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b(for)74 b Fo(n)27 b Fk(2)i Fp(N)e Fo(;)236 1154 y Fq(where)37 b Fo(X)602 1169 y Fl(0)677 1154 y Fq(is)e(a)g(real-v)-5 b(alued)34 b(random)g(v)-5 b(ariable)34 b(and)i Fo(H)2364 1169 y Fn(n)2411 1154 y Fo(;)17 b(n)32 b Fk(2)h Fp(N)i Fq(are)h(random)e (trans-)236 1275 y(formations)41 b(from)h Fp(R)g Fq(to)h Fp(R)p Fq(.)g(T)-8 b(o)43 b(obtain)e(the)j(Mark)m(o)m(v)g(prop)s(ert)m (y)g(the)f(v)-5 b(ariables)42 b Fo(X)3477 1290 y Fl(0)3516 1275 y Fo(;)236 1395 y(H)317 1410 y Fl(1)356 1395 y Fo(;)17 b(H)481 1410 y Fl(2)520 1395 y Fo(;)g(:)g(:)g(:)36 b Fq(are)g(supp)s(osed)h(to)f(b)s(e)h(indep)s(enden)m(t,)h(to)e(obtain)f (a)g(stationary)h(transition)236 1516 y(k)m(ernel)j(the)f(v)-5 b(ariables)37 b Fo(H)1193 1531 y Fl(1)1232 1516 y Fo(;)17 b(H)1357 1531 y Fl(2)1396 1516 y Fo(;)g(:)g(:)g(:)37 b Fq(are)h(supp)s(osed)h(to)f(b)s(e)g(iden)m(tically)d(distributed.)59 b(If)236 1636 y(their)39 b(common)e(distribution)g Fo(\027)45 b Fq(is)39 b(carried)f(b)m(y)i(a)e(\014nite)h(family)d(of)j 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Fn(n)1331 2819 y Fq(=)28 b(\()p Fo(X)1554 2834 y Fn(n)p Fm(\000)p Fl(1)1713 2819 y Fq(+)22 b Fo(U)1877 2834 y Fn(n)1924 2819 y Fq(\))1962 2778 y Fl(+)2146 2819 y Fq(for)73 b Fo(n)28 b Fk(2)g Fp(N)236 3039 y Fq(with)48 b(i.i.d.)90 b(v)-5 b(ariables)47 b Fo(U)1241 3054 y Fn(n)1288 3039 y Fo(;)17 b(n)54 b Fk(2)h Fp(N)p Fq(.)91 b(Here,)53 b Fo(X)2129 3054 y Fn(n)p Fm(\000)p Fl(1)2314 3039 y Fq(denotes)d(the)f(w)m(aiting)e(time)g(of)236 3159 y(customer)29 b Fo(n)14 b Fk(\000)g Fq(1)29 b(and)f Fo(U)1142 3174 y Fn(n)1217 3159 y Fq(=)g Fo(S)1381 3174 y Fn(n)p Fm(\000)p Fl(1)1532 3159 y Fk(\000)14 b Fq(\()p Fo(T)1718 3174 y Fn(n)1779 3159 y Fk(\000)g Fo(T)1927 3174 y Fn(n)p Fm(\000)p Fl(1)2065 3159 y Fq(\))28 b(is)g(the)h(balance) f(b)s(et)m(w)m(een)j(his)d(service)236 3280 y(time)j(and)i(the)g(arriv) -5 b(al)30 b(times)i(of)g(customers)h Fo(n)23 b Fk(\000)f Fq(1)33 b(and)f Fo(n)p Fq(.)236 3400 y(\(2\))41 b(A)33 b(\\sa)m(vings)g(pro)s(cess")g(is)g(de\014ned)h(b)m(y)1191 3620 y Fo(X)1272 3635 y Fn(n)1347 3620 y Fq(=)27 b Fo(U)1516 3635 y Fn(n)1564 3620 y Fo(X)1645 3635 y Fn(n)p Fm(\000)p Fl(1)1804 3620 y Fq(+)22 b Fo(V)1959 3635 y Fn(n)2130 3620 y Fq(for)74 b Fo(n)28 b Fk(2)g Fp(N)236 3840 y Fq(with)43 b(i.i.d.)74 b(v)-5 b(ariables)42 b(\()p Fo(U)1253 3855 y Fn(n)1300 3840 y Fo(;)17 b(V)1401 3855 y Fn(n)1448 3840 y Fq(\))p Fo(;)g(n)45 b Fk(2)i Fp(N)p Fq(.)75 b(Here,)46 b Fo(X)2287 3855 y Fn(n)p Fm(\000)p Fl(1)2468 3840 y Fq(denotes)e(the)g(balance)f(of)g(a)236 3961 y(sa)m(vings)34 b(accoun)m(t)g(at)g(time)e Fo(n)23 b Fk(\000)g Fq(1,)33 b Fo(U)1635 3976 y Fn(n)1716 3961 y Fq(the)h(in)m(terest)16 b(/)g(in\015ation)32 b(rate)i(during)f(p)s(erio)s(d)f Fo(n)236 4081 y Fq(and)c Fo(V)478 4096 y Fn(n)552 4081 y Fq(the)g(dep)s(osit)g(made)f(at)g(time)f Fo(n)i Fq(\(this)g(and)f(a)h (surv)m(ey)h(of)f(other)f(a\016ne)i(recursions)236 4201 y(in)j(biology)-8 b(,)31 b(economics,)h(ph)m(ysics)i(etc.)44 b(can)33 b(b)s(e)g(found)f(in)g(V)-8 b(erv)j(aat)33 b([29]\).)236 4322 y(\(3\))41 b(An)33 b(\\exc)m(hange)h(pro)s(cess")g(is)e(de\014ned) i(b)m(y)1097 4542 y Fo(X)1178 4557 y Fn(n)1253 4542 y Fq(=)27 b(\()p Fo(X)1475 4557 y Fn(n)p Fm(\000)p Fl(1)1634 4542 y Fk(\000)c Fo(U)1800 4557 y Fn(n)1847 4542 y Fq(\))f Fk(_)h Fo(V)2053 4557 y Fn(n)2224 4542 y Fq(for)74 b Fo(n)28 b Fk(2)g Fp(N)236 4762 y Fq(with)35 b(i.i.d.)49 b(v)-5 b(ariables)34 b(\()p Fo(U)1212 4777 y Fn(n)1259 4762 y Fo(;)17 b(V)1360 4777 y Fn(n)1406 4762 y Fq(\))p Fo(;)g(n)32 b Fk(2)g Fp(N)p Fq(.)50 b(Here,)36 b Fo(X)2182 4777 y Fn(n)p Fm(\000)p Fl(1)2354 4762 y Fq(is)f(the)g(utilit)m(y)e(of) i(some)f(equip-)236 4882 y(men)m(t)j(in)g(use)h(at)f(time)f Fo(n)25 b Fk(\000)h Fq(1,)38 b Fo(U)1496 4897 y Fn(n)1581 4882 y Fq(its)e(loss)h(in)g(utilit)m(y)e(during)i(p)s(erio)s(d)f Fo(n)p Fq(,)i(and)g Fo(V)3324 4897 y Fn(n)3408 4882 y Fq(the)236 5002 y(utilit)m(y)33 b(of)i(a)f(new)i(equipmen)m(t)g(a)m(v) -5 b(ailable)32 b(at)j(time)e Fo(n)j Fq(\(for)e(this)h(and)g(related)f (examples)236 5123 y(see,)g(for)e(instance,)h(Helland)e(and)i(Nilsen)f ([13]\).)383 5243 y(The)e(\014rst)g(example)f(di\013ers)g(from)f(the)i (second)h(and)e(the)h(third)f(one)h(b)m(y)g(a)f(particular)236 5364 y(c)m(haracteristic:)40 b(the)25 b(state)h(0)f(is)g(a)g (regeneration)f(p)s(oin)m(t,)i(and)g(th)m(us)g(the)g(queuing)f(pro)s (cess)236 5484 y(\014ts)33 b(in)m(to)f(the)h(Do)s(eblin-Harris)c (theory)k(for)f(not)h(necessarily)g(discrete)g(Mark)m(o)m(v)h(c)m (hains.)1865 5753 y(1)p eop %%Page: 2 3 2 2 bop 236 154 a Fq(Since)41 b(the)g(dominating)e(measure,)k(whose)f (existence)g(is)f(p)s(ostulated)f(in)g(this)h(theory)-8 b(,)236 274 y(fails)40 b(to)i(exist)g(in)f(general,)j(there)e(are)g (attempts,)j(for)c(instance)h(b)m(y)h(Rosen)m(blatt)f([27])236 395 y(or)37 b(Tw)m(eedie)h([28],)f(to)g(classify)f(Mark)m(o)m(v)i(c)m (hains)f(b)m(y)g(stressing)h(the)f(top)s(ological)c(struc-)236 515 y(ture)f(of)g(the)g(state)g(space.)44 b(Ev)m(en)34 b(under)e(con)m(tin)m(uit)m(y)g(assumptions)g(on)g(the)g(underlying)236 636 y(k)m(ernel,)41 b(ho)m(w)m(ev)m(er,)h(this)d(results)f(in)g(a)g(v) -5 b(ariet)m(y)39 b(of)f(notions)g(of)g(transience)h(and)f(n)m(ull)g (or)236 756 y(p)s(ositiv)m(e)i(recurrence,)k(b)s(eing)c(th)m(us)h(less) f(con)m(vincing)g(than)h(the)f(classical)f(notions)h(for)236 876 y(discrete)h(Mark)m(o)m(v)h(c)m(hains.)66 b(The)42 b(presen)m(t)g(pap)s(er,)g(therefore,)h(emphasizes)d(the)h(order)236 997 y(structure)30 b(of)d(the)i(state)f(space,)j(motiv)-5 b(ated)26 b(b)m(y)j(t)m(w)m(o)g(observ)-5 b(ations)28 b(holding)e(for)i(v)-5 b(arious)236 1117 y(applications.)41 b(First,)30 b(the)g(transformations)f Fo(h)f Fq(:)f Fo(x)h Fk(!)g Fq(\()p Fo(x)17 b Fq(+)g Fo(u)p Fq(\))2591 1081 y Fl(+)2650 1117 y Fq(,)31 b Fo(h)c Fq(:)h Fo(x)g Fk(!)g Fo(ux)17 b Fq(+)g Fo(v)t Fq(,)30 b(and)236 1237 y Fo(h)h Fq(:)h Fo(x)f Fk(!)g Fq(\()p Fo(x)24 b Fk(\000)g Fo(u)p Fq(\))e Fk(_)i Fo(v)39 b Fq(app)s(earing)33 b(in)h(the)h(examples)f(ab) s(o)m(v)m(e)i(are)e(all)f(order-preserving)236 1358 y(\(observing)k Fo(U)781 1373 y Fn(n)863 1358 y Fk(\025)e Fq(0)i(in)f(\(2\)\).)55 b(Second,)39 b(in)d(these)i(examples)f(the)g(prop)s(er)g(state)g(space) 236 1478 y(is)c Fp(R)419 1493 y Fl(+)511 1478 y Fq(\(observing)g Fo(V)1043 1493 y Fn(n)1119 1478 y Fk(\025)d Fq(0)j(in)f(\(2\))h(and)h (\(3\)\).)45 b(As)34 b(it)e(turns)i(out,)g(accepting)f(these)i(t)m(w)m (o)236 1599 y(restrictions)27 b(presen)m(ts)j(an)e(appropriate)f (compromise)f(in)h(order)h(to)f(obtain)g(a)g(satisfying)236 1719 y(theory)33 b(as)g(w)m(ell)f(as)h(substan)m(tial)f(applications.) 383 1839 y(While)f(there)i(exists)g(an)g(extensiv)m(e)h(literature)d (based)i(on)g(the)g(metric)e(structure)j(of)236 1960 y(the)42 b(state)g(space)h(b)m(y)f(requiring,)h(for)e(instance,)j(an)d (\\a)m(v)m(erage)i(con)m(tractivit)m(y")e(of)g(the)236 2080 y(underlying)29 b(transformations,)g(there)h(are)g(only)f(a)g(few) h(pap)s(ers)h(with)e(sp)s(ecial)f(emphasis)236 2200 y(on)22 b(the)g(order)h(structure.)41 b(The)23 b(earliest)e(one)h(dates)g(bac)m (k)h(to)f(Dubins)g(and)g(F)-8 b(reedman)21 b([7],)236 2321 y(who)31 b(limit,)d(ho)m(w)m(ev)m(er,)33 b(their)d(in)m(v)m (estigations)g(to)g(the)h(compact)f(state)h(space)g([0,1].)43 b(This)236 2441 y(restriction)29 b(is)g(giv)m(en)h(up)g(in)f(Y)-8 b(aha)m(v)30 b([30],)g(who)g(studies)g(conca)m(v)m(e)h(increasing)e (mappings)236 2562 y(from)34 b Fp(R)553 2577 y Fl(+)646 2562 y Fq(to)g Fp(R)851 2577 y Fl(+)910 2562 y Fq(.)50 b(Extending)35 b(the)g(state)g(space)h(further)f(to)f Fp(R)p Fq(,)h(Bhattac)m(hara)m(y)m(a)g(and)236 2682 y(W)-8 b(a)m(ymire)40 b(in)h(Section)f(I)s(I.14)h(of)f([4])h(tak)m(e)h(up)f(a) g(\\splitting")d(condition)i(from)f([7].)69 b(In)236 2802 y(all)30 b(this)h(treatmen)m(ts)h(the)g(main)e(in)m(terest)i (concerns)h(the)f(existence)h(and)f(uniqueness)i(of)236 2923 y(stationary)f(distributions.)45 b(This)33 b(holds)g(as)h(w)m(ell) f(for)g(Brandt)g(et)h(al.,)e(who)i(consider)g(in)236 3043 y(Section)f(1.3)d(of)g([5])g(order-preserving)h(mappings)e(in)h (partially)e(ordered)j(P)m(olish)f(spaces)236 3164 y(requiring,)i(ho)m (w)m(ev)m(er,)j(appropriate)d(compactness)h(and)g(con)m(traction)f (prop)s(erties.)383 3284 y(T)-8 b(o)34 b(see)i(\015uctuation)e(asp)s (ects)h(to)g(b)s(e)f(as)h(in)m(teresting)f(as)h(equilibrium)c(results,) k(con-)236 3404 y(sider)25 b(an)g(exc)m(hange)h(pro)s(cess)g(with)e (deterministic)f(loss)i(of)f(utilit)m(y)-8 b(,)25 b(sa)m(y)g Fo(U)2936 3419 y Fn(n)3011 3404 y Fq(=)j(1.)40 b(Then)26 b(it)236 3525 y(is)h(easily)g(established)h(that)f(a)g(\(unique\))h (stationary)f(distribution)f(exists)i(if)f(and)g(only)g(if)236 3645 y(the)g(utilit)m(y)e Fo(V)742 3660 y Fn(n)816 3645 y Fq(of)h(the)h(substitute)h(has)f(a)f(\014nite)h(exp)s(ectation.)41 b(This)27 b(fails,)g(for)f(instance,)236 3765 y(if)32 b Fo(V)383 3780 y Fn(n)462 3765 y Fq(has)h(the)g(densit)m(y)669 3973 y Fo(f)717 3988 y Fl(1)756 3973 y Fq(\()p Fo(x)p Fq(\))28 b(=)g(\()p Fo(x)22 b Fq(+)g(1\))1319 3932 y Fm(\000)p Fl(2)1496 3973 y Fq(or)83 b Fo(f)1714 3988 y Fl(2)1754 3973 y Fq(\()p Fo(x)p Fq(\))28 b(=)f(2)p Fo(x)p Fq(\()p Fo(x)c Fq(+)f(1\))2421 3932 y Fm(\000)p Fl(3)2639 3973 y Fq(for)74 b Fo(x)28 b Fk(\025)g Fq(0)17 b Fo(:)236 4180 y Fq(Due)29 b(to)f Fo(f)600 4195 y Fl(1)640 4180 y Fq(\()p Fo(x)p Fq(\))g Fk(\024)g Fo(f)952 4195 y Fl(2)992 4180 y Fq(\()p Fo(x)p Fq(\))g Fk(\024)g Fq(2)p Fo(f)1353 4195 y Fl(1)1392 4180 y Fq(\()p Fo(x)p Fq(\))h(for)f Fo(x)g Fk(\025)h Fq(1,)g(in)f(b)s(oth)g(cases)j Fo(V)2626 4195 y Fn(n)2701 4180 y Fq(b)s(eha)m(v)m(es)g(similarly)25 b(as)236 4300 y(far)j(as)g(it)f(concerns)i(the)f(existence)i(of)d (momen)m(ts.)42 b(Nev)m(ertheless,)31 b(there)e(is)e(a)h(signi\014can)m (t)236 4421 y(di\013erence:)62 b(while)40 b(in)h(the)h(second)g(case)g (\()p Fo(X)1948 4436 y Fn(n)1995 4421 y Fq(\))2033 4436 y Fn(n)p Fm(\025)p Fl(0)2212 4421 y Fq(escap)s(es)g(to)f(in\014nit)m(y) -8 b(,)43 b(the)f(pro)s(cess)236 4541 y(is)36 b(uniformly)e (distributed)h(on)h Fp(R)1511 4556 y Fl(+)1606 4541 y Fq(in)f(the)i(\014rst)f(case)h(\(this)f(example)f(app)s(ears)i(in)e (the)236 4661 y(con)m(text)f(of)e(\(2.4\),)g(\(6.3\))p Fk(\000)p Fq(\(6.4\),)g(\(9.7\)\).)383 4782 y(A)47 b(detailed)f(surv)m (ey)k(of)d(the)h(principal)d(results)j(of)f(the)h(presen)m(t)h(pap)s (er)f(app)s(ears)236 4902 y(disp)s(ensable,)40 b(b)s(ecause)f(the)f (headings)g(of)f(the)i(di\013eren)m(t)f(sections)g(pro)m(vide)h(a)e (\014rst)h(in-)236 5023 y(formation)c(ab)s(out)j(the)g(con)m(ten)m(ts.) 58 b(Instead,)39 b(the)e(main)e(feature)i(will)d(b)s(e)j(summarized)236 5143 y(as)30 b(follo)m(ws:)40 b(the)30 b(order-preserving)g(random)e (dynamical)g(systems)j(as)e(considered)h(here)236 5263 y(represen)m(t)41 b(one)e(of)g(the)g(b)s(est)g(suited)g(mo)s(dels)f (for)g(extending)i(discrete)f(Mark)m(o)m(v)h(c)m(hain)236 5384 y(theory)26 b(to)e(an)h(uncoun)m(table)g(state)h(space.)42 b(That)25 b(is)f(b)s(ecause)j(the)e(fundamen)m(tal)f(criteria)236 5504 y(for)30 b(p)s(ositiv)m(e)16 b(/)g(n)m(ull)29 b(recurrence)j(or)e (transience)h Fk(\000)g Fq(b)m(y)g(means)f(of)g(the)h(n-step)g (transition)1865 5753 y(2)p eop %%Page: 3 4 3 3 bop 236 154 a Fq(k)m(ernels)28 b(resp.)43 b(the)27 b(p)s(oten)m(tial)f(k)m(ernel,)i(b)m(y)g(means)f(of)g(hitting)e (probabilities)f(resp.)43 b(mean)236 274 y(passage)27 b(times,)f(or)g(b)m(y)h(means)e(of)h(a)f(unique)i(in)m(v)-5 b(arian)m(t)24 b(measure)i Fk(\000)h Fq(all)d(\014nd)i(their)f(coun-) 236 395 y(terpart)36 b(in)f(the)i(presen)m(t)g(pap)s(er)f(\(for)f(a)h (\014rst)g(orien)m(tation)f(see)i(the)f(remarks)g(follo)m(wing)236 515 y(\(6.5\),)c(\(9.8\),)g(\(10.1\))g(and)h(preceding)g(\(11.4\)\).) 383 636 y(Finally)-8 b(,)28 b(a)j(historical)e(remark)i(is)g(in)f (order.)43 b(This)32 b(w)m(ork)g(originated)d(from)h(a)h(three-)236 756 y(part)i(pap)s(er)f(dev)m(oted)i(to)f(the)g(sp)s(ecial)e(case)j(of) e(recursions)770 952 y Fo(X)851 967 y Fn(n)926 952 y Fq(=)27 b Fo(U)1095 967 y Fn(n)1143 952 y Fo(X)1224 967 y Fn(n)p Fm(\000)p Fl(1)1383 952 y Fq(+)22 b Fo(V)1538 967 y Fn(n)1668 952 y Fq(with)82 b Fo(U)2006 967 y Fn(n)2054 952 y Fo(;)17 b(V)2155 967 y Fn(n)2229 952 y Fk(\025)28 b Fq(0)124 b(for)74 b Fo(n)28 b Fk(2)g Fp(N)17 b Fo(:)236 1149 y Fq(This)25 b(a\016ne)g(mo)s(del,)g(whic)m(h)h(for)e(constan)m(t) h Fo(U)1878 1164 y Fn(n)1950 1149 y Fq(con)m(tains)g(in)f(particular)f (\014rst-order)i(auto-)236 1269 y(regressiv)m(e)34 b(pro)s(cesses,)g (is)d(of)h(sp)s(ecial)f(imp)s(ortance,)g(b)s(ecause)i(it)e(ma)m(y)h (serv)m(e)i(to)d(appro)m(x-)236 1390 y(imate)f(more)g(complex)g (situations)g(b)m(y)i(linearization.)40 b(During)29 b(the)j(refereeing) f(pro)s(cess)236 1510 y(of)h([16],)g(ho)m(w)m(ev)m(er,)j(it)c(b)s (ecame)h(clear)f(that)h(most)g(results)g(rely)g(on)g(the)h(top)s (ological)28 b(and)236 1630 y(order)40 b(structure)h(of)e(the)h(state)g (space)h(only)e(and)h(mak)m(e)g(no)f(use)i(of)e(the)h(linear)e(struc-) 236 1751 y(ture.)47 b(Moreo)m(v)m(er,)35 b(it)d(turned)i(out)g(that)f 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Fq(b)s(e)g(the)h(random)e(w)m(alk)i(with)e(incremen)m(ts)i(log)17 b Fo(U)2527 2608 y Fn(n)2591 2593 y Fq(\()p Fk(\025)28 b(\0001)p Fq(\).)43 b(Then,)32 b(under)236 2714 y(a)42 b(w)m(eak)h(b)s(oundedness)i(condition)40 b(on)i Fo(V)1802 2729 y Fn(n)1891 2714 y Fq(\(and)g(excluding)g(the)h(degenerate)g(case) g(of)236 2834 y(a)36 b(common)e(\014xed)j(p)s(oin)m(t)e(of)g(the)i (underlying)e(mappings\),)h(the)g(follo)m(wing)d(tric)m(hotom)m(y)236 2954 y(holds:)40 b(the)27 b(pro)s(cess)g(\()p Fo(X)1140 2969 y Fn(n)1187 2954 y Fq(\))1225 2969 y Fn(n)p Fm(\025)p Fl(0)1388 2954 y Fq(is)f(p)s(ositiv)m(e)f(recurren)m(t)j(resp.)42 b(n)m(ull)25 b(recurren)m(t)j(resp.)42 b(tran-)236 3075 y(sien)m(t,)d(if)d(the)i(asso)s(ciated)f(random)f(w)m(alk)h(\()p Fo(S)1928 3090 y Fn(n)1975 3075 y Fq(\))2013 3090 y Fn(n)p Fm(\025)p Fl(0)2187 3075 y Fq(div)m(erges)h(to)f Fk(\0001)h Fq(resp.)58 b(oscillates)236 3195 y(b)s(et)m(w)m(een)35 b Fk(\0001)e Fq(and)f(+)p Fk(1)g Fq(resp.)45 b(div)m(erges)33 b(to)f(+)p Fk(1)p 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b(b)s(e)g(em)m(b)s(edded)h(in)m(to)d Fk(H)q Fq([)p Fo(E)6 b Fq(],)37 b(the)f(mapping)e(\()p Fo(x;)17 b(h)p Fq(\))33 b Fk(!)g Fo(h)p Fq(\()p Fo(x)p Fq(\))236 5242 y(is)f(con)m(tin)m(uous,)i(to)s(o.)383 5384 y Fk(M)p Fq(\()p Fo(E)6 b Fq(\))40 b(denotes)i(the)g(class)f(of)f(lo)s(cally)f (\014nite)h(measures)i(on)f Fo(E)47 b Fq(and)41 b Fk(M)3174 5399 y Fl(1)3213 5384 y Fq(\()p Fo(E)6 b Fq(\))41 b(the)236 5504 y(sub)s(class)50 b(consisting)f(of)g(probabilit)m(y)e(measures.)95 b(If)49 b Fo(\026f)60 b Fq(denotes)51 b(the)f Fo(\026)p Fk(\000)p Fq(in)m(tegral)1865 5753 y(3)p eop %%Page: 4 5 4 4 bop 236 154 a Fq(of)47 b(a)g(function)g Fo(f)11 b Fq(,)51 b(then)d Fk(M)p Fq(\()p Fo(E)6 b Fq(\))47 b(is)g(endo)m(w)m(ed) i(with)e(the)h(v)-5 b(ague)48 b(\(w)m(eak)3030 118 y Fm(\003)3071 154 y Fq(\))f(top)s(ology)-8 b(,)236 274 y(i.e.)74 b(the)43 b(initial)c(top)s(ology)i(with)h(resp)s(ect)i(to)f (the)g(mappings)e Fo(\026)k Fk(!)g Fo(\026f)5 b(;)33 b(f)56 b Fk(2)45 b(K)q Fq(\()p Fo(E)6 b Fq(\);)236 395 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y(this)i(space)i(a)e(top)s(ological)d(semigroup)i(itself.)75 b(Corresp)s(onding)43 b(p)s(o)m(w)m(ers)i(are)f(simply)236 1237 y(denoted)34 b(b)m(y)f Fo(\027)794 1201 y Fn(n)842 1237 y Fq(,)f(i.e.)560 1337 y Fi(Z)659 1455 y Fo(f)11 b Fq(\()p Fo(h)p Fq(\()p Fo(x)p Fq(\)\))p Fo(\027)1035 1413 y Fn(n)1083 1455 y Fq(\()p Fo(dh)p Fq(\))27 b(=)1397 1337 y Fi(Z)1497 1455 y Fo(:)17 b(:)g(:)1628 1337 y Fi(Z)1727 1455 y Fo(f)11 b Fq(\()p Fo(h)1880 1470 y Fl(1)1942 1455 y Fk(\016)22 b Fo(:)17 b(:)g(:)k Fk(\016)h Fo(h)2278 1470 y Fn(n)2325 1455 y Fq(\()p Fo(x)p Fq(\)\))17 b Fo(\027)6 b Fq(\()p Fo(dh)2710 1470 y Fl(1)2750 1455 y Fq(\))17 b Fo(:)g(:)g(:)f(\027)6 b Fq(\()p Fo(dh)3135 1470 y Fn(n)3182 1455 y Fq(\))236 1681 y(for)33 b Fo(x)c Fk(2)g Fo(E)6 b(;)33 b(f)39 b Fk(2)29 b(C)6 b Fq(\()p Fo(E)g Fq(\))29 b(and)g Fo(n)g Fk(2)g Fp(N)p Fq(,)j(while)h Fo(\027)1951 1645 y Fl(0)2024 1681 y Fq(is)f(the)i(unit)e(measure)i Fo(")2924 1696 y Fn(h)3001 1681 y Fq(with)f Fo(h)g Fq(b)s(eing)236 1802 y(the)28 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Fk(L)p Fq(\()p Fo(X)1272 3733 y Fl(0)1312 3718 y Fq(\).)42 b(Here,)32 b(the)e(primary)f(comp)s(onen)m(t)h(is)g Fo(\027)6 b Fq(,)31 b(and)g(all)d(notions)236 3839 y(to)g(b)s(e)g(de\014ned)i (will)25 b(dep)s(end)30 b(on)e(that)f(distribution.)40 b(This)29 b(dep)s(endence)h(will,)d(ho)m(w)m(ev)m(er,)236 3959 y(b)s(e)33 b(suppressed)i(in)d(the)h(related)f(notations,)g(b)s (ecause)i Fo(\027)39 b Fq(is)32 b(supp)s(osed)i(to)e(b)s(e)h(\014xed.) 383 4079 y(As)46 b(usual,)k(the)c(initial)d(la)m(w)j(is)f(largely)g(of) h(secondary)h(imp)s(ortance)e(only)-8 b(.)84 b(If)46 b(in)236 4200 y(particular)31 b Fo(X)767 4215 y Fl(0)834 4200 y Fq(=)d Fo(x)p Fq(,)33 b(this)f(will)e(b)s(e)j(expressed)i(b)m(y) f(the)f(notation)e(\()p Fo(X)2818 4164 y Fn(x)2810 4224 y(n)2862 4200 y Fq(\))2900 4215 y Fn(n)p Fm(\025)p Fl(0)3037 4200 y Fq(,)h(i.e.)828 4392 y Fo(X)917 4351 y Fn(x)909 4417 y(n)989 4392 y Fq(=)27 b Fo(H)1173 4407 y Fn(n)1242 4392 y Fk(\016)22 b Fo(:)17 b(:)g(:)22 b Fk(\016)g Fo(H)1604 4407 y 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Fq(giv)m(en)32 b(b)m(y)1007 5449 y Fo(P)14 b(f)d Fq(\()p Fo(x)p Fq(\))27 b(=)1405 5332 y Fi(Z)1451 5521 y Fm(H)p Fl([)p Fn(E)t Fl(])1626 5449 y Fo(f)11 b Fq(\()p Fo(h)p Fq(\()p Fo(x)p Fq(\)\))17 b Fo(\027)6 b Fq(\()p Fo(dh)p Fq(\))125 b(for)73 b Fo(x)28 b Fk(2)g Fo(E)1865 5753 y Fq(4)p eop %%Page: 5 6 5 5 bop 236 154 a Fq(and)33 b(a)f(measure)h Fo(\026)f Fq(on)h Fo(E)38 b Fq(in)m(to)32 b Fo(\026P)46 b Fq(giv)m(en)33 b(b)m(y)839 390 y Fo(\026P)14 b Fq(\()p Fo(B)5 b Fq(\))27 b(=)1260 273 y Fi(Z)1306 461 y Fn(E)1383 390 y Fo(\027)6 b Fq(\()p Fo(h)p Fq(\()p Fo(x)p Fq(\))28 b Fk(2)g Fo(B)5 b Fq(\))17 b Fo(\026)p Fq(\()p Fo(dx)p Fq(\))124 b(for)74 b Fo(B)32 b Fk(2)d(B)s Fq(\()p Fo(E)6 b Fq(\))17 b Fo(;)236 630 y Fq(whic)m(h)33 b(for)f(a)h Fo(\033)t Fk(\000)p Fq(\014nite)f(measure)h Fo(\026)g Fq(b)m(y)g(F)-8 b(ubini)31 b(equals)789 866 y Fo(\026P)14 b Fq(\()p Fo(B)5 b Fq(\))27 b(=)1210 748 y Fi(Z)1256 937 y Fm(H)p Fl([)p Fn(E)t Fl(])1432 866 y Fo(\026)p Fq(\()p Fo(h)p Fq(\()p Fo(x)p Fq(\))h 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Fl(+)2108 2638 y Fq(=)g Fo(E)2315 2601 y Fm(0)2386 2638 y Fq(let)47 b Fo(h)2598 2653 y Fl(+)2710 2638 y Fk(2)54 b(H)q Fq([)p Fo(E)6 b Fq(])47 b(b)s(e)h(strictly)236 2758 y(increasing)25 b(with)g Fo(h)956 2773 y Fl(+)1015 2758 y Fq(\()p Fo(x)p Fq(\))j Fo(>)g(x)e Fq(for)e(all)g Fo(x)k Fk(2)g Fo(E)k Fq(and)25 b(de\014ne)i Fo(h)2423 2773 y Fm(\000)2510 2758 y Fk(2)h(H)q Fq([)p Fo(E)6 b Fq(])26 b(b)m(y)g Fo(h)3031 2773 y Fm(\000)3090 2758 y Fq(\()p Fo(x)p Fq(\))i(=)g(0)d(for)236 2878 y Fo(x)j Fk(\024)g Fo(h)480 2893 y Fl(+)540 2878 y Fq(\(0\))k(and)g Fo(h)942 2893 y Fm(\000)1001 2878 y Fq(\()p Fo(x)p Fq(\))c(=)g Fo(h)1320 2842 y Fm(\000)p Fl(1)1320 2903 y(+)1414 2878 y Fq(\()p Fo(x)p Fq(\))33 b(otherwise.)44 b(Then)33 b(a)f(distribution) e Fo(\027)35 b Fk(2)28 b(N)15 b Fq([)p Fo(E)6 b Fq(])32 b(with)236 2999 y(supp)s(ort)38 b Fo(N)46 b Fq(=)35 b Fk(f)p Fo(h)943 3014 y Fm(\000)1002 2999 y Fo(;)17 b(h)1102 3014 y Fl(+)1161 2999 y Fk(g)37 b Fq(is)g(conjugate)g(to)g(the)h (distribution)d 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b(t)p Fq([)36 b(and)h Fo(E)1983 5199 y Fl(2)2057 5184 y Fq(=)d Fo(E)d Fk(n)25 b Fo(E)2417 5199 y Fl(1)2493 5184 y Fq(suc)m(h)38 b(that)f Fo(h)p Fq([)p Fo(E)3088 5199 y Fn(i)3116 5184 y Fq(])e Fk(\032)g Fo(E)3362 5199 y Fn(i)3427 5184 y Fq(for)236 5305 y Fo(\027)6 b Fk(\000)p Fq(almost)37 b(all)e Fo(h)h Fk(2)g(H)q Fq([)p Fo(E)6 b Fq(])38 b(and)f Fo(i)f Fq(=)g(1)p Fo(;)17 b Fq(2.)57 b(With)37 b(0)e(=)h(min)15 b Fo(E)44 b Fq(as)37 b(reference)i(state)f(the)236 5425 y(corresp)s(onding)33 b(\\class")f(can)h(b)s(e)g(c)m(haracterized)g(explitely)f(as)g(w)m(ell) g(as)h(implicitly:)1865 5753 y(5)p eop %%Page: 6 7 6 6 bop 236 154 a Fp(\(1.1\))41 b(Prop)s(osition)75 b Fh(F)-7 b(or)34 b Fo(\027)g Fk(2)28 b(N)15 b Fq([)p Fo(E)6 b Fq(])34 b Fh(the)h(set)1124 374 y Fo(E)1196 389 y Fl(0)1263 374 y Fq(:=)1427 291 y Fi([)1394 481 y Fn(n)p Fm(2)p Fj(N)1569 374 y Fk(f)p Fo(x)28 b Fk(2)g Fo(E)34 b Fq(:)28 b Fp(P)p Fq(\()p Fo(X)2161 333 y Fl(0)2153 399 y Fn(n)2227 374 y Fk(\025)g Fo(x)p Fq(\))h Fo(>)e 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Fk(2)g(H)q Fq([)p Fo(E)6 b Fq(])28 b(:)g Fo(h)p Fq(\()p Fo(x)p Fq(\))g Fk(2)g Fo(E)1208 2877 y Fl(0)1248 2862 y Fk(g)82 b(\033)i(f)p Fo(h)28 b Fk(2)g(H)q Fq([)p Fo(E)6 b Fq(])28 b(:)f Fp(P)p Fq(\()p Fo(X)2272 2821 y Fl(0)2264 2887 y Fn(n)p Fl(+1)2429 2862 y Fk(\025)h Fo(h)p Fq(\()p Fo(x)p Fq(\)\))g Fo(>)f Fq(0)p Fk(g)1380 3007 y(\033)84 b(f)p Fo(h)28 b Fk(2)g(H)q Fq([)p Fo(E)6 b Fq(])28 b(:)f Fp(P)p Fq(\()p Fo(X)2272 2966 y Fl(0)2264 3032 y Fn(n)2339 3007 y Fk(\025)h Fo(x;)17 b(H)2624 3022 y Fn(n)p Fl(+1)2761 3007 y Fq(\()p Fo(x)p Fq(\))28 b Fk(\025)g Fo(h)p Fq(\()p Fo(x)p Fq(\)\))g Fo(>)g Fq(0)p Fk(g)1381 3153 y Fq(=)84 b Fk(f)p Fo(h)28 b Fk(2)g(H)q Fq([)p Fo(E)6 b Fq(])28 b(:)f Fp(P)p Fq(\()p Fo(H)2264 3168 y Fn(n)p Fl(+1)2401 3153 y Fq(\()p Fo(x)p Fq(\)\))h Fk(\025)g Fo(h)p Fq(\()p Fo(x)p Fq(\)\))g Fo(>)g Fq(0)p Fk(g)1381 3298 y Fq(=)84 b Fk(f)p Fo(h)28 b Fk(2)g(H)q Fq([)p Fo(E)6 b Fq(])28 b(:)f Fo(\027)6 b Fq(\()p Fk(H)2244 3313 y Fn(h)p Fl(\()p Fn(x)p Fl(\))2384 3298 y Fq(\))28 b 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Fo(x)c Fq(whenev)m(er)j(the)d(condition)p 2607 2073 V 22 w Fo(x)k Fk(2)h Fo(E)g Fq(is)24 b(satis\014ed.)124 b Fg(2)383 2325 y Fq(The)34 b(assumption)p 1101 2272 V 33 w Fo(x)d Fk(2)f Fo(E)39 b Fq(is)34 b(no)f(real)g(restriction,)g(b) s(ecause)j Fo(E)j Fq(b)s(eing)33 b(replaced)h(b)m(y)p 236 2367 79 4 v 236 2445 a Fo(E)44 b Fq(:=)38 b Fo(E)32 b Fk([)26 b(f)p 739 2393 56 4 v Fo(x)q Fk(g)38 b Fq(mappings)f Fo(h)h Fk(2)g(H)q Fq([)p Fo(E)6 b Fq(])39 b(ha)m(v)m(e)h(unique)f (extensions)p 2812 2366 57 4 v 40 w Fo(h)f Fk(2)g(H)q Fq([)p 3122 2367 79 4 v Fo(E)6 b Fq(],)40 b(where)236 2566 y(p)s(ossibly)33 b Fk(1)f Fq(has)i(to)e(b)s(e)i(adjoined)e(to)h Fp(R)1769 2581 y Fl(+)1827 2566 y Fq(.)45 b(Moreo)m(v)m(er,)35 b(if)p 2445 2513 56 4 v 32 w Fo(x)29 b Fk(2)g Fo(E)39 b Fq(and)33 b(the)g(supp)s(ort)h Fo(N)236 2686 y Fq(of)e Fo(\027)39 b Fq(is)32 b(\014nite,)p 809 2633 V 33 w Fo(x)h Fq(clearly)e(is)h(a)h(\014xed)g(p)s(oin)m(t)f(of)g(some)h Fo(h)28 b Fk(2)g Fo(N)10 b Fq(.)383 2831 y(The)33 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y(trivial)28 b(case)i Fo(E)k Fq(=)28 b Fk(f)p Fq(0)p Fk(g)p Fq(,)i(to)f(treat)h (exclusiv)m(ely)h(the)f(t)m(w)m(o)g(cases)i Fo(E)i Fq(=)27 b([0)p Fo(;)17 b Fq(1])29 b(and)h Fo(E)k Fq(=)28 b Fp(R)3485 3726 y Fl(+)236 3831 y Fk(\000)33 b Fq(at)f(the)h(exp)s(ense,)i(ho)m(w) m(ev)m(er,)g(of)d(p)s(ermanen)m(t)h(rep)s(etitions.)383 3976 y(The)g(notion)f(of)g(irreducibilit)m(y)e(is)i(also)f(compatible)g (with)h(con)m(v)m(olution)g(p)s(o)m(w)m(ers:)236 4175 y Fp(\(1.4\))41 b(Prop)s(osition)75 b Fh(If)34 b Fo(\027)1309 4139 y Fn(k)1380 4175 y Fk(2)29 b(N)15 b Fq([)p Fo(E)6 b Fq(])35 b Fh(is)f(irr)-5 b(e)g(ducible)35 b(for)g(one)f Fo(k)d Fk(2)e Fp(N)p Fh(,)34 b(this)h(holds)g(for)236 4295 y(al)5 b(l)35 b Fo(k)30 b Fk(2)f Fp(N)p Fh(.)236 4494 y(Pr)-5 b(o)g(of.)41 b Fq(The)25 b(assertion)e(is)f(an)h (immediate)e(consequence)26 b(of)d(the)h(fact)f(that,)i(for)d (arbitrary)236 4615 y Fo(x)28 b Fk(2)g Fo(E)6 b Fq(,)33 b(the)g(sequence)i(\()p Fp(P)p Fq(\()p Fo(X)1365 4579 y Fl(0)1357 4639 y Fn(n)1432 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Fq(turn)e(out)g(to)g(b)s(e)h(constan)m(ts,)g (whic)m(h)g(in)1865 5753 y(7)p eop %%Page: 8 9 8 8 bop 236 154 a Fq(addition)39 b(do)i(not)g(dep)s(end)h(on)f(the)g (initial)c(v)-5 b(ariable)40 b Fo(X)2368 169 y Fl(0)2407 154 y Fq(.)68 b(The)42 b(\014rst)g(result)f(is)f(easily)236 274 y(established:)236 474 y Fp(\(1.5\))h(Theorem)76 b Fh(If)35 b Fo(\027)f Fk(2)28 b(N)15 b Fq([)p Fo(E)6 b Fq(])34 b Fh(is)h(irr)-5 b(e)g(ducible,)34 b(the)h(c)-5 b(onstant)p 2722 421 56 4 v 34 w Fo(x)29 b Fq(=)e(sup)18 b Fo(E)41 b Fh(satis\014es)1428 694 y Fq(lim)17 b(sup)1486 763 y Fn(n)p Fm(!1)1760 694 y Fo(X)1841 709 y Fn(n)1916 694 y Fq(=)p 2020 641 V 28 w Fo(x)83 b Fh(a.s.)16 b Fo(;)236 948 y Fh(r)-5 b(e)g(gar)g(d)5 b(less)34 b(of)h(the)g(initial)f(law.)236 1147 y(Pr)-5 b(o)g(of.)41 b Fq(F)-8 b(or)32 b Fo(x)c Fk(2)g Fo(E)39 b Fq(there)33 b(exists)h Fo(l)29 b Fk(2)g Fp(N)j Fq(suc)m(h)i(that)1280 1367 y Fp(P)p Fq(\()p Fo(H)1476 1382 y Fn(l)1524 1367 y Fk(\016)22 b Fo(:)17 b(:)g(:)k Fk(\016)h Fo(H)1885 1382 y Fl(1)1925 1367 y Fq(\(0\))27 b Fk(\025)h Fo(x)p Fq(\))g Fo(>)g Fq(0)17 b Fo(:)236 1587 y Fq(Then)34 b(b)m(y)f(monotonicit)m(y)e(and)i(indep)s(endence)347 1808 y Fp(P)p Fq(\()p Fo(X)543 1823 y Fn(n)617 1808 y Fk(\025)28 b Fo(x)42 b Fq(in\014nitely)31 b(often\))83 b Fk(\025)h Fp(P)p Fq(\(lim)17 b(sup)1898 1887 y Fn(k)r Fm(!1)2154 1808 y Fk(f)p Fo(H)2285 1823 y Fl(\()p Fn(k)r Fl(+1\))p Fn(l)2516 1808 y Fk(\016)22 b Fo(:)17 b(:)g(:)22 b Fk(\016)g Fo(H)2878 1823 y Fl(1)2917 1808 y Fq(\(0\))27 b Fk(\025)h Fo(x)p Fk(g)p Fq(\))1563 2003 y Fk(\025)84 b Fp(P)p Fq(\(lim)17 b(sup)1898 2082 y Fn(k)r Fm(!1)2154 2003 y Fk(f)p Fo(H)2285 2019 y Fl(\()p Fn(k)r Fl(+1\))p Fn(l)2516 2003 y Fk(\016)22 b Fo(:)17 b(:)g(:)22 b Fk(\016)g Fo(H)2878 2018 y Fn(k)r(l)q Fl(+1)3032 2003 y Fq(\(0\))27 b Fk(\025)i Fo(x)p Fk(g)p Fq(\))1564 2188 y(=)84 b(1)p Fo(:)236 2408 y Fq(Since)38 b Fo(x)f Fk(2)g Fo(E)44 b Fq(is)37 b(arbitrary)-8 b(,)38 b(lim)17 b(sup)1665 2433 y Fn(n)p Fm(!1)1886 2408 y Fo(X)1967 2423 y Fn(n)2050 2408 y Fk(\025)p 2164 2355 V 37 w Fo(x)38 b Fq(holds)f(almost)f(surely) -8 b(,)40 b(while)d(the)236 2529 y(in)m(v)m(erse)d(inequalit)m(y)e(is)g (ob)m(vious.)127 b Fg(2)383 2728 y Fq(The)33 b(second)h(result)e(is)h (less)f(immediate:)236 2927 y Fp(\(1.6\))41 b(Theorem)79 b Fh(If)37 b Fo(\027)i Fk(2)32 b(N)15 b Fq([)p Fo(E)6 b Fq(])38 b Fh(is)f(irr)-5 b(e)g(ducible,)37 b(ther)-5 b(e)37 b(is)h(a)f(c)-5 b(onstant)37 b Fo(x)p 3026 2940 V 33 w Fk(\024)p 3224 2874 V 33 w Fo(x)h Fh(satis-)236 3047 y(fying)1443 3168 y Fq(lim)17 b(inf)1486 3218 y Fn(n)p Fm(!1)1746 3168 y Fo(X)1827 3183 y Fn(n)1902 3168 y Fq(=)27 b Fo(x)p 2005 3181 V 84 w Fh(a.s.)16 b Fo(;)236 3357 y Fh(r)-5 b(e)g(gar)g(d)5 b(less)34 b(of)h(the)g(initial)f(law.) 236 3557 y(Pr)-5 b(o)g(of.)41 b Fq(1.)j(F)-8 b(or)31 b Fo(x)d Fk(2)h Fo(E)38 b Fq(and)33 b Fo(n)28 b Fk(\025)g Fq(0)k(de\014ne)1133 3777 y Fo(X)p 1133 3790 89 4 v 1221 3735 a Fn(x)1221 3801 y(n)1296 3777 y Fq(:=)c(lim)17 b(inf)1472 3837 y Fn(k)r Fm(!1)1730 3777 y 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2060 4647 a Fl(0)2060 4713 y Fn(n)2107 4689 y Fq(\))33 b(yields)1213 4909 y Fo(X)p 1213 4922 V 1302 4867 a Fl(0)1302 4933 y(0)1369 4909 y Fq(=)27 b Fo(X)p 1472 4922 V 1561 4867 a Fl(0)1561 4933 y Fn(n)1691 4909 y Fq(a.s.)125 b(for)32 b(all)72 b Fo(n)28 b Fk(\025)g Fq(0)17 b Fo(;)236 5129 y Fq(hence)32 b Fo(X)p 505 5142 V 594 5087 a Fl(0)594 5153 y(0)664 5129 y Fq(is)e(measurable)g(with)g (resp)s(ect)i(to)f(the)g(completed)f(tail)e Fo(\033)t Fq(-\014eld)i(of)g(\()p Fo(H)3268 5144 y Fn(n)3315 5129 y Fq(\))3353 5146 y Fn(n)p Fm(2)p Fj(N)3516 5129 y Fq(.)236 5249 y(Th)m(us)k(there)g(is)e(a)g(constan)m(t)h Fo(x)p 1305 5262 56 4 v 33 w Fq(satisfying)236 5469 y(\(1\))1237 b Fo(X)p 1598 5482 89 4 v 1687 5427 a Fl(0)1687 5494 y(0)1754 5469 y Fq(=)28 b Fo(x)p 1858 5482 56 4 v 83 w Fq(a.s.)17 b Fo(:)1865 5753 y Fq(8)p eop %%Page: 9 10 9 9 bop 383 154 a Fq(2.)47 b(F)-8 b(or)33 b(\014xed)i Fo(x)c Fk(2)f Fo(E)40 b Fq(c)m(ho)s(ose)35 b Fo(n)30 b Fk(2)h Fp(N)i Fq(suc)m(h)j(that)d Fp(P)p Fq(\()p Fo(A)p Fq(\))d Fo(>)g Fq(0)k(for)f Fo(A)d Fq(:=)g Fk(f)p Fo(X)3234 118 y Fl(0)3226 179 y Fn(n)3303 154 y Fk(\025)h Fo(x)p Fk(g)p Fq(.)236 274 y(With)h(the)h(notation)e Fo(\026)1105 289 y Fn(n)1180 274 y Fq(:=)c Fk(L)p Fq(\()p Fo(X)1506 238 y Fl(0)1498 299 y Fn(n)1545 274 y Fq(\))33 b(it)e(follo)m(ws)h (from)f(\(1\))h(that)692 493 y(1)83 b(=)h Fp(P)p Fq(\(\()p Fo(X)p 1137 506 89 4 v 1225 451 a Fl(0)1225 517 y(0)1292 493 y Fk(\024)29 b Fo(x)p 1398 506 56 4 v Fq(\))824 668 y(=)984 551 y Fi(Z)1030 739 y Fn(E)1106 668 y Fp(P)p Fq(\()p Fo(X)p 1221 681 89 4 v 1310 626 a Fn(y)1310 692 y(n)1384 668 y Fk(\024)f Fo(x)p 1489 681 56 4 v 1 w Fq(\))17 b Fo(\026)1659 683 y Fn(n)1705 668 y Fq(\()p Fo(dy)t Fq(\))824 890 y Fk(\024)984 773 y Fi(Z)1030 961 y Fn(y)r Fm(\025)p Fn(x)1183 890 y Fp(P)p Fq(\()p Fo(X)p 1298 903 89 4 v 1386 848 a Fn(x)1386 914 y(n)1461 890 y Fk(\024)28 b Fo(x)p 1566 903 56 4 v Fq(\))17 b Fo(\026)1735 905 y Fn(n)1782 890 y Fq(\()p Fo(dy)t Fq(\))37 b(+)2113 773 y Fi(Z)2159 961 y Fn(y)r()h Fq(0)k(and)h(the)g(equation)f Fk(L)p Fq(\()p Fo(X)p 1995 1313 89 4 v 2084 1258 a Fn(x)2084 1325 y Fl(0)2128 1300 y Fq(\))27 b(=)h Fk(L)p Fq(\()p Fo(X)p 2404 1313 V 2492 1258 a Fn(x)2492 1325 y(n)2539 1300 y Fq(\))33 b(this)f(implies)236 1519 y(\(2\))881 b Fo(X)p 1242 1532 V 1331 1477 a Fn(x)1331 1543 y Fl(0)1403 1519 y Fk(\024)28 b Fo(x)p 1508 1532 56 4 v 83 w Fq(a.s.)125 b(for)32 b(all)72 b Fo(x)28 b Fk(2)g Fo(E)23 b(:)383 1737 y Fq(3.)43 b(T)-8 b(ogether,)33 b(\(1\))f(and)h(\(2\))f(yield)1018 1955 y Fo(x)p 1018 1968 V 29 w Fq(=)27 b Fo(X)p 1205 1968 89 4 v 1294 1913 a Fl(0)1294 1980 y(0)1361 1955 y Fk(\024)h Fo(X)p 1466 1968 V 1555 1913 a Fn(x)1555 1980 y Fl(0)1627 1955 y Fk(\024)g Fo(x)p 1732 1968 56 4 v 83 w Fq(a.s.)125 b(for)32 b(all)72 b Fo(x)28 b Fk(2)g Fo(E)23 b(;)236 2173 y Fq(and)33 b(the)g(assertion)f(follo)m(ws)g(b)m (y)h(applying)f(F)-8 b(ubini.)124 b Fg(2)383 2372 y Fq(The)33 b(preceding)g(results)g(suggest)h(the)f(follo)m(wing)c(terminology:)236 2572 y Fp(\(1.7\))41 b(De\014nition)78 b Fh(If)37 b Fo(\027)j Fk(2)33 b(N)15 b Fq([)p Fo(E)6 b Fq(])38 b Fh(is)f(irr)-5 b(e)g(ducible,)38 b(the)g(c)-5 b(onstants)37 b Fo(x)p 2842 2585 V 38 w Fh(and)p 3127 2519 V 37 w Fo(x)h Fh(in)g Fq(\(1.5\))236 2692 y Fh(and)d Fq(\(1.6\))f Fh(ar)-5 b(e)35 b(c)-5 b(al)5 b(le)-5 b(d)34 b(\\lower)g(limit")g(and)g(\\upp)-5 b(er)35 b(limit")f(of)h Fo(\027)6 b Fh(,)35 b(r)-5 b(esp)g(e)g (ctively.)383 2891 y Fq(No)m(w)33 b(a)f(coun)m(terpart)h(of)f(\(1.2\))h (can)f(b)s(e)h(deriv)m(ed:)236 3091 y Fp(\(1.8\))41 b(Prop)s(osition)75 b Fh(If)34 b Fo(\027)g Fk(2)28 b(N)15 b Fq([)p Fo(E)6 b Fq(])35 b Fh(is)f(irr)-5 b(e)g(ducible,)34 b(its)h(lower)g(limit)f Fo(x)p 2931 3104 V 36 w Fh(is)g(given)g(by)834 3309 y Fo(x)p 834 3322 V 28 w Fq(=)28 b(max)o Fk(f)p Fo(x)g Fk(2)g Fo(E)34 b Fq(:)28 b Fo(h)p Fq(\()p Fo(x)p Fq(\))g Fk(\025)1835 3367 y Fn(\027)1902 3309 y Fo(x)p Fk(g)125 b Fh(whenever)e Fo(x)p 2646 3322 V 28 w Fk(2)29 b Fo(E)22 b(:)236 3602 y Fh(Pr)-5 b(o)g(of.)41 b Fq(F)-8 b(or)24 b(eac)m(h)h Fo(x)k Fk(2)f Fo(E)i Fq(with)24 b Fo(h)p Fq(\()p Fo(x)p Fq(\))k Fk(\025)1651 3660 y Fn(\027)1718 3602 y Fo(x)d Fq(iteration)e(yields)h Fo(X)2541 3566 y Fn(x)2533 3626 y(n)2612 3602 y Fk(\025)k Fo(x)42 b Fq(a.s.)25 b(for)f(all)f Fo(n)k Fk(2)i Fp(N)p Fq(,)236 3743 y(whic)m(h)40 b(b)m(y)g(\(1.6\))f(implies)e Fo(x)p 1242 3756 V 40 w Fk(\025)j Fo(x)p Fq(.)64 b(On)39 b(the)h(other)f (hand,)j(whenev)m(er)g Fo(x)p 2920 3756 V 40 w Fk(2)d Fo(E)6 b Fq(,)41 b(c)m(ho)s(ose)236 3863 y Fo(x)28 b(>)g(x)p 423 3876 V 29 w Fq(in)h(the)h(case)g Fo(x)p 986 3876 V 28 w(<)p 1172 3810 V 27 w(x)g Fq(and)f Fo(x)f Fq(=)g Fo(x)p 1630 3876 V 30 w Fq(in)g(the)i(case)g Fo(x)p 2193 3876 V 28 w Fq(=)p 2379 3810 V 27 w Fo(x)q Fq(.)42 b(In)30 b(b)s(oth)f(cases)h(the)g(hitting)236 3984 y(times)e Fo(T)550 3999 y Fl(1)618 3984 y Fo(<)f(T)778 3999 y Fl(2)845 3984 y Fo(<)h(:)17 b(:)g(:)29 b Fq(of)f([0)p Fo(;)17 b(x)p Fq(])29 b(b)m(y)h(\()p Fo(X)1690 3948 y Fl(0)1682 4008 y Fn(n)1729 3984 y Fq(\))1767 3999 y Fn(n)p Fm(\025)p Fl(0)1933 3984 y Fq(are)f(de\014ned)i(almost)c(surely)-8 b(,)30 b(where)h(again)236 4104 y(b)m(y)j(\(1.6\))1374 4322 y Fo(x)p 1374 4335 V 84 w Fq(=)84 b(lim)17 b(inf)1716 4372 y Fn(n)p Fm(!1)1976 4322 y Fo(X)2065 4281 y Fl(0)2057 4347 y Fn(n)1512 4498 y Fk(\024)84 b Fq(lim)17 b(inf)1718 4558 y Fn(k)r Fm(!1)1976 4498 y Fo(X)2065 4456 y Fl(0)2057 4522 y Fn(T)2098 4534 y Ff(k)2136 4522 y Fl(+1)1512 4674 y Fk(\024)84 b Fq(lim)17 b(inf)1718 4734 y Fn(k)r Fm(!1)1976 4674 y Fo(H)2057 4689 y Fn(T)2098 4701 y Ff(k)2136 4689 y Fl(+1)2231 4674 y Fq(\()p Fo(x)p Fq(\))g Fo(:)236 4923 y Fq(Since)45 b Fo(T)560 4938 y Fn(k)602 4923 y Fo(;)17 b(k)34 b Fk(2)d Fp(N)p Fo(;)44 b Fq(are)g(stopping)g(times)f(with)h (resp)s(ect)h(to)f(\()p Fo(H)2670 4938 y Fn(n)2717 4923 y Fq(\))2755 4940 y Fn(n)p Fm(2)p Fj(N)2918 4923 y Fq(,)j(the)e(v)-5 b(ariables)236 5043 y Fo(H)317 5058 y Fn(T)358 5070 y Ff(k)396 5058 y Fl(+1)491 5043 y Fo(;)17 b(k)31 b Fk(2)e Fp(N)p Fo(;)k Fq(are)g(again)f(indep)s(enden)m(t)j(with)d(distribution) g Fo(\027)6 b Fq(.)45 b(Th)m(us)35 b Fo(H)3048 5058 y Fn(T)3089 5070 y Ff(k)3127 5058 y Fl(+1)3222 5043 y Fq(\()p Fo(x)p Fq(\))29 b Fk(\025)g Fo(x)p 3488 5056 V 236 5164 a Fq(a.s.)q(,)34 b(or)f(equiv)-5 b(alen)m(tly)d(,)34 b Fo(h)p Fq(\()p Fo(x)p Fq(\))c Fk(\025)1369 5222 y Fn(\027)1438 5164 y Fo(x)p 1438 5177 V 1 w Fq(,)k(whic)m(h)g(for)f Fo(x)e Fk(#)e Fo(x)p 2150 5177 V 34 w Fq(\(if)k(necessary\))j(implies) 31 b Fo(h)p Fq(\()p Fo(x)p 3265 5177 V 1 w Fq(\))f Fk(\025)3419 5222 y Fn(\027)3488 5164 y Fo(x)p 3488 5177 V 236 5305 a Fq(b)m(y)k(the)f(con)m(tin)m(uit)m(y)f(of)g Fo(h)c Fk(2)g(H)q Fq([)p Fo(E)6 b Fq(].)127 b Fg(2)383 5504 y Fq(As)40 b(stated)h(for)e(the)h(upp)s(er)h(limit,)36 b(if)j Fo(x)p 1817 5517 V 28 w Fk(2)28 b Fo(E)46 b Fq(and)40 b(the)h(supp)s(ort)f Fo(N)50 b Fq(of)40 b Fo(\027)46 b Fq(is)39 b(\014nite,)1865 5753 y(9)p eop %%Page: 10 11 10 10 bop 236 154 a Fo(x)p 236 167 56 4 v 33 w Fq(clearly)32 b(is)g(a)g(\014xed)i(p)s(oin)m(t)e(of)g(some)g Fo(h)c Fk(2)g Fo(N)10 b Fq(.)383 274 y(The)37 b(upp)s(er)g(limit)d(of)i(an)g (irreducible)g(distribution)f Fo(\027)43 b Fq(is)36 b(alw)m(a)m(ys)h (uniquely)g(deter-)236 395 y(mined)32 b(b)m(y)h(its)f(supp)s(ort)h Fo(N)10 b Fq(;)33 b(indeed:)p 910 562 V 910 615 a Fo(x)28 b Fq(=)f(sup)q Fk(f)p Fo(h)1349 630 y Fn(n)1418 615 y Fk(\016)22 b Fo(:)17 b(:)g(:)22 b Fk(\016)g Fo(h)1755 630 y Fl(1)1794 615 y Fq(\(0\))28 b(:)g Fo(n)f Fk(2)i Fp(N)e Fq(and)h Fo(h)2538 630 y Fn(i)2594 615 y Fk(2)g Fo(N)10 b Fk(g)17 b Fo(:)236 835 y Fq(A)31 b(corresp)s(onding)g(result) h(for)e(the)i(lo)m(w)m(er)f(limit)d(fails)h(to)i(hold;)g(in)f(fact,)h (ev)m(en)i(the)f(alter-)236 955 y(nativ)m(e)f Fo(x)p 527 968 V 28 w Fk(2)d Fo(E)37 b Fq(or)31 b Fo(x)p 931 968 V 39 w(=)-60 b Fk(2)28 b Fo(E)37 b Fq(is)30 b(not)h(a)f(question)h (on)g Fo(N)41 b Fq(alone,)31 b(but)g(will)d(lead)j(to)f(the)h(basic)236 1076 y(distinction)g(b)s(et)m(w)m(een)k(recurrence)f(and)f(transience)g 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4287 y Fn(n)3130 4272 y Fq(\))j Fk(!)g Fo(h)p Fq(\()p Fo(X)8 b Fq(\))236 4392 y(in)32 b(probabilit)m(y)f(due)i(to)f(the)h(con)m(tin)m(uit)m(y)g(of)f Fo(h)c Fk(2)g(H)q Fq([)p Fo(E)6 b Fq(].)44 b(Therefore)912 4612 y Fo(d)p Fq(\()p Fo(x;)17 b(h)p Fq(\()p Fo(x)p Fq(\)\))28 b(=)g(0)124 b(for)74 b Fo(\026)21 b Fk(\012)i Fo(\027)6 b Fk(\000)p Fq(almost)32 b(all)39 b(\()p Fo(x;)17 b(h)p Fq(\))g Fo(;)236 4832 y Fq(hence)34 b(b)m(y)g(applying)d(F)-8 b(ubini)1095 5052 y Fo(h)p Fq(\()p Fo(x)p Fq(\))28 b(=)1353 5110 y Fn(\027)1420 5052 y Fo(x)124 b Fq(for)74 b Fo(\026)p Fk(\000)p Fq(almost)31 b(all)40 b Fo(x)28 b Fk(2)g Fo(E)22 b(:)383 5293 y Fq(3.)42 b(The)30 b(implication)25 b(\(c\))30 b Fk(\))f Fq(\(a\))g(is)g(a)g(consequence)k(of)c(\(1.2\),)h(yielding)p 3085 5240 V 27 w Fo(x)e Fk(\024)h Fo(x)p Fq(,)h(and)236 5413 y(\(1.8\),)i(yielding)f Fo(x)p 862 5426 V 28 w Fk(\025)d Fo(x)p Fq(.)127 b Fg(2)1841 5753 y Fq(10)p eop %%Page: 11 12 11 11 bop 236 154 a Fp(2.)50 b(Recurrence)37 b(and)i(transience)236 353 y Fq(Besides)c(the)f(prop)s(er)g(con)m(v)m(ergence)i Fo(x)p 1606 366 56 4 v 30 w Fq(=)p 1796 300 V 29 w Fo(x)30 b Fk(2)g Fo(E)40 b Fq(considered)34 b(in)f(\(1.9\))g(there)i(is)e(an)g (im-)236 474 y(prop)s(er)g(con)m(v)m(ergence)j Fo(x)p 1088 487 V 28 w Fq(=)p 1276 421 V 29 w Fo(x)k(=)-61 b Fk(2)29 b Fo(E)6 b Fq(,)33 b(generalizing)e(the)i(almost)e(sure)j(div)m (ergence)g(of)f(the)236 594 y(pro)s(cess)k(\()p Fo(X)698 609 y Fn(n)745 594 y Fq(\))783 609 y Fn(n)p Fm(\025)p Fl(0)955 594 y Fq(to)e Fk(1)g Fq(in)g(the)h(sp)s(ecial)e(case)j Fo(E)h Fq(=)33 b Fp(R)2333 609 y Fl(+)2392 594 y Fq(.)52 b(This)35 b(is)g(a)g(\014rst)h(motiv)-5 b(ation)236 714 y(for)32 b(the)h(follo)m(wing)d(notion,)i(used)h(similarly)c(in)j([19]) g(in)g(related)g(con)m(text:)236 914 y Fp(\(2.1\))41 b(De\014nition)72 b Fh(If)31 b Fo(\027)j Fk(2)28 b(N)15 b Fq([)p Fo(E)6 b Fq(])31 b Fh(is)h(irr)-5 b(e)g(ducible,)31 b(the)h(distribution)f Fo(\027)38 b Fh(\(or)31 b(the)h(kernel)236 1034 y Fo(P)49 b Fh(or)34 b(the)h(pr)-5 b(o)g(c)g(ess)34 b Fq(\()p Fo(X)1084 1049 y Fn(n)1131 1034 y Fq(\))1169 1049 y Fn(n)p Fm(\025)p Fl(0)1306 1034 y Fh(\))h(is)f(c)-5 b(al)5 b(le)-5 b(d)236 1217 y(\()p Fq(a)p Fh(\))58 b(\\r)-5 b(e)g(curr)g(ent")66 b(if)41 b Fo(x)p 1072 1230 V 29 w Fk(2)28 b Fo(E)6 b Fh(,)236 1399 y(\()p Fq(b)p Fh(\))58 b(\\tr)-5 b(ansient")83 b(if)41 b Fo(x)p 1087 1412 V 40 w(=)-61 b Fk(2)28 b Fo(E)6 b Fh(.)383 1599 y Fq(T)-8 b(o)29 b(b)s(egin)f(with)h(the)h(simplest)e(example,)h(a)g (deterministic)e(system)j(\()p Fo(E)6 b(;)17 b(h)p Fq(\))29 b(is)g(easily)236 1719 y(seen)i(to)e(b)s(e)h(recurren)m(t)h(if)d(and)i (only)f(if)f Fo(E)34 b Fq(=)27 b([0)p Fo(;)p 2007 1666 V 17 w(x)p Fq(])j(with)p 2338 1666 V 29 w Fo(x)g Fq(b)s(eing)f(the)h (maxim)m(um)d(of)i(the)236 1839 y(increasing)e(sequence)k(\()p Fo(h)1181 1803 y Fn(n)1228 1839 y Fq(\(0\)\))1391 1854 y Fn(n)p Fm(\025)p Fl(0)1527 1839 y Fq(.)42 b(Th)m(us)29 b(c)m(ho)s(osing)f Fo(E)33 b Fq(=)28 b([0)p Fo(;)17 b Fq(1[)27 b(and)h 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eop %%Page: 12 13 12 12 bop 236 154 a Fq(the)32 b(corresp)s(onding)g(distribution)d Fo(\027)38 b Fq(is)31 b(carried)g(b)m(y)h(the)g(mappings)f Fo(h)d Fq(:)f Fo(x)h Fk(!)g Fq(\()p Fo(x)20 b Fq(+)g Fo(u)p Fq(\))3458 118 y Fl(+)3516 154 y Fo(;)236 274 y(u)36 b Fk(2)g Fp(R)p Fo(;)i Fq(and)f(th)m(us)i(irreducible)d(whenev)m (er)k Fp(P)p Fq(\()p Fo(U)2094 289 y Fn(n)2177 274 y Fo(>)c Fq(0\))g Fo(>)g Fq(0.)58 b(In)38 b(this)f(case)h(the)g(con-)236 395 y(dition)30 b(in)i(\(2.2\))f(is)g(not)h(satis\014ed,)h(while)e(it)g (is)g(w)m(ell-kno)m(wn)h(that)g Fo(X)2781 410 y Fn(n)2856 395 y Fk(!)27 b(1)41 b Fq(a.s.)32 b(if)f(and)236 515 y(only)h(if)g(the)h(random)e(w)m(alk)i(with)f(incremen)m(ts)h Fo(U)2072 530 y Fn(n)2152 515 y Fq(do)s(es)g(not)f(div)m(erge)h(to)g(+) 17 b Fk(1)p Fq(.)383 660 y(In)29 b(the)h(case)h(of)e(transience)h(the)g (pro)s(cess)g(\()p Fo(X)2056 675 y Fn(n)2103 660 y Fq(\))2141 675 y Fn(n)p Fm(\025)p Fl(0)2308 660 y Fq(div)m(erges)g(exp)s(onen)m (tially)f(fast)g(in)236 780 y(the)k(follo)m(wing)d(sense:)236 979 y Fp(\(2.3\))41 b(Prop)s(osition)75 b Fh(If)34 b Fo(\027)g Fk(2)28 b(N)15 b Fq([)p Fo(E)6 b Fq(])35 b Fh(is)f(tr)-5 b(ansient,)35 b(the)g(r)-5 b(andom)34 b(c)-5 b(ar)g(dinality)1387 1194 y Fo(Z)35 b Fq(:=)27 b Fk(jf)p Fo(n)h Fk(\025)g Fq(0)f(:)h Fo(X)2100 1209 y Fn(n)2175 1194 y Fk(\024)g Fo(t)p Fk(gj)236 1409 y Fh(for)35 b(arbitr)-5 b(ary)35 b(initial)g(law)f(and)g(every)h Fo(t)28 b Fk(2)g Fo(E)41 b Fh(satis\014es)1115 1624 y Fp(E)p Fq(\(exp\()p Fo(uZ)7 b Fq(\)\))27 b Fo(<)h Fk(1)124 b Fh(for)35 b(some)40 b Fo(u)28 b(>)f Fq(0)17 b Fo(:)236 1913 y Fh(Pr)-5 b(o)g(of.)41 b Fq(De\014ne)33 b(recursiv)m(ely)727 2128 y Fo(T)784 2143 y Fl(0)851 2128 y Fq(:=)28 b(0)124 b(and)h Fo(T)1494 2143 y Fn(k)1564 2128 y Fq(:=)28 b(inf)22 b Fk(f)p Fo(n)28 b(>)g(T)2126 2143 y Fn(k)r Fm(\000)p Fl(1)2286 2128 y Fq(:)g Fo(X)2422 2143 y Fn(n)2497 2128 y Fk(\024)g Fo(t)p Fk(g)41 b Fq(\()p Fk(\024)29 b(1)p Fq(\))17 b Fo(:)236 2343 y Fq(Then)34 b(b)m(y)f(transience)752 2557 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Ff(k)q(l)1918 3531 y Fn(<)p Fm(1g)2099 3459 y Fo(g)t Fq(\()p Fo(X)2269 3474 y Fn(T)2310 3486 y Ff(k)q(l)2372 3459 y Fq(\))41 b Fo(d)p Fp(P)2596 3418 y Fl(0)1577 3698 y Fk(\024)1737 3581 y Fi(Z)1783 3769 y Fm(f)p Fn(T)1859 3781 y Ff(k)q(l)1918 3769 y Fn(<)p Fm(1g)2099 3698 y Fo(g)t Fq(\(0\))g Fo(d)p Fp(P)2461 3657 y Fl(0)1578 3903 y Fq(=)83 b Fo(#)42 b Fp(P)1930 3862 y Fl(0)1969 3903 y Fq(\()p Fo(T)2064 3918 y Fn(k)r(l)2156 3903 y Fo(<)28 b Fk(1)p Fq(\))124 b(for)73 b Fo(k)31 b Fk(\025)d Fq(0)17 b Fo(:)236 4118 y Fq(This)33 b(yields)f(the)h(b)s (ound)1493 4238 y Fp(P)1587 4197 y Fl(0)1625 4238 y Fq(\()p Fo(T)1720 4253 y Fn(k)r(l)1813 4238 y Fo(<)27 b Fk(1)p Fq(\))g Fk(\024)i Fo(#)2244 4197 y Fn(k)236 4410 y Fq(and)k(th)m(us)h (b)m(y)f(monotonicit)m(y)1198 4625 y Fp(P)p Fq(\()p Fo(Z)i Fk(\025)28 b Fo(k)s(l)r Fq(\))83 b Fk(\024)g Fp(P)1980 4584 y Fl(0)2019 4625 y Fq(\()p Fo(T)2114 4640 y Fn(k)r(l)2206 4625 y Fo(<)28 b Fk(1)p Fq(\))1726 4770 y Fk(\024)83 b Fo(#)1943 4729 y Fn(k)2111 4770 y Fq(for)74 b Fo(k)30 b Fk(\025)f Fq(0)17 b Fo(:)236 4985 y Fq(P)m(artial)31 b(in)m(tegration)g(sho)m(ws)j(that)e(eac)m(h)i Fo(u)27 b(<)h Fk(\000)2041 4946 y Fl(1)p 2041 4962 36 4 v 2048 5019 a Fn(l)2103 4985 y Fq(log)17 b Fo(#)33 b Fq(satis\014es)g(the)g (assertion.)126 b Fg(2)383 5184 y Fq(It)26 b(is)h(a)f(fundamen)m(tal)g (consequence)k(of)d(\(2.3\))f(that)h(recurrence)i(and)d(transience)i (can)236 5305 y(b)s(e)36 b(c)m(haracterized)h(b)m(y)g(applying)d(the)j (p)s(oten)m(tial)d(k)m(ernel)i Fo(G)e Fq(:=)2633 5238 y Fi(P)2737 5330 y Fn(n)p Fm(\025)p Fl(0)2903 5305 y Fo(P)2980 5269 y Fn(n)3063 5305 y Fq(to)h(in)m(terv)-5 b(als)236 5425 y([0)p Fo(;)17 b(t)p Fq(])33 b(with)f(the)h(righ)m(t)f (endp)s(oin)m(ts:)1841 5753 y(12)p eop %%Page: 13 14 13 13 bop 236 154 a Fp(\(2.4\))41 b(Theorem)71 b Fh(If)30 b Fo(\027)k Fk(2)28 b(N)15 b Fq([)p Fo(E)6 b Fq(])30 b Fh(is)g(irr)-5 b(e)g(ducible,)30 b(the)g(fol)5 b(lowing)29 b(dichotomy)g(holds)h(for)236 274 y(arbitr)-5 b(ary)35 b(initial)g(law:)236 457 y(\()p Fq(a)p Fh(\))41 b(if)35 b Fo(\027)41 b Fh(is)35 b(r)-5 b(e)g(curr)g(ent,)35 b(then)1180 574 y Fi(X)1173 756 y Fn(n)p Fm(\025)p Fl(0)1340 657 y Fp(P)p Fq(\()p Fo(X)1536 672 y Fn(n)1610 657 y Fk(\024)28 b Fo(t)p Fq(\))g(=)f Fk(1)124 b Fh(for)77 b Fo(t)27 b(>)h(x)p 2507 670 56 4 v 17 w(;)236 938 y Fh(\()p Fq(b)p Fh(\))42 b(if)34 b Fo(\027)42 b Fh(is)34 b(tr)-5 b(ansient,)35 b(then)1180 1055 y Fi(X)1173 1237 y Fn(n)p Fm(\025)p Fl(0)1340 1138 y Fp(P)p Fq(\()p Fo(X)1536 1153 y Fn(n)1610 1138 y Fk(\024)28 b Fo(t)p Fq(\))g Fo(<)f Fk(1)124 b Fh(for)77 b Fo(t)27 b(<)p 2507 1085 V 28 w(x)17 b(:)236 1412 y Fh(Pr)-5 b(o)g(of.)41 b Fq(Both)33 b(assertions)g(follo)m(w)e(b) m(y)i(taking)f(the)h(exp)s(ectation)g(of)1494 1612 y Fo(Z)i Fq(=)1706 1529 y Fi(X)1699 1711 y Fn(n)p Fm(\025)p Fl(0)1865 1612 y Fq(1)1914 1627 y Fl([0)p Fn(;t)p Fl(])2038 1612 y Fq(\()p Fo(X)2157 1627 y Fn(n)2204 1612 y Fq(\))17 b Fo(;)236 1893 y Fq(whic)m(h)47 b(in)f(case)i(\(a\))e(is)g(almost)f (surely)i(in\014nite)e(and)i(in)f(case)h(\(b\))g(is)f(in)m(tegrable)f (b)m(y)236 2014 y(\(2.3\).)126 b Fg(2)383 2213 y Fq(F)-8 b(or)31 b(an)i(application)d(consider)j(an)f(exc)m(hange)i(pro)s(cess)g (with)e Fo(U)2757 2228 y Fn(n)2832 2213 y Fq(=)c(1,)k(i.e.)1107 2412 y Fo(X)1188 2427 y Fn(n)1263 2412 y Fq(=)c(\()p Fo(X)1486 2427 y Fn(n)p Fm(\000)p Fl(1)1645 2412 y Fk(\000)22 b Fq(1\))g Fk(_)h Fo(V)1999 2427 y Fn(n)2170 2412 y Fq(for)74 b Fo(n)28 b Fk(2)g Fp(N)17 b Fo(;)236 2612 y Fq(where)34 b(the)f(i.i.d.)42 b(v)-5 b(ariables)31 b Fo(V)1380 2627 y Fn(n)1427 2612 y Fo(;)17 b(n)28 b Fk(2)g Fp(N)p Fo(;)k Fq(are)h(nonnegativ)m(e.)44 b(With)32 b(state)h(space)1263 2812 y Fo(E)h Fq(=)28 b Fk(f)p Fo(x)f Fk(\025)i Fq(0)e(:)h Fp(P)p Fq(\()p Fo(V)2014 2827 y Fn(n)2088 2812 y Fk(\025)g Fo(x)p Fq(\))g Fo(>)g Fq(0)p Fk(g)236 3011 y Fq(the)f(corresp)s(onding) f(distribution)e Fo(\027)32 b Fq(is)26 b(carried)g(b)m(y)h(the)f (mappings)f Fo(h)j Fq(:)g Fo(x)g Fk(!)f Fq(\()p Fo(x)9 b Fk(\000)g Fq(1\))g Fk(_)g Fo(v)t(;)236 3132 y(v)52 b Fk(2)c Fo(E)6 b(;)44 b Fq(and)h(th)m(us)g(irreducible)e(b)m(y)i (\(1.1\).)78 b(Moreo)m(v)m(er)46 b(it)d(is)h(recurren)m(t)i(in)e(the)g (case)p 236 3199 V 236 3252 a Fo(x)28 b(<)g Fk(1)p Fq(.)43 b(Indeed,)34 b(in)e(this)g(case)858 3452 y(sup)861 3530 y Fn(x)p Fm(2)p Fn(E)1022 3452 y Fq(\(\()p Fo(x)22 b Fk(\000)h Fq(1\))f Fk(_)g Fo(v)t Fq(\))28 b(=)f(\()p 1730 3399 V Fo(x)c Fk(\000)f Fq(1\))g Fk(_)h Fo(v)31 b Fk(2)d Fo(E)131 b Fq(for)73 b Fo(v)32 b Fk(2)c Fo(E)236 3705 y Fq(and)40 b(th)m(us)h(\(2.2\))e(applies.)65 b(As)40 b(already)g(indicated)f(in)g(the)h(in)m(tro)s(duction,)g(recurrence)236 3825 y(as)i(w)m(ell)f(as)h(transience)g(can)g(o)s(ccur)g(in)e(the)j (case)p 2146 3772 V 42 w Fo(x)g Fq(=)g Fk(1)p Fq(.)71 b(T)-8 b(o)41 b(exhibit)g(appropriate)236 3946 y(examples,)34 b(denote)g(b)m(y)g Fo(F)46 b Fq(the)34 b(common)e(distribution)f (function)i(of)g Fo(V)2889 3961 y Fn(n)2935 3946 y Fo(;)17 b(n)29 b Fk(2)g Fp(N)p Fq(.)45 b(Then)236 4066 y(the)33 b(explicit)e(represen)m(tation)674 4266 y Fo(X)763 4224 y Fl(0)755 4290 y Fn(n)830 4266 y Fq(=)c(\()p Fo(V)1028 4281 y Fl(1)1090 4266 y Fk(\000)22 b Fq(\()p Fo(n)h Fk(\000)f Fq(1\)\))g Fk(_)h Fo(:)17 b(:)g(:)k Fk(_)i Fq(\()p Fo(V)1963 4281 y Fn(n)p Fm(\000)p Fl(1)2122 4266 y Fk(\000)g Fq(1\))e Fk(_)i Fo(V)2476 4281 y Fn(n)2647 4266 y Fq(for)74 b Fo(n)28 b Fk(2)g Fp(N)236 4465 y Fq(yields)k(b)m(y)i(indep)s(endence) 1240 4665 y Fp(P)p Fq(\()p Fo(X)1444 4624 y Fl(0)1436 4689 y Fn(n)1510 4665 y Fk(\024)28 b Fo(t)p Fq(\))g(=)1892 4582 y Fi(Y)1819 4765 y Fl(0)p Fm(\024)p Fn(m)f Fq(0)17 b Fo(;)1841 5753 y Fq(13)p eop %%Page: 14 15 14 14 bop 236 154 a Fq(i.e.)43 b(the)33 b(pro)s(cess)h(is)e(recurren)m (t;)236 274 y(\(2\))41 b(if)54 b(the)h(densit)m(y)h(is)e(replaced)h(b)m (y)g(the)g(function)f Fo(f)11 b Fq(\()p Fo(x)p Fq(\))66 b(=)f(2)p Fo(x)p Fq(\()p Fo(x)37 b Fq(+)g(1\))3172 238 y Fm(\000)p Fl(3)3266 274 y Fq(,)61 b(then)236 436 y Fo(F)14 b Fq(\()p Fo(t)p Fq(\))28 b(=)f(\()688 369 y Fo(t)p 603 413 205 4 v 603 505 a(t)22 b Fq(+)h(1)817 436 y(\))855 395 y Fl(2)895 436 y Fq(,)32 b(and)h(it)e(follo)m(ws)h (similarly)d(that)j(the)h(pro)s(cess)h(is)e(transien)m(t.)383 590 y(More)46 b(profound)f(criteria)f(for)h(recurrence)j(and)e (transience)g(can)g(b)s(e)g(deriv)m(ed)g(b)m(y)236 710 y(linearization,)c(i.e.)74 b(comparing)41 b(the)i(underlying)f (mappings)g(with)h(a\016ne)g(ones.)75 b(As)236 830 y(already)25 b(men)m(tioned)f(in)g(the)i(in)m(tro)s(duction,)f(ho)m(w)m(ev)m(er,)k (this)24 b(topic)g(is)h(p)s(ostp)s(oned)g(to)f([17].)383 972 y(This)37 b(section)h(concludes)g(with)g(a)f(consequence)k(of)c (\(2.4\))g(b)m(y)h(whic)m(h)g(some)g(pro)s(ofs)236 1092 y(can)33 b(b)s(e)g(simpli\014ed:)236 1292 y Fp(\(2.5\))41 b(Prop)s(osition)78 b Fh(If)38 b Fo(\027)1316 1256 y Fn(k)1393 1292 y Fk(2)c(N)15 b Fq([)p Fo(E)6 b Fq(])38 b Fh(is)g(r)-5 b(e)g(curr)g(ent)38 b(for)g(one)g Fo(k)f Fk(2)d Fp(N)p Fh(,)39 b(this)f(holds)g(for)236 1412 y(al)5 b(l)35 b Fo(k)30 b Fk(2)f Fp(N)p Fh(.)44 b(Mor)-5 b(e)g(over,)34 b(the)h(asso)-5 b(ciate)g(d)34 b(limits)h Fo(x)p 2062 1425 56 4 v 21 x Fn(k)2195 1412 y Fh(and)p 2384 1359 V 34 w Fo(x)2439 1427 y Fn(k)2517 1412 y Fh(ar)-5 b(e)34 b(indep)-5 b(endent)34 b(of)g Fo(k)s Fh(.)236 1611 y(Pr)-5 b(o)g(of.)41 b Fq(The)51 b(case)f Fo(x)p 975 1624 V 57 w Fq(=)p 1220 1558 V 57 w Fo(x)f Fq(is)g(settled)h(b)m(y)h(\(1.4\),)i (b)s(ecause)e Fo(X)2664 1626 y Fn(n)2767 1611 y Fk(!)56 b Fo(X)50 b Fq(a.s.)g(implies)236 1732 y Fo(X)317 1747 y Fn(k)r(n)446 1732 y Fk(!)42 b Fo(X)49 b Fq(a.s.)42 b(and)g(th)m(us)h Fo(x)p 1324 1745 V 20 x Fn(k)1465 1732 y Fq(=)f Fo(x)p 1583 1745 V 44 w Fq(=)p 1801 1679 V 43 w Fo(x)h Fq(=)p 2018 1679 V 43 w Fo(x)2073 1747 y Fn(k)2116 1732 y Fq(.)70 b(T)-8 b(o)42 b(settle)f(the)h(case)h Fo(x)p 3028 1745 V 43 w(<)p 3245 1679 V 43 w(x)p Fq(,)h(it)d(is)236 1852 y(su\016cien)m(t)30 b(to)f(apply)f(\(2.4\))g(with)h Fo(X)1551 1867 y Fl(0)1618 1852 y Fq(=)e(0,)j(b)s(ecause)g(it)e(follo)m (ws)f(as)i(in)f(the)i(pro)s(of)d(of)i(\(1.4\))236 1972 y(that)k(the)g(sequence)i(\()p Fp(P)p Fq(\()p Fo(X)1262 1936 y Fl(0)1254 1997 y Fn(n)1328 1972 y Fk(\024)28 b Fo(t)p Fq(\)\))1544 1987 y Fn(n)p Fm(\025)p Fl(0)1714 1972 y Fq(decreases.)129 b Fg(2)236 2213 y Fp(3.)50 b(A)37 b(fundamen)m(tal)h(inequalit)m(y)236 2412 y Fq(The)27 b(follo)m(wing)c(result)j(will)e(pla)m(y)i(a)f(k)m(ey)j(role)d(in)g (deriving)g(the)h(main)e(results)j(in)e(Sections)236 2533 y(4)33 b(and)f(6:)236 2732 y Fp(\(3.1\))41 b(Lemma)75 b Fh(If)34 b Fo(\027)g Fk(2)28 b(N)15 b Fq([)p Fo(E)6 b Fq(])34 b Fh(is)f(irr)-5 b(e)g(ducible,)34 b(ther)-5 b(e)34 b(exists)f(an)h(incr)-5 b(e)g(asing)33 b(function)236 2852 y Fo(c)28 b Fq(:)g Fo(E)33 b Fk(!)28 b Fp(R)678 2867 y Fl(+)771 2852 y Fh(such)35 b(that)947 2964 y Fi(X)940 3146 y Fn(n)p Fm(\025)p Fl(0)1106 3047 y Fp(E)p Fq(\()p Fo(f)11 b Fq(\()p Fo(X)1404 3005 y Fn(x)1396 3071 y(n)1447 3047 y Fq(\))22 b Fk(\000)h Fo(f)11 b Fq(\()p Fo(X)1793 3005 y Fl(0)1785 3071 y Fn(n)1832 3047 y Fq(\)\))27 b Fk(\024)i Fo(c)p Fq(\()p Fo(x)p Fq(\))k(sup)2254 3124 y Fn(n)p Fm(\025)p Fl(0)2427 3047 y Fp(E)p Fq(\()p Fo(f)11 b Fq(\()p Fo(X)2725 3005 y Fl(0)2717 3071 y Fn(n)2763 3047 y Fq(\)\))236 3317 y Fh(for)35 b(e)-5 b(ach)34 b(incr)-5 b(e)g(asing)34 b(function)g Fo(f)39 b Fq(:)27 b Fo(E)34 b Fk(!)27 b Fp(R)1912 3332 y Fl(+)2006 3317 y Fh(and)34 b(every)h Fo(x)28 b Fk(2)g Fo(E)6 b Fh(.)236 3516 y(Pr)-5 b(o)g(of.)41 b Fq(First)f(of)h(all,)g(the)h(left-hand)e(side)h(is)g(w)m (ell-de\014ned,)i(b)s(ecause)g(the)e(di\013erences)236 3637 y Fo(f)11 b Fq(\()p Fo(X)422 3601 y Fn(x)414 3661 y(n)466 3637 y Fq(\))22 b Fk(\000)g Fo(f)11 b Fq(\()p Fo(X)811 3601 y Fl(0)803 3661 y Fn(n)850 3637 y Fq(\))33 b(are)f(nonnegativ)m(e.)44 b(No)m(w,)33 b(\014x)h Fo(x)28 b Fk(2)g Fo(E)38 b Fq(and)33 b(c)m(ho)s(ose)g Fo(k)e Fk(2)d Fp(N)k Fq(suc)m(h)i(that)1260 3831 y Fk(H)1345 3790 y Fm(0)1396 3831 y Fq(:=)28 b Fk(f)p Fo(h)1633 3790 y Fm(0)1684 3831 y Fk(2)g(H)q Fq([)p Fo(E)6 b Fq(])28 b(:)f 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Fn(n)2193 5311 y Fq(\()p Fo(dh)p Fq(\))17 b Fo(\027)2447 5270 y Fn(k)2489 5311 y Fq(\()p Fo(dh)2634 5270 y Fm(0)2657 5311 y Fq(\))1136 5504 y(=)83 b Fo(\015)1351 5463 y Fm(\000)p Fl(1)1462 5504 y Fp(E)p Fq(\()p Fo(f)11 b Fq(\()p Fo(X)1760 5463 y Fl(0)1752 5529 y Fn(n)p Fl(+)p Fn(k)1892 5504 y Fq(\))22 b Fk(\000)g Fo(f)11 b Fq(\()p Fo(X)2237 5463 y Fl(0)2229 5529 y Fn(n)2276 5504 y Fq(\)\))125 b(for)73 b Fo(n)28 b Fk(\025)g Fq(0)17 b Fo(;)1841 5753 y Fq(14)p eop %%Page: 15 16 15 15 bop 236 154 a Fq(where)34 b(the)f(\014rst)g(inequalit)m(y)f (follo)m(ws)f(from)952 372 y Fo(f)11 b Fq(\()p Fo(h)22 b Fk(\016)g Fo(h)1255 331 y Fm(0)1279 372 y Fq(\(0\)\))27 b Fk(\025)h Fo(f)11 b Fq(\()p Fo(h)p Fq(\()p Fo(x)p Fq(\)\))125 b(for)73 b Fo(h)28 b Fk(2)g(H)q Fo(;)17 b(h)2574 331 y Fm(0)2625 372 y Fk(2)28 b(H)2804 331 y Fm(0)236 590 y Fq(and)33 b(the)g(second)h(one)f(from)844 809 y Fo(f)11 b Fq(\()p Fo(h)22 b Fk(\016)g Fo(h)1147 768 y Fm(0)1170 809 y Fq(\(0\)\))28 b Fk(\025)g Fo(f)11 b Fq(\()p Fo(h)p 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17 18 17 17 bop 236 154 a Fo(f)11 b Fq(\()p Fo(y)t Fq(\))29 b(=)h(0)k(for)g Fo(y)f Fk(2)970 88 y Fi(S)1056 179 y Fn(k)r Fm(\025)p Fl(0)1215 154 y Fo(B)1289 169 y Fn(k)1365 154 y Fq(and)i Fo(f)11 b Fq(\()p Fo(y)t Fq(\))29 b(=)h(1)k(for)f Fo(y)h Fk(2)2290 88 y Fi(S)2376 179 y Fn(k)r Fm(\025)p Fl(0)2535 154 y Fo(C)2605 169 y Fn(k)2647 154 y Fq(.)48 b(By)35 b(\(2\))f(and)g(\(3\))g(this)236 274 y(implies)789 395 y(lim)17 b(sup)849 474 y Fn(k)r Fm(!1)1105 395 y Fk(f)p Fo(T)1212 410 y Fn(k)1282 395 y Fk(\024)29 b Fo(n)1446 410 y Fn(k)1488 395 y Fk(g)f(\032)g(f)p Fq(lim)17 b(sup)1778 464 y Fn(n)p Fm(!1)2053 395 y Fk(j)p Fo(f)11 b Fq(\()p Fo(X)2267 354 y Fn(x)2259 419 y(n)2310 395 y Fq(\))22 b Fk(\000)h Fo(f)11 b Fq(\()p Fo(X)2656 354 y Fl(0)2648 419 y Fn(n)2695 395 y Fq(\))p Fk(j)27 b Fq(=)g(1)p Fk(g)236 614 y Fq(almost)k(surely)-8 b(,)33 b(whic)m(h)g(b)m(y)h(\(1\))e(yields) 1146 821 y(lim)17 b(sup)1204 890 y Fn(n)p Fm(!1)1478 821 y Fk(j)p Fo(f)11 b Fq(\()p Fo(X)1692 780 y Fn(x)1684 845 y(n)1735 821 y Fq(\))22 b Fk(\000)h Fo(f)11 b Fq(\()p Fo(X)2081 780 y Fl(0)2073 845 y Fn(n)2120 821 y Fq(\))p Fk(j)27 b Fq(=)h(1)82 b(a.s.)17 b Fo(;)236 1062 y Fq(pro)m(viding)32 b(the)h(desired)g(coun)m(terexample.)383 1183 y(As)g(a)f(corollary)f (assertion)i(\(b\))f(of)h(\(3.3\))f(yields)g(a)h(result)f(on)h (functions)f Fo(f)44 b Fq(that)32 b(are)236 1303 y(regular)c(with)g (resp)s(ect)i(to)e Fo(P)14 b Fq(,)29 b(i.e.)42 b(satisfy)28 b(0)g Fk(\024)g Fo(f)38 b Fq(=)28 b Fo(P)14 b(f)d Fq(:)40 b(in)28 b(the)h(irreducible)e(case)j(suc)m(h)236 1424 y(a)k(function,)h(pro)m(vided)g(it)e(is)h(con)m(tained)h(in)e Fk(R)p Fq(\()p Fo(E)6 b Fq(\),)36 b(has)f(to)f(b)s(e)g(constan)m(t.)50 b(Indeed,)37 b(the)236 1544 y(equation)c Fo(P)712 1508 y Fn(n)758 1544 y Fo(f)38 b Fq(=)28 b Fo(f)43 b Fq(for)32 b Fo(n)c Fk(\025)g Fq(0)33 b(and)f(the)h(b)s(oundedness)i(of)d Fo(f)43 b Fq(com)m(bine)32 b(b)m(y)i(\(3.3b\))e(to)1034 1751 y Fo(f)11 b Fq(\()p Fo(x)p Fq(\))22 b Fk(\000)h Fo(f)11 b Fq(\(0\))94 b(=)h 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Fk(\025)h Fp(E)p Fq(\()p Fo(f)1798 5474 y Fl(1)1837 5459 y Fq(\()p Fo(X)1964 5418 y Fl(0)1956 5484 y Fn(n)2004 5459 y Fq(\)\))22 b Fk(\000)g Fp(E)p Fq(\()p Fo(f)2361 5474 y Fl(2)2400 5459 y Fq(\()p Fo(X)2527 5418 y Fn(t)2519 5484 y(n)2566 5459 y Fq(\)\))28 b(=:)f Fo(\016)2843 5474 y Fn(n)2907 5459 y Fo(:)1841 5753 y Fq(17)p eop %%Page: 18 19 18 18 bop 236 154 a Fq(No)m(w)33 b(the)g(iden)m(tit)m(y)972 348 y Fo(\016)1015 363 y Fn(n)1090 348 y Fq(=)28 b Fp(E)p Fq(\()p Fo(f)11 b Fq(\()p Fo(X)1492 307 y Fl(0)1484 373 y Fn(n)1530 348 y Fq(\)\))22 b Fk(\000)h Fq(\()p Fp(E)p Fq(\()p Fo(f)1926 363 y Fl(2)1965 348 y Fq(\()p Fo(X)2092 307 y Fn(t)2084 373 y(n)2131 348 y Fq(\)\))f Fk(\000)g Fp(E)p Fq(\()p Fo(f)2488 363 y Fl(2)2527 348 y Fq(\()p Fo(X)2654 307 y Fl(0)2646 373 y Fn(n)2693 348 y Fq(\)\)\))236 542 y(ensures)34 b(the)f(existence)h(of)e(some)g Fo(m)c Fk(2)g Fp(N)j Fq(with)h Fo(\016)2107 557 y Fn(m)2202 542 y Fo(>)27 b Fq(0,)32 b(b)s(ecause)i(otherwise)f(b)m(y)g(\(3.1\))789 653 y Fi(X)782 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b Fk(\024)29 b Fo(\026P)1841 5753 y Fq(18)p eop %%Page: 19 20 19 19 bop 236 154 a Fq(for)30 b(excessiv)m(e)k(measures,)e(whic)m(h)f (follo)m(ws)e(as)i(the)g(dual)f(result)h(for)f(excessiv)m(e)j (functions)236 274 y(\(see)50 b(e.g.)91 b(Prop)s(osition)47 b(2.2.6)h(in)f([26]\).)91 b(Applied)48 b(to)g Fo(A)54 b Fq(=)h([0)p Fo(;)17 b(t)p Fq(])48 b(this)g(inequalit)m(y)236 395 y(concerns)632 359 y Fn(t)645 395 y Fo(P)e Fq(and)33 b(implies)236 595 y(\()p Fk(\003)p Fq(\))631 b(\()1031 553 y Fn(t)1044 595 y Fo(\026)1120 553 y Fn(t)1132 595 y Fo(P)14 b Fq(\)\()p Fo(B)5 b Fq(\))27 b Fk(\024)i Fq(\()p Fo(\026P)14 b Fq(\)\()p Fo(B)5 b Fq(\))123 b(for)74 b Fo(B)32 b Fk(2)d(B)s Fq(\([0)p Fo(;)17 b(t)p Fq(]\))g Fo(:)236 794 y Fq(Therefore)36 b Fo(\026P)45 b Fk(\024)31 b Fo(\026)k Fq(yields)1325 758 y Fn(t)1338 794 y Fo(\026)1414 758 y Fn(t)1427 794 y Fo(P)44 b Fk(\024)1643 758 y Fn(t)1656 794 y Fo(\026)35 b Fq(pro)m(ving)f(\(a\),)h(b)s(ecause)2654 758 y Fn(t)2667 794 y Fo(\026)g 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b(coun)m(terexample)g(at)g(the)g(end) 236 5248 y(of)41 b(the)g(preceding)g(section)g(implies,)f(on)h(the)g (other)g(hand,)i(that)e(a)f(pro)s(cess)i(\()p Fo(X)3321 5263 y Fn(n)3368 5248 y Fq(\))3406 5263 y Fn(n)p Fm(\025)p Fl(0)236 5368 y Fq(recurren)m(t)36 b(according)d(to)g(\(2.1\))h(need)h (not)e(b)s(e)h(recurren)m(t)i(in)d(the)h(restricted)g(sense,)j(b)s(e-) 236 5489 y(cause)26 b(in)f(this)g(case)h(there)g(is)e(alw)m(a)m(ys)i (an)f Fk(\000)h Fq(up)f(to)g(a)g(constan)m(t)h(factor)f Fk(\000)g Fq(unique)h Fo(\033)t Fq(-\014nite)1841 5753 y(23)p eop %%Page: 24 25 24 24 bop 236 154 a Fq(in)m(v)-5 b(arian)m(t)32 b(measure)h(\(see)g (e.g.)44 b([23,)33 b(26]\).)43 b(There)34 b(is,)e(ho)m(w)m(ev)m(er,)j (a)e(particular)e(situation)236 274 y(that)g(\014ts)h(in)m(to)f(this)g (framew)m(ork)g(and)h(will)d(b)s(e)j(needed)h(in)d(the)i(follo)m(wing)d (section.)43 b(This)236 395 y(\\regenerativ)m(e)30 b(case")f(mak)m(es)h (no)f(use)h(of)e(the)i(top)s(ological)25 b(structure)30 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Fq(\))948 4938 y Fn(n)p Fm(\025)p Fl(0)1085 4923 y Fq(,)f(some)f(preparations)g(are)h(necessary:)236 5122 y Fp(\(6.1\))41 b(Lemma)82 b Fh(If)40 b Fo(\027)k Fk(2)39 b(N)15 b Fq([)p Fo(E)6 b Fq(])40 b Fh(is)g(r)-5 b(e)g(curr)g(ent,)42 b(then)f(for)f Fo(x)p 2472 5135 V 39 w(<)e(t)g Fk(2)g Fo(E)47 b Fh(and)40 b(arbitr)-5 b(ary)236 5243 y(initial)35 b(law)1013 5280 y Fi(X)948 5463 y Fl(0)p Fm(\024)p Fn(m)f Fq(0)p Fh(.)45 b(Then)34 b(for)h Fo(x)p 1185 2992 V 28 w(<)27 b(t)h Fk(2)g Fo(E)41 b Fh(and)34 b(arbitr)-5 b(ary)36 b(initial)e(law)706 3108 y Fi(X)641 3291 y Fl(0)p Fm(\024)p Fn(m)h(k)s Fq(\))124 b(for)74 b Fo(k)30 b Fk(\025)f Fq(0)17 b Fo(:)236 524 y Fq(Then)34 b(decomp)s(osition)d(according)h(to)g(the)h(last)f(sta)m (y)h(in)f Fo(z)37 b Fq(yields)1128 627 y Fi(X)1063 809 y Fl(0)p Fm(\024)p Fn(m)g(x)p 1250 3803 56 4 v 41 w Fq(or)f Fo(t)h Fq(=)p 1665 3737 V 40 w Fo(x)q Fq(.)66 b(Moreo)m(v)m(er,)44 b Fo(\026)p Fq(\()p Fk(f)p Fo(t)p Fk(g)p Fq(\))c(=)h(0)f(ma)m(y)g(b)s(e)h(supp)s (osed)236 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Fm(!1)1627 154 y Fo(\045)1677 169 y Fn(n)1725 154 y Fq(\([)p Fo(a;)g(b)p Fq(]\))28 b Fk(\024)g Fo(\045)p Fq(\([)p Fo(a;)17 b(b)p Fq(]\))g Fo(:)236 385 y Fq(In)35 b(particular,)e(\(2\))h(implies)e(that)i(the)h(con)m(v)m(ergence)i Fo(\045)2305 400 y Fn(n)2352 385 y Fq(\()p Fk(f)p Fo(z)t Fk(g)p Fq(\))31 b Fk(!)g Fo(\045)p Fq(\()p Fk(f)p Fo(z)t Fk(g)p Fq(\),)k(established)236 506 y(in)40 b(\(6.2\))f(for)h(the)h (case)g Fo(\026)p Fq(\()p Fk(f)p Fo(z)t Fk(g)p Fq(\))f Fo(>)h Fq(0,)h(extends)g(to)e(the)g(case)h Fo(\026)p Fq(\()p Fk(f)p Fo(z)t Fk(g)p Fq(\))g(=)g(0.)66 b(T)-8 b(ogether)236 626 y(with)32 b(\(1\))h(this)f(leads)g(to)632 817 y(lim)17 b(inf)675 867 y Fn(n)p Fm(!1)935 817 y Fo(\045)985 832 y Fn(n)1033 817 y Fq(\([)p Fo(a;)g(b)p Fq(]\))84 b(=)g(lim)17 b(inf)1586 867 y Fn(n)p Fm(!1)1829 817 y Fq(\()p Fo(\045)1917 832 y Fn(n)1965 817 y Fq(\(])p Fo(a;)g(b)p Fq([\))22 b(+)g Fo(\045)2401 832 y Fn(n)2448 817 y Fq(\()p Fk(f)p Fo(a)p Fk(g)p Fq(\))g(+)g Fo(\045)2845 832 y Fn(n)2893 817 y Fq(\()p Fk(f)p Fo(b)p Fk(g)p Fq(\)\))1383 983 y(=)84 b(lim)17 b(inf)1586 1033 y Fn(n)p Fm(!1)1846 983 y Fo(\045)1896 998 y Fn(n)1943 983 y Fq(\(])p Fo(a;)g(b)p Fq([\))23 b(+)f Fo(\045)p Fq(\()p Fk(f)p Fo(a)p Fk(g)p Fq(\))g(+)g Fo(\045)p Fq(\()p Fk(f)p Fo(b)p Fk(g)p Fq(\))1382 1149 y Fk(\025)84 b Fo(\045)p Fq(\(])p Fo(a;)17 b(b)p Fq([\))23 b(+)f Fo(\045)p Fq(\()p Fk(f)p Fo(a)p Fk(g)p Fq(\))g(+)g Fo(\045)p Fq(\()p Fk(f)p Fo(b)p Fk(g)p Fq(\))17 b Fo(;)236 1340 y Fq(hence)34 b(to)236 1531 y(\(3\))949 b(lim)17 b(inf)1353 1581 y Fn(n)p Fm(!1)1613 1531 y Fo(\045)1663 1546 y Fn(n)1710 1531 y Fq(\([)p Fo(a;)g(b)p Fq(]\))28 b Fk(\025)g Fo(\045)p Fq(\([)p Fo(a;)17 b(b)p Fq(]\))g Fo(:)383 1743 y Fq(4.)43 b(Com)m(bined,)32 b(equations)h(\(2\))f(and)h (\(3\))f(yield)865 1934 y Fo(\045)915 1949 y Fn(n)962 1934 y Fo(f)38 b Fk(!)28 b Fo(\045f)38 b Fq(=)28 b Fo(\026f)11 b(=\026g)127 b Fq(for)32 b(all)72 b Fo(f)38 b Fq(=)28 b(1)2381 1949 y Fl([)p Fn(a;b)p Fl(])2539 1934 y Fk(2)g(R)p Fq(\()p Fo(E)6 b 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Fn(n)1035 2997 y Fk(\024)28 b Fo(s)p Fq(\))17 b Fo(=)g Fp(P)p Fq(\()p Fo(X)1511 2956 y Fl(0)1503 3021 y Fn(n)1576 2997 y Fk(\024)28 b Fo(t)p Fq(\))g(=)1958 2914 y Fi(Y)1886 3097 y Fl(0)p Fm(\024)p Fn(m)g(x)p 1940 607 56 4 v 29 w Fq(or)g Fo(t)g Fq(=)p 2305 541 V 27 w Fo(x)q Fq(.)42 b(Since)29 b(in)f(a)g(self-explanatory)236 714 y(notation)1145 797 y Fo(S)6 b Fq(\()p Fo(f)f(;)17 b(x)p Fq(\))p 1145 842 295 4 v 1149 933 a Fo(S)6 b Fq(\()p Fo(g)t(;)17 b(y)t Fq(\))1493 865 y(=)1624 797 y Fo(S)6 b Fq(\()p Fo(f)f(;)17 b(x)p Fq(\))p 1624 842 V 1626 933 a Fo(S)6 b Fq(\()p Fo(g)t(;)17 b(x)p Fq(\))1965 797 y Fo(S)6 b Fq(\()p Fo(g)t(;)17 b(x)p Fq(\))p 1965 842 292 4 v 1968 933 a Fo(S)6 b Fq(\()p Fo(g)t(;)17 b Fq(0\))2305 797 y Fo(S)6 b Fq(\()p Fo(g)t(;)17 b Fq(0\))p 2304 842 287 4 v 2304 933 a Fo(S)6 b Fq(\()p Fo(g)t(;)17 b(y)t Fq(\))2617 865 y Fo(;)236 1089 y Fq(only)32 b(the)h(follo)m(wing)d(t)m(w)m(o)j(assertions)g(ha)m(v)m(e)h(to)f(b)s (e)f(v)m(eri\014ed:)236 1310 y(\(1\))978 1226 y Fi(X)912 1409 y Fl(0)p Fm(\024)p Fn(m)f Fq(0)17 b Fo(:)236 5240 y Fq(Applying)39 b(\(6.4\))f(to)h Fo(f)50 b Fq(=)39 b(1)1293 5255 y Fn(I)1324 5267 y Ff(k)1405 5240 y Fq(and)h Fo(g)i Fq(=)d(1)1855 5255 y Fl([0)p Fn(;t)p Fl(])1979 5240 y Fq(,)i(where)f Fo(t)g Fk(2)f Fo(E)45 b Fq(with)39 b Fo(t)h(>)f(x)p 3051 5253 56 4 v 39 w Fq(or)g Fo(t)h Fq(=)p 3461 5187 V 39 w Fo(x)p Fq(,)236 5360 y(sho)m(ws)34 b(that)f(this)f(holds)g(indeed)h(with)f(probabilit)m(y)f(1.)1841 5753 y(31)p eop %%Page: 32 33 32 32 bop 383 154 a Fq(2.)41 b(T)-8 b(o)29 b(pro)m(v)m(e)h(the)f(in)m (v)m(erse)g(inclusion,)f(denote)i(b)m(y)f Fo(L)2320 169 y Fn(t)2350 154 y Fq(\()p Fo(!)t Fq(\))f(the)g(analogue)g(of)g Fo(L)p Fq(\()p Fo(!)t Fq(\))g(for)236 274 y(the)34 b(pro)s(cess)h(\() 784 238 y Fn(t)797 274 y Fo(X)878 289 y Fn(n)925 274 y Fq(\))963 289 y Fn(n)p Fm(\025)p Fl(0)1100 274 y Fq(.)46 b(Moreo)m(v)m(er,)35 b(let)e Fo(D)j Fq(consist)d(of)p 2321 222 56 4 v 33 w Fo(x)h Fq(in)f(the)g(case)p 2901 222 V 35 w Fo(x)c Fk(2)h Fo(E)39 b Fq(and)34 b(of)f(a)236 395 y(coun)m(table)g(subset)h(of)e Fk(f)p Fo(t)c Fk(2)g Fo(E)34 b Fq(:)27 b Fo(t)h(>)g(x)p 1622 408 V Fk(g)33 b Fq(with)f(sup)17 b Fo(D)31 b Fq(=)p 2360 342 V 27 w Fo(x)i Fq(otherwise.)44 b(Then)34 b(clearly)1498 605 y Fo(L)p Fq(\()p Fo(!)t Fq(\))27 b(=)1856 522 y Fi([)1836 706 y Fn(t)p Fm(2)p Fn(D)2002 605 y Fo(L)2068 620 y Fn(t)2098 605 y Fq(\()p Fo(!)t Fq(\))17 b Fo(;)236 887 y Fq(hence)34 b(it)e(su\016ces)i(to)f(v)m(erify)g(that)f(with)g(probabilit)m(y)f(1) 1267 1097 y Fo(L)1333 1112 y Fn(t)1363 1097 y Fq(\()p Fo(!)t Fq(\))c Fk(\032)h Fq(supp)18 b Fo(\026)124 b Fq(for)74 b Fo(t)28 b Fk(2)g Fo(D)19 b(:)236 1308 y Fq(T)-8 b(o)37 b(this)f(end)i(let)e Fo(X)993 1323 y Fl(0)1069 1308 y Fq(\014rst)h(b)s(e)g(distributed)f(according)g(to)g(\(the)h(trivial)d (extension)k(of)7 b(\))236 1428 y(the)34 b(normalized)e(restriction)g (of)h Fo(\026)g Fq(to)g([0)p Fo(;)17 b(t)p Fq(].)46 b(Then)35 b(\()2245 1392 y Fn(t)2258 1428 y Fo(X)2339 1443 y Fn(n)2386 1428 y Fq(\))2424 1443 y Fn(n)p Fm(\025)p Fl(0)2594 1428 y Fq(is)e(stationary)g(b)m(y)h(\(4.2a\))236 1549 y(and)f(th)m(us)236 1759 y(\()p Fk(\003)p Fq(\))304 b Fp(P)p Fq(\()781 1718 y Fn(t)793 1759 y Fo(X)874 1774 y Fn(n)960 1759 y Fo(=)-60 b Fk(2)28 b Fo(I)1086 1774 y Fn(k)1170 1759 y Fq(ev)m(en)m(tually)q(\)) f(=)h(1)83 b(whenev)m(er)i Fo(I)2426 1774 y Fn(k)2508 1759 y Fk(\\)39 b Fq(supp)17 b Fo(\026)28 b Fq(=)f Fk(;)17 b Fo(;)236 1970 y Fq(as)36 b(desired.)53 b(Finally)-8 b(,)33 b(an)j(application)d(of)i(\(3.3b\))g(to)g Fo(f)43 b Fq(=)33 b(1)2512 1985 y Fn(I)2543 1997 y Ff(k)2620 1970 y Fq(sho)m(ws)k(that)e(the)h(distri-)236 2090 y(bution)c(of)g Fo(X)737 2105 y Fl(0)809 2090 y Fq(in)g(fact)g(is)g(irrelev)-5 b(an)m(t)32 b(for)g(\()p Fk(\003)p Fq(\).)126 b Fg(2)383 2290 y Fq(T)-8 b(ogether,)48 b(\(2.4\))c(and)h(\(6.5\))f(imply)f(that)i (the)g(t)m(w)m(o)g(familiar)c(criteria)i(for)h(recur-)236 2410 y(rence)16 b(/)g(transience)33 b(from)c(discrete)j(Mark)m(o)m(v)g (c)m(hain)e(theory)h(carry)g(o)m(v)m(er)h(to)e(the)i(presen)m(t)236 2531 y(setting)h(in)e(the)i(follo)m(wing)d(form:)236 2705 y(\(1\))41 b(If)33 b Fo(\027)39 b Fq(is)32 b(recurren)m(t,)i(then) f(for)f Fo(x)c Fk(2)h Fq(supp)17 b Fo(\026)32 b Fq(alw)m(a)m(ys)1176 2916 y Fp(P)1270 2874 y Fn(x)1313 2916 y Fq(\()p Fo(X)1432 2931 y Fn(n)1506 2916 y Fk(2)c Fo(G)42 b Fq(in\014nitely)31 b(often\))d(=)f(1)17 b Fo(;)236 3126 y Fq(hence)1254 3247 y Fp(E)1328 3206 y Fn(x)1371 3247 y Fq(\()p Fk(jf)p Fo(n)28 b Fk(\025)g Fq(0)f(:)h Fo(X)1890 3262 y Fn(n)1965 3247 y Fk(2)g Fo(G)p Fk(gj)p Fq(\))f(=)g Fk(1)17 b Fo(;)236 3417 y Fq(pro)m(vided)33 b Fo(G)g Fq(is)f(an)g(op)s(en)h(neigh)m(b)s (orho)s(o)s(d)e(of)i Fo(x)p Fq(.)236 3591 y(\(2\))41 b(If)33 b Fo(\027)39 b Fq(is)32 b(transien)m(t,)h(then)g(for)f Fo(x)c Fk(2)g Fo(E)39 b Fq(alw)m(a)m(ys)1247 3802 y Fp(E)1321 3761 y Fn(x)1365 3802 y Fq(\()p Fk(jf)p Fo(n)27 b Fk(\025)h Fq(0)g(:)f Fo(X)1883 3817 y Fn(n)1958 3802 y Fk(2)h Fo(K)7 b Fk(gj)p Fq(\))27 b Fo(<)h Fk(1)17 b Fo(;)236 4013 y Fq(hence)1169 4133 y Fp(P)1263 4092 y Fn(x)1306 4133 y Fq(\()p Fo(X)1425 4148 y Fn(n)1500 4133 y Fk(2)28 b Fo(K)48 b Fq(in\014nitely)32 b(often\))27 b(=)h(0)17 b Fo(;)236 4304 y Fq(pro)m(vided)33 b Fo(K)40 b Fq(is)32 b(a)g(compact)h(subset)h(of)e Fo(E)6 b Fq(.)383 4447 y(The)33 b(\014nal)f(result)g(of)g(this)h(section)f(is)g(related)h(to)f (\(5.5\))g(and)h(\(5.6\):)236 4647 y Fp(\(6.6\))41 b(Prop)s(osition)87 b Fh(L)-5 b(et)48 b Fo(\027)57 b Fk(2)51 b(N)15 b Fq([)p Fo(E)6 b Fq(])47 b Fh(b)-5 b(e)47 b(r)-5 b(e)g(curr)g(ent)48 b(with)f(invariant)g(me)-5 b(asur)g(e)47 b Fo(\026)p Fh(.)236 4767 y(Then)36 b(for)g(e)-5 b(ach)36 b(op)-5 b(en)36 b(subset)h Fo(G)g Fh(of)f Fo(E)43 b Fh(satisfying)36 b Fo(\026)p Fq(\()p Fo(G)p Fq(\))30 b Fo(>)h Fq(0)36 b Fh(and)h(every)f Fo(t)31 b Fk(2)h Fo(E)42 b Fh(ther)-5 b(e)236 4888 y(exists)35 b Fo(n)561 4903 y Fl(0)628 4888 y Fk(2)28 b Fp(N)35 b Fh(such)f(that)924 5098 y Fp(P)p Fq(\()p Fo(X)1128 5057 y Fn(x)1120 5123 y(n)1198 5098 y Fk(2)28 b Fo(G)45 b Fh(for)f Fq(0)27 b Fk(\024)h Fo(x)h Fk(\024)f Fo(t)p Fq(\))g Fo(>)f Fq(0)124 b Fh(for)77 b Fo(n)27 b Fk(\025)i Fo(n)2773 5113 y Fl(0)2829 5098 y Fo(:)236 5384 y Fh(Pr)-5 b(o)g(of.)41 b Fq(Since)36 b Fo(G)g Fq(ma)m(y)f(b)s(e)h(assumed)g(to)f(b)s(e)h(a)g(b)s(ounded)g (in)m(terv)-5 b(al,)35 b Fo(f)44 b Fq(=)32 b(1)3025 5399 y Fn(G)3120 5384 y Fq(in)j(view)h(of)236 5504 y(\(6.3\))27 b(satis\014es)g(all)e(conditions)h(in)g(\(3.4\).)41 b(Accordingly)26 b(there)i(exists)g Fo(m)g Fk(2)g Fp(N)e Fq(suc)m(h)i(that)1841 5753 y(32)p eop %%Page: 33 34 33 33 bop 1116 154 a Fo(#)28 b Fq(:=)f Fp(P)p Fq(\()p Fo(X)1535 113 y Fn(x)1527 179 y(m)1621 154 y Fk(2)h Fo(G)44 b Fq(for)g(0)28 b Fk(\024)g Fo(x)g Fk(\024)g Fo(t)p Fq(\))g Fo(>)f Fq(0)17 b Fo(;)236 360 y Fq(where)33 b(the)e(assumption)g Fo(t)d(>)f(x)p 1364 373 56 4 v 32 w Fq(or)k Fo(t)d Fq(=)p 1735 307 V 27 w Fo(x)k Fq(means)f(no)g(loss)g(of)g(generalit)m(y)-8 b(.)42 b(By)31 b(applying)236 480 y(F)-8 b(ubini)31 b(it)h(follo)m(ws)f (that)330 686 y Fp(P)p Fq(\()p Fo(X)534 645 y Fn(x)526 710 y(l)q Fl(+)p Fn(m)696 686 y Fk(2)d Fo(G)44 b Fq(for)g(0)28 b Fk(\024)g Fo(x)g Fk(\024)g Fo(t)p Fq(\))84 b(=)1759 569 y Fi(Z)1805 757 y Fm(H)p Fl([)p Fn(E)t Fl(])1980 686 y Fp(P)p Fq(\()p Fo(X)2184 645 y Fn(h)p Fl(\()p Fn(x)p Fl(\))2176 710 y Fn(m)2351 686 y Fk(2)28 b Fo(G)44 b Fq(for)g(0)27 b Fk(\024)i Fo(x)f Fk(\024)g Fo(t)p Fq(\))17 b Fo(\027)3241 645 y Fn(l)3267 686 y Fq(\()p Fo(dh)p Fq(\))1598 894 y Fk(\025)84 b Fo(#)17 b(\027)1887 853 y Fn(l)1913 894 y Fq(\()p Fo(h)p Fq(\()p Fo(t)p Fq(\))28 b Fk(\024)g Fo(t)p Fq(\))125 b(for)74 b Fo(l)30 b Fk(2)e Fp(N)17 b Fo(;)236 1100 y Fq(b)s(ecause)33 b Fo(x)28 b Fk(\024)g Fo(t)j Fq(and)h Fo(h)p Fq(\()p Fo(t)p Fq(\))c Fk(\024)g Fo(t)j Fq(imply)e Fo(h)p Fq(\()p Fo(x)p Fq(\))g Fk(\024)f Fo(t)p Fq(.)43 b(No)m(w)32 b Fo(\027)2378 1064 y Fn(l)2404 1100 y Fq(\()p Fo(h)p Fq(\()p Fo(t)p Fq(\))c Fk(\024)g Fo(t)p Fq(\))g Fo(>)g Fq(0)j(follo)m(ws)e(from)236 1220 y(\(2.5\))38 b(and)h(\(1.8\))g(in)f(the)h(case)g Fo(t)g(>)f(x)p 1605 1233 V 39 w Fq(and)h(is)f(trivial)e(in)i(the)i (case)f Fo(t)g Fq(=)p 2990 1168 V 38 w Fo(x)p Fq(.)62 b(Therefore)236 1341 y Fo(n)294 1356 y Fl(0)362 1341 y Fq(=)27 b Fo(m)33 b Fq(satis\014es)g(the)g(assertion.)127 b Fg(2)236 1582 y Fp(7.)50 b(Prop)s(erties)36 b(of)i(the)f(attractor) 236 1781 y Fq(While)i(in)h(the)h(transien)m(t)f(case)p 1458 1728 V 42 w Fo(x)g Fq(attracts)h(the)g(pro)s(cess)g(\()p Fo(X)2567 1796 y Fn(n)2614 1781 y Fq(\))2652 1796 y Fn(n)p Fm(\025)p Fl(0)2789 1781 y Fq(,)h(in)e(the)h(recurren)m(t)236 1901 y(case)34 b(\(6.5\))e(suggests)i(the)f(follo)m(wing)d (terminology:)236 2100 y Fp(\(7.1\))41 b(De\014nition)75 b Fh(If)34 b Fo(\027)g Fk(2)28 b(N)15 b Fq([)p Fo(E)6 b Fq(])34 b Fh(is)g(r)-5 b(e)g(curr)g(ent)35 b(with)f(invariant)g(me)-5 b(asur)g(e)34 b Fo(\026)p Fh(,)g(the)g(set)236 2221 y Fo(M)39 b Fq(:=)27 b(supp)18 b Fo(\026)35 b Fh(is)f(c)-5 b(al)5 b(le)-5 b(d)34 b(the)h(\\attr)-5 b(actor")35 b(of)g Fo(\027)6 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b(e)-5 b(quivalently,)34 b(the)h(c)-5 b(ondition)236 3912 y Fq(\(b\))940 b Fo(h)p Fq([)p Fo(F)14 b Fq(])28 b Fk(\032)g Fo(F)138 b Fh(for)35 b(al)5 b(l)76 b Fo(h)27 b Fk(2)h Fo(N)g(:)236 4192 y Fh(Pr)-5 b(o)g(of.)41 b Fq(1.)55 b(Since)36 b(the)h(set)g(of)f(mappings)f Fo(h)f Fk(2)g(H)q Fq([)p Fo(E)6 b Fq(])37 b(with)f Fo(h)p Fq([)p Fo(F)14 b Fq(])34 b Fk(\032)g Fo(F)50 b Fq(is)36 b(closed,)h(b)s(oth)236 4312 y(conditions)e(are)g(clearly)g(equiv)-5 b(alen)m(t.)52 b(Moreo)m(v)m(er,)38 b(an)m(y)e(nonempt)m(y)g(closed)g(set)g Fo(F)46 b Fk(\032)33 b Fo(E)236 4433 y Fq(satisfying)g(\(a\))g (satis\014es)i(the)f(corresp)s(onding)f(condition)f(with)i(resp)s(ect)h (to)e Fo(\027)3149 4397 y Fn(n)3230 4433 y Fq(as)h(w)m(ell.)236 4553 y(F)-8 b(or)30 b(an)m(y)i Fo(x)c Fk(2)g Fo(F)45 b Fq(this)30 b(implies)f Fp(P)p Fq(\()p Fo(X)1599 4517 y Fn(x)1591 4578 y(n)1670 4553 y Fk(2)f Fo(F)14 b Fq(\))27 b(=)h(1)i(for)h(all)e Fo(n)e Fk(\025)i Fq(0,)i(and)g(th)m(us)h(the)f (pro)s(cess)236 4674 y(\()p Fo(X)363 4637 y Fn(x)355 4698 y(n)407 4674 y Fq(\))445 4689 y Fn(n)p Fm(\025)p Fl(0)607 4674 y Fq(with)25 b(probabilit)m(y)e(1)i(has)h(all)d(its)i (limit)d(p)s(oin)m(ts)j(in)f Fo(F)14 b Fq(.)41 b(Therefore)26 b(the)g(inclusion)236 4794 y Fo(M)39 b Fk(\032)28 b Fo(F)46 b Fq(is)32 b(a)g(consequence)k(of)c(\(6.5\).)383 4937 y(2.)41 b(It)28 b(remains)f(to)g(v)m(erify)h(that)f(\(b\))h(is)f (satis\014ed)h(for)f Fo(F)42 b Fq(=)27 b Fo(M)10 b Fq(.)43 b(T)-8 b(o)27 b(this)h(end)g(consider)236 5058 y Fo(x)g Fq(=)g Fo(h)479 5073 y Fl(0)518 5058 y Fq(\()p Fo(x)611 5073 y Fl(0)651 5058 y Fq(\))h(with)f Fo(x)991 5073 y Fl(0)1059 5058 y Fk(2)g Fo(M)39 b Fq(and)29 b Fo(h)1528 5073 y Fl(0)1595 5058 y Fk(2)f Fo(N)10 b Fq(,)30 b(and)f(let)f Fo(G)g Fk(\032)g Fo(E)35 b Fq(b)s(e)29 b(an)m(y)g(op)s(en)g(neigb)s (orho)s(o)s(d)236 5178 y(of)d Fo(x)p Fq(.)42 b(Then,)29 b(due)e(to)f(the)h(con)m(tin)m(uit)m(y)g(of)f(the)h(mapping)e(\()p Fo(x;)17 b(h)p Fq(\))28 b Fk(!)f Fo(h)p Fq(\()p Fo(x)p Fq(\),)h(there)g(are)e(op)s(en)236 5298 y(sets)34 b Fo(G)504 5313 y Fl(0)571 5298 y Fk(\032)28 b Fo(E)39 b Fq(and)32 b Fk(H)1060 5313 y Fl(0)1128 5298 y Fk(\032)c(H)q Fq([)p Fo(E)6 b Fq(])33 b(suc)m(h)h(that)614 5504 y(\()p Fo(x)707 5519 y Fl(0)747 5504 y Fo(;)17 b(h)847 5519 y Fl(0)886 5504 y Fq(\))28 b Fk(2)g Fo(G)1123 5519 y Fl(0)1185 5504 y Fk(\002)22 b(H)1368 5519 y Fl(0)1532 5504 y Fq(and)125 b Fo(h)p Fq(\()p Fo(x)p Fq(\))28 b Fk(2)g Fo(G)83 b Fq(for)g Fo(x)28 b Fk(2)g Fo(G)2737 5519 y Fl(0)2776 5504 y Fo(;)17 b(h)28 b Fk(2)g(H)3082 5519 y Fl(0)3138 5504 y Fo(:)1841 5753 y Fq(33)p eop %%Page: 34 35 34 34 bop 236 154 a Fq(By)33 b(the)g(in)m(v)-5 b(ariance)32 b(of)g Fo(\026)g Fq(this)g(yields)1107 372 y Fo(\026)p Fq(\()p Fo(G)p Fq(\))83 b(=)h(\()p Fo(\026)22 b Fk(\012)h Fo(\027)6 b Fq(\))p Fk(f)p Fq(\()p Fo(x;)17 b(h)p Fq(\))28 b(:)f Fo(h)p Fq(\()p Fo(x)p Fq(\))i Fk(2)f Fo(G)p Fk(g)1402 518 y(\025)83 b Fq(\()p Fo(\026)22 b Fk(\012)h Fo(\027)6 b Fq(\))17 b(\()p Fo(G)2005 533 y Fl(0)2066 518 y Fk(\002)23 b(H)2250 533 y Fl(0)2289 518 y Fq(\))69 b Fo(>)g Fq(0)17 b Fo(:)236 736 y Fq(Therefore)34 b Fo(x)f Fq(has)g(to)f(b)s(e)h(con)m (tained)g(in)e(the)i(supp)s(ort)g(of)f Fo(\026)p Fq(.)127 b Fg(2)383 935 y Fq(It)37 b(is)g(a)g(consequence)j(of)d(\(b\))g(that)g (the)h(attractor)f(of)g(a)g(recurren)m(t)h(distribution)e Fo(\027)236 1056 y Fq(dep)s(ends)e(on)f(it)e(only)i(through)f(its)g (supp)s(ort)h Fo(N)10 b Fq(.)383 1201 y(In)29 b(the)g(follo)m(wing)d(t) m(w)m(o)j(prop)s(ositions)p 1800 1123 80 4 v 27 w Fo(B)34 b Fq(denotes)c(the)f(closure)g(of)f(a)h(subset)h Fo(B)k Fq(of)28 b Fo(E)6 b Fq(.)236 1321 y(Then)33 b Fo(M)42 b Fq(and)32 b Fo(N)42 b Fq(are)31 b(related)g(b)m(y)h(an)g(equation)f (that)g(is)g(basic)h(in)e(the)i(con)m(text)h(of)e(self-)236 1442 y(similar)f(sets)j(\(see)h(e.g.)44 b([12,)32 b(14]\))o(:)236 1641 y Fp(\(7.3\))41 b(Prop)s(osition)75 b Fh(If)34 b Fo(\027)g Fk(2)28 b(N)15 b Fq([)p Fo(E)6 b Fq(])35 b Fh(is)f(r)-5 b(e)g(curr)g(ent,)35 b(its)g(attr)-5 b(actor)36 b Fo(M)46 b Fh(satis\014es)1534 1880 y Fo(M)55 b Fq(=)p 1803 1779 400 4 v 1832 1797 a Fi([)1803 1981 y Fn(h)p Fm(2)p Fn(N)1987 1880 y Fo(h)p Fq([)p Fo(M)10 b Fq(])18 b Fo(:)236 2173 y Fh(Pr)-5 b(o)g(of.)41 b Fq(If)26 b Fo(F)38 b Fq(denotes)27 b(the)f(set)g(on)f(the)g(righ)m(t-hand)g(side,) i(the)e(inclusion)f Fo(F)41 b Fk(\032)29 b Fo(M)36 b Fq(follo)m(ws)236 2294 y(from)c(.)43 b(Con)m(v)m(ersely)-8 b(,)35 b(the)e(con)m(tin)m(uit)m(y)f(of)g Fo(h)c Fk(2)g(H)q Fq([)p Fo(E)6 b Fq(])33 b(yields)1422 2533 y Fo(h)p Fq([)p Fo(F)14 b Fq(])28 b Fk(\032)p 1742 2431 572 4 v 28 w Fo(h)p Fq([)1882 2450 y Fi([)1842 2635 y Fn(h)1883 2616 y Fe(0)1905 2635 y Fm(2)p Fn(N)2048 2533 y Fo(h)2104 2504 y Fm(0)2128 2533 y Fq([)p Fo(M)10 b Fq(]])17 b Fo(;)236 2828 y Fq(where)34 b(for)e Fo(h)723 2792 y Fm(0)774 2828 y Fk(2)c Fo(N)10 b Fq(,)33 b(again)f(b)m(y)h(\(7.2\),)f Fo(h)1728 2792 y Fm(0)1752 2828 y Fq([)p Fo(M)10 b Fq(])28 b Fk(\032)g Fo(M)10 b Fq(.)45 b(Therefore)1137 3047 y Fo(h)p Fq([)p Fo(F)14 b Fq(])28 b Fk(\032)p 1457 2962 216 4 v 28 w Fo(h)p Fq([)p Fo(M)10 b Fq(])29 b Fk(\032)f Fo(F)138 b Fq(for)32 b(all)72 b Fo(h)28 b Fk(2)g Fo(N)f(;)236 3265 y Fq(and)33 b(the)g(inclusion)e Fo(M)38 b Fk(\032)29 b Fo(F)46 b Fq(follo)m(ws,)31 b(once)i(more)f(from)g(\(7.2\).)126 b Fg(2)383 3464 y Fq(Whenev)m(er)24 b(b)s(oth)e Fo(M)32 b Fq(and)23 b Fo(N)32 b Fq(are)22 b(compact,)i(due)f(to)e(the)i(con)m (tin)m(uit)m(y)f(of)f(the)i(mapping)236 3585 y(\()p Fo(x;)17 b(h)p Fq(\))56 b Fk(!)f Fo(h)p Fq(\()p Fo(x)p Fq(\),)e(this)48 b(result)h(holds)g(without)f(taking)g(the)h(closure.)93 b(T)-8 b(o)48 b(see)i(that,)236 3705 y(without)32 b(assuming)f Fo(M)43 b Fq(to)31 b(b)s(e)i(compact,)e(this)h(ma)m(y)g(fail)e(ev)m(en) j(if)e Fo(N)42 b Fq(is)32 b(\014nite,)g(consider)236 3825 y(the)h(autoregressiv)m(e)h(pro)s(cess)1192 4083 y Fo(X)1273 4098 y Fn(n)1347 4083 y Fq(=)1461 4015 y(1)p 1461 4059 49 4 v 1461 4151 a(3)1519 4083 y Fo(X)1600 4098 y Fn(n)p Fm(\000)p Fl(1)1760 4083 y Fq(+)22 b Fo(V)1915 4098 y Fn(n)2086 4083 y Fq(for)74 b Fo(n)28 b Fk(2)g Fp(N)17 b Fo(;)236 4339 y Fq(where)42 b(the)f(i.i.d.)67 b(v)-5 b(ariables)39 b Fo(V)1429 4354 y Fn(n)1476 4339 y Fo(;)17 b(n)42 b Fk(2)g Fp(N)p Fo(;)e Fq(attain)g(the)h(v)-5 b(alues)40 b(0)h(and)2951 4300 y Fl(2)p 2951 4316 36 4 v 2951 4374 a(3)3037 4339 y Fq(with)f(proba-)236 4460 y(bilit)m(y)509 4421 y Fl(1)p 509 4437 V 509 4494 a(2)555 4460 y Fq(.)69 b(With)40 b Fo(E)48 b Fq(=)42 b([0)p Fo(;)17 b Fq(1[)41 b(as)g(state)g(space)i(the)e(corresp)s(onding)g (distribution)e Fo(\027)48 b Fq(is)236 4580 y(supp)s(orted)39 b(b)m(y)f(the)g(t)m(w)m(o)g(mappings)f Fo(h)1703 4595 y Fl(1)1779 4580 y Fq(:)f Fo(x)h Fk(!)e Fo(x=)p Fq(3)j(and)g Fo(h)2511 4595 y Fl(2)2586 4580 y Fq(:)f Fo(x)f Fk(!)g Fq(\()p Fo(x)26 b Fq(+)f(2\))p Fo(=)p Fq(3.)58 b(It)38 b(is)236 4701 y(clearly)33 b(irreducible)g(and)h(b)m(y)h(\(2.2\))f (recurren)m(t.)50 b(In)34 b(view)g(of)g(sup)18 b Fo(M)41 b Fq(=)30 b(1)k(b)m(y)h(\(7.2\))e(also)246 4782 y Fl(1)p 246 4798 V 246 4855 a(3)319 4821 y Fq(=)28 b(sup)17 b Fo(h)642 4836 y Fl(1)682 4821 y Fq([)p Fo(M)10 b Fq(])28 b Fk(2)g Fo(M)10 b Fq(,)34 b(while)e(on)g(the)h(other)g(hand)2194 4782 y Fl(1)p 2194 4798 V 2194 4855 a(3)2278 4821 y Fo(=)-60 b Fk(2)28 b Fo(h)2417 4836 y Fl(1)2456 4821 y Fq([)p Fo(E)6 b Fq(])39 b Fk([)g Fo(h)2788 4836 y Fl(2)2828 4821 y Fq([)p Fo(E)6 b Fq(].)383 4966 y(Next,)31 b Fo(M)42 b Fq(will)28 b(b)s(e)j(describ)s(ed)g(b)m(y)g(means)g(of)f(the)h (semigroup)e Fo(N)2781 4930 y Fm(\003)2852 4966 y Fq(generated)i(b)m(y) h Fo(N)10 b Fq(:)236 5286 y Fp(\(7.4\))41 b(Prop)s(osition)75 b Fh(If)34 b Fo(\027)g Fk(2)28 b(N)15 b Fq([)p Fo(E)6 b Fq(])35 b Fh(is)f(r)-5 b(e)g(curr)g(ent,)35 b(its)g(attr)-5 b(actor)36 b Fo(M)46 b Fh(satis\014es)999 5504 y Fo(M)38 b Fq(=)p 1235 5419 676 4 v 28 w Fk(f)p Fo(h)p Fq(\()p Fo(x)p Fq(\))28 b(:)g Fo(h)f Fk(2)i Fo(N)1821 5475 y Fm(\003)1861 5504 y Fk(g)124 b Fh(for)35 b(every)41 b Fo(x)28 b Fk(2)g Fo(M)g(:)1841 5753 y Fq(34)p eop %%Page: 35 36 35 35 bop 236 229 a Fh(Pr)-5 b(o)g(of.)41 b Fq(If)26 b Fo(F)38 b Fq(denotes)27 b(the)f(set)g(on)f(the)g(righ)m(t-hand)g (side,)i(the)e(inclusion)f Fo(F)41 b Fk(\032)29 b Fo(M)36 b Fq(follo)m(ws)236 349 y(from)c(\(7.2\))o(.)44 b(Con)m(v)m(ersely)-8 b(,)34 b(as)f(in)f(the)h(pro)s(of)e(of)i(\(7.3\),)1238 569 y Fo(h)p Fq([)p Fo(F)14 b Fq(])83 b Fk(\032)p 1669 484 873 4 v 84 w(f)p Fo(h)22 b Fk(\016)g Fo(h)1925 540 y Fm(0)1948 569 y Fq(\()p Fo(x)p Fq(\))28 b(:)g Fo(h)2218 540 y Fm(0)2269 569 y Fk(2)g Fo(N)2451 540 y Fm(\003)2491 569 y Fk(g)1508 714 y(\032)p 1669 630 769 4 v 84 w(f)p Fo(h)1775 686 y Fm(0)t(0)1821 714 y Fq(\()p Fo(x)p Fq(\))g(:)g Fo(h)2091 686 y Fm(0)5 b(0)2166 714 y Fk(2)28 b Fo(N)2348 686 y Fm(\003)2388 714 y Fk(g)1509 860 y Fq(=)84 b Fo(F)138 b Fq(for)32 b(all)72 b Fo(h)28 b Fk(2)g Fo(N)f(;)236 1080 y Fq(and)33 b(the)g(inclusion)e Fo(M)38 b Fk(\032)29 b Fo(F)46 b Fq(follo)m(ws)31 b(again)g(from)g(\(7.2\).)126 b Fg(2)383 1279 y Fq(The)34 b(assumption)e Fo(x)e Fk(2)f Fo(M)44 b Fq(is)33 b(clearly)f(essen)m(tial)h(for)g(the)g(inclusion)f Fo(F)43 b Fk(\032)29 b Fo(M)10 b Fq(,)34 b(while)236 1399 y(the)f(pro)s(of)f(sho)m(ws)i(that)e(the)h(inclusion)e Fo(M)39 b Fk(\032)28 b Fo(F)46 b Fq(holds)33 b(for)f(an)m(y)h Fo(x)28 b Fk(2)g Fo(E)6 b Fq(.)383 1545 y(The)33 b(most)f(explicit)f(c) m(haracterization)h(of)g Fo(M)43 b Fq(uses)34 b(the)f(closure)g Fo(N)2916 1508 y Fm(\003\003)3024 1545 y Fq(of)f Fo(N)3223 1508 y Fm(\003)3263 1545 y Fq(:)236 1744 y Fp(\(7.5\))41 b(Theorem)76 b Fh(If)35 b Fo(\027)f Fk(2)28 b(N)15 b Fq([)p Fo(E)6 b Fq(])34 b Fh(is)h(r)-5 b(e)g(curr)g(ent,)35 b(its)g(attr)-5 b(actor)35 b Fo(M)46 b Fh(satis\014es)1094 1964 y Fo(x)28 b Fk(2)g Fo(M)135 b Fh(if)35 b(and)f(only)h(if)124 b Fo(j)6 b Fq(\()p Fo(x)p Fq(\))28 b Fk(2)g Fo(N)2567 1923 y Fm(\003\003)2659 1964 y Fo(;)236 2184 y Fh(wher)-5 b(e)34 b Fo(j)41 b Fh(is)35 b(the)g(c)-5 b(anonic)g(al)33 b(inje)-5 b(ction)34 b(of)h Fo(E)41 b Fh(into)34 b Fk(H)q Fq([)p Fo(E)6 b Fq(])p Fh(.)236 2383 y(Pr)-5 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Fq(\))28 b Fo(>)f Fq(0)33 b(for)f(these)i(sets,)f(\(6.6\))f(applies)g(and)h(pro)m(vides)g Fo(n)28 b Fk(2)g Fp(N)k Fq(with)1211 3284 y Fp(P)p Fq(\()p Fo(X)1415 3243 y Fn(x)1407 3309 y(n)1486 3284 y Fk(2)c Fo(G)1657 3299 y Fl(0)1741 3284 y Fq(for)44 b(0)27 b Fk(\024)h Fo(x)g Fk(\024)g Fo(t)p Fq(\))g Fo(>)g Fq(0)17 b Fo(:)236 3504 y Fq(Th)m(us)34 b(there)g(exist)f Fo(h)1020 3519 y Fl(1)1059 3504 y Fo(;)17 b(:)g(:)g(:)f(;)h(h)1334 3519 y Fn(n)1408 3504 y Fk(2)28 b Fo(N)44 b Fq(suc)m(h)34 b(that)1072 3724 y Fo(h)1128 3739 y Fn(n)1198 3724 y Fk(\016)22 b Fo(:)17 b(:)g(:)k Fk(\016)h Fo(h)1534 3739 y Fl(1)1574 3724 y Fq(\()p Fo(x)p Fq(\))28 b Fk(2)g Fo(G)1904 3739 y Fl(0)2068 3724 y Fq(for)73 b(0)28 b Fk(\024)g Fo(x)g Fk(\024)g Fo(t)17 b(;)236 3944 y Fq(as)33 b(had)g(to)f(b)s(e)h (sho)m(wn.)383 4089 y(2.)59 b(T)-8 b(o)38 b(pro)m(v)m(e)h(the)f(con)m (v)m(erse,)k(let)37 b Fo(h)1732 4104 y Fn(n)1816 4089 y Fk(2)g Fo(N)2007 4053 y Fm(\003)2085 4089 y Fq(con)m(v)m(erge)j(to)d Fo(h)2669 4104 y Fl(0)2745 4089 y Fq(=)g Fo(j)6 b Fq(\()p Fo(x)2997 4104 y Fl(0)3037 4089 y Fq(\))38 b(and)g(apply)236 4210 y(\(7.4\))32 b(to)g Fo(x)d Fq(=)e Fo(x)p 775 4223 56 4 v 28 w Fk(2)h Fo(M)10 b Fq(,)34 b(leading)d(to)1158 4430 y Fo(x)1213 4445 y Fl(0)1281 4430 y Fq(=)c Fo(h)1440 4445 y Fl(0)1480 4430 y Fq(\()p Fo(x)p 1518 4443 V Fq(\))h(=)51 b(lim)1742 4479 y Fn(n)p Fm(!1)1959 4430 y Fo(h)2015 4445 y Fn(n)2063 4430 y Fq(\()p Fo(x)p 2101 4443 V Fq(\))27 b Fk(2)i Fo(M)e(:)83 b Fg(2)383 4766 y Fq(It)42 b(is)g(a)g(consequence) k(of)c(this)g(result,)j(that)d(the)h(\014xed)g(p)s(oin)m(ts)f(of)g(the) h(mappings)236 4886 y Fo(h)28 b Fk(2)g Fo(N)502 4850 y Fm(\003)575 4886 y Fq(are)k(dense)i(in)e Fo(M)10 b Fq(.)44 b(Indeed,)34 b(since)f(this)g(is)f(trivial)e(for)i Fo(x)p 2603 4899 V 28 w Fq(=)p 2789 4833 V 27 w Fo(x)q Fq(,)g(assume)1096 5106 y Fo(M)27 b Fk(\\)17 b Fq(])p Fo(s;)g(t)p Fq([)g Fk(6)p Fq(=)44 b Fk(;)124 b Fq(with)74 b Fo(x)p 2054 5119 V 28 w(<)28 b(s)f(<)h(t)g(<)p 2584 5053 V 27 w(x)18 b(:)236 5326 y Fq(Then)31 b(an)m(y)f Fo(h)e Fk(2)g Fo(N)935 5290 y Fm(\003)1005 5326 y Fq(with)h Fo(s)f(<)f(h)p Fq(\()p Fo(x)p Fq(\))h Fo(<)g(t)i Fq(for)f(0)e Fk(\024)h Fo(x)g Fk(\024)h Fo(t)g Fq(satis\014es)i Fo(s)c(<)h(h)p Fq(\()p Fo(s)p Fq(\))f Fk(\024)i Fo(h)p Fq(\()p Fo(t)p Fq(\))f Fo(<)f(t)236 5447 y Fq(and)33 b(th)m(us)h(b)m(y)f(its)f(con)m (tin)m(uit)m(y)h Fo(h)p Fq(\()p Fo(x)p Fq(\))28 b(=)f Fo(x)33 b Fq(for)f(some)h Fo(x)28 b Fk(2)p Fq(])p Fo(s;)17 b(t)p Fq([.)1841 5753 y(35)p eop %%Page: 36 37 36 36 bop 383 154 a Fq(On)31 b(the)h(other)g(hand,)g(\014xed)h(p)s(oin) m(ts)e(of)g(mappings)g(in)g Fo(N)2512 118 y Fm(\003)2583 154 y Fq(need)i(not)e(b)s(elong)g(to)g Fo(M)10 b Fq(.)236 274 y(This)43 b(is)e(seen,)46 b(for)c(instance,)j(replacing)c Fo(\027)48 b Fq(b)m(y)c Fo(\027)2128 238 y Fm(0)2195 274 y Fq(=)g(\()p Fo(\027)35 b Fq(+)29 b Fo(\027)2589 289 y Fl(0)2628 274 y Fq(\))p Fo(=)p Fq(2,)45 b(where)e Fo(\027)3175 289 y Fl(0)3257 274 y Fq(is)e(con-)236 395 y(cen)m(trated)j(on)f(the)g(iden)m(tit)m(y)f(map;)47 b(for)42 b(it)g(is)g(ob)m(vious)h(that,)i(passing)e(from)e Fo(\027)49 b Fq(to)42 b Fo(\027)3492 359 y Fm(0)3516 395 y Fq(,)236 515 y(neither)d(recurrence)i(nor)e(in)m(v)-5 b(arian)m(t)38 b(measure)h(are)g(concerned.)65 b(It)39 b(is,)i(ho)m(w)m(ev)m(er,)i(true)236 636 y(that)30 b(a)g(\014xed)h(p)s (oin)m(t)f Fo(x)g Fq(of)g(a)g(mapping)f Fo(h)e Fk(2)h Fo(N)1947 599 y Fm(\003)2018 636 y Fq(b)s(elongs)h(to)h Fo(M)10 b Fq(,)32 b(if)d(it)g(is)g(minimal)d(under)236 756 y(the)37 b(condition)f Fo(x)f Fk(\025)h Fo(x)p 1044 769 56 4 v Fq(.)56 b(Indeed,)40 b(in)c(this)g(case)i(the)f(increasing)f (sequence)k(\()p Fo(h)3190 720 y Fn(n)3237 756 y Fq(\()p Fo(x)p 3275 769 V Fq(\)\))3406 771 y Fn(n)p Fm(\025)p Fl(0)236 876 y Fq(con)m(v)m(erges)35 b(to)d Fo(x)h Fq(and)g(\(7.4\))f (applies.)383 1021 y(The)c(rest)h(of)e(this)h(section)g(concerns)h (conditions)e(under)i(whic)m(h)f(the)h(attractor)e(is)g(an)236 1141 y(in)m(terv)-5 b(al.)42 b(The)34 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Fo(t)28 b Fk(2)g Fo(E)23 b(:)1841 5753 y Fq(36)p eop %%Page: 37 38 37 37 bop 236 154 a Fh(Pr)-5 b(o)g(of.)41 b Fq(1.)j(The)33 b(crucial)e(p)s(oin)m(t)h(is)g(the)h(implication)236 368 y(\()p Fk(\003)p Fq(\))568 b([)p Fo(s;)17 b(t)p Fq(])28 b Fk(\032)g Fo(M)94 b Fq(if)40 b Fo(s)28 b(<)g(t)41 b Fq(satis\014es)i Fo(\027)6 b Fq(\()p Fo(h)p Fq(\()p Fo(t)p Fq(\))28 b Fo(<)g(s)p Fq(\))f Fo(>)h Fq(0)17 b Fo(:)236 582 y Fq(T)-8 b(o)34 b(deriv)m(e)h(it,)f(consider)g(a)g(nonempt)m(y)g (op)s(en)h(subset)g Fo(G)f Fq(of)g(])p Fo(s;)17 b(t)p Fq([.)48 b(Then)35 b(condition)e(\(a\))236 702 y(and)g(the)g (assumption)f Fo(\027)6 b Fq(\()p Fo(h)p Fq(\()p Fo(t)p Fq(\))28 b Fo(<)g(s)p Fq(\))f Fo(>)h Fq(0)k(yield)652 917 y Fo(\027)6 b Fq(\()p Fk(H)828 932 y Fn(s)865 917 y Fq(\))28 b Fo(>)f Fq(0)83 b(for)g Fk(H)1450 932 y Fn(s)1514 917 y Fq(:=)28 b Fk(f)p Fo(h)g Fk(2)g(H)q Fq([)p Fo(E)6 b Fq(])28 b(:)f Fo(h)p Fq(\()p Fo(t)p Fq(\))h Fo(<)g(s)g(<)f(t)h(<)g Fq(sup)17 b Fo(h)p Fk(g)g Fo(:)236 1131 y Fq(F)-8 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4268 y Fn(i)2186 4253 y Fo(;)17 b(z)32 b Fk(2)c Fo(E)33 b Fh(with)28 b Fo(x)2766 4268 y Fl(1)2834 4253 y Fo(<)f(x)2992 4268 y Fl(2)3049 4253 y Fo(:)236 4542 y Fh(Pr)-5 b(o)g(of.)41 b Fq(1.)k(If)33 b Fo(D)i Fq(is)e(a)f(coun)m(table)h(dense)i(subset)f (of)f Fo(E)6 b Fq(,)33 b(application)d(of)j(\()p Fk(\003)p Fq(\))f(to)h(all)e(pairs)236 4662 y Fo(x)291 4677 y Fl(1)359 4662 y Fo(<)c(x)517 4677 y Fl(2)590 4662 y Fq(with)32 b Fo(x)867 4677 y Fn(i)923 4662 y Fk(2)c Fo(D)35 b Fq(yields)922 4876 y Fk(j)p Fo(h)1006 4835 y Fm(\000)p Fl(1)1100 4876 y Fq([)p Fk(f)p Fo(z)t Fk(g)p Fq(])p Fk(j)28 b(\024)g Fq(1)124 b(for)74 b Fo(\027)6 b Fk(\000)p Fq(almost)32 b(all)39 b Fo(h)28 b Fk(2)g(H)q Fq([)p Fo(E)6 b Fq(])17 b Fo(:)236 5090 y Fq(Consequen)m(tly)-8 b(,)41 b(the)d(k)m(ernel)g Fo(P)50 b Fq(transforms)37 b(nonatomic)f(measures)i Fo(\045)e Fk(2)g(M)3148 5105 y Fl(1)3187 5090 y Fq(\()p Fo(E)6 b Fq(\))37 b(in)m(to)236 5211 y(measures)d(of)e(the)h(same)f(t)m(yp)s (e,)i(b)s(ecause)1141 5449 y Fo(\045P)14 b Fq(\()p Fk(f)p Fo(z)t Fk(g)p Fq(\))28 b(=)1624 5332 y Fi(Z)1671 5521 y Fm(H)p Fl([)p Fn(E)t Fl(])1846 5449 y Fo(\045)p Fq(\()p Fo(h)p Fq(\()p Fo(x)p Fq(\))h(=)e Fo(z)t Fq(\))17 b Fo(\027)6 b Fq(\()p Fo(dh)p Fq(\))17 b Fo(;)1841 5753 y Fq(37)p eop %%Page: 38 39 38 38 bop 236 154 a Fq(where)34 b(the)f(in)m(tegrand)f(v)-5 b(anishes)33 b(for)f Fo(\027)6 b Fk(\000)p Fq(almost)32 b(all)f Fo(h)d Fk(2)g(H)q Fq([)p Fo(E)6 b Fq(].)383 298 y(2.)88 b(Starting)47 b(with)g(an)m(y)h(nonatomic)e(initial)e(la)m(w)k Fo(\026)2418 313 y Fl(0)2504 298 y Fq(on)g Fo(E)60 b Fk(6)p Fq(=)53 b Fk(f)p Fq(0)p Fk(g)p Fq(,)e(it)c(follo)m(ws)236 419 y(from)38 b(part)h(1)g(of)g(the)h(pro)s(of)e(that)h Fo(\026)1610 434 y Fn(n)1696 419 y Fq(:=)g Fk(L)p Fq(\()p Fo(X)2026 434 y Fn(n)2072 419 y Fq(\))g(is)g(nonatomic)f(for)g(all)f Fo(n)j Fk(2)f Fp(N)p Fq(.)63 b(An)236 539 y(application)41 b(of)h(\(6.3\))g(to)h Fo(f)56 b Fq(=)45 b(1)1519 554 y Fm(f)p Fn(z)s Fm(g)1672 539 y Fq(and)e(an)g(admissible)e(function)h Fo(g)47 b Fq(with)42 b Fo(\026g)48 b Fk(6)p Fq(=)e(0)236 659 y(under)34 b Fo(X)594 674 y Fl(0)661 659 y Fq(=)27 b Fo(X)853 623 y Fm(0)845 684 y Fl(0)917 659 y Fq(yields)898 884 y Fo(\026)p Fq(\()p Fk(f)p Fo(z)t Fk(g)p Fq(\))p Fo(=\026g)j Fq(=)52 b(lim)1471 934 y Fn(n)p Fm(!1)1753 801 y Fi(X)1688 984 y Fl(0)p Fm(\024)p Fn(m)f Fq(1)p Fo(=k)h Fq(and)e(op)s(en)f(sets)i Fo(G)e Fq(with)g Fo(\045)p Fq(\()p Fo(G)p Fq(\))i Fo(<)h Fq(1)p Fo(=l)r Fq(,)f(hence)g(is)e(an)g (op)s(en)h(subset)h(of)236 4955 y Fk(H)q Fq([)p Fo(E)6 b Fq(].)44 b(Therefore)34 b Fk(H)1050 4970 y Fl(0)1122 4955 y Fq(itself)d(is)h(of)g(t)m(yp)s(e)i(F)1851 4970 y Fn(\033)r(\016)1931 4955 y Fq(.)383 5099 y(2.)42 b(Let)31 b(no)m(w)h Fo(\026)e Fq(b)s(e)h(decomp)s(osed)g(in)m(to)f(its)h (absolutely)f(con)m(tin)m(uous)h(part)g Fo(\026)3187 5114 y Fn(c)3252 5099 y Fq(and)g(its)236 5220 y(singular)g(part)i Fo(\026)876 5235 y Fn(s)945 5220 y Fq(with)f(resp)s(ect)i(to)e Fo(\045)p Fq(.)44 b(Then)33 b(the)g(equation)858 5457 y Fo(\026)917 5472 y Fn(c)952 5457 y Fo(P)14 b Fq(\()p Fo(B)5 b Fq(\))27 b(=)1314 5339 y Fi(Z)1361 5528 y Fm(H)1421 5537 y Fd(0)1476 5457 y Fo(\026)1535 5472 y Fn(c)1569 5457 y Fq(\()p Fo(h)1663 5415 y Fm(\000)p Fl(1)1758 5457 y Fq([)p Fo(B)5 b Fq(]\))17 b Fo(\027)6 b Fq(\()p Fo(dh)p Fq(\))124 b(for)74 b Fo(B)33 b Fk(2)28 b(B)s Fq(\()p Fo(E)6 b Fq(\))1841 5753 y(38)p eop %%Page: 39 40 39 39 bop 236 154 a Fq(implies)30 b Fo(\026)626 169 y Fn(c)661 154 y Fo(P)41 b Fk(\034)27 b Fo(\045)p Fq(,)33 b(hence)h(the)f(equation)1211 337 y Fo(\026)1270 352 y Fn(c)1304 337 y Fo(P)j Fq(+)22 b Fo(\026)1560 352 y Fn(s)1596 337 y Fo(P)41 b Fq(=)28 b Fo(\026P)40 b Fq(=)28 b Fo(\026)f Fq(=)h Fo(\026)2319 352 y Fn(c)2375 337 y Fq(+)22 b Fo(\026)2532 352 y Fn(s)236 520 y Fq(implies)42 b Fo(\026)638 535 y Fn(c)673 520 y Fo(P)61 b Fk(\034)48 b Fo(\026)1004 535 y Fn(c)1038 520 y Fq(.)80 b(By)45 b(the)g(uniqueness)h(of)e(the)h(in)m(v)-5 b(arian)m(t)44 b(measure,)k(therefore,)236 640 y Fo(\026)295 655 y Fn(c)359 640 y Fq(=)29 b Fo(\015)515 655 y Fn(c)549 640 y Fo(\026)k Fq(with)g(some)g(constan)m(t)i Fo(\015)1555 655 y Fn(c)1589 640 y Fq(,)f(hence)h Fo(\026)1981 655 y Fn(s)2046 640 y Fq(=)29 b Fo(\015)2202 655 y Fn(s)2239 640 y Fo(\026)k Fq(with)g(some)g(constan)m(t)h Fo(\015)3244 655 y Fn(s)3281 640 y Fq(.)46 b(No)m(w)236 760 y Fo(\026)295 775 y Fn(c)352 760 y Fk(^)22 b Fo(\026)499 775 y Fn(s)563 760 y Fq(=)28 b(0)k(implies)e Fo(\015)1130 775 y Fn(c)1187 760 y Fk(^)22 b Fo(\015)1326 775 y Fn(s)1391 760 y Fq(=)27 b(0)32 b(and)h(pro)m(v)m (es)h(the)f(assertion.)127 b Fg(2)383 960 y Fq(In)36 b(general,)g(there)g(is)g(no)f(further)i(information)32 b(a)m(v)-5 b(ailable)33 b(ab)s(out)j(the)g(alternativ)m(e)236 1080 y(in)d(this)g(prop)s(osition.)44 b(An)34 b(ob)m(vious)f(exception) h(is)f(the)h(case,)h(where)f(the)g(common)e(dis-)236 1200 y(tribution)g(of)g(the)i(v)-5 b(ariables)32 b Fo(H)1415 1215 y Fn(n)1462 1200 y Fq(\()p Fo(x)p Fq(\))p Fo(;)17 b(n)29 b Fk(2)g Fp(N)p Fo(;)k 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Fn(k)1556 2382 y Fh(is)35 b(exc)-5 b(essive)33 b(for)42 b Fo(\027)2280 2397 y Fn(k)2339 2382 y Fo(;)236 2577 y Fq(\(b\))993 b Fo(\027)1407 2592 y Fn(k)1478 2577 y Fk(!)1494 2602 y Fn(w)1608 2577 y Fo(\027)131 b Fh(and)124 b Fo(\026)2125 2592 y Fn(k)2195 2577 y Fk(!)2219 2602 y Fn(v)2318 2577 y Fo(\026)17 b(;)236 2760 y Fh(then)35 b Fo(\026)f Fh(is)h(exc)-5 b(essive)34 b(for)g Fo(\027)6 b Fh(.)236 2959 y(Pr)-5 b(o)g(of.)41 b Fq(Let)33 b Fo(g)e Fk(2)d(K)q Fq(\()p Fo(E)6 b Fq(\))33 b(satisfy)g(0)27 b Fk(\024)h Fo(g)j Fk(\024)d Fq(1)33 b(and)f(de\014ne)i Fo(\045)2417 2974 y Fn(k)2460 2959 y Fo(;)17 b(\045)28 b Fk(2)g(M)p Fq(\()p Fo(E)6 b Fq(\))32 b(b)m(y)1410 3142 y Fo(d\045)1511 3157 y Fn(k)1553 3142 y Fo(=d\026)1712 3157 y Fn(k)1782 3142 y Fq(=)27 b Fo(g)k Fq(=)d Fo(d\045=d\026)17 b(:)236 3325 y Fq(Since)25 b Fo(f)39 b Fk(2)28 b(C)6 b Fq(\()p Fo(E)g Fq(\))25 b(implies)d Fo(f)11 b(g)31 b Fk(2)d(K)q Fq(\()p Fo(E)6 b Fq(\),)27 b(the)e(v)-5 b(ague)25 b(con)m(v)m(ergence)j(in)c(\(b\))h(yields)f Fo(\045)3275 3340 y Fn(k)3346 3325 y Fk(!)3362 3350 y Fn(w)3466 3325 y Fo(\045)p Fq(.)236 3445 y(T)-8 b(ogether)33 b(with)g(the)g(w)m(eak)g(con)m(v)m(ergence)j(in)c(\(b\))g(this)g(leads) h(to)1524 3628 y Fo(\045)1574 3643 y Fn(k)1639 3628 y Fk(\012)23 b Fo(\027)1787 3643 y Fn(k)1857 3628 y Fk(!)1873 3653 y Fn(w)1985 3628 y Fo(\045)g Fk(\012)f Fo(\027)h(;)236 3811 y Fq(b)s(ecause)46 b(the)g(m)m(ultiplication)40 b(of)k(measures)i(is)f(con)m(tin)m(uous)g(in)f(the)i(w)m(eak)g(top)s (ology)-8 b(.)236 3931 y(No)m(w)33 b(for)f(0)c Fk(\024)g Fo(f)38 b Fk(2)29 b(K)q Fq(\()p Fo(E)6 b Fq(\))32 b(the)h(mapping)e(\() p Fo(x;)17 b(h)p Fq(\))28 b Fk(!)f Fo(f)11 b Fq(\()p Fo(h)p Fq(\()p Fo(x)p Fq(\)\))33 b(is)f(con)m(tin)m(uous,)i(hence)263 4027 y Fi(Z)309 4215 y Fn(E)386 4027 y Fi(Z)432 4215 y Fm(H)p Fl([)p Fn(E)t Fl(])607 4144 y Fo(f)11 b Fq(\()p Fo(h)p Fq(\()p Fo(x)p Fq(\)\))p Fo(g)t Fq(\()p Fo(x)p Fq(\))17 b Fo(\026)p Fq(\()p Fo(dx)p Fq(\))g Fo(\027)6 b Fq(\()p Fo(dh)p Fq(\))83 b(=)106 b(lim)1866 4204 y Fn(k)r Fm(!1)2078 4027 y Fi(Z)2125 4215 y Fn(E)2201 4027 y Fi(Z)2247 4215 y Fm(H)p Fl([)p Fn(E)t Fl(])2422 4144 y Fo(f)11 b Fq(\()p Fo(h)p Fq(\()p Fo(x)p Fq(\)\))p Fo(g)t Fq(\()p Fo(x)p Fq(\))17 b Fo(\026)3002 4159 y Fn(k)3044 4144 y Fq(\()p Fo(dx)p Fq(\))g Fo(\027)3291 4159 y Fn(k)3333 4144 y Fq(\()p Fo(dh)p Fq(\))1705 4382 y Fk(\024)84 b Fq(lim)17 b(inf)1911 4442 y Fn(k)r Fm(!1)2169 4265 y Fi(Z)2215 4454 y Fn(E)2291 4265 y Fi(Z)2337 4454 y Fm(H)p Fl([)p Fn(E)t Fl(])2513 4382 y Fo(f)11 b Fq(\()p Fo(h)p Fq(\()p Fo(x)p Fq(\)\))17 b Fo(\026)2911 4397 y Fn(k)2953 4382 y Fq(\()p Fo(dx)p Fq(\))g Fo(\027)3200 4397 y Fn(k)3242 4382 y Fq(\()p Fo(dh)p Fq(\))1705 4621 y Fk(\024)84 b Fq(lim)17 b(inf)1911 4681 y Fn(k)r Fm(!1)2169 4504 y Fi(Z)2215 4692 y Fn(E)2291 4621 y Fo(f)28 b(d\026)2477 4636 y Fn(k)1706 4843 y Fq(=)1866 4726 y Fi(Z)1912 4914 y Fn(E)1988 4843 y Fo(f)f(d\026)17 b(;)236 5060 y Fq(where)34 b(the)f(second)h(inequalit)m(y)d(uses)j(\(a\).)44 b(By)33 b(letting)e Fo(g)k Fq(increase)e(to)g(1,)f(therefore)1192 5242 y Fo(\026P)14 b(f)37 b Fk(\024)28 b Fo(\026f)135 b Fq(for)74 b(0)27 b Fk(\024)h Fo(f)39 b Fk(2)28 b(K)q Fq(\()p Fo(E)6 b Fq(\))17 b Fo(;)236 5425 y Fq(as)33 b(had)g(to)f(b)s(e)h(sho)m(wn.)128 b Fg(2)1841 5753 y Fq(39)p eop %%Page: 40 41 40 40 bop 383 154 a Fq(If)31 b Fo(\027)527 169 y Fn(k)569 154 y Fo(;)17 b(\027)34 b Fk(2)28 b(N)15 b Fq([)p Fo(E)6 b Fq(])31 b(satisfy)g Fo(\027)1400 169 y Fn(k)1471 154 y Fk(!)1487 179 y Fn(w)1597 154 y Fo(\027)6 b Fq(,)32 b(all)d(that)i(can)g(b)s(e)g(said)g(ab)s(out)f(their)h(restrictions)236 274 y(in)h(the)h(sense)h(of)f(\(1.1\))f(is)g(the)h(inequalit)m(y)1595 492 y(lim)17 b(inf)1640 552 y Fn(k)r Fm(!1)p 1898 439 56 4 v 1898 492 a Fo(x)1954 507 y Fn(k)2024 492 y Fk(\025)p 2129 439 V 28 w Fo(x)236 734 y Fq(for)35 b(the)h(resp)s(ectiv)m(e)g (upp)s(er)g(limits.)48 b(F)-8 b(or)35 b(simplicit)m(y)-8 b(,)33 b(therefore,)k(in)d(the)i(sequel)g(all)d(o)s(c-)236 855 y(curring)24 b(distributions)f(will)f(b)s(e)j(assumed)g(to)f(b)s(e) h(irreducible)e(with)i(resp)s(ect)g(to)f(the)h(same)236 975 y(state)33 b(space.)45 b(Then)34 b(the)f(follo)m(wing)c(stabilit)m (y)i(criterion)h(holds:)236 1174 y Fp(\(8.4\))41 b(Theorem)75 b Fh(L)-5 b(et)33 b Fo(\027)1235 1189 y Fn(k)1278 1174 y Fo(;)17 b(\027)34 b Fk(2)28 b(N)15 b Fq([)p Fo(E)6 b Fq(])33 b Fh(b)-5 b(e)33 b(r)-5 b(e)g(curr)g(ent)33 b(with)g(invariant)g(me)-5 b(asur)g(es)33 b Fo(\026)3399 1189 y Fn(k)3441 1174 y Fo(;)17 b(\026)236 1295 y Fh(and)36 b(denote)f(the)i(r)-5 b(esp)g(e)g(ctive)35 b(lower)h(limits)g(by)g Fo(x)p 2004 1308 V 20 x Fn(k)2102 1295 y Fo(;)17 b(x)p 2146 1308 V Fh(.)49 b(Then)35 b(the)h(c)-5 b(onver)g(genc)g(e)35 b Fo(\027)3282 1310 y Fn(k)3355 1295 y Fk(!)3371 1320 y Fn(w)3489 1295 y Fo(\027)236 1415 y Fh(implies)f(the)h(existenc)-5 b(e)34 b(of)g(norming)g(c)-5 b(onstants)35 b Fo(\015)2127 1430 y Fn(k)2204 1415 y Fh(with)f 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2382 y Fo(\015)1530 2397 y Fn(k)1572 2382 y Fo(\026)1631 2397 y Fn(k)1673 2382 y Fq(\([0)p Fo(;)g(t)p Fq([\))28 b Fk(\025)g Fo(\026)p Fq(\([0)p Fo(;)17 b(t)p Fq([\))27 b Fo(>)h Fq(0)17 b Fo(;)236 2631 y Fq(hence)34 b Fo(x)p 507 2644 V 563 2651 a Fn(k)633 2631 y Fk(\024)28 b Fo(t)33 b Fq(for)f(almost)f (all)f Fo(k)h Fk(2)d Fp(N)k Fq(b)m(y)i(\(4.4\),)e(as)h(had)f(to)h(b)s (e)f(sho)m(wn.)383 2776 y(2.)72 b(T)-8 b(o)42 b(pro)m(v)m(e)h (su\016ciency)-8 b(,)47 b(c)m(ho)s(ose)c Fo(t)h Fk(2)g Fo(E)k Fq(with)42 b Fo(t)i(>)g(x)p 2516 2789 V 43 w Fq(or)e Fo(t)i Fq(=)p 2942 2723 V 44 w Fo(x)e Fq(and)h(assume)236 2896 y Fo(\026)p Fq(\()p Fk(f)p Fo(t)p Fk(g)p Fq(\))32 b(=)f(0)k(unless)h Fo(t)c Fq(=)p 1195 2843 V 32 w Fo(x)q Fq(.)51 b(Then)36 b(\(4.4\))e(and)i(the)f(inequalit)m(y)f(\()p Fk(\003)p Fq(\))h(yield)f Fo(\026)p Fq(\([0)p Fo(;)17 b(t)p Fq(]\))32 b Fo(>)g Fq(0)236 3016 y(and)26 b Fo(\026)478 3031 y Fn(k)521 3016 y Fq(\([0)p Fo(;)17 b(t)p Fq(]\))27 b Fo(>)h Fq(0)d(for)h(almost)e(all)g Fo(k)31 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Fo(s)p Fq(\))g Fo(<)g(t)p Fq(\))f Fo(>)h Fq(0)17 b Fo(;)1841 5753 y Fq(40)p eop %%Page: 41 42 41 41 bop 236 154 a Fq(and)33 b(th)m(us)h(it)d(is)h(no)h(real)e (restriction)h(to)g(assume)1315 370 y Fo(#)c Fq(:=)46 b(inf)1531 435 y Fn(k)r Fm(2)p Fj(N)1719 370 y Fo(\027)1773 329 y Fn(n)1767 395 y(k)1820 370 y Fq(\()p Fo(h)p Fq(\()p Fo(s)p Fq(\))28 b Fo(<)f(t)p Fq(\))h Fo(>)g Fq(0)17 b Fo(:)236 621 y Fq(Since)33 b Fo(\026)550 636 y Fn(k)625 621 y Fq(is)f(in)m(v)-5 b(arian)m(t)31 b(for)h Fo(\027)1335 585 y Fn(n)1329 645 y(k)1415 621 y Fq(as)h(w)m(ell,)f(this)g(yields)869 837 y Fo(\026)928 852 y Fn(k)970 837 y Fq(\([0)p Fo(;)17 b(s)p Fq(]\))83 b Fk(\024)g Fo(#)1539 796 y Fm(\000)p Fl(1)1651 837 y Fo(\026)1710 852 y Fn(k)1752 837 y Fq(\([0)p Fo(;)17 b(s)p Fq(]\))g Fo(\027)2092 796 y Fn(n)2086 861 y(k)2139 837 y Fq(\()p Fo(h)p Fq(\()p Fo(s)p Fq(\))27 b Fo(<)h(t)p Fq(\))1322 982 y Fk(\024)83 b Fo(#)1539 941 y Fm(\000)p Fl(1)1651 982 y Fq(\()p Fo(\026)1748 997 y Fn(k)1812 982 y Fk(\012)23 b 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2022 y Fn(k)2342 2088 y Fl(2)2385 2063 y Fq(\()p Fo(x)p Fq(\))28 b(=)g(\(1)21 b Fk(\000)2869 1996 y Fq(1)p 2866 2040 55 4 v 2866 2132 a Fo(k)2930 2063 y Fq(\))c Fo(x)22 b Fk(^)3161 1996 y Fq(1)p 3161 2040 49 4 v 3161 2132 a(2)3236 2063 y Fo(:)236 2318 y Fq(If)k Fo(\027)375 2333 y Fn(k)444 2318 y Fq(and)h Fo(\027)32 b Fq(assign)26 b(mass)1234 2279 y Fl(1)p 1234 2294 36 4 v 1234 2352 a(2)1305 2318 y Fq(to)g Fo(h)1474 2333 y Fl(1)1514 2318 y Fo(;)17 b(h)1614 2282 y Fn(k)1614 2342 y Fl(2)1682 2318 y Fq(and)27 b Fo(h)1922 2333 y Fl(1)1961 2318 y Fo(;)17 b(h)2061 2333 y Fl(2)2100 2318 y Fq(,)28 b(resp)s(ectiv)m(ely)-8 b(,)29 b(then)d(clearly)g Fo(\027)3270 2333 y Fn(k)3340 2318 y Fk(!)3356 2343 y Fn(w)3462 2318 y Fo(\027)6 b Fq(.)236 2438 y(Moreo)m(v)m(er,)49 b Fo(\027)754 2453 y Fn(k)842 2438 y Fq(and)c Fo(\027)51 b Fq(are)44 b(recurren)m(t)i(b)m(y)g(\(2.2\),)h(their)d(lo)m(w)m(er)h (limits,)f(ho)m(w)m(ev)m(er,)50 b(are)236 2559 y Fo(x)p 236 2572 56 4 v 292 2579 a Fn(k)362 2559 y Fq(=)28 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b(satisfy)236 3536 y Fo(\026)295 3551 y Fn(k)392 3536 y Fk(!)416 3561 y Fn(v)555 3536 y Fo(\026)p Fq(.)90 b(In)49 b(this)f(case)h Fo(\026)1356 3551 y Fn(k)1399 3536 y Fq(\()p Fk(f)p Fq(0)p Fk(g)p Fq(\))k Fo(>)i Fq(0)48 b(b)m(y)h(\(5.6\))f (or)g(\(5.7\),)j(and)e(th)m(us)g Fo(\026)3214 3551 y Fn(k)3305 3536 y Fq(has)g(a)236 3657 y(represen)m(tation)34 b(according)e(to)g(\(5.2\).)383 3777 y(Finally)-8 b(,)36 b(the)i(sp)s(ecial)f(case)p 1447 3724 V 38 w Fo(x)g Fk(2)g Fo(E)44 b Fq(has)38 b(to)f(b)s(e)h(men)m(tioned,)h(where)g(the)f(in)m (v)-5 b(arian)m(t)236 3898 y(measures)37 b(can)f(b)s(e)g(normalized)e (to)i Fo(\026)1660 3913 y Fn(k)1702 3898 y Fo(;)17 b(\026)33 b Fk(2)g(M)2057 3913 y Fl(1)2096 3898 y Fq(\()p Fo(E)6 b Fq(\).)54 b(Since)36 b(in)f(this)g(case)i(v)-5 b(ague)36 b(and)236 4018 y(w)m(eak)42 b(con)m(v)m(ergence)h(coincide,)e(the)g (constan)m(ts)h Fo(\015)2112 4033 y Fn(k)2194 4018 y Fq(are)e(needless)i(and)e(th)m(us)i Fo(\026)3226 4033 y Fn(k)3309 4018 y Fk(!)3325 4043 y 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b(le)-5 b(d)236 5243 y(\()p Fq(a)p Fh(\))58 b(\\p)-5 b(ositive)34 b(r)-5 b(e)g(curr)g(ent")58 b(if)41 b Fo(\026)p Fq(\()p Fo(E)6 b Fq(\))28 b Fo(<)11 b Fk(1)p Fh(,)236 5425 y(\()p Fq(b)p Fh(\))58 b(\\nul)5 b(l)35 b(r)-5 b(e)g(curr)g(ent")183 b(if)41 b Fo(\026)p Fq(\()p Fo(E)6 b Fq(\))27 b(=)h Fk(1)p Fh(.)1841 5753 y Fq(41)p eop %%Page: 42 43 42 42 bop 383 154 a Fq(As)34 b(the)g(alternativ)m(e)f(recurren)m(t)16 b(/)g(transien)m(t,)37 b(considered)e(in)e(\(2.5\),)h(this)f (classi\014ca-)236 274 y(tion)28 b(is)h(in)m(v)-5 b(arian)m(t)27 b(under)j(a)f(passage)h(from)e Fo(\027)35 b Fq(to)29 b Fo(\027)2114 238 y Fn(n)2161 274 y Fq(,)h(b)s(ecause)g(this)f(do)s (es)h(not)f(a\013ect)g(the)236 395 y(in)m(v)-5 b(arian)m(t)31 b(measure.)383 540 y(The)h(follo)m(wing)e(criterion,)h(where)i(again)d Fo(\027)39 b Fq(is)31 b(iden)m(ti\014ed)h(with)g(the)g(corresp)s (onding)236 660 y(k)m(ernel)h Fo(P)14 b Fq(,)32 b(includes)h(the)g (transien)m(t)g(case:)236 860 y Fp(\(9.2\))41 b(Prop)s(osition)79 b 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Fl(0)2402 1490 y Fn(n)2502 1465 y Fo(=)-61 b Fk(2)28 b Fo(E)48 b Fq(a.s.)17 b Fo(:)236 1786 y Fp(10.)50 b(F)-9 b(urther)38 b(ergo)s(dic)e(theorems)236 1985 y Fq(The)e(con)m(v)m (ergence)h(in)d(\(9.7\))g(is)g(unnecessarily)h(restricted)h(to)e (functions)g Fo(f)39 b Fk(2)28 b(C)6 b Fq(\()p Fo(E)g Fq(\):)236 2184 y Fp(\(10.1\))41 b(Theorem)79 b Fh(L)-5 b(et)39 b Fo(\027)h Fk(2)34 b(N)15 b Fq([)p Fo(E)6 b Fq(])38 b Fh(b)-5 b(e)38 b(p)-5 b(ositive)37 b(r)-5 b(e)g(curr)g(ent)38 b(with)g(stationary)h(distri-)236 2305 y(bution)c Fo(\026)p Fh(.)45 b(Then)34 b(the)g(c)-5 b(onver)g(genc)g(e)1232 2509 y Fo(\026)1291 2524 y Fn(n)1337 2509 y Fo(f)39 b Fk(!)27 b Fo(\026f)93 b Fh(with)83 b Fo(\026)2070 2524 y Fn(n)2145 2509 y Fq(:=)27 b Fk(L)p Fq(\()p Fo(X)2463 2524 y Fn(n)2510 2509 y Fq(\))236 2713 y Fh(holds)34 b(in)h(e)-5 b(ach)34 b(of)h(the)f(fol)5 b(lowing)34 b(c)-5 b(ases:)236 2896 y(\()p Fq(a)p Fh(\))41 b Fo(f)e Fk(2)28 b(R)p Fq(\()p Fo(E)6 b Fq(\))p Fh(,)236 3078 y(\()p Fq(b)p Fh(\))42 b Fo(f)c Fq(:)28 b Fo(E)34 b Fk(!)27 b Fp(R)870 3093 y Fl(+)963 3078 y Fh(incr)-5 b(e)g(asing)40 b(and)h Fq(supp)18 b Fo(\026)1902 3093 y Fl(0)1976 3078 y Fh(c)-5 b(omp)g(act.)236 3277 y(Pr)g(o)g(of.)41 b Fq(\(a\))h(Since)53 b Fo(f)65 b Fq(is)53 b(b)s(ounded,)60 b(application)51 b(of)i(\(3.3b\))g(with)h(initial)c(v)-5 b(ariables)236 3398 y Fo(X)317 3413 y Fl(0)384 3398 y Fq(=)28 b Fo(x)543 3413 y Fl(0)615 3398 y Fq(resp.)45 b Fo(X)941 3413 y Fl(0)1008 3398 y Fq(=)27 b Fo(x)33 b Fq(yields)643 3626 y Fo(\026)702 3641 y Fn(n)749 3626 y Fo(f)f Fk(\000)23 b Fo(\026f)38 b Fq(=)1178 3509 y Fi(Z)1224 3698 y Fn(E)1300 3509 y Fi(Z)1346 3698 y Fn(E)1422 3626 y Fp(E)p Fq(\()p Fo(f)11 b Fq(\()p Fo(X)1720 3585 y Fn(x)1760 3594 y Fd(0)1712 3651 y Fn(n)1798 3626 y Fq(\))22 b Fk(\000)g Fo(f)11 b Fq(\()p Fo(X)2143 3585 y Fn(x)2135 3651 y(n)2187 3626 y Fq(\)\))17 b Fo(\026)2339 3641 y Fl(0)2377 3626 y Fq(\()p Fo(dx)2521 3641 y Fl(0)2561 3626 y Fq(\))g Fo(\026)p Fq(\()p Fo(dx)p Fq(\))43 b Fk(!)h Fq(0)17 b 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Fq(\()p Fo(X)1740 4865 y Fn(t)1732 4931 y(n)1778 4906 y Fq(\)\))1282 5051 y Fk(\024)83 b Fo(\026f)33 b Fq(+)22 b(\()p Fp(E)p Fq(\()p Fo(f)11 b Fq(\()p Fo(X)2016 5010 y Fn(t)2008 5076 y(n)2054 5051 y Fq(\)\))22 b Fk(\000)g Fp(E)p Fq(\()p Fo(f)11 b Fq(\()p Fo(X)2549 5010 y Fl(0)2541 5076 y Fn(n)2588 5051 y Fq(\)\)\))17 b Fo(:)236 5256 y Fq(In)33 b(view)g(of)f(\()p Fk(\003)p Fq(\))g(the)h(fundamen)m(tal)f(inequalit)m(y)f(\(3.1\))i(yields)1416 5460 y(lim)17 b(sup)1473 5529 y Fn(n)p Fm(!1)1748 5460 y Fo(\026)1807 5475 y Fn(n)1854 5460 y Fo(f)38 b Fk(\024)28 b Fo(\026f)f(:)83 b Fg(2)1841 5753 y Fq(47)p eop %%Page: 48 49 48 48 bop 383 270 a Fq(T)-8 b(o)45 b(see)i(the)f(regularit)m(y)e (condition)g(in)h(case)h(\(a\))f(to)g(b)s(e)h(essen)m(tial,)j(consider) d(the)236 391 y(autoregressiv)m(e)d(pro)s(cess)g(follo)m(wing)d (\(3.3\).)71 b(It)41 b(is)h(p)s(ositiv)m(e)f(recurren)m(t)i(with)f(the) g(uni-)236 511 y(form)g(distribution)e(on)j([0,1[)f(as)h(stationary)f (distribution.)72 b(Here,)46 b(the)d(con)m(v)m(ergence)236 631 y Fo(\026)295 646 y Fn(n)342 631 y Fo(f)38 b Fk(!)28 b Fo(\026f)40 b Fq(fails,)29 b(for)h(instance,)i(if)d Fo(X)1653 646 y Fl(0)1720 631 y Fq(=)f Fo(x)1879 646 y Fl(0)1946 631 y Fk(2)g Fo(D)33 b Fq(and)e Fo(f)38 b Fq(=)28 b(1)2581 646 y Fn(D)2675 631 y Fq(with)i Fo(D)j Fq(denoting)d(the)236 752 y(set)k(of)e(dy)m(adic)h(n)m(um)m(b)s(ers)g (in)f Fo(E)6 b Fq(.)383 872 y(The)28 b(compactness)h(condition)d(in)h (case)i(\(b\))f(is)f(essen)m(tial)h(as)g(w)m(ell.)41 b(Indeed,)30 b(consider)236 993 y(an)43 b(exc)m(hange)i(pro)s(cess)f (with)f Fo(E)51 b Fq(=)45 b Fp(R)1721 1008 y Fl(+)1823 993 y Fq(and)e Fp(E)p Fq(\()p Fo(V)2192 1008 y Fn(n)2238 993 y Fq(\))i Fo(<)h Fk(1)p Fq(,)f(pro)m(v)m(ed)f(to)f(b)s(e)g(p)s (ositiv)m(e)236 1113 y(recurren)m(t)38 b(b)m(y)f(\(9.7\).)54 b(Assume)37 b(in)e(addition)1952 1042 y Fi(R)1991 1138 y Fn(E)2067 1113 y Fo(x)17 b(d\026)34 b(<)f Fk(1)j Fq(for)g(its)g (stationary)f(distri-)236 1233 y(bution)f Fo(\026)h Fk(\000)g 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1848 56 4 v 31 w(<)c(t)h Fk(2)f Fo(E)41 b Fq(carries)34 b(o)m(v)m(er)h(in)236 1956 y(the)e(follo)m(wing)d(form:) 236 2130 y(\(1\))41 b Fo(\027)48 b Fq(p)s(ositiv)m(e)32 b(recurren)m(t)71 b Fk(,)93 b Fq(lim)1485 2180 y Fn(n)p Fm(!1)1685 2130 y Fp(P)p Fq(\()p Fo(X)1881 2145 y Fn(n)1955 2130 y Fk(\024)29 b Fo(t)p Fq(\))11 b Fo(>)19 b Fq(0,)236 2396 y(\(2\))41 b Fo(\027)48 b Fq(n)m(ull)32 b(recurren)m(t)236 b Fk(,)94 b Fq(lim)1482 2445 y Fn(n)p Fm(!1)1683 2396 y Fp(P)p Fq(\()p Fo(X)1879 2411 y Fn(n)1953 2396 y Fk(\024)28 b Fo(t)p Fq(\))g(=)g(0)55 b(and)2586 2313 y Fi(X)2579 2495 y Fn(n)p Fm(\025)p Fl(0)2729 2396 y Fp(P)p Fq(\()p Fo(X)2925 2411 y Fn(n)2999 2396 y Fk(\024)28 b Fo(t)p Fq(\))g(=)f Fk(1)p Fq(,)236 2636 y(while)32 b(in)g(the)h(transien)m(t)f (case)1456 2553 y Fi(X)1449 2736 y Fn(n)p Fm(\025)p Fl(0)1615 2636 y Fp(P)p Fq(\()p Fo(X)1811 2651 y Fn(n)1885 2636 y Fk(\024)d Fo(t)p Fq(\))19 b Fo(<)11 b Fk(1)92 b Fq(for)32 b(all)f Fo(t)d Fk(2)g Fo(E)6 b Fq(.)383 2839 y(T)-8 b(o)45 b(pro)m(v)m(e)i(a)e(la)m(w)g(of)g(large)g(n)m(um)m(b)s(ers)h(not)g (restricted)g(to)f(functions)g Fo(f)61 b Fk(2)50 b(C)6 b Fq(\()p Fo(E)g Fq(\),)236 2959 y(ergo)s(dicit)m(y)32 b(will)e(b)s(e)j(established)f(\014rst.)44 b(More)33 b(generally)f(the)h(follo)m(wing)d(holds:)236 3158 y Fp(\(10.2\))41 b(Theorem)79 b Fh(L)-5 b(et)39 b Fo(\027)h Fk(2)34 b(N)15 b Fq([)p Fo(E)6 b Fq(])38 b Fh(b)-5 b(e)38 b(p)-5 b(ositive)37 b(r)-5 b(e)g(curr)g(ent)38 b(with)g(stationary)h (distri-)236 3279 y(bution)c Fo(\026)p Fh(.)45 b(Then)34 b(the)g(pr)-5 b(o)g(c)g(ess)34 b Fq(\()p Fo(X)1533 3294 y Fn(n)1580 3279 y Fq(\))1618 3294 y Fn(n)p Fm(\025)p Fl(0)1790 3279 y Fh(with)h Fk(L)p Fq(\()p Fo(X)2190 3294 y Fl(0)2229 3279 y Fq(\))28 b(=)f Fo(\026)35 b Fh(is)f(mixing.)236 3478 y(Pr)-5 b(o)g(of.)41 b Fq(1.)47 b(Extending)35 b Fo(H)1210 3493 y Fn(n)1256 3478 y Fo(;)17 b(n)30 b Fk(2)g Fp(N)p Fo(;)k Fq(let)f Fo(H)1856 3493 y Fn(n)1903 3478 y Fo(;)17 b(n)30 b Fk(2)g Fp(Z)p Fo(;)k Fq(b)s(e)g(indep)s(enden)m(t)h 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4553 y Fn(n)1568 4538 y Fq(\))1606 4555 y Fn(n)p Fm(2)p Fj(N)1769 4538 y Fq(,)h(the)g(pro)s(cesses)i(\()p Fo(X)2535 4553 y Fn(n)2582 4538 y Fq(\))2620 4553 y Fn(n)p Fm(\025)p Fl(0)2789 4538 y Fq(and)e(\()p Fo(X)3105 4502 y Fm(0)3097 4563 y Fn(n)3144 4538 y Fq(\))3182 4553 y Fn(n)p Fm(\025)p Fl(0)3351 4538 y Fq(ha)m(v)m(e)236 4658 y(the)h(same)g(distribution,)d (and)j(it)f(su\016ces)i(to)f(pro)m(v)m(e)g(the)g(assertion)g(for)f(\()p Fo(X)3060 4622 y Fm(0)3052 4683 y Fn(n)3099 4658 y Fq(\))3137 4673 y Fn(n)p Fm(\025)p Fl(0)3274 4658 y Fq(.)383 4804 y(2.)42 b(Denote)29 b(b)m(y)i Fo(\033)i Fq(resp.)44 b Fo(\033)1356 4768 y Fm(0)1409 4804 y Fq(the)30 b(shift)f(in)g Fo(W)41 b Fq(:=)2165 4737 y Fi(Q)2260 4829 y Fn(n)p Fm(2)p Fj(Z)2432 4804 y Fk(H)q Fq([)p Fo(E)6 b Fq(])30 b(resp.)43 b Fo(W)3028 4768 y Fm(0)3079 4804 y Fq(:=)3210 4737 y Fi(Q)3305 4829 y Fn(n)p Fm(\025)p Fl(0)p 3465 4726 79 4 v 3465 4804 a Fo(E)236 4933 y Fq(with)p 458 4855 V 32 w Fo(E)34 b Fq(=)28 b Fo(E)44 b Fk([)c(f)p 940 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Fj(Z)3179 154 y Fo(\027)39 b Fq(carries)236 274 y(o)m(v)m(er)e(to)f Fo(\033)630 238 y Fm(0)689 274 y Fq(with)g(resp)s(ect)h(to)f(its)f(image)f(b)m(y)j Fo(\034)11 b Fq(.)54 b(Since)36 b(this)g(apparen)m(tly)g(is)g(the)g (distri-)236 395 y(bution)c(of)g(\()p Fo(X)783 359 y Fm(0)775 419 y Fn(n)822 395 y Fq(\))860 410 y Fn(n)p Fm(\025)p Fl(0)997 395 y Fq(,)h(the)g(assertion)f(follo)m(ws.)126 b Fg(2)383 594 y Fq(In)44 b(con)m(trast)i(to)e(the)h(result)f(of)g (\(5.4\),)j(in)d(general)g(the)h(tail)d Fo(\033)t Fq(-\014eld)i(of)g (\()p Fo(X)3294 609 y Fn(n)3341 594 y Fq(\))3379 609 y Fn(n)p Fm(\025)p Fl(0)3516 594 y Fq(,)236 714 y(ev)m(en)34 b(under)f(stationarit)m(y)-8 b(,)31 b(need)i(not)f(b)s(e)g(trivial.)41 b(A)32 b(coun)m(terexample)g(is)g(pro)m(vided)g(b)m(y)236 835 y(an)m(y)k(distribution)d Fo(\027)38 b Fk(2)32 b(N)15 b Fq([)p Fo(E)6 b Fq(],)35 b(whose)h(supp)s(ort)f(consists)h(of)e(t)m (w)m(o)i(injectiv)m(e)e(mappings)236 955 y Fo(h)292 970 y Fl(0)332 955 y Fo(;)17 b(h)432 970 y Fl(1)514 955 y Fk(2)43 b(H)q Fq([)p Fo(E)6 b Fq(])42 b(with)f(disjoin)m(t)g(images)f Fo(h)1855 970 y Fn(i)1883 955 y Fq([)p Fo(E)6 b Fq(],)45 b(as)c(considered)i(in)e(Section)g(9.)70 b(Since)236 1076 y(in)39 b(this)h(case)h Fo(X)850 1091 y Fn(n)p Fm(\000)p Fl(1)1027 1076 y Fq(can)f(b)s(e)g(reconstructed)i(from)d Fo(X)2287 1091 y Fn(n)2374 1076 y Fq(with)g(probabilit)m(y)f(1,)k(the)e (tail)236 1196 y Fo(\033)t Fq(-\014eld)c(of)g(\()p Fo(X)777 1211 y Fn(n)824 1196 y Fq(\))862 1211 y Fn(n)p Fm(\025)p Fl(0)1036 1196 y Fq(coincides)g(with)g(the)h(full)e Fo(\033)t Fq(-\014eld)g(generated)j(b)m(y)f(the)g(pro)s(cess)h(up)236 1316 y(to)32 b(sets)i(of)e(probabilit)m(y)f(0.)383 1462 y(If)40 b(the)h(underlying)g(mappings)e Fo(h)j Fk(2)g(H)q Fq([)p Fo(E)6 b Fq(])41 b(satisfy)g(a)f(Lipsc)m(hitz)h(condition,)g(la) m(ws)236 1582 y(of)35 b(large)f(n)m(um)m(b)s(ers)i(regarding)e (functions)h Fo(f)42 b Fk(2)33 b(C)6 b Fq(\()p Fo(E)g Fq(\))35 b(can)g(b)s(e)h(found)f(in)f([3,)i(9].)50 b(In)36 b(the)236 1702 y(order)d(con)m(text)h(more)e(general)g(results)h(are)f (a)m(v)-5 b(ailable:)236 1902 y Fp(\(10.3\))41 b(Theorem)79 b Fh(L)-5 b(et)39 b Fo(\027)h Fk(2)34 b(N)15 b Fq([)p Fo(E)6 b Fq(])38 b Fh(b)-5 b(e)38 b(p)-5 b(ositive)37 b(r)-5 b(e)g(curr)g(ent)38 b(with)g(stationary)h(distri-)236 2022 y(bution)c Fo(\026)p Fh(.)45 b(Then)34 b(for)g(arbitr)-5 b(ary)35 b(initial)g(law)f(the)h(c)-5 b(onver)g(genc)g(e)1322 2213 y Fq(1)p 1317 2258 59 4 v 1317 2349 a Fo(n)1467 2198 y Fi(X)1402 2381 y Fl(0)p Fm(\024)p Fn(m)f Fq(0)17 b Fo(;)236 1456 y Fh(henc)-5 b(e)34 b(in)h(p)-5 b(articular)35 b Fp(E)1145 1420 y Fl(0)1184 1456 y Fq(\()p Fo(T)1293 1420 y Fn(t)1322 1456 y Fq(\))28 b Fo(<)f Fk(1)p Fh(.)236 1655 y(Pr)-5 b(o)g(of.)41 b Fq(Cho)s(ose)36 b Fo(l)d Fk(2)f Fp(N)i Fq(suc)m(h)i(that)f Fo(#)c Fq(:=)h Fp(P)p Fq(\()p Fo(X)2021 1619 y Fl(0)2013 1680 y Fn(l)2091 1655 y Fk(\025)f Fo(t)p Fq(\))h Fo(>)f Fq(0.)50 b(Then)35 b(monotonicit)m(y)e(and)236 1775 y(indep)s(endence)h (imply)598 1986 y Fp(P)692 1945 y Fl(0)730 1986 y Fq(\()p Fo(T)839 1945 y Fn(t)896 1986 y Fo(>)28 b(k)s(l)r Fq(\))83 b Fk(\024)h Fp(P)p Fq(\()p Fo(H)1563 2002 y Fl(\()p Fn(i)p Fl(+1\))p Fn(l)1779 1986 y Fk(\016)22 b Fo(:)17 b(:)g(:)22 b Fk(\016)g Fo(H)2141 2001 y Fn(il)q Fl(+1)2281 1986 y Fq(\(0\))27 b Fo(<)h(t)g Fq(for)f(0)g Fk(\024)h Fo(i)g(<)g(k)s Fq(\))1207 2132 y(=)84 b(\(1)21 b Fk(\000)i Fo(#)p Fq(\))1670 2090 y Fn(k)1838 2132 y Fq(for)32 b(all)72 b Fo(k)31 b Fk(\025)d Fq(0)17 b Fo(:)236 2342 y Fq(P)m(artial)48 b(in)m(tegration)f(sho)m(ws)k(that)e(eac)m(h)h Fo(u)55 b(<)h Fk(\000)2180 2303 y Fl(1)p 2180 2319 36 4 v 2187 2377 a Fn(l)2242 2342 y Fq(log\(1)33 b Fk(\000)h Fo(#)p Fq(\))49 b(satis\014es)h(the)g(asser-)236 2463 y(tion.)126 b Fg(2)383 2662 y Fq(The)30 b(follo)m(wing)e(coun)m(terpart)i(of)g (\(11.1\))f(separates)i(the)g(t)m(w)m(o)f(kinds)h(of)e(recurrence:)236 2861 y Fp(\(11.2\))41 b(Theorem)76 b Fh(L)-5 b(et)35 b Fo(\027)f Fk(2)29 b(N)15 b Fq([)p Fo(E)6 b Fq(])34 b Fh(b)-5 b(e)34 b(r)-5 b(e)g(curr)g(ent)36 b(and)e(\014x)g Fo(t)28 b Fk(2)g Fo(E)41 b Fh(with)35 b Fo(t)27 b(>)h(x)p 3194 2874 56 4 v Fh(.)45 b(Then)236 2982 y(the)35 b(hitting)g(time)g Fo(T)986 2997 y Fn(t)1050 2982 y Fh(of)g Fk(f)p Fo(x)28 b Fk(2)g Fo(E)34 b Fq(:)27 b Fo(x)h Fk(\024)h Fo(t)p Fk(g)35 b Fh(by)g Fq(\()p Fo(X)2107 2997 y Fn(n)2153 2982 y Fq(\))2191 2997 y Fn(n)p Fm(\025)p Fl(0)2363 2982 y Fh(satis\014es)236 3164 y(\()p Fq(a)p Fh(\))41 b Fp(E)480 3128 y Fn(x)524 3164 y Fq(\()p Fo(T)619 3179 y Fn(t)648 3164 y Fq(\))28 b Fo(<)g Fk(1)41 b Fh(for)34 b(al)5 b(l)35 b Fo(x)28 b Fk(2)g Fo(E)48 b Fh(whenever)40 b Fo(\027)h Fh(is)35 b(p)-5 b(ositive)34 b(r)-5 b(e)g(curr)g(ent,)236 3347 y(\()p Fq(b)p Fh(\))42 b Fp(E)486 3311 y Fn(x)529 3347 y Fq(\()p Fo(T)624 3362 y Fn(t)654 3347 y Fq(\))27 b(=)h Fk(1)41 b Fh(for)35 b Fo(t)28 b Fk(\024)g Fo(x)g Fk(2)g Fo(E)48 b Fh(whenever)40 b Fo(\027)h Fh(is)35 b(nul)5 b(l)35 b(r)-5 b(e)g(curr)g(ent.)236 3546 y(Pr)g(o)g(of.)41 b Fq(\(a\))h(Clearly)-8 b(,)54 b(the)d(pro)s(cess)g(\()p Fo(X)1743 3561 y Fn(n)1790 3546 y Fq(\))1828 3561 y Fn(n)p Fm(\025)p Fl(0)2016 3546 y Fq(ma)m(y)f(b)s(e)h(assumed)g(to)f(b)s(e)h (stationary)-8 b(.)236 3667 y(Moreo)m(v)m(er,)54 b(the)49 b(assumption)e Fo(x)55 b(>)f(t)49 b Fq(means)f(no)g(loss)g(of)g (generalit)m(y)-8 b(,)51 b(b)s(ecause)f(the)236 3787 y(assertion)31 b(is)e(trivial)f(in)i(the)g(case)i Fo(t)c Fq(=)p 1673 3734 V 27 w Fo(x)q Fq(.)42 b(Since)31 b Fp(P)p Fq(\()p Fo(X)2247 3802 y Fl(0)2313 3787 y Fk(\024)e Fo(t)p Fq(\))e Fo(>)h Fq(0)i(b)m(y)h(\(4.4\),)g(an)m(y)g Fo(n)d Fk(2)g Fp(N)236 3907 y Fq(with)k Fp(P)p Fq(\()p Fo(X)662 3871 y Fl(0)654 3932 y Fn(n)729 3907 y Fk(\025)c Fo(x)p Fq(\))g Fo(>)g Fq(0)k(satis\014es)236 4118 y(\()p Fk(\003)p Fq(\))1016 b Fp(P)p Fq(\()p Fo(X)1574 4133 y Fl(0)1640 4118 y Fk(\024)28 b Fo(t;)17 b(X)1905 4133 y Fn(n)1980 4118 y Fk(\025)28 b Fo(x)p Fq(\))g Fo(>)g Fq(0)17 b Fo(:)236 4329 y Fq(If)33 b Fo(n)f Fq(is)h(c)m(hosen)h(minimal)28 b(with)k(resp)s(ect)i(to)e(\()p 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Fn(y)1751 5504 y Fq(\()p Fo(T)1846 5519 y Fn(t)1875 5504 y Fq(\))125 b(for)73 b Fo(y)31 b Fk(2)d Fo(E)1841 5753 y Fq(51)p eop %%Page: 52 53 52 52 bop 236 154 a Fq(the)33 b(recurrence)i(theorem)d(b)m(y)i(Kac)e (and)h(the)g(Mark)m(o)m(v)h(prop)s(ert)m(y)f(imply)1155 384 y Fp(P)p Fq(\()p Fo(T)1327 399 y Fn(t)1384 384 y Fo(<)28 b Fk(1)p Fq(\))83 b(=)1869 267 y Fi(Z)1915 456 y Fm(f)p Fn(X)2008 465 y Fd(0)2043 456 y Fm(\024)p Fn(t)p Fm(g)2179 384 y Fo(T)2236 399 y Fn(t)2283 384 y Fo(d)p Fp(P)1708 623 y Fk(\025)1869 506 y Fi(Z)1915 694 y Fn(A)1989 623 y Fo(T)2046 638 y Fn(t)2092 623 y Fo(d)p Fp(P)1709 845 y Fq(=)1869 728 y Fi(Z)1915 916 y Fn(A)1972 845 y Fq(\()p Fo(n)22 b Fq(+)g Fo(g)t Fq(\()p Fo(X)2358 860 y Fn(n)2405 845 y Fq(\)\))17 b Fo(d)p Fp(P)1708 1024 y Fk(\025)84 b Fp(P)p Fq(\()p Fo(A)p Fq(\))17 b(\()p Fo(n)k Fq(+)h Fo(g)t Fq(\()p Fo(x)p Fq(\)\))17 b Fo(:)236 1230 y Fq(Therefore)34 b Fo(g)t Fq(\()p Fo(x)p Fq(\))27 b Fo(<)h Fk(1)p Fq(,)k(as)h(had)f(to)h(b)s(e)f(sho)m(wn.)383 1373 y(\(b\))41 b(If)33 b Fo(\026)f Fq(is)g(the)h(in)m(v)-5 b(arian)m(t)31 b(measure)i(and)g Fo(t)28 b(<)f(y)k Fk(2)d Fo(E)6 b Fq(,)33 b(then)924 1579 y Fo(\026)983 1538 y Fm(0)1006 1579 y Fq(\()p Fo(B)5 b Fq(\))28 b(:=)f(\()p Fo(\026)p Fq(\([0)p Fo(;)17 b(y)t Fq(]\))1691 1538 y Fm(\000)p Fl(1)1784 1579 y Fo(\026)p Fq(\()p Fo(B)5 b Fq(\))124 b(for)74 b Fo(B)32 b Fk(2)c(B)s Fq(\([0)p Fo(;)17 b(y)t Fq(]\))236 1785 y(b)m(y)36 b(\(4.2a\))e(and)h(\(4.4\))g (de\014nes)h(a)f(stationary)f(distribution)f(with)i(resp)s(ect)h(to) 3175 1749 y Fn(y)3200 1785 y Fo(P)14 b Fq(.)50 b(No)m(w)236 1905 y(let)24 b Fo(X)450 1920 y Fl(0)514 1905 y Fq(b)s(e)g(distributed) h(according)e(to)h(\(the)h(trivial)d(extension)j(of)7 b(\))24 b Fo(\026)2781 1869 y Fm(0)2829 1905 y Fq(and)g(let)g Fo(T)3214 1869 y Fm(0)3200 1930 y Fn(t)3261 1905 y Fq(denote)236 2026 y(the)36 b(hitting)e(time)g(of)h([0)p Fo(;)17 b(t)p Fq(])35 b(b)m(y)i(\()1463 1990 y Fn(y)1487 2026 y Fo(X)1568 2041 y Fn(n)1615 2026 y Fq(\))1653 2041 y Fn(n)p Fm(\025)p Fl(0)1790 2026 y Fq(.)53 b(Then)36 b(it)f(is)g(ob)m(vious)h(that)f Fo(T)2970 1990 y Fm(0)2956 2050 y Fn(t)3025 2026 y Fk(\024)f Fo(T)3193 2041 y Fn(t)3222 2026 y Fq(,)j(and)e(it)236 2146 y(follo)m(ws,)d(again)f(from)g(the)i(recurrence)i(theorem)d(b)m(y) h(Kac,)g(that)1098 2377 y Fp(P)p Fq(\()p Fo(T)1284 2335 y Fm(0)1270 2401 y Fn(t)1334 2377 y Fo(<)27 b Fk(1)p Fq(\))84 b(=)1818 2259 y Fi(Z)1865 2448 y Fm(f)1900 2429 y Ff(y)1926 2448 y Fn(X)1984 2457 y Fd(0)2018 2448 y Fm(\024)p Fn(t)p Fm(g)2155 2377 y Fo(T)2226 2335 y Fm(0)2212 2401 y Fn(t)2265 2377 y Fo(d)p Fp(P)1659 2615 y Fq(=)1818 2498 y Fi(Z)1865 2687 y Fn(x)1905 2696 y Fd(0)1939 2687 y Fm(\024)p Fn(t)2040 2615 y Fp(E)2114 2574 y Fn(x)2154 2583 y Fd(0)2192 2615 y Fq(\()p Fo(T)2301 2574 y Fm(0)2287 2640 y Fn(t)2324 2615 y Fq(\))17 b Fo(\026)2438 2574 y Fm(0)2461 2615 y Fq(\()p Fo(dx)2605 2630 y Fl(0)2644 2615 y Fq(\))1658 2848 y Fk(\024)1818 2731 y Fi(Z)1865 2920 y Fn(x)1905 2929 y Fd(0)1939 2920 y Fm(\024)p Fn(t)2040 2848 y Fp(E)2114 2807 y Fn(x)2154 2816 y Fd(0)2192 2848 y Fq(\()p Fo(T)2287 2863 y Fn(t)2317 2848 y Fq(\))g Fo(\026)2431 2807 y Fm(0)2453 2848 y Fq(\()p Fo(dx)2597 2863 y Fl(0)2637 2848 y Fq(\))1658 3046 y Fk(\024)83 b Fo(\026)1877 3005 y Fm(0)1900 3046 y Fq(\([0)p Fo(;)17 b(t)p Fq(]\))g Fp(E)2249 3005 y Fn(t)2278 3046 y Fq(\()p Fo(T)2373 3061 y Fn(t)2403 3046 y Fq(\))g Fo(:)236 3252 y Fq(Since)33 b Fo(T)562 3216 y Fm(0)548 3277 y Fn(t)617 3252 y Fq(is)g(almost)d(surely)j (\014nite,)g(therefore)1336 3458 y Fo(\026)p Fq(\([0)p Fo(;)17 b(y)t Fq(]\))26 b Fk(\024)i Fo(\026)p Fq(\([0)p Fo(;)17 b(t)p Fq(]\))g Fp(E)2209 3417 y Fn(t)2237 3458 y Fq(\()p Fo(T)2332 3473 y Fn(t)2362 3458 y Fq(\))g Fo(:)236 3664 y Fq(Letting)43 b Fo(y)k Fq(increase)e(to)p 1197 3611 56 4 v 44 w Fo(x)f Fq(in)f(view)h(of)g Fo(\026)p Fq(\([0)p Fo(;)17 b(t)p Fq(]\))46 b Fo(<)h Fk(1)d Fq(yields)f Fp(E)2769 3628 y Fn(t)2799 3664 y Fq(\()p Fo(T)2894 3679 y Fn(t)2923 3664 y Fq(\))k(=)g Fk(1)p Fq(,)g(hence)236 3784 y Fp(E)310 3748 y Fn(x)354 3784 y Fq(\()p Fo(T)449 3799 y Fn(t)478 3784 y Fq(\))28 b(=)g Fk(1)k Fq(for)g(all)e Fo(x)e Fk(2)g Fo(E)39 b Fq(with)32 b Fo(x)c Fk(\025)g Fo(t)p Fq(.)127 b Fg(2)383 3983 y Fq(Since)43 b Fp(P)742 3947 y Fn(t)771 3983 y Fq(\()p Fo(T)866 3998 y Fn(t)943 3983 y Fo(<)j Fk(1)p Fq(\))h(=)f(1)e(clearly)f(implies)e Fo(x)p 2130 3996 V 47 w Fk(\024)47 b Fo(t)p Fq(,)g(in)c(the)h(transien) m(t)g(case)h(the)236 4104 y(equation)33 b Fp(E)709 4068 y Fn(t)738 4104 y Fq(\()p Fo(T)833 4119 y Fn(t)862 4104 y Fq(\))28 b(=)g Fk(1)k Fq(holds)g(for)g(all)f Fo(t)c Fk(2)i Fo(E)6 b Fq(.)383 4247 y(Next,)33 b(the)g(top)s(ological)c (structure)34 b(of)e(the)h(state)g(space)h(is)e(tak)m(en)h(in)m(to)f (accoun)m(t:)236 4446 y Fp(\(11.3\))41 b(Prop)s(osition)78 b Fh(L)-5 b(et)39 b Fo(\027)25 b Fk(2)18 b(N)d Fq([)p Fo(E)6 b Fq(])39 b Fh(b)-5 b(e)38 b(p)-5 b(ositive)38 b(r)-5 b(e)g(curr)g(ent)39 b(with)f(stationary)h(dis-)236 4567 y(tribution)d Fo(\026)p Fh(.)47 b(Then)34 b(for)i(e)-5 b(ach)35 b(op)-5 b(en)35 b(subset)g Fo(G)h Fh(of)f Fo(E)42 b Fh(satisfying)35 b Fo(\026)p Fq(\()p Fo(G)p Fq(\))28 b Fo(>)h Fq(0)35 b Fh(and)g(every)236 4687 y Fo(t)28 b Fk(2)g Fo(E)41 b Fh(the)35 b(stopping)f(time)1035 4893 y Fo(T)41 b Fq(:=)28 b(inf)6 b Fk(f)p Fo(n)28 b Fk(2)g Fp(N)f Fq(:)h Fo(X)1871 4852 y Fn(x)1863 4918 y(n)1942 4893 y Fk(2)g Fo(G)g Fh(for)g Fq(0)f Fk(\024)h Fo(x)g Fk(\024)g Fo(t)p Fk(g)236 5099 y Fh(has)35 b(a)f(\014nite)h(exp)-5 b(e)g(ctation.)236 5298 y(Pr)g(o)g(of.)41 b Fq(1.)j(By)33 b(\(6.6\))f(there)h(exists)g Fo(n)28 b Fk(2)g Fp(N)k Fq(suc)m(h)j(that)1140 5504 y Fo(#)28 b Fq(:=)f Fp(P)p Fq(\()p Fo(X)1559 5463 y Fn(x)1551 5529 y(n)1630 5504 y Fk(2)h Fo(G)g Fq(for)f(0)h Fk(\024)g Fo(x)g Fk(\024)g Fo(t)p Fq(\))g Fo(>)f Fq(0)17 b Fo(:)1841 5753 y Fq(52)p eop %%Page: 53 54 53 53 bop 236 154 a Fq(If)39 b Fo(T)411 118 y Fm(0)472 154 y Fq(denotes)g(the)g(analogous)e(stopping)h(time)f(with)h(resp)s (ect)i(to)e Fo(\027)2837 118 y Fm(0)2898 154 y Fq(=)g Fo(\027)3066 118 y Fn(n)3113 154 y Fq(,)i(then)f(ap-)236 274 y(paren)m(tly)33 b Fo(T)41 b Fk(\024)29 b Fo(nT)954 238 y Fm(0)977 274 y Fq(,)j(hence)i Fo(n)28 b Fq(=)g(1)k(ma)m(y)g(b)s (e)h(assumed)h(in)d(view)i(of)f(\(2.5\).)383 420 y(2.)43 b(Moreo)m(v)m(er,)34 b Fo(t)28 b(>)f(x)p 1123 433 56 4 v 33 w Fq(or)33 b Fo(t)27 b Fq(=)p 1497 367 V 28 w Fo(x)33 b Fq(means)g(no)f(loss)g(of)h(generalit)m(y)-8 b(.)42 b(Then)579 640 y Fo(S)639 655 y Fl(0)706 640 y Fq(:=)28 b(0)124 b(and)h Fo(S)1352 655 y Fn(k)r Fl(+1)1512 640 y Fq(:=)28 b(inf)6 b Fk(f)p Fo(n)27 b(>)h(S)2060 655 y Fn(k)2130 640 y Fq(:)g Fo(H)2266 655 y Fn(n)2335 640 y Fk(\016)22 b Fo(:)17 b(:)g(:)22 b Fk(\016)g Fo(H)2697 655 y Fn(S)2740 667 y Ff(k)2777 655 y Fl(+1)2872 640 y Fq(\()p Fo(t)p Fq(\))28 b Fk(\024)g Fo(t)p Fk(g)236 860 y Fq(are)47 b(almost)d(surely)j(\014nite)f(stopping)g(times)f(with) h(resp)s(ect)i(to)e(\()p Fo(H)2823 875 y Fn(n)2870 860 y Fq(\))2908 877 y Fn(n)p Fm(2)p Fj(N)3071 860 y Fq(,)j(whic)m(h)e(b)m (y)236 980 y(\(11.2a\))32 b(satisfy)236 1200 y(\(1\))620 b Fp(E)p Fq(\()p Fo(S)1153 1215 y Fn(k)1218 1200 y Fk(\000)22 b Fo(S)1377 1215 y Fn(k)r Fm(\000)p Fl(1)1510 1200 y Fq(\))28 b(=)f Fp(E)1753 1159 y Fn(t)1782 1200 y Fq(\()p Fo(T)1877 1215 y Fn(t)1907 1200 y Fq(\))h Fo(<)f Fk(1)124 b Fq(for)74 b Fo(k)31 b Fk(2)d Fp(N)17 b Fo(:)236 1420 y Fq(Finally)-8 b(,)30 b(the)j(ev)m(en)m(ts)1121 1640 y Fo(A)1194 1655 y Fn(k)1265 1640 y Fq(:=)27 b Fk(f)p Fo(H)1526 1655 y Fn(S)1569 1667 y Ff(k)1607 1655 y Fl(+1)1701 1640 y Fq(\()p Fo(x)p Fq(\))h Fk(2)g Fo(G)g Fq(for)f(0)h Fk(\024)g Fo(x)g Fk(\024)g Fo(t)p Fk(g)236 1860 y Fq(b)m(y)34 b(the)f(assumption)f Fo(n)c Fq(=)f(1)32 b(satisfy)236 2080 y(\(2\))913 b Fp(P)p Fq(\()p Fo(A)1462 2095 y Fn(k)1504 2080 y Fq(\))28 b(=)g Fo(#)g(>)f Fq(0)125 b(for)73 b Fo(k)31 b Fk(\025)d Fq(0)17 b Fo(:)383 2300 y Fq(3.)42 b(By)32 b(construction)f(the)g(v)-5 b(ariables)30 b(1)1828 2315 y Fn(A)1881 2324 y Fd(0)1919 2300 y Fo(;)17 b(:)g(:)g(:)f(;)h Fq(1)2187 2315 y Fn(A)2240 2327 y Ff(k)q 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Fo(k)h Fq(the)e(factor)f(with)h Fo(i)d Fq(=)f Fo(k)34 b Fq(is)c(cancelled,)h(the)g(b)s(ound)f(for)h Fo(T)44 b Fq(is)30 b(increased)h(and)g(the)236 3305 y(summands)i(are)f (comp)s(osed)h(of)f(indep)s(enden)m(t)i(factors.)43 b(By)33 b(\(1\))g(and)f(\(2\))g(this)h(yields)1179 3525 y Fp(E)p Fq(\()p Fo(T)14 b Fq(\))27 b Fk(\024)h Fq(1)39 b(+)f Fo(#)1791 3484 y Fm(\000)p Fl(1)1903 3525 y Fp(E)1977 3484 y Fn(t)2006 3525 y Fq(\()p Fo(T)2101 3540 y Fn(t)2131 3525 y Fq(\))27 b Fo(<)h Fk(1)17 b Fo(:)82 b Fg(2)383 3862 y Fq(If)35 b(in)h(particular)p 1054 3809 V 34 w Fo(x)e Fk(2)f Fo(E)6 b Fq(,)37 b(this)f(result)g(implies)d(sup)2351 3886 y Fn(x)p Fm(2)p Fn(E)2526 3862 y Fp(E)2600 3825 y Fn(x)2644 3862 y Fq(\()p Fo(T)2739 3877 y Fn(G)2798 3862 y Fq(\))g Fo(<)g Fk(1)p Fq(,)k(where)g Fo(T)3484 3877 y Fn(G)236 3982 y Fq(denotes)45 b(the)e(hitting)f(time)f(of)i Fo(G)g Fq(b)m(y)h(\()p Fo(X)1848 3997 y Fn(n)1895 3982 y Fq(\))1933 3997 y Fn(n)p Fm(\025)p Fl(0)2070 3982 y Fq(.)76 b(With)42 b(this)h(notation)f(the)h(familiar)236 4102 y(criterion)28 b(for)h(p)s(ositiv)m(e)16 b(/)g(n)m(ull)28 b(recurrence)k(b)m(y)e(mean)f(passage)h(times)f(carries)g(o)m(v)m(er)h (from)236 4223 y(discrete)j(Mark)m(o)m(v)h(c)m(hain)f(theory)g(in)f (the)h(follo)m(wing)d(form:)236 4422 y Fp(\(11.4\))41 b(Theorem)69 b Fh(L)-5 b(et)28 b Fo(\027)35 b Fk(2)28 b(N)15 b Fq([)p Fo(E)6 b Fq(])27 b Fh(b)-5 b(e)28 b(r)-5 b(e)g(curr)g(ent)28 b(with)g(attr)-5 b(actor)28 b Fo(M)39 b Fh(and)27 b(\014x)g Fo(x)i Fk(2)f Fo(M)10 b Fh(.)236 4542 y(Then)34 b Fo(\027)42 b Fh(is)34 b(p)-5 b(ositive)34 b(r)-5 b(e)g(curr)g(ent)36 b(if)e(and)h(only)f(if)686 4762 y Fp(E)760 4721 y Fn(x)803 4762 y Fq(\()p Fo(T)898 4777 y Fn(G)957 4762 y Fq(\))28 b Fo(<)g Fk(1)124 b Fh(for)34 b(al)5 b(l)35 b(op)-5 b(en)34 b(subsets)h(G)g(of)g(E)g(c)-5 b(ontaining)33 b(x)17 b Fo(:)236 5057 y Fh(Pr)-5 b(o)g(of.)41 b Fq(According)36 b(to)h(\(11.3\))e(only)h(the)h(su\016ciency)h(of)e (the)h(condition)e(has)h(to)g(b)s(e)h(es-)236 5177 y(tablished.)58 b(T)-8 b(o)37 b(this)h(end)g(assume)g Fo(\027)44 b Fq(to)37 b(b)s(e)h(n)m(ull)e(recurren)m(t.)60 b(Then)p 2866 5125 V 39 w Fo(x)48 b(=)-61 b Fk(2)37 b Fo(E)43 b Fq(b)m(y)c(\(9.3\),)236 5298 y(hence)h Fo(x)e(<)p 719 5245 V 37 w(x)h Fq(and)g(th)m(us)g Fo(\027)6 b Fq(\()p Fo(h)p Fq(\()p Fo(x)p Fq(\))39 b Fo(>)e(x)p Fq(\))h Fo(>)g Fq(0)g(b)m(y)h(\(1.1\).)60 b(This)39 b(implies)d(the)j(existence)1841 5753 y(53)p eop %%Page: 54 55 54 54 bop 236 154 a Fq(of)43 b Fo(t)i Fk(2)h Fo(E)j Fq(with)43 b Fo(t)j(>)f(x)e Fq(suc)m(h)i(that)d Fo(#)k Fq(:=)g Fo(\027)6 b Fq(\()p Fo(h)p Fq(\()p Fo(x)p Fq(\))46 b Fo(>)f(t)p Fq(\))h Fo(>)f Fq(0.)75 b(With)42 b(the)i(notation)236 274 y Fo(\026)295 289 y Fl(1)362 274 y Fq(:=)28 b Fk(L)p Fq(\()p Fo(X)689 238 y Fn(x)681 299 y Fl(1)732 274 y Fq(\))k(an)h(application)d(of)i(\(11.2b\))g(yields)908 524 y Fp(E)982 483 y Fn(x)1025 524 y Fq(\()p Fo(T)1120 539 y Fn(t)1150 524 y Fq(\))84 b(=)g(\(1)21 b Fk(\000)i Fo(#)p Fq(\))39 b(+)1889 407 y Fi(Z)1935 596 y Fn(y)r(>t)2057 524 y Fq(\(1)22 b(+)g Fp(E)2338 483 y Fn(y)2379 524 y Fq(\()p Fo(T)2474 539 y Fn(t)2503 524 y Fq(\)\))17 b Fo(\026)2655 539 y Fl(1)2694 524 y Fq(\()p Fo(dy)t Fq(\))1271 759 y Fk(\025)1432 642 y Fi(Z)1478 831 y Fn(y)r(>t)1616 759 y Fp(E)1690 718 y Fn(t)1719 759 y Fq(\()p Fo(T)1814 774 y Fn(t)1844 759 y Fq(\))g Fo(\026)1958 774 y Fl(1)1997 759 y Fq(\()p Fo(dy)t Fq(\))1272 959 y(=)84 b Fo(#)28 b Fp(E)1591 918 y Fn(t)1620 959 y Fq(\()p Fo(T)1715 974 y Fn(t)1745 959 y Fq(\))69 b(=)g Fk(1)17 b Fo(;)236 1179 y Fq(i.e.)43 b(the)33 b(condition)e(is)h(violated)g(for)g Fo(G)27 b Fq(=)h([0)p Fo(;)17 b(t)p Fq([)g Fk(3)27 b Fo(x)p Fq(.)127 b Fg(2)236 1412 y Fp(References)282 1608 y Fc([1])67 b(Alpuim,)32 b(M.:)48 b(An)33 b(extremal)h(Mark)m(o)m(vian) h(sequence.)51 b(J.)34 b(Appl.)49 b(Prob.)i Fb(26)p Fc(,)35 b(219)p Fa(\000)p Fc(232)444 1721 y(\(1989\))282 1859 y([2])67 b(Barnsley)-8 b(,)33 b(M.,)g(Elton,)f(J.:)45 b(A)32 b(new)g(class)g(of)g(Mark)m(o)m(v)i(pro)s(cesses)e(for)h(image)f (enco)s(ding.)444 1972 y(Adv.)40 b(Appl.)f(Prob.)h Fb(20)p Fc(,)31 b(14)p Fa(\000)p Fc(32)h(\(1988\))282 2109 y([3])67 b(Barnsley)-8 b(,)27 b(M.,)g(Elton,)g(J.,)g(Hardin,)f(D.:)39 b(Recurren)m(t)26 b(iterated)g(function)f(systems.)39 b(Con-)444 2222 y(str.)h(Appro)m(x.)h Fb(5)p Fc(,)30 b(3)p Fa(\000)p Fc(31)i(\(1989\))282 2360 y([4])67 b(Bhattac)m(hary)m (a,)38 b(R.,)e(W)-8 b(a)m(ymire,)36 b(E.:)49 b(Sto)s(c)m(hastic)35 b(pro)s(cesses)f(with)f(applications.)51 b(New)444 2473 y(Y)-8 b(ork:)41 b(Wiley)30 b(1990)282 2611 y([5])67 b(Brandt,)33 b(A.,)i(F)-8 b(rank)m(en,)34 b(P)-8 b(.,)34 b(Lisek,)f(B.:)47 b(Stationary)33 b(sto)s(c)m(hastic)g(mo)s(dels.)48 b(Chic)m(hester:)444 2724 y(Wiley)29 b(1990)282 2862 y([6])67 b(Chama)m(y)m(ou,)31 b(J.,)f(Letac,)h(G.:)41 b(Explicit)28 b(stationary)i(distributions)c(for)k(comp)s(ositions)e (of)444 2975 y(random)i(functions)f(and)h(pro)s(ducts)f(of)i(random)f (matrices.)41 b(J.)30 b(Theor.)41 b(Prob.)g Fb(4)p Fc(,)31 b(3)p Fa(\000)p Fc(36)444 3087 y(\(1991\))282 3225 y([7])67 b(Dubins,)23 b(L.,)i(F)-8 b(reedman,)26 b(D.:)38 b(In)m(v)-5 b(arian)m(t)23 b(probabilities)d(for)j(certain)h(Mark)m(o)m(v)h(pro)s (cesses.)444 3338 y(Ann.)40 b(Math.)h(Stat.)g Fb(37)p Fc(,)31 b(837)p Fa(\000)p Fc(848)i(\(1966\))282 3476 y([8])67 b(Elton,)33 b(J.:)45 b(An)32 b(ergo)s(dic)g(theorem)h(for)f (iterated)h(maps.)47 b(Ergo)s(dic)32 b(Theory)g(Dyn.)47 b(Syst.)444 3589 y Fb(7)p Fc(,)31 b(481)p Fa(\000)p Fc(488)h(\(1987\)) 282 3727 y([9])67 b(Elton,)44 b(J.:)64 b(A)42 b(m)m(ultiplicativ)m(e)e (ergo)s(dic)h(theorem)i(for)e(Lipsc)m(hitz)g(maps.)75 b(Sto)s(c)m(hastic)444 3840 y(Pro)s(cesses)30 b(Appl.)39 b Fb(34)p Fc(,)32 b(39)p Fa(\000)p Fc(47)f(\(1990\))237 3977 y([10])67 b(F)-8 b(oguel,)36 b(S.:)49 b(The)33 b(ergo)s(dic)h (theory)h(of)f(p)s(ositiv)m(e)g(op)s(erators)g(on)g(con)m(tin)m(uous)g (functions.)444 4090 y(Ann.)40 b(Sc.)g(Norm.)h(Sup)s(er.)e(Pisa)29 b Fb(27)p Fc(,)i(19)p Fa(\000)p Fc(51)h(\(1973\))237 4228 y([11])67 b(Goldie,)28 b(C.:)39 b(Implicit)25 b(renew)m(al)j (theory)g(and)g(tails)f(of)h(solutions)e(of)i(random)f(equations.)444 4341 y(Ann.)40 b(Appl.)f(Prob.)h Fb(1)p Fc(,)31 b(126)p Fa(\000)p Fc(166)h(\(1991\))237 4479 y([12])67 b(Hata,)46 b(M.:)63 b(On)41 b(the)g(structure)g(of)h(self-similar)c(sets.)75 b(Japan)41 b(J.)g(Appl.)72 b(Math.)j Fb(2)p Fc(,)444 4592 y(381)p Fa(\000)p Fc(414)33 b(\(1985\))237 4730 y([13])67 b(Helland,)52 b(I.,)i(Nilsen,)f(T.:)78 b(On)48 b(a)i(general)f(random)f(exc)m(hange)j(mo)s(del.)96 b(J.)49 b(Appl.)444 4842 y(Prob.)40 b Fb(13)p Fc(,)31 b(781)p Fa(\000)p Fc(790)i(\(1976\))237 4980 y([14])67 b(Hutc)m(hinson,)47 b(J.:)68 b(F)-8 b(ractals)46 b(and)e(self)f(similarit)m(y)-8 b(.)81 b(Indiana)42 b(Univ.)82 b(Math.)h(J.)45 b Fb(30)p Fc(,)444 5093 y(713)p Fa(\000)p Fc(747)33 b(\(1981\))237 5231 y([15])67 b(Karlin,)47 b(S.:)71 b(Random)45 b(w)m(alks)g(arising)f (in)g(learning)g(mo)s(dels.)84 b(P)m(ac.)k(J.)45 b(Math.)87 b Fb(3)p Fc(,)444 5344 y(725)p Fa(\000)p Fc(756)33 b(\(1953\))237 5482 y([16])67 b(Kellerer,)29 b(H.:)41 b(Ergo)s(dic)29 b(b)s(eha)m(viour)g(of)i(a\016ne)f(recursions)f(I,)i(I)s(I,)e(I)s(I)s (I.)g(Preprin)m(ts)g(\(1992\))1841 5753 y Fq(54)p eop %%Page: 55 56 55 55 bop 237 154 a Fc([17])67 b(Kellerer,)46 b(H.:)69 b(Order-preserving)41 b(random)j(dynamical)e(systems:)68 b(the)45 b(a\016ne)f(case.)444 267 y(T)-8 b(o)31 b(app)s(ear)237 405 y([18])67 b(Kifer,)28 b(Y.:)41 b(Ergo)s(dic)28 b(theory)h(of)h (random)e(transformations.)39 b(Boston)31 b(Basel)e(Stuttgart:)444 518 y(Birkh\177)-45 b(auser)29 b(1986)237 655 y([19])67 b(Lamp)s(erti,)30 b(J.:)43 b(Criteria)30 b(for)h(the)g(recurrence)g(or) h(transience)f(of)g(sto)s(c)m(hastic)h(pro)s(cesses.)444 768 y(J.)e(Math.)41 b(Anal.)f(Appl.)g Fb(1)p Fc(,)30 b(314)p Fa(\000)p Fc(330)j(\(1960\))237 906 y([20])67 b(Letac,)37 b(G.:)49 b(A)34 b(con)m(traction)h(principle)c(for)j (certain)g(Mark)m(o)m(v)i(c)m(hains)e(and)g(its)f(applica-)444 1019 y(tions.)40 b(Con)m(temp.)h(Math.)g Fb(50)p Fc(,)31 b(263)p Fa(\000)p Fc(273)i(\(1986\))237 1157 y([21])67 b(Lindley)-8 b(,)30 b(D.:)44 b(The)31 b(theory)h(of)g(a)g(queue)f(with) g(a)h(single)e(serv)m(er.)45 b(Pro)s(c.)f(Cam)m(b.)h(Philos.)444 1270 y(So)s(c.)40 b Fb(48)p Fc(,)31 b(277)p Fa(\000)p Fc(289)i(\(1952\))237 1408 y([22])67 b(Lo)m(ynes,)40 b(R.:)56 b(The)38 b(stabilit)m(y)e(of)i(a)h(queue)e(with)g(non-indep)s (enden)m(t)f(in)m(ter-arriv)-5 b(al)36 b(and)444 1521 y(service)30 b(times.)40 b(Pro)s(c.)h(Cam)m(b.)f(Philos.)f(So)s(c.)i Fb(58)p Fc(,)31 b(497)p Fa(\000)p Fc(520)i(\(1962\))237 1658 y([23])67 b(Nummelin,)25 b(E.:)39 b(General)27 b(irreducible)c (Mark)m(o)m(v)29 b(c)m(hains)d(and)g(non-negativ)m(e)i(op)s(erators.) 444 1771 y(Cam)m(bridge:)40 b(Univ)m(ersit)m(y)29 b(Press)h(1984)237 1909 y([24])67 b(Rac)m(hev,)26 b(S.,)g(Samoro)s(dnitsky)-8 b(,)24 b(G.:)38 b(Limit)23 b(la)m(ws)g(for)h(a)h(sto)s(c)m(hastic)g (pro)s(cess)e(and)h(random)444 2022 y(recursion)37 b(arising)f(in)h (probabilit)m(y)f(mo)s(delling.)62 b(Adv.)j(Appl.)e(Prob.)h Fb(27)p Fc(,)41 b(185)p Fa(\000)p Fc(202)444 2135 y(\(1995\))237 2273 y([25])67 b(Rac)m(hev,)30 b(S.,)g(T)-8 b(o)s(doro)m(vic,)29 b(P)-8 b(.:)41 b(On)28 b(the)h(rate)h(of)f(con)m(v)m(ergence)j(of)d (some)g(functionals)f(of)h(a)444 2386 y(sto)s(c)m(hastic)i(pro)s(cess.) 40 b(J.)31 b(Appl.)39 b(Prob.)h Fb(27)p Fc(,)31 b(805)p Fa(\000)p Fc(814)i(\(1990\))237 2523 y([26])67 b(Revuz,)29 b(D.:)41 b(Mark)m(o)m(v)31 b(c)m(hains,)d(2nd)h(edition.)39 b(Amsterdam)28 b(NewY)-8 b(ork:)41 b(North-Holland)444 2636 y(1984)237 2774 y([27])67 b(Rosen)m(blatt,)25 b(M.:)37 b(Recurren)m(t)23 b(p)s(oin)m(ts)f(and)g(transition)f(functions)h (acting)h(on)g(con)m(tin)m(uous)444 2887 y(functions.)39 b(Z.)30 b(W)-8 b(ahrsc)m(heinlic)m(hk)m(eitstheorie)29 b(V)-8 b(erw.)41 b(Geb.)g Fb(30)p Fc(,)31 b(173)p Fa(\000)p Fc(183)i(\(1974\))237 3025 y([28])67 b(Tw)m(eedie,)51 b(R.:)75 b(Criteria)45 b(for)i(classifying)e(general)j(Mark)m(o)m(v)g (c)m(hains.)91 b(Adv.)g(Appl.)444 3138 y(Prob.)40 b Fb(8)p Fc(,)31 b(737)p Fa(\000)p Fc(771)h(\(1976\))237 3276 y([29])67 b(V)-8 b(erv)j(aat,)32 b(W.:)42 b(On)29 b(a)i(sto)s(c)m (hastic)g(di\013erence)f(equation)h(and)f(a)g(represen)m(tation)h(of)g (non-)444 3389 y(negativ)m(e)g(in\014nitely)d(divisible)f(random)i(v)-5 b(ariables.)40 b(Adv.)h(Appl.)e(Prob.)i Fb(11)p Fc(,)31 b(750)p Fa(\000)p Fc(783)444 3501 y(\(1979\))237 3639 y([30])67 b(Y)-8 b(aha)m(v,)42 b(J.:)57 b(On)37 b(a)i(Mark)m(o)m(v)h (pro)s(cess)e(generated)i(b)m(y)e(non-decreasing)g(conca)m(v)m(e)j (func-)444 3752 y(tions.)f(Sto)s(c)m(hastic)31 b(Pro)s(cesses)f(Appl.) 39 b Fb(4)p Fc(,)31 b(41)p Fa(\000)p Fc(54)h(\(1976\))1841 5753 y Fq(55)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF