.)j(By)c(the)g(h)o(yp)q(othesis)268 431 y(lim)8 b(inf)411 438 y Fn(t)p Fm(!1)507 431 y Fo(t)g Fp(P)p Fq(\(log)i Fo(Z)18 b(>)13 b(s)e Fq(+)g Fo(t)p Fq(\))j Fo(>)g Fq(log)1028 397 y(1)p 1026 419 V 1026 465 a Fo(\015)1146 431 y Fq(for)j(all)24 b Fo(s)14 b Fk(\025)f Fq(0)p Fo(;)118 548 y Fq(whic)o(h)j(for)g(\014xed)g Fo(s)g Fq(implies)e(the)i(existence)e(of)j Fo(\013)d(>)g Fq(1)j(and)f Fo(l)f Fk(2)f Fp(N)i Fq(suc)o(h)g(that)268 662 y Fo(m)j Fq(log)408 629 y(1)p 406 651 V 406 696 a Fo(\015)447 662 y Fp(P)p Fq(\(log)10 b Fo(Z)18 b(>)13 b(s)e Fq(+)g Fo(m)d Fq(log)890 629 y(1)p 889 651 V 889 696 a Fo(\015)922 662 y Fq(\))13 b Fk(\025)h Fo(\013)9 b Fq(log)1125 629 y(1)p 1123 651 V 1123 696 a Fo(\015)1243 662 y Fq(for)25 b Fo(m)14 b Fk(\025)f Fo(l)q(:)118 779 y Fq(Therefore,)j(b)o(y)f(indep)q(endence,)268 874 y Fp(P)p Fq(\()p Fo(X)365 881 y Fn(n)403 874 y Fk(\024)e Fo(e)478 854 y Fn(s)496 874 y Fq(\))42 b Fk(\024)f Fp(P)p Fq(\()694 841 y Fi(T)745 881 y Fn(l)p Fm(\024)p Fn(m)g(l)q(;)118 1549 y Fq(where)i(the)g(last)g(inequalit)o(y)f(mak)o(es)f(use)j(of)f Fo(\013)e Fk(\025)g Fq(1,)i(implying)268 1644 y(\(1)11 b Fk(\000)g Fo(x)p Fq(\))419 1623 y Fn(\013)457 1644 y Fk(\025)j Fq(1)d Fk(\000)g Fo(\013x)87 b Fq(for)25 b(0)15 b Fk(\024)e Fo(x)h Fk(\024)f Fq(1)p Fo(:)118 1738 y Fq(Summation)h(o)o(v)o(er)i Fo(n)g Fq(yields)268 1819 y Fi(P)320 1859 y Fn(n)p Fm(\025)p Fl(0)397 1852 y Fp(P)p Fq(\()p Fo(X)494 1859 y Fn(n)532 1852 y Fk(\024)d Fo(e)607 1832 y Fn(s)625 1852 y Fq(\))h Fk(\024)g Fq(\()p Fo(l)d Fq(+)g(1\))h(+)f(\()p Fo(l)g Fk(\000)g Fq(1\))1047 1832 y Fn(\013)1083 1819 y Fi(P)1135 1859 y Fn(n)p Fm(\025)p Fn(l)1228 1819 y Fq(1)p 1213 1841 54 2 v 1213 1887 a Fo(n)1242 1872 y Fn(\013)1286 1852 y Fo(<)j Fk(1)p Fo(;)118 1963 y Fq(and)j(the)f(transience)g(follo)o(ws)g(from)f(\(2.2a\),)h(b)q (ecause)h Fo(s)f Fq(is)g(arbitrary)l(.)191 2023 y(\(b\))f(Again,)f Fo(Y)25 b Fq(=)14 b Fo(\015)j Fq(ma)o(y)c(b)q(e)i(assumed.)20 b(In)14 b(addition)h Fo(Z)j Fq(ma)o(y)13 b(b)q(e)i(replaced)e(b)o(y)h (0)h(on)118 2083 y(the)i(set)g Fk(f)p Fo(Z)h Fk(\024)d Fo(z)r Fk(g)i Fq(for)g(\014xed)g Fo(z)f(<)f Fk(1)p Fq(,)i(b)q(ecause)g (this)g(is)f(irrelev)m(an)o(t)g(for)h(the)g(h)o(yp)q(othesis)118 2143 y(and)g(c)o(hanges)f(the)f(v)m(alues)h(of)h Fo(X)717 2150 y Fn(n)757 2143 y Fq(b)o(y)e Fo(z)r(=)p Fq(\(1)c Fk(\000)f Fo(\015)s Fq(\))16 b(at)g(most.)21 b(By)15 b(an)h(appropriate)h(c)o(hoice)118 2203 y(of)i Fo(\013)g(<)g Fq(1)g(and)h Fo(z)g(<)f Fk(1)p Fq(,)g(therefore,)g(the)g(follo)o(wing)f (conditions)h(can)h(b)q(e)f(satis\014ed)g(\(in)118 2264 y(the)d(giv)o(en)f(order\):)118 2381 y(\(1\))88 b Fo(t)8 b Fp(P)p Fq(\(log)h Fo(Z)18 b(>)c(t)p Fq(\))f Fk(\024)h Fo(\013)j Fq(log)754 2347 y(1)p 752 2369 29 2 v 752 2415 a Fo(\015)872 2381 y Fq(for)g(all)24 b Fo(t)13 b Fk(\025)h Fq(0)p Fo(;)118 2515 y Fq(\(2\))88 b Fp(P)p Fq(\()p Fo(Z)18 b Fq(=)13 b(0\))i Fk(\025)e Fo(\016)j Fq(:=)d Fo(\015)668 2495 y Fl(1)p Fm(\000)p Fn(\013)738 2515 y Fo(:)118 2620 y Fq(With)j(the)g(abbreviation)268 2725 y Fo(m)311 2732 y Fn(n)348 2725 y Fq(:=)d(log)c Fo(n=)f Fq(log)624 2691 y(1)p 622 2713 V 622 2759 a Fo(\015)675 2725 y Fq(+)19 b(2)87 b(for)26 b Fo(n)14 b Fk(2)g Fp(N)920 2877 y Fq(15)p eop %%Page: 16 17 16 16 bop 118 77 a Fq(this)16 b(yields)f(b)o(y)h(indep)q(endence)268 196 y Fp(P)p Fq(\()p Fo(X)369 175 y Fl(0)365 208 y Fn(n)403 196 y Fk(\024)d Fq(1\))42 b Fk(\025)f Fp(P)p Fq(\()677 162 y Fi(T)729 203 y Fl(0)p Fm(\024)p Fn(m)c Fq([)p Fo(m)8 b Fq(log)1152 643 y(1)p 1151 665 29 2 v 1151 710 a Fo(\015)1195 676 y Fk(\000)i Fq(log)g Fo(n)p Fq(]\)\))376 832 y Fk(\025)457 798 y Fi(Q)504 839 y Fn(m)535 843 y Fg(n)557 839 y Fn()f(m)524 963 y Fn(n)564 956 y Fq(the)j(di\013erence)f([)p Fo(:)8 b(:)g(:)f Fq(])16 b(satis\014es)g(the)g(conditions)268 1077 y([)p Fo(:)8 b(:)g(:)n Fq(])14 b Fo(>)g Fq(0)50 b(and)g Fo(\013)9 b Fq(log)738 1044 y(1)p 736 1066 V 736 1112 a Fo(\015)777 1077 y(=)f Fq([)p Fo(:)g(:)g(:)p Fq(])13 b Fk(\024)h Fq(1)p Fo(:)118 1199 y Fq(The)i(estimation)f(of)513 1166 y Fi(Q)560 1206 y Fl(2)596 1199 y Fq(can)i(b)q(e)f(con)o(tin)o(ued)f(b) o(y)268 1265 y Fi(Q)315 1305 y Fl(2)376 1298 y Fk(\025)457 1265 y Fi(Q)504 1305 y Fn(m)535 1309 y Fg(n)557 1305 y Fn()h Fq(0.)118 2511 y(|)25 b(If)19 b Fo(t)8 b Fp(P)p Fq(\(log)h Fo(Z)23 b(>)c(t)p Fq(\))f Fk(!)h(1)p Fq(,)g(then)g(the)g(sequence)f(\() p Fo(X)1183 2518 y Fn(n)1207 2511 y Fq(\))1226 2518 y Fn(n)p Fm(\025)p Fl(0)1314 2511 y Fq(is)h(transien)o(t)g(whenev)o(er) 118 2571 y Fo(Y)25 b Fq(=)14 b Fo(\015)j(>)c Fq(0;)k(if)e Fo(t)8 b Fp(P)p Fq(\(log)i Fo(Z)18 b(>)13 b(t)p Fq(\))h Fk(!)f Fq(0,)k(then)f(it)g(is)g(recurren)o(t)f(whenev)o(er)g Fo(Y)25 b Fq(=)13 b Fo(\015)k(<)d Fq(1.)191 2632 y(The)e(rest)g(of)h (this)f(section)g(is)f(dev)o(oted)h(to)g(recurrence)f(criteria)g(for)h (the)g(case)g(that)h(the)118 2692 y(underlying)19 b(a\016ne)g(maps)f (are)i(not)f(necessarily)f(con)o(tractions.)31 b(Here)18 b(the)h(p)q(ossibilit)o(y)118 2752 y Fo(S)148 2759 y Fn(n)194 2752 y Fk(!)j Fq(+)p Fk(1)g Fq(is)f(ruled)f(out)i(b)o(y)f (\(2.4\),)i(i.e.)35 b(inf)1027 2759 y Fn(n)p Fm(\025)p Fl(0)1104 2752 y Fo(S)1134 2759 y Fn(n)1180 2752 y Fq(=)23 b Fk(\0001)d Fq(has)j(to)e(b)q(e)h(assumed.)920 2877 y(16)p eop %%Page: 17 18 17 17 bop 118 77 a Fq(Then)14 b(the)f(follo)o(wing)g(auxiliary)g (result)g(can)h(b)q(e)f(obtained)h(b)o(y)f(a)h(partition)g(in)o(to)f (random)118 137 y(blo)q(c)o(ks:)118 226 y Fp(\(3.2\))28 b(Lemma.)22 b Fh(If)28 b Fq(inf)612 233 y Fn(n)p Fm(\025)p Fl(0)689 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y(Moreo)o(v)o(er,)f(de\014ne)271 801 y Fo(Y)311 781 y Fm(0)299 814 y Fn(k)364 801 y Fq(:=)41 b Fo(Y)485 808 y Fn(T)506 814 y Fg(k)q Fd(\000)p Fe(1)572 808 y Fl(+1)627 801 y Fo(:)8 b(:)g(:)g(Y)721 808 y Fn(T)742 814 y Fg(k)763 801 y Fo(;)268 903 y(Z)305 882 y Fm(0)301 915 y Fn(k)364 903 y Fq(:=)41 b Fo(Z)490 910 y Fn(T)511 916 y Fg(k)q Fd(\000)p Fe(1)577 910 y Fl(+1)632 903 y Fo(Y)660 910 y Fn(T)681 916 y Fg(k)q Fd(\000)p Fe(1)747 910 y Fl(+2)803 903 y Fo(:)8 b(:)g(:)f(Y)896 910 y Fn(T)917 916 y Fg(k)949 903 y Fq(+)k Fo(:)d(:)g(:)j Fq(+)g Fo(Z)1149 910 y Fn(T)1170 916 y Fg(k)1191 903 y Fo(:)118 1005 y Fq(Since)16 b(\()p Fo(T)294 1012 y Fn(k)315 1005 y Fq(\))334 1012 y Fn(k)q Fm(\025)p Fl(0)418 1005 y Fq(is)h(a)g(pro)q(cess)h(with)f (indep)q(enden)o(t)g(and)g(iden)o(tically)e(distributed)h(incre-)118 1065 y(men)o(ts,)d(the)h(random)g(v)m(ariables)g(\()p Fo(Y)790 1047 y Fm(0)779 1077 y Fn(k)802 1065 y Fo(;)8 b(Z)861 1047 y Fm(0)857 1077 y Fn(k)879 1065 y Fq(\))p Fo(;)16 b(k)g Fk(2)e 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Fl(0)543 1490 y Fq(=)e(0.)21 b(Then,)16 b(clearly)268 1592 y Fo(X)312 1572 y Fm(0)308 1604 y Fn(k)343 1592 y Fq(=)e Fo(X)439 1572 y Fl(0)435 1604 y Fn(T)456 1610 y Fg(k)564 1592 y Fq(for)25 b Fo(k)16 b Fk(\025)e Fq(0)p Fo(;)118 1694 y Fq(and)j(this)f(implies)e(b)o(y)h(F) l(atou)268 1796 y Fo(x)p 268 1803 28 2 v 41 w Fq(=)42 b Fp(E)p Fq(\(lim)8 b(inf)616 1803 y Fn(n)p Fm(!1)719 1796 y Fo(X)763 1775 y Fl(0)759 1808 y Fn(n)783 1796 y Fq(\))337 1897 y Fk(\024)41 b Fp(E)p Fq(\(lim)8 b(inf)616 1904 y Fn(k)q Fm(!1)716 1897 y Fo(X)760 1877 y Fm(0)756 1910 y Fn(k)778 1897 y Fq(\))337 1999 y Fk(\024)41 b Fq(lim)8 b(inf)560 2006 y Fn(k)q Fm(!1)661 1999 y Fp(E)p Fq(\()p Fo(X)761 1978 y Fm(0)757 2011 y Fn(k)778 1999 y Fq(\))337 2101 y(=)42 b(lim)8 b(inf)560 2108 y Fn(k)q Fm(!1)661 2101 y Fq(\()p Fo(\016)r(\015)732 2080 y Fn(k)q Fm(\000)p Fl(1)809 2101 y Fq(+)j Fo(:)d(:)g(:)i Fq(+)h Fo(\016)r Fq(\))337 2202 y(=)42 b Fo(\016)10 b(=)e Fq(\(1)k Fk(\000)f Fo(\015)s Fq(\))j Fo(<)f Fk(1)p Fo(:)49 b Ff(2)191 2304 y Fq(No)o(w)16 b(the)g(t)o(w)o(o)g(main)f(recurrence)g(criteria)g (can)i(b)q(e)f(deriv)o(ed)f(sim)o(ultaneously:)118 2393 y Fp(\(3.3\))28 b(Theorem.)23 b Fh(If)41 b Fq(inf)664 2400 y Fn(n)p Fm(\025)p Fl(0)741 2393 y Fo(S)771 2400 y Fn(n)822 2393 y Fq(=)27 b Fk(\0001)p Fh(,)f(then)f Fo(\027)j Fh(is)d(r)n(e)n(curr)n(ent)f(in)h(e)n(ach)g(of)f(the)118 2453 y(fol)r(lowing)c(c)n(ases:)118 2555 y Fq(\(a\))88 b Fp(E)p Fq(\()p Fo(Z)17 b Fk(j)d Fo(Y)d Fq(\))j Fk(\024)f Fo(\015)53 b(f)5 b(or)20 b(some)d(\015)g(<)d Fk(1)p Fo(;)118 2694 y Fq(\(b\))87 b Fp(P)p Fq(\()p Fo(Y)26 b Fq(=)13 b(0\))i(=)e(0)88 b Fo(and)f Fp(E)857 2645 y Fi(\020)888 2660 y Fo(Z)p 887 2682 40 2 v 887 2728 a(Y)931 2645 y Fi(\021)969 2694 y Fo(<)14 b Fk(1)p Fo(:)920 2877 y Fq(17)p eop %%Page: 18 19 18 18 bop 118 77 a Fh(Pr)n(o)n(of.)23 b Fq(1.)k(In)18 b(view)f(of)h(\(2.3\))h(the)e(h)o(yp)q(othesis)h Fp(P)p Fq(\()p Fo(Y)29 b Fq(=)16 b(0\))i(=)e(0)j(can)f(b)q(e)g(used)g(in)g(b)q (oth)118 137 y(cases.)32 b(Moreo)o(v)o(er,)19 b(the)h(assumptions)g Fo(\015)j Fk(\024)d Fq(1)g(in)f(\(a\))i(and)f Fp(E)p Fq(\()p Fo(Z)q(=)s(Y)11 b Fq(\))20 b Fk(\024)g Fq(1)g(in)g(\(b\))g(are) 118 197 y(admissible)15 b(simpli\014cations,)g(b)q(ecause)i(a)g(scalar) g(m)o(ultipli)o(cation)d(has)k(the)e(same)g(e\013ect)118 258 y(on)h Fo(X)230 239 y Fl(0)226 270 y Fn(n)250 258 y Fo(;)f(n)e Fk(\025)g Fq(0)p Fo(;)i Fq(as)h(on)f Fo(Z)590 265 y Fn(n)614 258 y Fo(;)g(n)e Fk(2)g Fp(N)p Fo(:)191 318 y Fq(2.)22 b(Consider)16 b(no)o(w)h(\014rst)g(case)f(\(a\).)21 b(Then,)16 b(with)g Fo(T)23 b Fq(as)17 b(in)f(\(3.2\),)268 412 y Fp(E)p Fq(\(1)348 420 y Fm(f)p Fn(T)5 b Fl(=)p Fn(n)p Fm(g)459 412 y Fo(Z)492 419 y Fn(m)526 412 y Fo(Y)554 419 y Fn(m)p Fl(+1)641 412 y Fo(:)j(:)g(:)g(Y)735 419 y Fn(n)758 412 y Fq(\))42 b(=)g Fp(E)p Fq(\()p Fp(E)p Fq(\()p Fo(:)8 b(:)g(:)13 b Fk(j)g Fo(Y)1137 419 y Fl(1)1157 412 y Fo(;)8 b(:)g(:)g(:)g(;)g(Y)1295 419 y Fn(n)1318 412 y Fq(\)\))819 514 y(=)42 b Fp(E)p Fq(\(1)979 522 y Fm(f)p Fn(T)5 b Fl(=)p Fn(n)p Fm(g)1091 514 y Fo(Y)1119 521 y Fn(m)p Fl(+1)1206 514 y Fo(:)j(:)g(:)f(Y)1299 521 y Fn(n)1334 514 y Fp(E)p Fq(\()p Fo(Z)1423 521 y Fn(m)1470 514 y Fk(j)14 b Fo(Y)1526 521 y Fn(m)1560 514 y Fq(\)\))819 616 y Fk(\024)41 b Fp(E)p Fq(\(1)979 623 y Fm(f)p Fn(T)5 b Fl(=)p Fn(n)p Fm(g)1091 616 y Fo(Y)1119 623 y Fn(m)p Fl(+1)1206 616 y Fo(:)j(:)g(:)f(Y)1299 623 y Fn(n)1323 616 y Fq(\))87 b(for)25 b(1)15 b Fk(\024)e Fo(m)h Fk(\024)f Fo(n;)118 710 y Fq(where)25 b(the)f(last)h(equation)g(uses)g(that)g Fk(f)p Fo(T)h Fq(=)20 b Fo(n)p Fk(g)25 b Fq(is)g(measurable)e(with)i (resp)q(ect)f(to)118 770 y Fo(Y)146 777 y Fl(1)166 770 y Fo(;)8 b(:)g(:)g(:)g(;)g(Y)304 777 y Fn(n)345 770 y Fq(and)18 b(\()p Fo(Y)488 777 y Fn(m)521 770 y Fo(;)8 b(Z)576 777 y Fn(m)610 770 y Fq(\))17 b(is)h(indep)q(enden)o(t)e(of)i Fo(Y)1059 777 y Fn(l)1073 770 y Fo(;)e(l)g Fk(6)p Fq(=)g Fo(m)p Fq(.)25 b(In)17 b(view)g(of)h Fo(Y)1531 777 y Fl(1)1559 770 y Fo(:)8 b(:)g(:)g(Y)1653 777 y Fn(n)1693 770 y Fk(\024)15 b Fq(1)118 830 y(on)i Fk(f)p Fo(T)j Fq(=)14 b Fo(n)p Fk(g)i Fq(this)g(yields)268 925 y Fp(E)p Fq(\()p Fo(X)368 904 y Fl(0)364 937 y Fn(T)391 925 y Fq(\))42 b(=)532 892 y Fi(P)584 933 y Fn(n)p Fm(2)p Fj(N)674 892 y Fi(P)726 932 y Fl(1)p Fm(\024)p Fn(m)p Fm(\024)p Fn(n)862 925 y Fp(E)p Fq(\(1)942 933 y Fm(f)p Fn(T)5 b Fl(=)p Fn(n)p Fm(g)1053 925 y Fo(Z)1086 932 y Fn(m)1120 925 y Fo(Y)1148 932 y Fn(m)p Fl(+1)1235 925 y Fo(:)j(:)g(:)g(Y)1329 932 y Fn(n)1352 925 y Fq(\))452 1027 y Fk(\024)532 993 y Fi(P)584 1035 y Fn(m)p Fm(2)p Fj(N)684 993 y Fi(P)736 1034 y Fn(n>m)826 1027 y Fp(E)p Fq(\(1)906 1034 y Fm(f)p Fn(T)d Fl(=)p Fn(n)p Fm(g)1018 1027 y Fo(Y)1046 1034 y Fn(m)p Fl(+1)1133 1027 y Fo(:)j(:)g(:)g(Y)1227 1034 y Fn(n)1250 1027 y Fq(\))20 b(+)f(1)452 1154 y Fk(\024)532 1121 y Fi(P)584 1163 y Fn(m)p Fm(2)p Fj(N)684 1121 y Fi(P)736 1161 y Fn(n>m)826 1154 y Fp(E)p Fq(\(1)906 1162 y Fm(f)p Fn(T)5 b Fl(=)p Fn(n)p Fm(g)1035 1120 y Fq(1)p 1023 1142 49 2 v 1023 1188 a Fo(Y)1051 1195 y Fl(1)1084 1154 y Fo(:)j(:)g(:)1174 1120 y Fq(1)p 1155 1142 62 2 v 1155 1188 a Fo(Y)1183 1195 y Fn(m)1222 1154 y Fq(\))19 b(+)g(1)452 1307 y(=)532 1274 y Fi(P)584 1314 y Fn(m)p Fm(\025)p Fl(0)671 1307 y Fp(E)p Fq(\(1)751 1315 y Fm(f)p Fn(T)5 b(>m)p Fm(g)889 1273 y Fq(1)p 877 1296 49 2 v 877 1341 a Fo(Y)905 1348 y Fl(1)939 1307 y Fo(:)j(:)g(:)1028 1273 y Fq(1)p 1009 1296 62 2 v 1009 1341 a Fo(Y)1037 1348 y Fn(m)1076 1307 y Fq(\))p Fo(;)118 1425 y Fq(where)13 b(the)g(summand)f(1)i(stands)g(\014rst)g(for)913 1391 y Fi(P)965 1433 y Fn(n)p Fm(2)p Fj(N)1055 1425 y Fp(E)p Fq(\(1)1135 1432 y Fm(f)p Fn(T)5 b Fl(=)p Fn(n)p Fm(g)1247 1425 y Fq(\))13 b(and)h(then)f(for)h Fp(E)p Fq(\(1)1631 1432 y Fm(f)p Fn(T)5 b(>)p Fl(0)p Fm(g)1739 1425 y Fq(\).)118 1485 y(The)16 b(\014nal)h(result)f(can)g(b)q(e)g(rewritten)g(as)118 1579 y(\(1\))88 b Fp(E)p Fq(\()p Fo(X)368 1559 y Fl(0)364 1592 y Fn(T)391 1579 y Fq(\))14 b Fk(\024)g Fp(E)p Fq(\()533 1546 y Fi(P)585 1586 y Fl(0)p Fm(\024)p Fn(mm)p Fm(\000)p Fl(1)p Fm(g)947 2072 y Fq(1)p 935 2094 49 2 v 935 2140 a Fo(Y)963 2147 y Fl(1)997 2106 y Fo(:)j(:)g(:)1108 2072 y Fq(1)p 1067 2094 107 2 v 1067 2140 a Fo(Y)1095 2147 y Fn(m)p Fm(\000)p Fl(1)1179 2106 y Fq(\))j Fp(E)1246 2058 y Fi(\020)1276 2072 y Fo(Z)1309 2079 y Fn(m)p 1276 2094 67 2 v 1279 2140 a Fo(Y)1307 2147 y Fn(m)1347 2058 y Fi(\021)452 2262 y Fk(\024)532 2229 y Fi(P)584 2269 y Fn(m)p Fm(\025)p Fl(0)671 2262 y Fp(E)p Fq(\(1)751 2270 y Fm(f)p Fn(T)5 b(>m)p Fm(g)889 2229 y Fq(1)p 877 2251 49 2 v 877 2297 a Fo(Y)905 2304 y Fl(1)939 2262 y Fo(:)j(:)g(:)1028 2229 y Fq(1)p 1009 2251 62 2 v 1009 2297 a Fo(Y)1037 2304 y Fn(m)1076 2262 y Fq(\))p Fo(;)118 2380 y Fq(where)16 b(the)h(last)g(equation)f(uses)h (that)h Fk(f)p Fo(T)j(>)14 b(m)d Fk(\000)g Fq(1)p Fk(g)17 b Fq(is)g(measurable)e(with)i(resp)q(ect)f(to)118 2440 y Fo(Y)146 2447 y Fl(1)166 2440 y Fo(;)8 b(:)g(:)g(:)g(;)g(Y)304 2447 y Fn(m)p Fm(\000)p Fl(1)398 2440 y Fq(and)17 b Fo(Z)526 2447 y Fn(m)560 2440 y Fo(=)s(Y)615 2447 y Fn(m)665 2440 y Fq(is)f(indep)q(enden)o(t)f(of)i Fo(Y)1074 2447 y Fl(1)1094 2440 y Fo(;)8 b(:)g(:)g(:)f(;)h(Y)1231 2447 y Fn(m)p Fm(\000)p Fl(1)1310 2440 y Fq(.)191 2500 y(4.)24 b(No)o(w)17 b(the)g(pro)q(of)h(can)f(b)q(e)h(join)o(tly)e(completed.)21 b(The)c(essen)o(tial)f(to)q(ol)i(is)f(Spitzer's)118 2560 y(iden)o(tit)o(y)l(,)g(whic)o(h)g(will)h(b)q(e)g(applied)g(in)g(its)h (real)f(form)f(\(see)h(e.g.)27 b([28],)19 b(p.395\).)29 b(Letting)118 2621 y(there)16 b Fo(t)d Fk(")h Fq(1)j(leads)f(to)118 2734 y(\(2\))88 b Fp(E)p Fq(\()324 2701 y Fi(P)375 2741 y Fl(0)p Fm(\024)p Fn(m)d Fq(0)p Fh(.)118 1812 y(Pr)n(o)n(of.)23 b Fq(Both)f(assertions)h(are)f(immedi)o(ate)d(consequences)i(of)i (their)e(coun)o(terparts)h(in)118 1873 y(\(3.3\))p Fo(:)50 b Ff(2)118 2047 y Fp(References)141 2158 y Fc(1.)24 b(Alpuim,)d(M.:)26 b(An)20 b(extremal)e(Mark)o(o)o(vian)g(sequence.)32 b(J.)19 b(Appl.)32 b(Prob.)e Fb(26)p Fc(,)20 b(219{232)201 2214 y(\(1989\))141 2273 y(2.)k(Barnsley)l(,)19 b(M.,)e(Elton,)h(J.:)25 b(A)18 b(new)f(class)h(of)g(Mark)o(o)o(v)e(pro)q(cesses)i(for)f(image)h (enco)q(ding.)201 2330 y(Adv.)i(Appl.)h(Prob.)f Fb(20)p Fc(,)15 b(14{32)f(\(1988\))141 2388 y(3.)24 b(Barnsley)l(,)16 b(M.,)f(Elton,)g(J.,)g(Hardin,)h(D.:)k(Recurren)o(t)c(iterated)f (function)i(systems.)j(Con-)201 2445 y(str.)f(Appro)o(x.)h Fb(5)p Fc(,)15 b(3{31)f(\(1989\))141 2503 y(4.)24 b(Boro)o(vk)o(o)o(v,) 11 b(A.:)18 b(On)13 b(the)f(ergo)q(dicit)o(y)h(and)f(stabilit)o(y)h(of) f(the)g(sequence)h Fa(w)1448 2510 y Fn(n)p Fl(+1)1529 2503 y Fc(=)g Fa(f)5 b Fc(\()p Fa(w)1655 2510 y Fn(n)1678 2503 y Fa(;)j(\030)1719 2510 y Fn(n)1741 2503 y Fc(\).)201 2560 y(Theory)15 b(Prob.)20 b(Appl.)h Fb(33)p Fc(,)15 b(595{611)e(\(1989\))141 2619 y(9.)24 b(Dubins,)15 b(L.,)g(F)l (reedman,)f(D.:)19 b(In)o(v)m(arian)o(t)14 b(probabilities)j(for)d (certain)h(Mark)o(o)o(v)e(pro)q(cesses.)201 2676 y(Ann.)21 b(Math.)e(Stat.)g Fb(37)p Fc(,)c(837{848)e(\(1966\))920 2877 y Fq(19)p eop %%Page: 20 21 20 20 bop 119 77 a Fc(10.)23 b(Elton,)18 b(J.:)25 b(An)18 b(ergo)q(dic)g(theorem)f(for)g(iterated)h(maps.)27 b(Ergo)q(dic)18 b(Theory)f(Dyn.)27 b(Syst.)201 133 y Fb(7)p Fc(,)15 b(481{488)e (\(1987\))119 192 y(11.)23 b(Elton,)14 b(J.:)19 b(A)14 b(m)o(ultiplicativ)o(e)j(ergo)q(dic)e(theorem)e(for)h(Lipsc)o(hitz)i (maps.)j(Sto)q(c)o(hastic)14 b(Pro-)201 249 y(cesses)i(Appl.)21 b Fb(34)p Fc(,)15 b(39{47)f(\(1990\))119 307 y(12.)23 b(Elton,)14 b(J.,)h(Y)l(an,)f(Z.:)19 b(Appro)o(ximation)14 b(of)g(measures)h(b)o(y)f(Mark)o(o)o(v)f(pro)q(cesses)h(and)h(homo-)201 364 y(geneous)h(a\016ne)f(iterated)g(function)h(systems.)k(Constr.)f (Appro)o(x.)g Fb(5)p Fc(,)c(69{87)f(\(1989\))119 423 y(17.)23 b(Grincevi)o(\024)-21 b(cius,)19 b(A.:)24 b(On)18 b(the)g(con)o(tin)o(uit)o(y)g(of)f(the)g(distribution)i(of)e(a)h(sum)f (of)g(dep)q(enden)o(t)201 480 y(v)m(ariables)h(connected)f(with)g (indep)q(enden)o(t)h(w)o(alks)e(on)h(lines.)24 b(Theory)17 b(Prob.)22 b(Appl.)j Fb(19)p Fc(,)201 536 y(163{168)13 b(\(1974\))119 595 y(25.)23 b(Lamp)q(erti,)15 b(J.:)20 b(Criteria)15 b(for)f(the)h(recurrence)g(or)f(transience)i(of)e(sto)q (c)o(hastic)h(pro)q(cesses)g(I.)201 652 y(J.)g(Math.)k(Anal.)i(Appl.)g Fb(1)p Fc(,)15 b(314{330)e(\(1960\))119 710 y(26.)23 b(Lamp)q(erti,)f(J.:)29 b(Criteria)21 b(for)e(sto)q(c)o(hastic)h(pro)q (cesses)h(I)q(I:)f(passage{time)g(momen)o(ts.)34 b(J.)201 767 y(Math.)19 b(Anal.)i(Appl.)g Fb(7)p Fc(,)15 b(127{145)e(\(1963\)) 119 825 y(27.)23 b(Lev,)g(G.:)31 b(Semi{Mark)o(o)o(v)20 b(pro)q(cesses)i(of)f(m)o(ultiplication)j(with)d(drift.)38 b(Theory)21 b(Prob.)201 882 y(Appl.)g Fb(17)p Fc(,)15 b(159{164)e(\(1972\))119 941 y(28.)23 b(Lo)o(\022)-21 b(ev)o(e,)23 b(M.:)33 b(Probabilit)o(y)23 b(theory)f(I,)g(4th)g (edition.)42 b(New)22 b(Y)l(ork{Heidelb)q(erg{Berlin:)201 998 y(Springer)16 b(1977)119 1056 y(29.)23 b(Masimo)o(v,)12 b(V.:)19 b(A)13 b(generalized)h(Bernoulli)h(sc)o(heme)f(and)f(its)g (limit)h(distribution.)21 b(Theory)201 1113 y(Prob.)f(Appl.)h Fb(18)p Fc(,)15 b(521{530)e(\(1973\))119 1172 y(30.)23 b(Mukherjea,)17 b(A.:)22 b(Recurren)o(t)17 b(random)f(w)o(alk)h(in)g (nonnegativ)o(e)g(matrices:)23 b(attractors)14 b(of)201 1229 y(certain)20 b(iterated)g(function)g(systems.)33 b(Prob.)f(Theory)20 b(Related)g(Fields)h Fb(91)p Fc(,)g(297{306)201 1285 y(\(1992\))119 1343 y(33.)i(Nummelin,)17 b(E.:)i(General)d (irreducible)i(Mark)o(o)o(v)13 b(c)o(hains)j(and)g(non{negativ)o(e)f (op)q(erators.)201 1400 y(Cam)o(bridge:)20 b(Univ)o(ersit)o(y)c(Press)f (1984)119 1459 y(34.)23 b(Rac)o(hev,)15 b(S.,)f(Samoro)q(dnitsky)l(,)g (G.:)k(Limit)e(la)o(ws)e(for)f(a)h(sto)q(c)o(hastic)g(pro)q(cess)h(and) f(random)201 1516 y(recursion)i(arising)g(in)g(probabilit)o(y)g(mo)q (delling.)22 b(Preprin)o(t)16 b(\(1992\))119 1574 y(35.)23 b(Revuz,)14 b(D.:)k(Mark)o(o)o(v)11 b(c)o(hains,)j(2nd)f(edition.)20 b(Amsterdam{New)13 b(Y)l(ork:)18 b(North{Holland)201 1631 y(1984)119 1690 y(36.)23 b(Ros)o(\023)-21 b(en,)18 b(B.:)25 b(On)18 b(the)g(asymptotic)g(distribution)h(of)f(sums)f(of)h (indep)q(enden)o(t)i(iden)o(tically)201 1747 y(distributed)d(random)e (v)m(ariables.)21 b(Ark.)f(Mat.)e Fb(4)p Fc(,)d(323{332)f(\(1961\))119 1805 y(37.)23 b(Rosen)o(blatt,)14 b(M.:)19 b(Recurren)o(t)14 b(p)q(oin)o(ts)g(and)h(transition)f(functions)g(acting)g(on)g(con)o (tin)o(uous)201 1862 y(functions.)21 b(Z.)14 b(W)l(ahrsc)o(heinlic)o (hk)o(eitstheorie)k(V)l(erw.)i(Geb.)g Fb(30)p Fc(,)15 b(173{183)f(\(1974\))119 1920 y(38.)23 b(Tw)o(eedie,)17 b(R.:)k(Criteria)16 b(for)f(classifying)i(general)g(Mark)o(o)o(v)d(c)o (hains.)22 b(Adv.)g(Appl.)h(Prob.)201 1977 y Fb(8)p Fc(,)15 b(737{771)e(\(1976\))119 2036 y(39.)23 b(V)l(erv)m(aat,)16 b(W.:)22 b(On)16 b(a)g(sto)q(c)o(hastic)h(di\013erence)g(equation)g (and)f(a)g(represen)o(tation)h(of)f(non{)201 2093 y(negativ)o(e)h (in\014nitely)j(divisible)g(random)d(v)m(ariables.)27 b(Adv.)f(Appl.)h(Prob.)f Fb(11)p Fc(,)18 b(750{783)201 2149 y(\(1979\))920 2877 y Fq(20)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF