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Fr(n)2935 380 y Fx({)g(the)g(follo)m(wing)236 493 y(tric)m(hotom)m(y)32 b(is)d(established)g(in)g(the)h(preprin)m(t)f ([24]:)236 642 y(\(a\))87 b(if)29 b Fs(S)577 656 y Fr(n)649 642 y Fq(!)c(\0001)p Fx(,)31 b(then)f(\()p Fs(X)1300 656 y Fr(n)1347 642 y Fs(;)h(n)25 b Fq(\025)g Fx(0\))31 b(is)e(\\)16 b(p)s(ositiv)m(e)29 b(recurren)m(t)15 b(",)236 797 y(\(b\))81 b(if)29 b(\()p Fs(S)612 811 y Fr(n)659 797 y Fs(;)i(n)25 b Fq(\025)g Fx(0\))31 b(oscillates,)f(then)g(\()p Fs(X)1729 811 y Fr(n)1777 797 y Fs(;)g(n)25 b Fq(\025)g Fx(0\))31 b(is)e(\\)16 b(n)m(ull)28 b(recurren)m(t)15 b(",)236 951 y(\(c\))92 b(if)29 b Fs(S)577 965 y Fr(n)649 951 y Fq(!)c Fx(+)p Fq(1)p Fx(,)31 b(then)f(\()p Fs(X)1300 965 y Fr(n)1347 951 y Fs(;)h(n)25 b Fq(\025)g Fx(0\))31 b(is)e(\\)16 b(transien)m(t)f(".)377 1101 y(In)35 b(the)h(existing)e (literature)h(\(see)h(in)e(particular)g(the)i(surv)m(eys)f(b)m(y)g(V)-8 b(erv)j(aat)38 b([38])e(and)f(b)m(y)236 1213 y(Em)m(brec)m(h)m (ts/Goldie)22 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b(in)g(the)g(preprin)m(t)f([25])j(pro)m(v)m(e)g(that)f(order)f(and)g (top)s(ology)h(com)m(bined)f(pro)m(vide)g(an)g(ideal)236 2117 y(framew)m(ork)31 b(for)f(a)h(fairly)d(complete)j(theory)g(of)f (random)g(dynamical)e(systems)j(as)f(considered)236 2230 y(in)f(this)h(pap)s(er.)377 2343 y(T)-8 b(o)24 b(co)m(v)m(er,)i(ho)m(w) m(ev)m(er,)g(autoregressiv)m(e)e(mo)s(dels)e(of)h(higher)e(dimension)g (or)i(sto)s(c)m(hastic)g(recur-)236 2455 y(sions)k(of)g(higher)g (order,)h(assuming)e Fs(E)33 b Fx(to)28 b(b)s(e)f(totally)h(ordered,)g (as)g(is)e(the)i(case)h(in)d([24])j(or)f([25],)236 2568 y(is)k(to)s(o)i(restrictiv)m(e.)48 b(In)32 b(the)h(sequel,)g (therefore,)h(these)f(preprin)m(ts)e(are)j(disregarded,)e(and)g(the)236 2681 y(state)i(space)e(is)f(supp)s(osed)f(to)j(b)s(e)e(an)m(y)h (\(partially\))f(ordered)g(top)s(ological)h(space.)46 b(In)31 b(essence,)236 2794 y(there)h(is)e(only)g(one)h(restriction:)42 b Fs(E)36 b Fx(is)30 b(supp)s(osed)f(to)j(b)s(e)f(b)s(ounded)e(from)h 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b(order-preserving)f(and)h(con)m(tin)m(uous.)377 3697 y(The)i(precise)g(assumptions)f(on)i(the)g(state)h(space)f Fs(E)38 b Fx(are)33 b(collected)g(at)g(the)g(b)s(eginning)d(of)236 3810 y(Section)38 b(0.)64 b(They)38 b(are)g(met)h(not)f(only)f(in)g (the)h(classical)g(case)h Fs(E)k Fx(=)38 b Fy(R)2797 3777 y Fr(d)2797 3833 y Fo(+)2856 3810 y Fx(,)j(but)c(as)h(w)m(ell,)h (for)236 3923 y(instance,)c(b)m(y)e(\(ro)s(oted\))h(tree)h(mo)s(dels.) 49 b(The)33 b(signi\014cance)f(of)i(the)g(order)f(is)f(re\015ected)j(b) m(y)e(the)236 4036 y(singular)39 b(role)h(of)g(order)h(con)m(v)m(ex)h (sets)f(and)f(functions)f(of)h(b)s(ounded)f(v)-5 b(ariation,)42 b(leading)d(to)236 4149 y(the)30 b(basic)f(classes)h Fm(V)p Fx(\()p Fs(E)5 b Fx(\))31 b(and)e Fq(V)7 b Fx(\()p Fs(E)e Fx(\),)31 b(resp)s(ectiv)m(ely)-8 b(.)41 b(The)29 b(extension)g(of)h(these)h(notions)e(from)236 4262 y(total)k(to)g (partial)e(order)g(is)g(straigh)m(tforw)m(ard,)i(apart)f(from)g(the)g (fact)h(that)g(monotone)g(sets)f(or)236 4375 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b(If)236 831 y(the)31 b(state)i(space)e(is)f(only)h(partially)e(ordered,)i(actually) f(\\)16 b(lo)s(cal)30 b(b)s(oundedness)13 b(")31 b(of)g Fs(E)37 b Fx(has)30 b(to)236 944 y(b)s(e)g(supp)s(osed)e(here)j(\(see)g (the)f(\014nal)g(section\).)377 1057 y(As)g(men)m(tioned)e(ab)s(o)m(v)m (e,)j(the)f(existing)e(literature)h(concerns)g(almost)g(exclusiv)m(ely) f(p)s(ositiv)m(e)236 1170 y(recurren)m(t)f(systems,)h(i.e.)39 b(the)27 b(case)h(of)f(a)g(\014nite)f(in)m(v)-5 b(arian)m(t)26 b(measure,)i(as)f(treated)g(in)f(Section)h(6.)236 1283 y(There)35 b(are)h(n)m(umerous)e(studies)g(considering)f Fs(E)39 b Fx(=)33 b Fy(R)2155 1297 y Fo(+)2250 1283 y Fx(with)h(its)g(total)j(order)d(and)h(random)236 1396 y(transformations)e(preserving)f(this)g(order)h(as,)h(for)f(instance,)h (Alpuim/A)m(tha)m(yde)f([1],)i(Goldie)236 1509 y([19],)28 b(Helland/Nilsen)c([21],)k(Lund)c(et)i(al.)39 b([30],)28 b(Y)-8 b(aha)m(v)27 b([40].)41 b(There)25 b(is,)h(moreo)m(v)m(er,)i(a)f (n)m(um)m(b)s(er)236 1622 y(of)35 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Fr(n)2705 4183 y Fs(;)15 b(n)26 b Fq(2)e Fy(N)p Fx(\).)236 4296 y(This)29 b(de\014nes)g(an)i Fk(or)-5 b(der-pr)g(eserving)34 b(r)-5 b(andom)35 b(dynamic)-5 b(al)35 b(system)c Fx(b)m(y)1275 4460 y Fs(X)1350 4474 y Fr(n)1423 4460 y Fx(=)25 b Fs(H)1595 4474 y Fr(n)1642 4460 y Fx(\()p Fs(X)1752 4474 y Fr(n)p Fp(\000)p Fo(1)1889 4460 y Fx(\))125 b(for)58 b Fs(n)25 b Fq(2)g Fy(N)15 b Fs(:)236 4623 y Fx(Therefore)38 b(the)g(distribution)c(of)k(\()p Fs(X)1544 4637 y Fr(n)1591 4623 y Fs(;)31 b(n)37 b Fq(\025)g Fx(0\))h(is)f(completely)h(determined)e(b)m(y)i Fs(\027)43 b Fx(and)37 b(the)236 4736 y(initial)j(la)m(w)j Fs(\026)743 4750 y Fo(0)828 4736 y Fx(=)i Fq(L)p Fx(\()p Fs(X)1117 4750 y Fo(0)1157 4736 y Fx(\).)78 b(If)42 b(in)g(particular)f Fs(X)2024 4750 y Fo(0)2109 4736 y Fx(=)46 b Fs(x)p Fx(,)g(this)41 b(will)f(b)s(e)i(expressed)g(b)m(y)h(the)236 4849 y(notation)31 b(\()p Fs(X)717 4816 y Fr(x)710 4872 y(n)762 4849 y Fs(;)f(n)25 b Fq(\025)g Fx(0\),)31 b(i.e.)930 5013 y Fs(X)1012 4975 y Fr(x)1005 5035 y(n)1082 5013 y Fx(=)25 b Fs(H)1254 5027 y Fr(n)1321 5013 y Fq(\016)20 b Fs(:)15 b(:)g(:)22 b Fq(\016)e Fs(H)1654 5027 y Fo(1)1708 5013 y Fx(\()p Fs(x)p Fx(\))126 b(for)58 b Fs(x)25 b Fq(2)g Fs(E)46 b Fx(and)41 b Fs(n)25 b Fq(\025)g Fx(0)15 b Fs(:)236 5177 y Fx(Th)m(us)30 b(for)g(general)g Fs(\026)974 5191 y Fo(0)1044 5177 y Fx(conditional)e(probabilities)f(are)k(giv)m(en)f(b) m(y)1009 5340 y Fy(P)1080 5303 y Fr(x)1124 5340 y Fx(\(\()p Fs(X)1269 5354 y Fr(n)1317 5340 y Fs(;)h(n)24 b Fq(\025)h Fx(0\))h Fq(2)f Fs(B)5 b Fx(\))25 b(=)g Fy(P)p Fx(\(\()p Fs(X)2193 5303 y Fr(x)2186 5363 y(n)2239 5340 y Fs(;)30 b(n)25 b Fq(\025)g Fx(0\))h Fq(2)f Fs(B)5 b Fx(\))236 5504 y(with)29 b(an)i(analogous)f(equation)g(for)h(conditional)d(exp)s (ectations.)1867 5753 y(6)p eop %%Page: 7 7 7 6 bop 377 154 a Fx(As)40 b(usual,)h(the)g(initial)c(la)m(w)i(is)g (largely)h(of)g(secondary)g(imp)s(ortance,)i(and)d(the)h(primary)236 267 y(comp)s(onen)m(t)h(is)f(the)h(distribution)d Fs(\027)6 b Fx(.)72 b(Therefore,)43 b(whenev)m(er)e(p)s(ossible,)g(it)g(will)d(b) s(e)i(brie\015y)236 380 y(referred)f(to)i(the)f(\()p Fk(dynamic)-5 b(al)18 b Fx(\))40 b Fk(system)h Fx(\()p Fs(E)5 b(;)15 b(\027)6 b Fx(\))16 b(.)70 b(All)38 b(notions)h(to)i(b)s (e)e(de\014ned)g(in)g(Sections)236 493 y(1)31 b({)g(9)f(will)e(dep)s (end)g(on)i Fs(\027)6 b Fx(,)31 b(but)e(this)g(dep)s(endence)h(will)d (b)s(e)j(suppressed)e(in)h(related)h(notations)236 606 y(\(as)h(for)f(the)h(supp)s(ort)e Fq(N)43 b Fx(ab)s(o)m(v)m(e\),)32 b(b)s(ecause)e Fs(\027)36 b Fx(is)30 b(supp)s(osed)e(to)j(b)s(e)f (\014xed.)377 719 y(Clearly)-8 b(,)39 b(\()p Fs(X)830 733 y Fr(n)878 719 y Fs(;)30 b(n)37 b Fq(\025)h Fx(0\))g(is)f(a)h (homogeneous)g(Mark)m(o)m(v)i(c)m(hain.)62 b(Its)38 b(transition)e(k)m (ernel)h Fs(P)236 831 y Fx(transforms)30 b(\(nonnegativ)m(e\))i (functions)c Fs(f)35 b Fq(2)25 b(B)s Fx(\()p Fs(E)5 b Fx(\))31 b(in)m(to)f Fs(P)13 b(f)40 b Fx(giv)m(en)30 b(b)m(y)1035 1057 y Fs(P)13 b(f)d Fx(\()p Fs(x)p Fx(\))26 b(=)1405 942 y Fi(Z)1478 1098 y Fp(H)p Fo([)p Fr(E)t Fo(])1681 1057 y Fs(f)10 b Fx(\()p Fs(h)p Fx(\()p Fs(x)p Fx(\)\))15 b Fs(\027)6 b Fx(\()p Fs(dh)p Fx(\))127 b(for)58 b Fs(x)25 b Fq(2)g Fs(E)236 1278 y Fx(and)30 b(a)h(measure)f Fs(\026)g Fx(on)g Fs(E)36 b Fx(in)m(to)30 b Fs(\026P)43 b Fx(giv)m(en)31 b(b)m(y)883 1498 y Fs(\026P)13 b Fx(\()p Fs(B)5 b Fx(\))25 b(=)1274 1383 y Fi(Z)1347 1534 y Fr(E)1436 1498 y Fs(\027)6 b Fx(\()p Fs(h)p Fx(\()p Fs(x)p Fx(\))26 b Fq(2)f Fs(B)5 b Fx(\))15 b Fs(\026)p Fx(\()p Fs(dx)p Fx(\))125 b(for)58 b Fs(B)30 b Fq(2)25 b Fm(B)p Fx(\()p Fs(E)5 b Fx(\))15 b Fs(;)236 1719 y Fx(whic)m(h)29 b(in)g(the)i Fs(\033)s Fq(\000)p Fx(\014nite)e(case)j(equals)833 1940 y Fs(\026P)13 b Fx(\()p Fs(B)5 b Fx(\))25 b(=)1224 1825 y Fi(Z)1297 1980 y Fp(H)p Fo([)p Fr(E)t Fo(])1485 1940 y Fs(\026)p Fx(\()p Fs(h)p Fx(\()p Fs(x)p Fx(\))i Fq(2)e Fs(B)5 b Fx(\))15 b Fs(\027)6 b Fx(\()p Fs(dh)p Fx(\))125 b(for)58 b Fs(B)30 b Fq(2)25 b Fm(B)p Fx(\()p Fs(E)5 b Fx(\))15 b Fs(:)377 2160 y(P)42 b Fx(is)27 b(a)i(F)-8 b(eller)28 b(k)m(ernel,)g(whic)m(h)f(in)g(addition)g(transforms)h (increasing)f(functions)f(in)m(to)j(func-)236 2273 y(tions)h(of)g(the)h (same)g(t)m(yp)s(e.)41 b(By)30 b(\(E2\))i(this)d(implies)f(in)h (particular:)377 2469 y Fg(\(0.1\))51 b(Lemma)78 b Fk(The)32 b(se)-5 b(quenc)g(e)33 b Fx(\()p Fs(")1652 2483 y Fo(0)1707 2469 y Fs(P)1778 2436 y Fr(n)1825 2469 y Fs(;)e(n)25 b Fq(\025)f Fx(0\))34 b Fk(is)e(sto)-5 b(chastic)g(al)5 b(ly)36 b(incr)-5 b(e)g(asing,)34 b(i.e.)943 2662 y Fy(E)p Fx(\()p Fs(f)10 b Fx(\()p Fs(X)1219 2624 y Fo(0)1212 2684 y Fr(n)p Fp(\000)p Fo(1)1349 2662 y Fx(\)\))26 b Fq(\024)f Fy(E)p Fx(\()p Fs(f)10 b Fx(\()p Fs(X)1817 2624 y Fo(0)1810 2684 y Fr(n)1857 2662 y Fx(\)\))125 b Fk(for)59 b Fx(0)26 b Fq(\024)e Fs(f)35 b Fq(2)25 b(B)2618 2616 y Fj(")2653 2662 y Fx(\()p Fs(E)5 b Fx(\))15 b Fs(:)377 2926 y Fg(Pr)n(oof.)61 b Fx(This)29 b(is)g(immediate)h(from)g(the)g (equations)1122 3146 y Fy(E)p Fx(\()p Fs(f)10 b Fx(\()p Fs(X)1398 3109 y Fo(0)1391 3169 y Fr(n)p Fp(\000)p Fo(1)1528 3146 y Fx(\)\))26 b(=)1720 3031 y Fi(Z)1793 3182 y Fr(E)1882 3146 y Fs(P)1953 3109 y Fr(n)p Fp(\000)p Fo(1)2090 3146 y Fs(f)e Fx(\(0\))15 b Fs(P)e Fx(\(0;)i Fs(dx)p Fx(\))g Fs(;)1164 3419 y Fy(E)p Fx(\()p Fs(f)10 b Fx(\()p Fs(X)1440 3381 y Fo(0)1433 3441 y Fr(n)1480 3419 y Fx(\)\))25 b(=)1671 3304 y Fi(Z)1745 3455 y Fr(E)1833 3419 y Fs(P)1904 3381 y Fr(n)p Fp(\000)p Fo(1)2041 3419 y Fs(f)g Fx(\()p Fs(x)p Fx(\))15 b Fs(P)e Fx(\(0;)i Fs(dx)p Fx(\))g Fs(:)860 b Ff(\003)377 3610 y Fx(Finally)-8 b(,)33 b(it)g(has)g(to)h(b)s(e)e (men)m(tioned)h(that)h(the)f(passage)h(from)f(the)h(distribution)29 b Fs(\027)39 b Fx(to)34 b(the)236 3722 y(k)m(ernel)c Fs(P)43 b Fx(in)30 b(general)g(is)f(not)i(injectiv)m(e.)377 3990 y Fy(1.)74 b(Irreducible)38 b(systems.)64 b Fx(Let)33 b(\()p Fs(E)5 b(;)15 b(\027)6 b Fx(\))31 b(b)s(e)h(a)i(dynamical)e (system)h(as)g(in)m(tro)s(duced)f(in)236 4103 y(the)j(preceding)f (section.)53 b(Then,)35 b(to)h(classify)d(it)h(as)h(recurren)m(t)f(or)h (transien)m(t,)h(requires)d(some)236 4216 y(comm)m(unication)44 b(structure)f(to)i(prev)m(en)m(t)g(the)f(state)i(space)e(from)g (splitting)e(in)m(to)i(di\013eren)m(t)236 4329 y(classes.)56 b(It)36 b(turns)e(out)h(to)i(b)s(e)d(su\016cien)m(t)h(that,)j(starting) d(from)g(the)g(minimal)e(state)k(0,)g(eac)m(h)236 4441 y(state)32 b Fs(x)e Fx(with)f(p)s(ositiv)m(e)h(probabilit)m(y)e(can)i (b)s(e)g(reac)m(hed)h(or)f(exceeded)i(in)d(the)h(follo)m(wing)f(sense:) 377 4637 y Fg(\(1.1\))56 b(Definition)90 b Fx(The)34 b(system)g(\()p Fs(E)5 b(;)15 b(\027)6 b Fx(\))33 b(is)h(called)g(\\)15 b(\()p Fk(upwar)-5 b(ds)p Fx(\))38 b Fk(irr)-5 b(e)g(ducible)16 b Fx(",)36 b(if)d(for)236 4750 y(an)m(y)e Fs(x)25 b Fq(2)g Fs(E)36 b Fx(there)30 b(is)g(some)g Fs(n)25 b Fq(2)g Fy(N)30 b Fx(suc)m(h)g(that)917 4943 y Fs(P)988 4905 y Fr(n)1035 4943 y Fx(\(0;)15 b([)p Fs(x;)g Fq(\001)g Fx(]\))43 b(=)d Fy(P)p Fx(\()p Fs(X)1714 4905 y Fo(0)1707 4965 y Fr(n)1780 4943 y Fq(\025)25 b Fs(x)p Fx(\))41 b(=)f Fs(\027)2166 4905 y Fr(n)2213 4943 y Fx(\()p Fs(h)p Fx(\(0\))27 b Fq(\025)e Fs(x)p Fx(\))41 b Fs(>)f Fx(0)15 b Fs(:)377 5165 y Fx(It)34 b(is)f(immediate)g(from)g(\(0.1\))i(that)f (the)g(system)g(\()p Fs(E)5 b(;)15 b(\027)2323 5132 y Fr(n)2371 5165 y Fx(\))34 b(is)e(irreducible)f(for)i(all)g Fs(n)d Fq(2)g Fy(N)p Fx(,)236 5278 y(whenev)m(er)g(this)g(holds)f(for)h (one)g Fs(n)25 b Fq(2)g Fy(N)p Fx(.)377 5391 y(The)35 b(question,)h(whether)f(irreducibilit)m(y)30 b(can)36 b(b)s(e)e(accomplished)g(b)m(y)h(suitably)f(reducing)236 5504 y(the)39 b(state)h(space,)h(is)d(p)s(ostp)s(oned)f(to)i(Section)f (9.)66 b(As)38 b(an)h(example)f(consider)f(the)i(\\)16 b(Can)m(tor)1867 5753 y(7)p eop %%Page: 8 8 8 7 bop 236 154 a Fx(system)16 b(",)49 b(assigning)43 b(mass)i Fs(\027)6 b Fx(\()p Fq(f)p Fs(h)1467 168 y Fr(i)1496 154 y Fq(g)p Fx(\))50 b(=)1756 118 y Fo(1)p 1756 133 36 4 v 1756 185 a(2)1846 154 y Fx(to)45 b(the)g(t)m(w)m(o)h(mappings)d Fs(h)2804 168 y Fo(1)2893 154 y Fx(:)50 b Fs(x)f Fq(7!)g Fs(x=)p Fx(3)d(and)236 267 y Fs(h)288 281 y Fo(2)353 267 y Fx(:)26 b Fs(x)f Fq(7!)g Fs(x=)p Fx(3)c(+)f(2)p Fs(=)p Fx(3)d(:)40 b(the)31 b(adequate)g(state)h(space)f(in)e(the)i (presen)m(t)f(setting)g(is)g Fs(E)g Fx(=)25 b([0)p Fs(;)15 b Fx(1[.)377 380 y(In)42 b(spite)f(of)i(its)e(w)m(eak)i(app)s(earance,) j(the)c(condition)f(in)g(\(1.1\))j(has)e(strong)g(recurrence)236 493 y(implications)28 b(for)i(increasing)f(in)m(terv)-5 b(als:)377 689 y Fg(\(1.2\))46 b(Pr)n(oposition)67 b Fk(If)27 b(the)h(system)h Fx(\()p Fs(E)5 b(;)15 b(\027)6 b Fx(\))26 b Fk(is)i(irr)-5 b(e)g(ducible,)29 b(then)f(for)h(arbitr)-5 b(ary)30 b(initial)236 802 y(law)k(and)f(al)5 b(l)34 b Fs(x)25 b Fq(2)g Fs(E)236 1004 y Fx(\(a\))1024 b Fy(P)p Fx(\(lim)15 b(sup)1529 1072 y Fr(n)p Fp(!1)1775 1004 y Fq(f)p Fs(X)1895 1018 y Fr(n)1968 1004 y Fq(\025)25 b Fs(x)p Fq(g)p Fx(\))h(=)f(1)15 b Fs(;)236 1190 y Fx(\(b\))1237 b Fy(E)p Fx(\()p Fs(T)1751 1208 y Fo([)p Fr(x;)11 b Fp(\001)h Fo(])1897 1190 y Fx(\))26 b Fs(<)f Fq(1)15 b Fs(;)236 1392 y Fk(wher)-5 b(e)34 b Fs(T)546 1406 y Fr(B)632 1392 y Fx(:=)25 b(inf)c Fq(f)p Fs(n)k Fq(2)g Fy(N)g Fx(:)h Fs(X)1323 1406 y Fr(n)1395 1392 y Fq(2)f Fs(B)5 b Fq(g)25 b Fx(\()p Fq(\024)h(1)p Fx(\))58 b Fk(for)33 b Fs(B)d Fq(2)24 b Fm(B)p Fx(\()p Fs(E)5 b Fx(\))p Fk(.)377 1588 y Fg(Pr)n(oof.)87 b Fx(By)38 b(assumption)f Fs(#)i Fx(:=)f Fs(\027)1695 1555 y Fr(n)1742 1588 y Fx(\()p Fs(h)p Fx(\(0\))i Fq(\025)f Fs(x)p Fx(\))g Fs(>)f Fx(0)h(for)f(some)h Fs(n)p Fx(,)h(whic)m(h)d(yields)f(b)m(y)236 1701 y(monotonicit)m(y)31 b(and)e(indep)s(endence)567 1904 y Fy(P)p Fx(\()p Fs(T)726 1922 y Fo([)p Fr(x;)11 b Fp(\001)h Fo(])898 1904 y Fs(>)25 b(k)s(n)p Fx(\))83 b Fq(\024)g Fy(P)1442 1866 y Fo(0)1482 1904 y Fx(\()p Fs(T)1570 1922 y Fo([)p Fr(x;)11 b Fp(\001)g Fo(])1741 1904 y Fs(>)25 b(k)s(n)p Fx(\))1217 2061 y Fq(\024)1371 1980 y Fi(Y)1492 2098 y Fo(0)p Fp(\024)p Fr(i)g Fx(0)31 b(therefore)916 4668 y Fq(f)p Fs(X)1043 4631 y Fo(0)1036 4691 y Fr(l)1108 4668 y Fq(\025)25 b Fs(x)p Fq(g)36 b(\\)1433 4574 y Fi(\020)1498 4587 y(\\)1605 4710 y Fr(m)p Fp(2)p Fh(N)1832 4587 y Fi([)1939 4710 y Fr(n)p Fp(2)p Fh(N)2131 4668 y Fq(f)p Fs(X)2258 4631 y Fo(0)2251 4691 y Fr(m)2344 4668 y Fq(\024)25 b Fs(y)2485 4682 y Fr(n)2532 4668 y Fq(g)2577 4574 y Fi(\021)2667 4668 y Fq(6)p Fx(=)40 b Fq(;)15 b Fs(;)236 4881 y Fx(hence)31 b(for)f Fs(m)25 b Fx(=)g Fs(l)32 b Fx(in)d(particular)1093 5094 y Fq(f)p Fs(X)1220 5056 y Fo(0)1213 5116 y Fr(l)1285 5094 y Fq(\025)c Fs(x)p Fq(g)36 b(\\)1610 5000 y Fi(\020)1675 5013 y([)1782 5136 y Fr(n)p Fp(2)p Fh(N)1974 5094 y Fq(f)p Fs(X)2101 5056 y Fo(0)2094 5116 y Fr(l)2167 5094 y Fq(\024)25 b Fs(y)2308 5108 y Fr(n)2354 5094 y Fq(g)2399 5000 y Fi(\021)2490 5094 y Fq(6)p Fx(=)40 b Fq(;)15 b Fs(:)236 5320 y Fx(This)29 b(implies)e Fs(x)f Fq(2)e Fs(A)983 5278 y Fj(#)1050 5320 y Fx(for)30 b Fs(A)25 b Fx(:=)h Fq(f)p Fs(y)1494 5334 y Fr(n)1566 5320 y Fx(:)f Fs(n)g Fq(2)g Fy(N)p Fq(g)p Fx(.)1539 b Ff(\003)377 5504 y Fx(The)30 b(follo)m(wing)f(inequalit)m (y)g(is)g(the)i(crucial)e(to)s(ol)h(in)f(the)i(sequel:)1867 5753 y(8)p eop %%Page: 9 9 9 8 bop 377 154 a Fg(\(1.4\))41 b(Theorem)57 b Fk(L)-5 b(et)24 b(the)g(system)g Fx(\()p Fs(E)5 b(;)15 b(\027)6 b Fx(\))26 b Fk(b)-5 b(e)23 b(irr)-5 b(e)g(ducible)24 b(and)h(supp)-5 b(ose)25 b Fx(0)g Fq(\024)g Fs(f)35 b Fq(2)25 b(B)3351 113 y Fj(")3386 154 y Fx(\()p Fs(E)5 b Fx(\))p Fk(.)236 267 y(Then)33 b(for)g(al)5 b(l)33 b Fs(x)26 b Fq(2)f Fs(E)797 390 y Fi(X)932 506 y Fr(n)p Fp(\025)p Fo(0)1099 471 y Fy(E)p Fx(\()p Fs(f)10 b Fx(\()p Fs(X)1375 434 y Fr(x)1368 494 y(n)1419 471 y Fx(\))20 b Fq(\000)g Fs(f)10 b Fx(\()p Fs(X)1737 434 y Fo(0)1730 494 y Fr(n)1777 471 y Fx(\)\))51 b Fq(\024)f Fy(E)p Fx(\()p Fs(T)2201 434 y Fo(0)2176 497 y([)p Fr(x;)11 b Fp(\001)h Fo(])2322 471 y Fx(\))31 b(sup)2390 547 y Fr(n)p Fp(\025)p Fo(0)2555 471 y Fy(E)p Fx(\()p Fs(f)10 b Fx(\()p Fs(X)2831 434 y Fo(0)2824 494 y Fr(n)2871 471 y Fx(\)\))15 b Fs(;)236 739 y Fk(wher)-5 b(e)34 b Fs(T)559 695 y Fr(y)546 766 y(B)632 739 y Fx(:=)25 b(inf)c Fq(f)p Fs(n)k Fq(2)g Fy(N)g Fx(:)h Fs(X)1330 706 y Fr(y)1323 761 y(n)1397 739 y Fq(2)f Fs(B)5 b Fq(g)25 b Fx(\()p Fq(\024)g(1)p Fx(\))58 b Fk(for)33 b Fs(y)28 b Fq(2)d Fs(E)38 b Fk(and)c Fs(B)c Fq(2)24 b Fm(B)p Fx(\()p Fs(E)5 b Fx(\))p Fk(.)377 935 y Fg(Pr)n(oof.)61 b Fx(If)30 b Fs(f)39 b Fx(is)29 b(replaced)g(b)m(y)h Fs(f)e Fq(^)19 b Fs(k)s(;)31 b(k)d Fq(2)d Fy(N)p Fx(,)30 b(then)f(the)h(corresp)s(onding)e(di\013erences)h(on)236 1048 y(the)35 b(left-hand)f(side)g(increase)g(for)h Fs(k)h Fq(!)c(1)j Fx(to)g(the)g(di\013erence)g(in)e(question,)j(i.e.)53 b Fs(f)44 b Fx(ma)m(y)36 b(b)s(e)236 1161 y(assumed)25 b(to)i(b)s(e)e(b)s(ounded.)37 b(In)25 b(addition,)g(b)m(y)h(\(1.2b\))h (the)f(stopping)f(time)h Fs(T)38 b Fx(:=)25 b Fs(T)3110 1128 y Fo(0)3086 1191 y([)p Fr(x;)11 b Fp(\001)g Fo(])3257 1161 y Fx(ma)m(y)27 b(b)s(e)236 1284 y(assumed)j(to)i(b)s(e)e(\014nite) g(ev)m(erywhere.)43 b(Then,)31 b(with)e(the)i(notation)h Fs(A)2641 1299 y Fr(l)2693 1284 y Fx(:=)26 b Fq(f)p Fs(T)40 b Fq(\024)26 b Fs(l)r Fq(g)p Fx(,)31 b(for)g Fs(n)26 b Fq(\025)g Fx(0)236 1397 y(\014xed)k(and)g Fs(m)25 b Fq(\024)g Fs(n)30 b Fx(arbitrary)236 1623 y(\()p Fq(\003)p Fx(\))639 1608 y Fy(E)p Fx(\(1)788 1622 y Fr(A)841 1631 y Fe(n)p Fj(\000)p Fe(m)1006 1608 y Fs(f)10 b Fx(\()p Fs(X)1178 1570 y Fo(0)1171 1630 y Fr(T)g Fo(+)p Fr(m)1343 1608 y Fx(\)\))84 b(=)f Fy(E)p Fx(\(1)1800 1622 y Fr(A)1853 1631 y Fe(n)p Fj(\000)p Fe(m)2017 1608 y Fs(f)10 b Fx(\()p Fs(H)2183 1622 y Fr(T)g Fo(+)p Fr(m)2375 1608 y Fq(\016)21 b Fs(:)15 b(:)g(:)21 b Fq(\016)g Fs(H)2709 1622 y Fr(T)10 b Fo(+1)2868 1608 y Fx(\()p Fs(X)2985 1570 y Fo(0)2978 1630 y Fr(T)3034 1608 y Fx(\)\)\))1497 1746 y Fq(\025)83 b Fy(E)p Fx(\(1)1800 1760 y Fr(A)1853 1769 y Fe(n)p Fj(\000)p Fe(m)2017 1746 y Fs(f)10 b Fx(\()p Fs(H)2183 1760 y Fr(T)g Fo(+)p Fr(m)2375 1746 y Fq(\016)21 b Fs(:)15 b(:)g(:)21 b Fq(\016)g Fs(H)2709 1760 y Fr(T)10 b Fo(+1)2868 1746 y Fx(\()p Fs(x)p Fx(\)\)\))1497 1883 y(=)83 b Fy(P)p Fx(\()p Fs(A)1825 1897 y Fr(n)p Fp(\000)p Fr(m)1990 1883 y Fx(\))15 b Fy(E)p Fx(\()p Fs(f)10 b Fx(\()p Fs(X)2316 1846 y Fr(x)2309 1906 y(m)2376 1883 y Fx(\)\))15 b Fs(;)236 2088 y Fx(where)35 b(the)h(inequalit)m(y)d(is)i(a)h(consequence)g(of)f Fs(f)45 b Fx(b)s(eing)34 b(increasing,)h(while)f(the)h(\014nal)f (equal-)236 2201 y(it)m(y)43 b(uses)e(the)i(fact)g(that)g(\()p Fs(H)1269 2215 y Fr(T)10 b Fo(+1)1414 2201 y Fs(;)15 b(:)g(:)g(:)i(;)e(H)1692 2215 y Fr(T)10 b Fo(+)p Fr(m)1864 2201 y Fx(\))43 b(is)e(indep)s(enden)m(t)f(of)j Fs(T)55 b Fx(and)42 b(distributed)d(as)236 2313 y(\()p Fs(H)347 2327 y Fo(1)387 2313 y Fs(;)15 b(:)g(:)g(:)h(;)f(H)664 2327 y Fr(m)731 2313 y Fx(\).)41 b(But,)31 b Fs(f)39 b Fx(is)30 b(nonnegativ)m(e)h(and)e(th)m(us)353 2437 y Fi(X)488 2553 y Fo(0)p Fp(\024)p Fr(m)p Fp(\024)p Fr(n)772 2518 y Fs(f)10 b Fx(\()p Fs(X)944 2480 y Fo(0)937 2540 y Fr(m)1003 2518 y Fx(\))41 b Fq(\025)1190 2437 y Fi(X)1325 2554 y Fo(0)p Fp(\024)p Fr(m;)12 b(T)e Fo(+)p Fr(m)p Fp(\024)p 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Fy(P)p Fx(\()p Fs(A)2343 3121 y Fr(n)p Fp(\000)p Fr(m)2508 3107 y Fx(\))15 b Fy(E)p Fx(\()p Fs(f)10 b Fx(\()p Fs(X)2834 3070 y Fr(x)2827 3130 y(m)2894 3107 y Fx(\)\))630 3276 y(=)784 3195 y Fi(X)919 3311 y Fo(0)p Fp(\024)p Fr(m)p Fp(\024)p Fr(n)1203 3276 y Fx(\(1)21 b Fq(\000)e Fy(P)p Fx(\()p Fs(A)1568 3290 y Fr(n)p Fp(\000)p Fr(m)1734 3276 y Fx(\)\))c(\()p Fy(E)p Fx(\()p Fs(f)10 b Fx(\()p Fs(X)2130 3239 y Fr(x)2123 3299 y(m)2190 3276 y Fx(\))21 b Fq(\000)f Fs(f)10 b Fx(\()p Fs(X)2509 3239 y Fo(0)2502 3299 y Fr(m)2568 3276 y Fx(\)\))21 b(+)f Fy(E)p Fx(\()p Fs(f)10 b Fx(\()p Fs(X)3026 3239 y Fo(0)3019 3299 y Fr(m)3086 3276 y Fx(\)\)\))15 b Fs(;)236 3490 y Fx(whic)m(h)29 b(b)m(y)i(\(0.1\))h(implies)1112 3614 y Fi(X)1247 3730 y Fo(0)p Fp(\024)p Fr(m)p Fp(\024)p Fr(n)1530 3695 y Fy(P)p Fx(\()p Fs(T)39 b Fq(\024)25 b Fs(n)20 b Fq(\000)g Fs(m)p Fx(\))15 b Fy(E)p Fx(\()p Fs(f)10 b Fx(\()p Fs(X)2396 3657 y Fr(x)2389 3717 y(m)2456 3695 y Fx(\))20 b Fq(\000)g Fs(f)10 b Fx(\()p Fs(X)2774 3657 y Fo(0)2767 3717 y Fr(m)2834 3695 y Fx(\)\))958 3863 y Fq(\024)1112 3782 y Fi(X)1247 3898 y Fo(0)p Fp(\024)p Fr(m)p Fp(\024)p Fr(n)1530 3863 y Fy(P)p Fx(\()p Fs(T)39 b(>)25 b(n)20 b Fq(\000)g Fs(m)p Fx(\))15 b Fy(E)p Fx(\()p Fs(f)10 b Fx(\()p Fs(X)2396 3826 y Fo(0)2389 3886 y Fr(m)2456 3863 y Fx(\)\))958 4023 y Fq(\024)83 b Fy(E)p Fx(\()p Fs(T)13 b Fx(\))i Fy(E)p Fx(\()p Fs(f)10 b Fx(\()p Fs(X)1608 3986 y Fo(0)1601 4046 y Fr(n)1648 4023 y Fx(\)\))15 b Fs(:)236 4227 y Fx(Since)30 b Fy(P)p Fx(\()p Fs(T)38 b Fq(\024)25 b Fs(n)20 b Fq(\000)g Fs(m)p Fx(\))25 b Fq(")h Fx(1)31 b(for)f Fs(n)25 b Fq(!)g(1)p Fx(,)31 b(the)f(assertion)g (follo)m(ws.)941 b Ff(\003)377 4412 y Fx(Apart)41 b(from)g(arising)e (measurabilit)m(y)g(problems)g(the)j(preceding)e(result)f(can)j(b)s(e)e (easily)236 4525 y(extended)31 b(to)g(a)f(larger)h(class)f(of)g (functions:)377 4721 y Fg(\(1.5\))46 b(Pr)n(oposition)67 b Fk(L)-5 b(et)29 b(the)f(system)g Fx(\()p Fs(E)5 b(;)15 b(\027)6 b Fx(\))27 b Fk(b)-5 b(e)28 b(irr)-5 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y(surable)k(functions)g Fs(f)987 269 y Fr(i)1041 255 y Fq(\025)c Fx(0)31 b(suc)m(h)f(that)h Fs(f)k Fx(=)25 b Fs(f)1836 269 y Fo(1)1896 255 y Fq(\000)20 b Fs(f)2032 269 y Fo(2)2071 255 y Fx(.)41 b(Since)30 b(the)g(v)-5 b(ariables)30 b Fs(X)2991 222 y Fr(x)2984 278 y(n)3060 255 y Fx(:)c(\012)g Fq(!)f Fs(E)36 b Fx(are)236 368 y(univ)m(ersally)29 b(measurable)h(as)i(w)m(ell,)e(pro)m(vided)g (the)i(underlying)c(probabilit)m(y)h(space)i(\(\012)p Fs(;)15 b Fm(A)p Fs(;)g Fy(P)p Fx(\))236 481 y(is)35 b(assumed)h(to)h(b)s(e)e(complete,)k(the)d(pro)s(of)g(of)g(\(1.4\))i(w) m(orks)e(for)g Fs(f)2559 495 y Fr(i)2587 481 y Fx(,)i(to)s(o.)59 b(In)35 b(view)h(of)g(\(1.2b\))236 594 y(this)29 b(pro)m(v)m(es)j (assertion)e(\(a\),)h(whic)m(h)e(b)m(y)i(F)-8 b(ubini)28 b(implies)f(assertion)j(\(b\).)722 b Ff(\003)377 778 y Fx(F)-8 b(or)30 b(some)f(applications,)f(where)g(summabilit)m(y)e(of) j(the)g(di\013erences)f(ma)m(y)h(b)s(e)f(replaced)g(b)m(y)236 891 y(con)m(v)m(ergence)33 b(to)e(zero,)h(the)e(class)g(of)h (admissible)c(functions)i(can)i(b)s(e)e(enlarged)h(once)h(more:)377 1087 y Fg(\(1.6\))44 b(Pr)n(oposition)65 b Fk(L)-5 b(et)27 b(the)g(system)g Fx(\()p Fs(E)5 b(;)15 b(\027)6 b Fx(\))26 b Fk(b)-5 b(e)27 b(irr)-5 b(e)g(ducible)27 b(and)h(supp)-5 b(ose)28 b Fs(f)34 b Fq(2)25 b(R)p Fx(\()p Fs(E)5 b Fx(\))p Fk(.)236 1200 y(Then)33 b(for)g(arbitr)-5 b(ary)35 b(initial)e(law)1359 1365 y Fs(f)10 b Fx(\()p Fs(X)1524 1379 y Fr(n)1571 1365 y Fx(\))21 b Fq(\000)f Fs(f)10 b Fx(\()p Fs(X)1890 1328 y Fo(0)1883 1388 y Fr(n)1930 1365 y Fx(\))40 b Fq(!)h Fx(0)58 b Fk(a.s.)16 b Fs(:)377 1602 y Fg(Pr)n(oof.)61 b Fx(This)29 b(is)g(immediate)h(from)g(\(1.5b\))i(and)d(the)i (de\014nition)d(of)i Fq(R)p Fx(\()p Fs(E)5 b Fx(\).)410 b Ff(\003)377 1786 y Fx(It)25 b(has)g(to)g(b)s(e)f(emphasized)g(that,)j (ev)m(en)e(under)e(a)j(total)f(ordering,)g(the)g(last)g(result)e(ma)m (y)j(fail)236 1899 y(for)i(functions)f Fs(f)34 b Fq(2)25 b(C)5 b Fx(\()p Fs(E)g Fx(\).)41 b(T)-8 b(o)29 b(construct)g(a)f (nondegenerate)h(\(i.e.)40 b(recurren)m(t\))29 b(coun)m(terexam-)236 2012 y(ple)g(consider)f(the)i(follo)m(wing)d(system:)41 b(on)29 b Fs(E)i Fx(=)25 b Fy(R)2007 2026 y Fo(+)2095 2012 y Fx(let)30 b(the)f(supp)s(ort)f Fq(N)42 b Fx(of)30 b Fs(\027)35 b Fx(consist)29 b(of)g(the)236 2125 y(fractional)h(linear) f(mappings)1045 2290 y Fs(h)1097 2304 y Fo(1)1162 2290 y Fx(:)c Fs(x)g Fq(7!)h Fs(x)20 b Fx(+)g(1)83 b(and)g Fs(h)1979 2304 y Fo(2)2044 2290 y Fx(:)25 b Fs(x)g Fq(7!)h Fs(x)15 b(=)g Fx(\()p Fs(x)21 b Fx(+)f(1\))15 b Fs(:)236 2455 y Fx(Starting)30 b(with)f Fs(n)852 2469 y Fo(0)916 2455 y Fx(:=)d(0)k(c)m(ho)s(ose)i(a)e(sequence)h(\()p Fs(n)1944 2470 y Fr(k)1987 2455 y Fs(;)15 b(k)29 b Fq(\025)c Fx(0\))31 b(suc)m(h)f(that)746 2621 y(lim)15 b(sup)796 2698 y Fr(k)r Fp(!1)1055 2621 y Fy(P)p Fx(\()p Fs(A)1229 2636 y Fr(k)1273 2621 y Fx(\))25 b(=)g(1)84 b(with)d Fs(A)1885 2636 y Fr(k)1954 2621 y Fx(:=)2075 2540 y Fi([)2182 2656 y Fr(n)2225 2668 y Fe(k)q Fj(\000)p Fd(1)2341 2656 y Fr()g Fx(1\))31 b(yields)1052 899 y Fs(A)1120 862 y Fo(0)1185 899 y Fx(=)25 b(lim)15 b(sup)1328 967 y Fr(n)p Fp(!1)1575 899 y Fq(f)p Fs(H)1696 913 y Fr(n)1763 899 y Fq(\016)20 b Fs(:)15 b(:)g(:)22 b Fq(\016)e Fs(H)2096 913 y Fo(2)2151 899 y Fx(\(0\))26 b Fq(2)f Fs(B)5 b Fq(g)58 b Fx(a.s.)16 b Fs(:)236 1142 y Fx(Rep)s(eating)31 b(the)g(argumen)m(t)h(sho)m(ws)f Fs(A)1565 1109 y Fo(0)1636 1142 y Fx(to)h(b)s(e)e(con)m(tained)i(in)d (the)j(completed)f(tail)f Fs(\033)s Fq(\000)p Fx(\014eld)g(of)236 1255 y(\()p Fs(H)347 1269 y Fr(n)394 1255 y Fs(;)15 b(n)25 b Fq(2)g Fy(N)p Fx(\),)31 b(and)f(the)g(assertion)g(follo)m(ws.)1693 b Ff(\003)377 1439 y Fx(The)30 b(preceding)g(result)f(justi\014es:)377 1635 y Fg(\(2.2\))59 b(Definition)96 b Fx(Let)38 b(the)g(system)g(\()p Fs(E)5 b(;)15 b(\027)6 b Fx(\))38 b(b)s(e)f(irreducible.)59 b(Then)36 b Fs(B)42 b Fq(2)37 b Fm(V)p Fx(\()p Fs(E)5 b Fx(\))38 b(is)236 1748 y(called)30 b(\\)15 b Fk(r)-5 b(e)g(curr)g(ent)17 b Fx(",)31 b(if)832 1953 y Fy(P)903 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Fs(>)f Fx(0)15 b Fs(:)377 3138 y Fg(Pr)n(oof.)61 b Fx(According)31 b(to)g(\(1.6\))1000 3333 y(1)1045 3347 y Fr(B)1106 3333 y Fx(\()p Fs(X)1223 3296 y Fr(x)1216 3356 y(n)1268 3333 y Fx(\))20 b Fq(\000)g Fx(1)1459 3347 y Fr(B)1520 3333 y Fx(\()p Fs(X)1637 3296 y Fo(0)1630 3356 y Fr(n)1678 3333 y Fx(\))26 b Fq(!)f Fx(0)58 b(a.s.)126 b(for)30 b(all)57 b Fs(x)25 b Fq(2)g Fs(A)15 b(:)236 3528 y Fx(Since)30 b Fs(A)g Fx(is)f(\014nite)h(and)g Fs(B)k Fx(is)c(recurren)m(t,)g(this)g (implies)653 3724 y(lim)15 b(sup)700 3791 y Fr(n)p Fp(!1)946 3724 y Fq(f)p Fs(X)1073 3686 y Fr(x)1066 3746 y(n)1143 3724 y Fq(2)25 b Fs(B)46 b Fx(for)30 b(all)41 b Fs(x)25 b Fq(2)g Fs(A)p Fq(g)h Fx(=)e(lim)15 b(sup)2066 3791 y Fr(n)p Fp(!1)2312 3724 y Fq(f)p Fs(X)2439 3686 y Fo(0)2432 3746 y Fr(n)2505 3724 y Fq(2)25 b Fs(B)5 b Fq(g)25 b Fx(=)g(\012)58 b(a.s.)16 b Fs(:)236 3953 y Fx(Therefore)30 b(the)h(probabilities)c(in)i(question)h(ev)m(en)h(sum)e(up)g(to)i Fq(1)p Fx(.)922 b Ff(\003)377 4137 y Fx(It)34 b(is)f(immediate)f(from)h (\(1.2\))i(that)g(eac)m(h)f(increasing)e(set)i Fs(B)h Fq(6)p Fx(=)c Fq(;)i Fx(is)g(recurren)m(t.)50 b(F)-8 b(or)34 b(de-)236 4250 y(creasing)c(sets)h(the)g(follo)m(wing)e (criterion)g(is)g(a)m(v)-5 b(ailable:)377 4446 y Fg(\(2.4\))59 b(Pr)n(oposition)91 b Fk(L)-5 b(et)40 b(the)f(system)h Fx(\()p Fs(E)5 b(;)15 b(\027)6 b Fx(\))39 b Fk(b)-5 b(e)39 b(irr)-5 b(e)g(ducible.)62 b(Then)40 b(for)g(arbitr)-5 b(ary)236 4559 y(initial)33 b(law)h(a)f(set)g Fs(B)c Fq(2)c Fm(B)1151 4518 y Fj(#)1188 4559 y Fx(\()p Fs(E)5 b Fx(\))33 b Fk(is)g(r)-5 b(e)g(curr)g(ent)34 b(if)e(and)i(only)f(if) 2449 4495 y Fi(P)2551 4594 y Fr(n)p Fp(\025)p Fo(0)2718 4559 y Fy(P)p Fx(\()p Fs(X)2899 4573 y Fr(n)2972 4559 y Fq(2)25 b Fs(B)5 b Fx(\))25 b(=)g Fq(1)p Fk(.)377 4765 y Fg(Pr)n(oof.)80 b Fx(Due)36 b(to)h(\(2.1\),)j(the)c(condition)f(is)g (certainly)g(necessary)-8 b(.)59 b(T)-8 b(o)36 b(pro)m(v)m(e)h(its)f (su\016-)236 4878 y(ciency)-8 b(,)40 b(let)e Fs(B)k Fx(b)s(e)c 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Fx(,)29 b(and)g(it)236 1870 y(follo)m(ws)h(similarly)d (that)k(the)f(pro)s(cess)g(is)g(transien)m(t.)377 2024 y(It)g(should)d(b)s(e)h(noted)h(that)h({)f(in)f(spite)g(of)i(the)f(con) m(trasting)h(asymptotic)f(b)s(eha)m(viour)f({)h(the)236 2137 y(v)-5 b(ariables)38 b Fs(U)684 2151 y Fr(n)771 2137 y Fx(b)s(eha)m(v)m(e)i(similarly)c(in)i(b)s(oth)h(cases)h(as)g (far)f(as)h(it)f(concerns)g(the)h(existence)g(of)236 2250 y(momen)m(ts,)34 b(due)d(to)j Fs(f)983 2264 y Fo(1)1022 2250 y Fx(\()p Fs(x)p Fx(\))29 b Fq(\024)f Fs(f)1317 2264 y Fo(2)1356 2250 y Fx(\()p Fs(x)p Fx(\))h Fq(\024)f Fx(2)p Fs(f)1696 2264 y Fo(1)1736 2250 y Fx(\()p Fs(x)p Fx(\))33 b(for)f Fs(x)c Fq(\025)g Fx(1)33 b(\(for)g(a)f(con)m(tin)m (uation)h(see)g(Sections)236 2363 y(3,)e(4,)g(and)f(6\).)377 2476 y(In)38 b(general,)j(as)d(in)f(discrete)h(Mark)m(o)m(v)i(c)m(hain) e(theory)-8 b(,)41 b(it)d(ma)m(y)h(demand)f(some)g(e\013ort)h(to)236 2589 y(decide)34 b(whether)g(a)h(system)g(\()p Fs(E)5 b(;)15 b(\027)6 b Fx(\))33 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b(treated)h(within)c(the)k(class)e Fl(M)p Fx(\()p Fs(E)5 b Fx(\))36 b(of)f(lo)s(cally)f(\014nite)g(measures.)54 b(The)35 b(easy)h(task)f(here)g(is)236 3533 y(existence,)i(where)e (general)g(results)f(b)m(y)h(F)-8 b(oguel)36 b([17])g(apply)-8 b(.)55 b(Since)34 b(the)h(argumen)m(ts)g(can)h(b)s(e)236 3646 y(simpli\014ed,)27 b(due)j(to)h(monotonicit)m(y)-8 b(,)31 b(the)g(pro)s(of)e(of)i(the)f(follo)m(wing)f(assertion)h(is)g (outlined:)377 3842 y Fg(\(3.1\))51 b(Pr)n(oposition)78 b Fk(L)-5 b(et)32 b(the)h(system)h Fx(\()p Fs(E)5 b(;)15 b(\027)6 b Fx(\))26 b Fk(b)-5 b(e)33 b(irr)-5 b(e)g(ducible)33 b(and)h(supp)-5 b(ose)963 4040 y Fx(0)26 b Fq(\024)f Fs(g)k Fq(2)c(K)1358 3994 y Fj(#)1394 4040 y Fx(\()p Fs(E)5 b Fx(\))84 b Fk(with)1883 3959 y Fi(X)2019 4075 y Fr(n)p Fp(\025)p Fo(0)2185 4040 y Fy(E)p Fx(\()p Fs(g)s Fx(\()p Fs(X)2452 4003 y Fo(0)2445 4063 y Fr(n)2493 4040 y Fx(\)\))26 b(=)f Fq(1)15 b Fs(:)236 4248 y Fk(Then)33 b(the)g(me)-5 b(asur)g(es)34 b Fs(\045)1060 4262 y Fr(n)1133 4248 y 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5504 y Fs(\026)p Fx(\(1)1688 5518 y Fr(K)1748 5530 y Fe(l)1777 5504 y Fs(f)10 b Fx(\))124 b(for)58 b(0)26 b Fq(\024)f Fs(f)34 b Fq(2)25 b(B)2553 5458 y Fj(#)2588 5504 y Fx(\()p Fs(E)5 b Fx(\))15 b Fs(:)1844 5753 y Fx(15)p eop %%Page: 16 16 16 15 bop 236 154 a Fx(Inserting)32 b Fs(f)40 b Fx(=)30 b(1)852 168 y Fr(K)912 176 y Fe(m)975 154 y Fs(;)15 b(m)31 b Fq(6)p Fx(=)f Fs(l)r(;)k Fx(pro)m(v)m(es)g Fs(\015)1646 169 y Fr(l)1706 154 y Fx(to)g(b)s(e)f(a)h(constan)m(t)g Fs(\015)39 b Fx(indep)s(enden)m(t)31 b(of)j Fs(l)h Fx(and)e(th)m(us,) 236 267 y(with)c Fq(K)512 281 y Fo(0)582 267 y Fx(as)i(de\014ned)e(in)g (\(10.6a\),)1168 463 y Fs(\026)1223 425 y Fp(0)1246 463 y Fx(\(1)1326 477 y Fr(K)1386 489 y Fe(l)1415 463 y Fs(f)10 b Fx(\))24 b(=)h Fs(\015)c(\026)p Fx(\(1)1828 477 y Fr(K)1888 489 y Fe(l)1916 463 y Fs(f)10 b Fx(\))124 b(for)58 b Fs(f)35 b Fq(2)25 b(K)2532 477 y Fo(0)2586 463 y Fs(:)236 658 y Fx(This)37 b(equation)i(extends)g(to)h Fs(f)49 b Fq(2)40 b(K)q Fx(\()p Fs(E)5 b Fx(\),)43 b(b)s(ecause)c(the)g (uniform)e(appro)m(ximation)h(of)h Fs(f)48 b Fx(b)m(y)236 771 y(a)39 b(sequence)f(\()p Fs(f)784 786 y Fr(k)827 771 y Fs(;)30 b(k)42 b Fq(2)37 b Fy(N)p Fx(\))i(from)e Fq(K)1516 785 y Fo(0)1594 771 y Fx(according)h(to)h(\(10.6a\))i(can)d (b)s(e)g(carried)f(out)i(with)d(all)236 884 y(functions)26 b(v)-5 b(anishing)24 b(outside)i(a)h(\014xed)f(compact)i(set)f (\(otherwise)g(m)m(ultiply)d(b)m(y)j(some)g Fs(g)3306 899 y Fr(l)3359 884 y Fx(from)236 997 y(\(10.7\)\).)44 b(The)29 b(assertion)h(follo)m(ws)g(for)g Fs(l)d Fq(!)e(1)p Fx(.)1613 b Ff(\003)377 1181 y Fx(As)37 b(usual)f(the)h(actually)g (one-dimensional)e(family)g(of)i(in)m(v)-5 b(arian)m(t)37 b(measures)g Fs(\026)f Fq(2)g Fl(M)p Fx(\()p Fs(E)5 b Fx(\))236 1294 y(will)28 b(b)s(e)i(brie\015y)e(called)i Fk(the)j(invariant)h(me)-5 b(asur)g(e)31 b Fx(in)e(the)i(sequel.)377 1407 y(It)f(has)e(to)i(b)s(e)f(emphasized)f(that,)i(ev)m(en)f(under)f (a)h(total)h(ordering,)f(the)g(uniqueness)e(state-)236 1520 y(men)m(t)38 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b(With)48 b Fs(\026)h Fx(as)h(initial)c (la)m(w)j(the)h(series)236 2259 y Fi(P)339 2358 y Fr(n)p Fp(\025)p Fo(0)505 2323 y Fy(P)p Fx(\()p Fs(X)686 2337 y Fr(n)780 2323 y Fq(\024)c Fs(y)s Fx(\))d(div)m(erges)g(for)f(all)g Fs(y)49 b(>)d Fx(0,)g(and)c(th)m(us)h(\()p Fs(E)5 b(;)15 b(\027)6 b Fx(\))47 b(is)42 b(recurren)m(t)g(b)m(y)h(\(2.6\).)236 2436 y(On)30 b(the)g(other)h(hand,)e(it)h(is)g(not)g(hard)g(to)h(c)m (hec)m(k)h(that)1119 2632 y Fs(\026)1174 2594 y Fp(0)1222 2632 y Fx(:=)1343 2551 y Fi(X)1478 2668 y Fr(x)p Fp(2)p Fr(D)1658 2632 y Fs(x)15 b(")1767 2646 y Fr(x)1895 2632 y Fx(with)81 b Fs(D)28 b Fx(:=)e Fs(E)40 b Fq(\\)35 b Fy(Q)236 2827 y Fx(de\014nes)j(another)g(in)m(v)-5 b(arian)m(t)38 b(measure,)i(whic)m(h,)f(ho)m(w)m(ev)m(er,)j(is)37 b Fs(\033)s Fq(\000)p Fx(\014nite)g(only)-8 b(.)64 b(Inciden)m(tally)-8 b(,)236 2940 y(the)28 b(existence)g(of)g(b)s(oth)f(a)h(\014nite)e(and)h (an)h(in\014nite,)e(but)h Fs(\033)s Fq(\000)p Fx(\014nite,)g(in)m(v)-5 b(arian)m(t)27 b(measure)g(in)f(this)236 3053 y(example)e(pro)m(v)m(es) h(the)g(Do)s(eblin-Harris)c(theory)k(of)f(recurren)m(t)g(Mark)m(o)m(v)i (c)m(hains)e(on)g(an)g(abstract)236 3166 y(state)32 b(space)f(to)g(b)s (e)f(inadequate)g(in)f(the)h(presen)m(t)h(setting.)377 3279 y(Finally)-8 b(,)30 b(it)h(has)f(to)i(b)s(e)e(men)m(tioned)h (that,)h(again)f(ev)m(en)g(under)f(a)h(total)h(ordering,)e(there)h(is) 236 3392 y(no)k(con)m(v)m(erse)i(of)e(\(3.4\),)k(i.e.)55 b(there)35 b(are)h(transien)m(t)f(systems)g(with)f(a)h(unique)f(non)m (trivial)f(and)236 3505 y(lo)s(cally)h(\014nite)g(in)m(v)-5 b(arian)m(t)34 b(measure.)55 b(F)-8 b(or)35 b(an)g(example)g(consider)f (the)h(transien)m(t)g(case)h(\(2\))g(of)236 3618 y(the)e(exc)m(hange)g (pro)s(cess)f(from)g(Section)g(2.)50 b(It)33 b(is)g(easily)f(seen)h (that)h(a)g(measure)f Fs(\026)d Fq(2)f Fl(M)p Fx(\()p Fs(E)5 b Fx(\))34 b(is)236 3731 y(in)m(v)-5 b(arian)m(t)30 b(if)f(and)h(only)f(if)h(the)g(function)f(\010\()p Fs(y)s Fx(\))d(:=)f Fs(\026)p Fx(\([0)p 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Fx(+)g Fs(n)p 2875 5115 246 4 v 2891 5198 a(y)j Fx(+)d Fs(n)3131 5136 y Fx(\))3166 5098 y Fo(2)3221 5136 y Fs(;)236 5350 y Fx(whic)m(h)29 b(for)h Fs(n)25 b Fq(!)g(1)31 b Fx(lead)f(to)h(\010\() p Fs(y)s Fx(\))25 b(=)g(\010\(1\))15 b Fs(y)1805 5317 y Fo(2)1876 5350 y Fx(and)29 b(th)m(us)h(to)i(the)e(asserted)h (uniqueness.)1844 5753 y(16)p eop %%Page: 17 17 17 16 bop 377 154 a Fy(4.)64 b(Ergo)s(dic)35 b(theorems.)55 b Fx(The)29 b(\014rst)g(theorem)h(in)e(this)h(section)g(requires)f (some)j(prepa-)236 267 y(ration,)f(concerning)g(again)h(asymptotic)f (indep)s(endence)e(of)j(the)g(initial)c(la)m(w:)377 463 y Fg(\(4.1\))51 b(Pr)n(oposition)78 b Fk(L)-5 b(et)33 b(the)g(system)h Fx(\()p Fs(E)5 b(;)15 b(\027)6 b Fx(\))26 b Fk(b)-5 b(e)33 b(r)-5 b(e)g(curr)g(ent)34 b(with)g(invariant)g(me)-5 b(asur)g(e)236 576 y Fs(\026)33 b Fk(and)g(supp)-5 b(ose)34 b Fx(0)26 b Fq(\024)f Fs(f)35 b Fq(2)24 b(B)1227 534 y Fj(#)1263 576 y Fx(\()p Fs(E)5 b Fx(\))33 b Fk(with)h Fs(\026f)g(>)25 b Fx(0)p Fk(.)42 b(Then)33 b(for)g(arbitr)-5 b(ary)35 b(initial)e(law)880 680 y Fi(X)1015 796 y Fo(0)p Fp(\024)p Fr(m)g Fx(0,)31 b(then)f(\(3.3\))i(implies)1269 1814 y Fi(X)1404 1930 y Fo(0)p Fp(\024)p Fr(m)c Fx(0)k(ma)m(y)g(b)s (e)f(assumed)236 3796 y(in)h(the)i(sequel.)236 3909 y(\(2\))59 b(Since)21 b(the)h(uniform)d(appro)m(ximation)i(of)h Fs(f)31 b Fx(b)m(y)21 b(a)h(sequence)h(\()p Fs(f)2521 3924 y Fr(k)2563 3909 y Fs(;)31 b(k)d Fq(2)d Fy(N)p Fx(\))d(from)f Fq(K)3194 3923 y Fo(0)3255 3909 y Fx(accord-)236 4022 y(ing)28 b(to)i(\(10.6a\))h(can)e(b)s(e)f(carried)g(out)h(from)f(b)s (elo)m(w)g(and)g(ab)s(o)m(v)m(e)i(with)e(all)f(functions)h(v)-5 b(anishing)236 4134 y(outside)31 b(a)i(\014xed)e(compact)i(set)g (\(otherwise)e(m)m(ultiply)e(b)m(y)j(some)g Fs(g)2565 4149 y Fr(l)2624 4134 y Fx(from)f(\(10.7\)\),)k Fs(f)41 b Fx(ma)m(y)33 b(b)s(e)236 4247 y(restricted)d(to)h Fq(K)821 4261 y Fo(0)861 4247 y Fx(.)236 4360 y(\(3\))59 b(Therefore)31 b(it)f(is)g(su\016cien)m(t)g(to)i(consider)e(the)h(case)h(0)26 b Fq(\024)g Fs(f)35 b Fq(2)26 b(K)2610 4319 y Fj(#)2646 4360 y Fx(\()p Fs(E)5 b Fx(\),)32 b(where)f(in)e(addition)236 4473 y Fs(\026f)35 b(>)25 b Fx(0)30 b(ma)m(y)h(b)s(e)f(assumed)g (\(otherwise)g(replace)g Fs(f)40 b Fx(b)m(y)30 b Fs(f)f Fx(+)20 b Fs(g)s Fx(\).)236 4586 y(\(4\))59 b(Th)m(us)34 b Fs(f)43 b Fx(and)34 b Fs(g)k Fx(b)s(oth)c(satisfy)g(the)g (assumptions)f(of)h(\(4.1\),)k(whic)m(h)33 b(justi\014es)g(\014nally)f (the)236 4699 y(restriction)e(to)h(the)f(initial)e(la)m(w)i Fs(")1408 4713 y Fo(0)1448 4699 y Fx(.)377 4842 y(2.)40 b(No)m(w)28 b(the)f(relativ)m(e)g(compactness)h(from)e(\(3.1a\))k(and)c (the)h(uniqueness)e(of)i(the)g(in)m(v)-5 b(arian)m(t)236 4955 y(measure)31 b(yield)d(a)j(constan)m(t)h Fs(\015)j Fx(suc)m(h)30 b(that)838 5054 y Fi(X)973 5170 y Fo(0)p Fp(\024)p Fr(m)g Fx(0.)42 b(Then)29 b(pro)s(ceed)h(as)h (follo)m(ws:)236 1768 y(\(1\))105 b(F)-8 b(or)39 b Fs(f)50 b Fq(2)39 b(V)7 b Fx(\()p Fs(E)e Fx(\))41 b(c)m(ho)s(ose)f(a)f (represen)m(tation)h Fs(f)49 b Fx(=)39 b Fs(f)2302 1782 y Fo(1)2367 1768 y Fq(\000)26 b Fs(f)2509 1782 y Fo(2)2587 1768 y Fx(according)39 b(to)h(\(10.8\))i(and)236 1881 y(a)i(constan)m(t)h Fs(\015)52 b Fq(\025)47 b Fs(f)967 1895 y Fo(1)1006 1881 y Fx(\(0\))30 b Fq(_)f Fs(f)1286 1895 y Fo(2)1325 1881 y Fx(\(0\),)48 b(so)c(that)g Fs(f)56 b Fx(=)47 b Fs(f)2122 1848 y Fp(0)2112 1904 y Fo(1)2180 1881 y Fq(\000)29 b Fs(f)2335 1848 y Fp(0)2325 1904 y Fo(2)2407 1881 y Fx(with)42 b(univ)m(ersally)f(measurable)236 1994 y(decreasing)46 b(functions)e Fs(f)1154 1961 y Fp(0)1144 2017 y Fo(1)1234 1994 y Fx(:=)52 b Fs(\015)35 b Fq(\000)c Fs(f)1611 2008 y Fo(2)1701 1994 y Fq(\025)50 b Fx(0)d(and)e Fs(f)2161 1961 y Fp(0)2151 2017 y Fo(2)2241 1994 y Fx(:=)51 b Fs(\015)36 b Fq(\000)30 b Fs(f)2617 2008 y Fo(1)2707 1994 y Fq(\025)51 b Fx(0.)88 b(If)45 b(in)g(addition)236 2107 y(supp)d Fs(f)522 2074 y Fp(0)512 2131 y Fr(i)593 2107 y Fq(\032)47 b Fs(K)7 b Fx(,)47 b(as)d(can)g(b)s(e)f(ac)m(hiev)m (ed)i(b)m(y)f(m)m(ultip)m(ying)d(with)i(1)2565 2121 y 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y(of)c(the)f(exc)m(hange)i(pro)s (cess)e(from)g(Section)g(2.)41 b(Here)31 b Fy(P)p Fx(\()p Fs(X)2249 2883 y Fo(0)2242 2939 y Fr(n)2316 2916 y Fq(\024)25 b Fs(y)s Fx(\))g(=)2709 2855 y Fs(y)p 2626 2895 214 4 v 2626 2979 a(y)e Fx(+)d Fs(n)2880 2916 y Fx(implies)866 3194 y Fy(E)p Fx(\()p Fs(f)10 b Fx(\()p Fs(X)1142 3157 y Fo(0)1135 3217 y Fr(n)1182 3194 y Fx(\)\))26 b(=)1388 3133 y(1)p 1384 3173 55 4 v 1384 3256 a Fs(n)1479 3079 y Fi(Z)1562 3105 y Fp(1)1525 3268 y Fo(0)1652 3194 y Fs(f)10 b Fx(\()p Fs(y)s Fx(\))1840 3100 y Fi(\020)1903 3133 y Fs(y)p 1899 3173 V 1899 3256 a(n)1984 3194 y Fx(+)20 b(1)2120 3100 y Fi(\021)2170 3123 y Fp(\000)p Fo(2)2279 3194 y Fs(dy)128 b Fx(for)58 b Fs(n)25 b Fq(2)g Fy(N)236 3364 y Fx(and)30 b(th)m(us)435 3560 y Fy(E)p Fx(\()p Fs(f)10 b Fx(\()p Fs(X)711 3522 y Fo(0)704 3582 y Fr(n)751 3560 y Fx(\)\))836 3466 y Fi(.)919 3560 y Fy(E)p Fx(\()p Fs(g)s Fx(\()p Fs(X)1186 3522 y Fo(0)1179 3582 y Fr(n)1227 3560 y Fx(\)\))41 b Fq(!)1470 3445 y Fi(Z)1553 3471 y Fp(1)1516 3633 y Fo(0)1643 3560 y Fs(f)10 b Fx(\()p Fs(y)s Fx(\))15 b Fs(dy)1941 3466 y Fi(.)2023 3445 y(Z)2106 3471 y Fp(1)2069 3633 y Fo(0)2196 3560 y Fs(g)s Fx(\()p Fs(y)s Fx(\))g Fs(dy)129 b Fx(for)58 b Fs(f)5 b(;)30 b(g)f Fq(2)c(K)q Fx(\()p Fy(R)3209 3574 y Fo(+)3269 3560 y Fx(\))15 b Fs(:)236 3779 y Fx(Since)21 b(this)g(con)m(v)m(ergence)k (carries)c(o)m(v)m(er)j(to)e(the)h(corresp)s(onding)d(quotien)m(ts)i (in)f(\(4.2\),)k(the)e(in)m(v)-5 b(ari-)236 3892 y(an)m(t)23 b(measure)f(is)f(simply)f(the)i(Leb)s(esgue)g(measure)g Fs(\025)g Fx(restricted)g(to)g Fy(R)2630 3906 y Fo(+)2690 3892 y Fx(.)38 b(By)22 b(\(4.3\))i(this)d(implies)236 4005 y(that,)35 b(regardless)d(of)i(the)f(initial)d(la)m(w,)k(the)f (pro)s(cess)g(\()p Fs(X)2198 4019 y Fr(n)2246 4005 y Fs(;)d(n)f Fq(\025)h Fx(0\))k(is)e(\\)15 b(equidistributed)d(")33 b(on)236 4118 y Fy(R)314 4132 y Fo(+)374 4118 y Fx(,)d(i.e.)688 4233 y Fi(X)823 4349 y Fo(0)p Fp(\024)p Fr(m)g Fx(0.)39 b(More)27 b(generally)-8 b(,)26 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y(Th)m(us)k(the)g(assertion)g(follo)m(ws)g(from)g Fs(h)p Fx([)p Fs(M)10 b Fx(])26 b(=)f Fq(f)p Fs(x)p Fq(g)31 b Fx(for)f Fs(h)c Fx(=)f Fs(j)5 b Fx(\()p Fs(x)p Fx(\).)377 5323 y(\(b\))31 b(By)g(de\014nition)d(of)i(the)h(top)s(ology)f(in)f Fq(H)q Fx([)p Fs(E)5 b Fx(])31 b(it)f(has)g(to)h(b)s(e)f(sho)m(wn)g (that)1388 5504 y Fq(f)p Fs(h)c Fq(2)f(N)1685 5466 y Fp(\016)1750 5504 y Fx(:)g Fs(h)p Fx([)p Fs(K)7 b Fx(])26 b Fq(\032)f Fs(G)p Fq(g)h(6)p Fx(=)f Fq(;)1844 5753 y Fx(22)p eop %%Page: 23 23 23 22 bop 236 154 a Fx(for)37 b Fs(G)f Fq(2)g Fm(G)p Fx(\()p Fs(E)5 b Fx(\))38 b(con)m(taining)f Fs(x)p Fx(,)h(hence)f (satisfying)f Fs(\026)p Fx(\()p Fs(G)p Fx(\))h Fs(>)f Fx(0,)j(and)d(arbitrary)g Fs(K)43 b Fq(2)35 b Fm(K)p Fx(\()p Fs(E)5 b Fx(\).)236 267 y(Since)32 b Fs(E)37 b Fx(is)32 b(lo)s(cally)f(b)s(ounded,)g Fs(K)39 b Fx(can)33 b(b)s(e)e(co)m(v)m(ered)j(b)m(y)f(a)g(\014nite)e(n)m(um)m(b)s(er)g(of)i (b)s(ounded)d(op)s(en)236 380 y(sets)40 b(and)e(th)m(us)h Fs(K)46 b Fq(\032)39 b Fs(A)1118 339 y Fj(#)1193 380 y Fx(for)g(some)g(\014nite)f(subset)h Fs(A)g Fx(of)g Fs(E)5 b Fx(.)67 b(Since,)40 b(moreo)m(v)m(er,)j(0)d(can)f(b)s(e)236 493 y(included)24 b(in)i Fs(A)h Fx(and)f Fs(G)h Fx(can)g(b)s(e)g(supp)s (osed)d(to)k(b)s(e)e(con)m(v)m(ex)j(b)m(y)d(\(10.4\),)k(it)d(is)f (su\016cien)m(t)g(to)h(pro)m(v)m(e)1108 682 y Fq(f)p Fs(h)f Fq(2)f(N)1405 644 y Fp(\016)1470 682 y Fx(:)g Fs(h)p Fx(\()p Fs(y)s Fx(\))h Fq(2)f Fs(G)42 b Fx(for)30 b(all)41 b Fs(y)28 b Fq(2)d Fs(A)p Fq(g)h(6)p Fx(=)f Fq(;)15 b Fs(:)236 871 y Fx(But)31 b(translated)f(in)m(to)g (probabilities)d(this)i(is)h(implied)d(b)m(y)906 1061 y Fy(P)p Fx(\()p Fs(X)1094 1023 y Fr(y)1087 1083 y(n)1162 1061 y Fq(2)e Fs(G)42 b Fx(for)30 b(all)40 b Fs(y)28 b Fq(2)d Fs(A)p Fx(\))h Fs(>)f Fx(0)125 b(for)30 b(some)59 b Fs(n)25 b Fq(2)f Fy(N)15 b Fs(;)236 1250 y Fx(whic)m(h)29 b(follo)m(ws)h(indeed)f(from)h(\(4.4a\))i(and)e(\(2.3\).)1526 b Ff(\003)377 1434 y Fx(Since)40 b(a)h(totally)f(ordered)g(space)i(is)d 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b(the)i(assumption) e(concerns)i(the)g(union)d(of)j(the)g(sets)236 5391 y Fq(f)p Fs(h)41 b Fq(2)f(H)q Fx([)p Fs(E)5 b Fx(])41 b(:)f Fs(h)p Fx([)p Fs(E)5 b Fx(])41 b Fq(\032)f Fs(K)1184 5406 y Fr(l)1210 5391 y Fq(g)p Fx(,)j(whic)m(h)38 b(due)g(to)i Fs(E)45 b Fx(and)39 b Fs(K)2267 5406 y Fr(l)2332 5391 y Fx(b)s(eing)f(of)h(t)m(yp)s(e)h Fs(K)2989 5405 y Fr(\033)3075 5391 y Fx(and)f Fs(G)3332 5406 y Fr(\016)3370 5391 y Fx(,)j(re-)236 5504 y(sp)s(ectiv)m(ely)-8 b(,)46 b(are)c(again)h(of)f (t)m(yp)s(e)h Fs(G)1510 5519 y Fr(\016)1548 5504 y Fx(.)77 b(This)41 b(settles)i(the)f(question)g(of)g(measurabilit)m(y)f(and)1844 5753 y(23)p eop %%Page: 24 24 24 23 bop 236 154 a Fx(ensures)30 b Fs(\015)g Fx(:=)25 b Fs(\027)6 b Fx(\()p Fs(h)p Fx([)p Fs(E)f Fx(])27 b Fq(\032)e Fs(K)1214 168 y Fr(m)1281 154 y Fx(\))g Fs(>)g Fx(0)31 b(for)f(some)h Fs(m)p Fx(,)f(whic)m(h)f(in)g(view)h(of)g(the)h (estimate)552 341 y Fy(P)p Fx(\()p Fs(X)733 355 y Fr(n)807 341 y Fq(2)25 b Fs(K)970 355 y Fr(m)1036 341 y Fx(\))h Fq(\025)f Fy(P)p Fx(\()p Fs(H)1375 355 y Fr(n)1422 341 y Fx(\()p Fs(x)p Fx(\))h Fq(2)f Fs(K)1733 355 y Fr(m)1841 341 y Fx(for)30 b(all)41 b Fs(x)25 b Fq(2)g Fs(E)5 b Fx(\))26 b(=)f Fs(\015)130 b Fx(for)30 b(all)57 b Fs(n)25 b Fq(2)f Fy(N)236 527 y Fx(pro)m(v)m(es)31 b(the)g(recurrence)f(of)h (\()p Fs(E)5 b(;)15 b(\027)6 b Fx(\))26 b(b)m(y)k(\(2.6\).)43 b(In)m(v)-5 b(ariance)30 b(of)g Fs(\026)p Fx(,)h(moreo)m(v)m(er,)h (implies)1025 747 y Fs(\026)p Fx(\()p Fs(K)1192 761 y Fr(m)1259 747 y Fx(\))25 b(=)1415 632 y Fi(Z)1489 783 y Fr(E)1577 747 y Fs(\027)6 b Fx(\()p Fs(h)p Fx(\()p Fs(x)p Fx(\))27 b Fq(2)d Fs(K)2026 761 y Fr(m)2093 747 y Fx(\))15 b Fs(\026)p Fx(\()p Fs(dx)p Fx(\))27 b Fq(\025)e Fs(\015)20 b(\026)p Fx(\()p Fs(E)5 b Fx(\))236 967 y(and)30 b(th)m(us)g Fs(\026)p Fx(\()p Fs(E)5 b Fx(\))26 b Fs(<)f Fq(1)30 b Fx(as)h(asserted.)377 1110 y(\(b\))h(The)f(maximalit)m(y)f (of)i Fs(x)f Fx(and)g(the)g(monotonicit)m(y)h(of)f Fs(h)d Fq(2)e(H)q Fx([)p Fs(E)5 b Fx(])32 b(pro)m(v)m(e)g Fs(h)p Fx(\(0\))d Fq(\025)e Fs(x)k Fx(and)236 1223 y Fs(h)p Fx([)p Fs(E)5 b Fx(])33 b(=)e Fq(f)p Fs(x)p Fq(g)k Fx(to)g(b)s(e)f (equiv)-5 b(alen)m(t)34 b(statemen)m(ts.)54 b(Since)33 b Fs(\027)2198 1190 y Fr(n)2244 1223 y Fx(\()p Fs(h)p Fx(\(0\))h Fq(\025)d Fs(x)p Fx(\))h Fs(>)g Fx(0)i(for)g(some)h Fs(n)c Fq(2)g Fy(N)p Fx(,)236 1336 y(b)m(y)f(\(a\))i(the)f(system)f(\() p Fs(E)5 b(;)15 b(\027)1165 1303 y Fr(n)1213 1336 y Fx(\))30 b({)h(and)f(th)m(us)g(also)g(\()p Fs(E)5 b(;)15 b(\027)6 b Fx(\))27 b({)j(is)g(p)s(ositiv)m(e)f(recurren)m(t.)410 b Ff(\003)377 1520 y Fx(It)25 b(is)e(a)i(consequence)g(of)g(\(6.3a\))i (and)d(\(6.2\))i(that)f(the)f(examples)g(follo)m(wing)f(\(1.6\))k(and)c (\(3.4\))236 1633 y(b)s(oth)30 b(are)h(ev)m(en)h(p)s(ositiv)m(e)d (recurren)m(t)i(coun)m(terexamples;)g(on)g(the)g(other)f(hand)g(the)h (latter)g(one)236 1746 y(pro)m(v)m(es)g(neither)f(condition)f(in)g (\(6.3\))j(to)f(b)s(e)f(necessary)g(for)h(p)s(ositiv)m(e)e(recurrence.) 377 1859 y(T)-8 b(o)29 b(obtain)f(a)h(necessary)h(and)e(su\016cien)m(t) g(condition)f(ideas)h(from)g(queuing)f(theory)i(will)d(b)s(e)236 1971 y(tak)m(en)32 b(up)d(that)i(can)g(b)s(e)e(traced)i(bac)m(k)h(to)f ([28,)g(29])h(and)e(lead)g(to)h(the)f(follo)m(wing)f(notion:)377 2167 y Fg(\(6.4\))54 b(Definition)87 b Fx(Let)33 b(\()p Fs(H)1483 2181 y Fr(n)1530 2167 y Fs(;)15 b(n)30 b Fq(2)f Fy(N)p Fx(\))k(b)s(e)f(the)h(generating)h(sequence)f(of)g(the)g(system) 236 2280 y(\()p Fs(E)5 b(;)15 b(\027)6 b Fx(\).)42 b(Then)29 b(the)i(random)f(v)-5 b(ariables)1175 2467 y Fs(Y)1248 2429 y Fr(x)1228 2490 y(n)1317 2467 y Fx(:=)25 b Fs(H)1514 2481 y Fo(1)1574 2467 y Fq(\016)20 b Fs(:)15 b(:)g(:)22 b Fq(\016)e Fs(H)1907 2481 y Fr(n)1969 2467 y Fx(\()p Fs(x)p Fx(\))125 b(for)58 b Fs(n)25 b Fq(\025)g Fx(0)236 2654 y(de\014ne)30 b(the)g(\\)16 b Fk(dual)33 b(pr)-5 b(o)g(c)g(ess)18 b Fx(")31 b(b)s(elonging)d(to)j Fs(x)26 b Fq(2)e Fs(E)5 b Fx(.)377 2838 y(Clearly)-8 b(,)48 b(the)c (distributions)d(of)j Fs(X)1644 2805 y Fr(x)1637 2860 y(n)1733 2838 y Fx(and)g Fs(Y)1997 2805 y Fr(x)1977 2860 y(n)2085 2838 y Fx(agree)i(for)e(\014xed)g Fs(n)p Fx(,)j(but)d(\()p Fs(Y)3140 2805 y Fr(x)3120 2860 y(n)3184 2838 y Fs(;)31 b(n)48 b Fq(\025)g Fx(0\))236 2951 y(need)34 b(not)g(b)s(e)f(a)i(Mark)m (o)m(v)g(c)m(hain.)51 b(Moreo)m(v)m(er,)37 b(while)32 b(the)j(join)m(t)e(distribution)d(of)35 b(the)f(pro)s(cess)236 3064 y(\()p Fs(X)353 3031 y Fr(x)346 3086 y(n)398 3064 y Fs(;)c(n)25 b Fq(\025)g Fx(0\))j(dep)s(ends)d(only)h(on)h(the)g(k)m (ernel)f Fs(P)40 b Fx(and)26 b(not)h(on)g(the)g(underlying)d (distribution)f Fs(\027)6 b Fx(,)236 3177 y(this)29 b(fails)g(in)g (general)i(for)f(the)h(pro)s(cess)e(\()p Fs(Y)1744 3144 y Fr(x)1723 3199 y(n)1788 3177 y Fs(;)h(n)25 b Fq(\025)g Fx(0\).)377 3290 y(Considered)f(in)f(the)j(compacti\014cation)f Fs(E)1843 3257 y Fp(\003)1908 3290 y Fx(of)g Fs(E)30 b Fx(all)24 b(dual)g(pro)s(cesses)h(con)m(v)m(erge)i(to)f(a)f(com-)236 3403 y(mon)30 b(limit:)377 3599 y Fg(\(6.5\))50 b(Pr)n(oposition)75 b Fk(If)31 b(the)h(system)g Fx(\()p Fs(E)5 b(;)15 b(\027)6 b Fx(\))27 b Fk(is)k(irr)-5 b(e)g(ducible,)33 b(ther)-5 b(e)32 b(is)f(a)h(r)-5 b(andom)34 b(vari-)236 3711 y(able)f Fs(Y)45 b Fx(:)26 b(\012)f Fq(!)g Fs(E)850 3678 y Fp(\003)922 3711 y Fk(such)33 b(that)236 3898 y Fx(\(a\))1139 b Fs(Y)1563 3861 y Fo(0)1543 3921 y Fr(n)1643 3898 y Fq(")41 b Fs(Y)103 b Fk(p)-5 b(ointwise)16 b Fs(;)236 4052 y Fx(\(b\))918 b Fs(Y)1348 4015 y Fr(x)1328 4075 y(n)1417 4052 y Fq(!)25 b Fs(Y)78 b Fk(a.s.)125 b(for)33 b(al)5 b(l)59 b Fs(x)25 b Fq(2)g Fs(E)c(:)377 4310 y Fg(Pr)n(oof.)61 b Fx(\(a\))29 b(Since)e(the)h(ordered)f(top)s(ological)h(space)g Fs(E)2383 4277 y Fp(\003)2451 4310 y Fx(is)f(sequen)m(tially)g(compact)i(and)236 4423 y(the)34 b(sequence)g(\()p Fs(Y)885 4390 y Fo(0)864 4446 y Fr(n)924 4423 y Fx(\()p Fs(!)s Fx(\))p Fs(;)d(n)g Fq(\025)g Fx(0\))j(is)f(increasing,)h(it)f(con)m(v)m(erges)j(in)c(view) i(of)g(\(10.4\))i(to)e(a)g(limit)236 4536 y Fs(Y)20 b Fx(\()p Fs(!)s Fx(\))37 b(for)f(eac)m(h)i 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Fq(\025)g Fx(0\),)31 b(due)f(to)h Fq(L)p Fx(\()p Fs(X)1330 5098 y Fr(x)1323 5153 y(n)1374 5131 y Fx(\))26 b(=)f Fq(L)p Fx(\()p Fs(Y)1702 5098 y Fr(x)1682 5153 y(n)1746 5131 y Fx(\),)31 b(this)e(implies)967 5317 y(1)1012 5331 y Fr(G)1067 5343 y Fe(k)1110 5317 y Fx(\()p Fs(Y)1218 5280 y Fr(x)1198 5340 y(n)1262 5317 y Fx(\))21 b Fq(\000)f Fx(1)1454 5331 y Fr(G)1509 5343 y Fe(k)1551 5317 y Fx(\()p Fs(Y)1660 5280 y Fo(0)1639 5340 y Fr(n)1699 5317 y Fx(\))26 b Fq(!)f Fx(0)58 b(a.s.)126 b(for)30 b(all)57 b Fs(k)28 b Fq(2)d Fy(N)15 b Fs(:)236 5504 y Fx(Therefore)30 b Fs(Y)721 5471 y Fr(x)701 5527 y(n)765 5504 y Fx(\()p Fs(!)s Fx(\))c Fq(!)f Fs(Y)20 b Fx(\()p Fs(!)s Fx(\))31 b(for)f(almost)h(all)e Fs(!)k Fx(satisfying)c Fs(Y)20 b Fx(\()p Fs(!)s Fx(\))26 b Fq(6)p Fx(=)f Fq(1)p Fx(.)676 b Ff(\003)1844 5753 y Fx(24)p eop %%Page: 25 25 25 24 bop 377 154 a Fx(The)30 b(v)-5 b(ariable)29 b Fs(Y)51 b Fx(ob)s(eys)30 b(a)g(zero-one)i(la)m(w)e(c)m(haracterizing)h(the)g(t) m(yp)s(e)f(of)h(recurrence:)377 350 y Fg(\(6.6\))55 b(Pr)n(oposition)85 b Fk(L)-5 b(et)37 b(the)f(system)h Fx(\()p Fs(E)5 b(;)15 b(\027)6 b Fx(\))33 b Fk(b)-5 b(e)36 b(irr)-5 b(e)g(ducible.)54 b(Then,)37 b(with)g Fs(Y)57 b Fk(b)-5 b(eing)236 463 y(de\014ne)g(d)34 b(by)e Fx(\(6.5\))p Fk(,)236 617 y Fx(\(a\))91 b Fy(P)p Fx(\()p Fs(Y)46 b Fx(=)25 b Fq(1)p Fx(\))h(=)f(0)83 b Fk(if)33 b Fx(\()p Fs(E)5 b(;)15 b(\027)6 b Fx(\))26 b Fk(is)33 b(p)-5 b(ositive)33 b(r)-5 b(e)g(curr)g(ent,)236 772 y Fx(\(b\))90 b Fy(P)p Fx(\()p Fs(Y)46 b Fx(=)25 b Fq(1)p Fx(\))h(=)f(1)83 b Fk(otherwise.)377 968 y Fg(Pr)n(oof.)60 b Fx(\(a\))28 b(Let)e Fs(\026)h Fx(b)s(e)e(the)i(stationary)f (distribution)d(and)j(c)m(ho)s(ose)h(functions)e Fs(g)3214 983 y Fr(l)3241 968 y Fs(;)30 b(l)d Fq(2)e Fy(N)p Fs(;)236 1081 y Fx(according)40 b(to)h(\(10.7\).)71 b(Iden)m(tifying)39 b Fs(g)1615 1096 y Fr(l)1681 1081 y Fx(with)f(its)i(con)m(tin)m(uous)f (extension)h(to)h Fs(E)3103 1048 y Fp(\003)3182 1081 y Fx(yields)d(b)m(y)236 1193 y(\(6.5b\))32 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Fx(\()p Fs(Y)20 b Fx(\)\))26 b(=)f(0)30 b(for)e(all)g Fs(l)g Fq(2)c Fy(N)p Fx(,)30 b(and)e(the)h(assertion)g(follo)m(ws)236 3207 y(again)i(for)f Fs(l)d Fq(!)e(1)p Fx(.)377 3320 y(\(b2\))31 b(If)f(\014nally)e(\()p Fs(E)5 b(;)15 b(\027)6 b Fx(\))27 b(is)i(transien)m(t,)i(then)f(\(6.5a\))i(and)e(\(2.6\))i (imply)428 3524 y Fy(P)p Fx(\()p Fs(Y)46 b Fq(2)25 b Fs(K)7 b Fx(\))26 b(=)53 b(lim)960 3574 y Fr(n)p Fp(!1)1159 3524 y Fy(P)p Fx(\()p Fs(Y)1338 3487 y Fo(0)1318 3547 y Fr(n)1403 3524 y Fq(2)25 b Fs(K)7 b Fx(\))25 b(=)54 b(lim)1729 3574 y Fr(n)p Fp(!1)1928 3524 y Fy(P)p Fx(\()p Fs(X)2116 3487 y Fo(0)2109 3547 y Fr(n)2183 3524 y Fq(2)25 b Fs(K)7 b Fx(\))40 b(=)g(0)125 b(for)58 b Fs(K)32 b Fq(2)25 b Fm(K)3132 3479 y Fj(#)3168 3524 y Fx(\()p Fs(E)5 b Fx(\))15 b Fs(;)236 3750 y Fx(hence)31 b(for)f(all)f Fs(K)j Fq(2)25 b Fm(K)p Fx(\()p Fs(E)5 b Fx(\).)2296 b Ff(\003)377 3935 y Fx(F)-8 b(or)44 b(an)f(application)f(of)h(this)f 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b(is)h(p)s(ositiv)m(e)g(recurren)m(t)h(if)f(and)h(only)f(if) g(the)h(v)-5 b(ariables)27 b Fs(U)2887 4921 y Fr(n)2934 4907 y Fs(;)k(n)25 b Fq(2)f Fy(N)p Fs(;)29 b Fx(ha)m(v)m(e)g(a)236 5020 y(\014nite)h(exp)s(ectation)g(\(for)h(extensions)f(see)h([22]\).) 377 5132 y(The)f(follo)m(wing)f(result)g(is)h(related)g(to)h(a)g(\\)16 b(con)m(traction)31 b(principle)12 b(")31 b(in)e([27]:)377 5328 y Fg(\(6.7\))49 b(Theorem)72 b Fk(If)30 b(the)h(system)g Fx(\()p Fs(E)5 b(;)15 b(\027)6 b Fx(\))26 b Fk(is)k(p)-5 b(ositive)32 b(r)-5 b(e)g(curr)g(ent,)32 b(its)e(stationary)j(distri-) 236 5441 y(bution)g(is)g(given)f(by)g Fq(L)p Fx(\()p Fs(Y)20 b Fx(\))p Fk(,)33 b(with)g Fs(Y)53 b Fk(b)-5 b(eing)32 b(de\014ne)-5 b(d)34 b(by)e Fx(\(6.5\))p Fk(.)1844 5753 y Fx(25)p eop %%Page: 26 26 26 25 bop 377 154 a Fg(Pr)n(oof.)91 b Fx(Application)37 b(of)j(\(6.5\))i(and)d(\(6.6\))i(to)g(the)f(sequence)g(\()p Fs(H)2819 168 y Fr(n)2865 154 y Fs(;)15 b(n)41 b(>)g Fx(1\))f(yields)e(a)236 267 y(v)-5 b(ariable)29 b Fs(Y)650 234 y Fp(0)704 267 y Fx(suc)m(h)h(that)929 457 y Fs(H)1005 471 y Fo(2)1064 457 y Fq(\016)20 b Fs(:)15 b(:)g(:)22 b Fq(\016)e Fs(H)1397 471 y Fr(n)1459 457 y Fx(\(0\))42 b Fq(")e Fs(Y)1774 419 y Fp(0)1922 457 y Fx(and)124 b Fy(P)p Fx(\()p Fs(Y)2373 419 y Fp(0)2422 457 y Fx(=)25 b Fq(1)p Fx(\))g(=)g(0)15 b Fs(:)236 646 y Fx(Therefore,)40 b(b)m(y)d(the)h(con)m(tin)m(uit)m(y)g(of)g(the)f(mappings)f Fs(H)2184 660 y Fo(1)2223 646 y Fx(\()p Fs(!)s Fx(\),)41 b(the)c(v)-5 b(ariables)37 b Fs(Y)57 b Fx(and)37 b Fs(H)3337 660 y Fo(1)3376 646 y Fx(\()p Fs(Y)3485 613 y Fp(0)3508 646 y Fx(\))236 759 y(agree)c(almost)e(surely)-8 b(.)41 b(Since)31 b Fs(Y)1376 726 y Fp(0)1430 759 y Fx(is)f(indep)s(enden)m(t) f(of)i Fs(H)2218 773 y Fo(1)2288 759 y Fx(and)g(distributed)d(as)j Fs(Y)20 b Fx(,)31 b(this)f(com-)236 872 y(mon)g(la)m(w)g(is)g(indeed)f (a)h(stationary)h(distribution.)1489 b Ff(\003)377 1057 y Fx(Finally)-8 b(,)29 b(the)i(criterion)e(\(2.6\))j(for)e(recurrence)h (resp.)40 b(transience)30 b(can)g(b)s(e)g(completed:)377 1252 y Fg(\(6.8\))53 b(Theorem)80 b Fk(L)-5 b(et)34 b(the)g(system)h Fx(\()p Fs(E)5 b(;)15 b(\027)6 b Fx(\))29 b Fk(b)-5 b(e)33 b(r)-5 b(e)g(curr)g(ent.)48 b(Then)34 b(for)g(arbitr)-5 b(ary)37 b(initial)236 1365 y(law)d(the)f(fol)5 b(lowing)34 b(c)-5 b(onditions)34 b(ar)-5 b(e)33 b(e)-5 b(quivalent:)236 1555 y Fx(\(1\))1070 b(\()p Fs(E)5 b(;)15 b(\027)6 b Fx(\))26 b Fk(is)33 b(nul)5 b(l)32 b(r)-5 b(e)g(curr)g(ent)17 b Fs(;)236 1715 y Fx(\(2\))783 b Fy(P)p Fx(\()p Fs(X)1315 1729 y Fr(n)1389 1715 y Fq(2)25 b Fs(K)7 b Fx(\))25 b Fq(!)g Fx(0)125 b Fk(for)33 b(al)5 b(l)59 b Fs(K)32 b Fq(2)25 b Fm(K)p Fx(\()p Fs(E)5 b Fx(\))15 b Fs(:)377 1977 y Fg(Pr)n(oof.)100 b Fx(1.)78 b(In)42 b(establishing)e(that)j (\(1\))h(implies)39 b(\(2\))44 b(clearly)e Fs(K)52 b Fq(2)45 b Fm(K)3019 1935 y Fj(#)3055 1977 y Fx(\()p Fs(E)5 b Fx(\))44 b(ma)m(y)f(b)s(e)236 2089 y(assumed.)d(By)31 b(\(6.5\))h(this)d(yields)809 2279 y Fy(P)p Fx(\()p Fs(X)990 2293 y Fr(n)1063 2279 y Fq(2)c Fs(K)7 b Fx(\))25 b Fq(\024)g Fy(P)p Fx(\()p Fs(X)1577 2242 y Fo(0)1570 2302 y Fr(n)1644 2279 y Fq(2)g Fs(K)7 b Fx(\))25 b(=)g Fy(P)p Fx(\()p Fs(Y)2150 2242 y Fo(0)2129 2302 y Fr(n)2214 2279 y Fq(2)g Fs(K)7 b Fx(\))26 b Fq(!)f Fy(P)p Fx(\()p Fs(Y)46 b Fq(2)25 b Fs(K)7 b Fx(\))236 2469 y(with)29 b Fy(P)p Fx(\()p Fs(Y)46 b Fq(2)25 b Fs(K)7 b Fx(\))26 b(=)e(0)31 b(b)m(y)f(\(6.6\).)377 2612 y(2.)41 b(Con)m(v)m(ersely)-8 b(,)31 b(let)g(condition)e(\(2\))i (b)s(e)f(satis\014ed.)40 b(Then)29 b(the)i(estimate)762 2801 y Fy(P)p Fx(\()p Fs(X)943 2815 y Fr(n)1016 2801 y Fq(2)25 b Fs(K)7 b Fx(\))26 b Fq(\025)e Fy(P)p Fx(\()p Fs(X)1523 2815 y Fo(0)1589 2801 y Fq(\024)h Fs(x)p Fx(\))15 b Fy(P)p Fx(\()p Fs(X)1975 2764 y Fr(x)1968 2824 y(n)2046 2801 y Fq(2)25 b Fs(K)7 b Fx(\))125 b(for)58 b Fs(K)31 b Fq(2)25 b Fm(K)2798 2756 y Fj(#)2834 2801 y Fx(\()p Fs(E)5 b Fx(\))15 b Fs(;)236 2991 y Fx(with)29 b Fy(P)p Fx(\()p Fs(X)624 3005 y Fo(0)690 2991 y Fq(\024)c Fs(x)p Fx(\))h Fs(>)f Fx(0)30 b(for)g(some)h Fs(x)25 b Fq(2)g Fs(E)36 b Fx(b)m(y)30 b(\(1.3\),)j(yields)894 3181 y Fy(P)p Fx(\()p Fs(Y)1074 3143 y Fr(x)1053 3203 y(n)1143 3181 y Fq(2)25 b Fs(K)7 b Fx(\))25 b(=)g Fy(P)p Fx(\()p Fs(X)1657 3143 y Fr(x)1650 3203 y(n)1727 3181 y Fq(2)g Fs(K)7 b Fx(\))26 b Fq(!)f Fx(0)125 b(for)58 b Fs(K)32 b Fq(2)24 b Fm(K)2666 3135 y Fj(#)2702 3181 y Fx(\()p Fs(E)5 b Fx(\))15 b Fs(;)236 3371 y Fx(hence)31 b(for)f(all)f Fs(K)j Fq(2)25 b Fm(K)p Fx(\()p Fs(E)5 b Fx(\).)41 b(Again)31 b(b)m(y)f(\(6.5\))i(and)e(\(6.6\),)i(this)e(v)m(eri\014es)f(\(1\).)606 b Ff(\003)377 3555 y Fx(No)m(w)31 b(the)g(results)e(of)i(\(2.6\))h(and) e(\(6.8\))i(can)e(b)s(e)g(summarized)f(as)i(follo)m(ws:)377 3731 y(\()p Fs(E)5 b(;)15 b(\027)6 b Fx(\))68 b(p)s(ositiv)m(e)30 b(recurren)m(t)113 b Fq(,)108 b Fy(P)p Fx(\()p Fs(X)1877 3698 y Fo(0)1870 3753 y Fr(n)1943 3731 y Fq(!)26 b(1)p Fx(\))f(=)g(0)83 b(and)g Fy(P)p Fx(\()p Fs(Y)2845 3698 y Fo(0)2824 3753 y Fr(n)2909 3731 y Fq(!)26 b(1)p Fx(\))f(=)g(0,)377 3927 y(\()p Fs(E)5 b(;)15 b(\027)6 b Fx(\))68 b(n)m(ull)29 b(recurren)m(t)270 b Fq(,)108 b Fy(P)p Fx(\()p Fs(X)1877 3894 y Fo(0)1870 3949 y Fr(n)1943 3927 y Fq(!)26 b(1)p Fx(\))f(=)g(0)83 b(and)g Fy(P)p Fx(\()p Fs(Y)2845 3894 y Fo(0)2824 3949 y Fr(n)2909 3927 y Fq(!)26 b(1)p Fx(\))f(=)g(1,)377 4122 y(\()p Fs(E)5 b(;)15 b(\027)6 b Fx(\))68 b(transien)m(t)460 b Fq(,)108 b Fy(P)p Fx(\()p Fs(X)1877 4089 y Fo(0)1870 4145 y Fr(n)1943 4122 y Fq(!)26 b(1)p Fx(\))f(=)g(1)83 b(and)g Fy(P)p Fx(\()p Fs(Y)2845 4089 y Fo(0)2824 4145 y Fr(n)2909 4122 y Fq(!)26 b(1)p Fx(\))f(=)g(1.)377 4411 y Fy(7.)108 b(Mean)49 b(passage)h(times.)92 b Fx(As)43 b(stated)g(in)f(\(1.2\),)48 b(the)43 b(exp)s(ected)g(time)g(to)h(en)m (ter)236 4524 y(an)e(increasing)f(in)m(terv)-5 b(al)41 b(is)g(\014nite)g(in)g(an)m(y)h(case.)77 b(In)41 b(con)m(trast,)47 b(decreasing)42 b(in)m(terv)-5 b(als)41 b(can)236 4637 y(serv)m(e)h(to)f(distinguish)c(p)s(ositiv)m(e)j(from)g(n)m(ull)f (recurrence.)72 b(This)39 b(relies)h(on)g(the)h(w)m(ell-kno)m(wn)236 4750 y(recurrence)35 b(theorem)h(of)f(Kac,)i(whic)m(h)d(is)g(mostly)h (stated)h(under)e(unnecessary)g(restrictions.)236 4863 y(In)c(the)g(presen)m(t)h(setting)f(the)h(follo)m(wing)e(dic)m(hotom)m (y)i(can)f(b)s(e)g(established:)377 5058 y Fg(\(7.1\))54 b(Theorem)81 b Fk(L)-5 b(et)35 b(the)g(system)h Fx(\()p Fs(E)5 b(;)15 b(\027)6 b Fx(\))30 b Fk(b)-5 b(e)35 b(r)-5 b(e)g(curr)g(ent)36 b(with)g(invariant)g(me)-5 b(asur)g(e)36 b Fs(\026)p Fk(.)236 5171 y(Then,)d(with)h(the)f(notations)h Fs(T)1302 5185 y Fr(B)1395 5171 y Fk(and)g Fs(T)1638 5138 y Fr(x)1625 5197 y(B)1718 5171 y Fk(as)f(de\014ne)-5 b(d)34 b(in)f(Se)-5 b(ction)33 b Fx(1)p Fk(,)236 5326 y Fx(\(a\))91 b Fy(E)p Fx(\()p Fs(T)612 5293 y Fr(x)599 5356 y Fo([0)p Fr(;x)p Fo(])737 5326 y Fx(\))26 b Fs(<)f Fq(1)p Fk(,)32 b(if)h Fx(\()p Fs(E)5 b(;)15 b(\027)6 b Fx(\))26 b Fk(is)33 b(p)-5 b(ositive)33 b(r)-5 b(e)g(curr)g(ent)34 b(and)g Fs(\026)p Fx(\([0)p Fs(;)15 b(x)p Fx(]\))27 b Fs(>)e Fx(0)p Fk(,)236 5491 y Fx(\(b\))90 b Fy(E)p Fx(\()p Fs(T)617 5458 y Fr(x)604 5521 y Fo([0)p Fr(;x)p Fo(])743 5491 y Fx(\))25 b(=)g Fq(1)p Fk(,)32 b(if)h Fx(\()p Fs(E)5 b(;)15 b(\027)6 b Fx(\))26 b Fk(is)33 b(nul)5 b(l)32 b(r)-5 b(e)g(curr)g(ent)34 b Fx(\()p Fk(and)g Fs(x)25 b Fq(2)g Fs(E)38 b Fk(arbitr)-5 b(ary)p Fx(\))p Fk(.)1844 5753 y Fx(26)p eop %%Page: 27 27 27 26 bop 377 154 a Fg(Pr)n(oof.)61 b Fx(\(a\))32 b(Consider)c(\014rst) i(the)h(case)g(that)g Fs(x)f Fx(is)g(a)g(maximal)g(elemen)m(t)g(of)h Fs(E)5 b Fx(.)41 b(Then)29 b(b)m(y)236 267 y(\(1.2b\))1154 380 y Fy(E)p Fx(\()p Fs(T)1324 342 y Fr(x)1311 405 y Fo([0)p Fr(;x)p Fo(])1450 380 y Fx(\))c Fq(\024)g Fy(E)p Fx(\()p Fs(T)1776 342 y Fr(x)1763 405 y Fp(f)p Fr(x)p Fp(g)1878 380 y Fx(\))g(=)g Fy(E)p Fx(\()p Fs(T)2204 342 y Fr(x)2191 405 y Fo([)p Fr(x;)11 b Fp(\001)g Fo(])2337 380 y Fx(\))26 b Fs(<)f Fq(1)15 b Fs(:)236 550 y Fx(Otherwise)43 b(c)m(ho)s(ose)j Fs(y)51 b Fq(2)d Fx([)p Fs(x;)31 b Fq(\001)15 b Fx(])30 b Fq(n)g(f)p Fs(x)p Fq(g)45 b Fx(and)f Fs(m)k Fq(2)g Fy(N)c Fx(suc)m(h)g(that)h Fy(P)p Fx(\()p Fs(X)2850 517 y Fo(0)2843 572 y Fr(m)2960 550 y Fq(\025)j Fs(y)s Fx(\))g Fs(>)h Fx(0.)83 b(If)236 662 y(\()p Fs(X)346 676 y Fr(n)394 662 y Fs(;)30 b(n)50 b Fq(\025)g Fx(0\))45 b(is)g(the)g(stationary)g(pro)s(cess)g(b)s(elonging)f(to)i(\()p Fs(E)5 b(;)15 b(\027)6 b Fx(\),)50 b(then)45 b(it)f(follo)m(ws)h(from) 236 775 y Fs(\026)p Fx(\([0)p Fs(;)15 b(x)p Fx(]\))27 b Fs(>)e Fx(0)31 b(b)m(y)f(the)h(Mark)m(o)m(v)h(prop)s(ert)m(y)e(and)f (monotonicit)m(y)i(that)1396 959 y Fy(P)p Fx(\()p Fs(X)1577 973 y Fo(0)1642 959 y Fq(\024)25 b Fs(x;)15 b(X)1905 973 y Fr(m)1998 959 y Fq(\025)25 b Fs(y)s Fx(\))g Fs(>)g Fx(0)15 b Fs(:)236 1142 y Fx(If)30 b Fs(m)g Fx(is)g(c)m(hosen)h (minimal)c(with)i(resp)s(ect)i(to)g(this)e(inequalit)m(y)-8 b(,)29 b(then)894 1325 y Fy(P)p Fx(\()p Fs(X)1075 1339 y Fo(0)1141 1325 y Fq(\024)c Fs(x;)15 b(X)1404 1340 y Fr(l)1456 1325 y Fq(\024)25 b Fs(x;)15 b(X)1719 1339 y Fr(m)1811 1325 y Fq(\025)25 b Fs(y)s Fx(\))h(=)f(0)125 b(for)57 b(0)26 b Fs(<)f(l)i(<)e(m)15 b(;)236 1509 y Fx(b)s(ecause)28 b(otherwise,)g(due)g(to)g(the)h(stationarit)m(y)-8 b(,)29 b Fs(m)f Fx(could)f(b)s(e)g(replaced)g(b)m(y)h Fs(m)16 b Fq(\000)f Fs(l)r Fx(.)39 b(Therefore)236 1622 y Fy(P)p Fx(\()p Fs(A)p Fx(\))27 b Fs(>)e Fx(0)31 b(for)f(the)g(ev)m (en)m(t)580 1805 y Fs(A)c Fx(:=)f Fq(f)p Fs(X)915 1819 y Fo(0)980 1805 y Fq(\024)g Fs(x;)15 b(X)1243 1820 y Fr(l)1305 1805 y Fs(=)-55 b Fq(2)25 b Fx([0)p Fs(;)15 b(x)p Fx(])43 b(for)e(0)26 b Fs(<)f(l)i(<)e(m;)15 b(X)2273 1819 y Fr(m)2365 1805 y Fq(\025)25 b Fs(y)s Fq(g)h(\032)f(f)p Fs(T)2774 1823 y Fo([0)p Fr(;x)p Fo(])2938 1805 y Fs(>)g(m)p Fq(g)15 b Fs(:)236 1988 y Fx(With)30 b(the)h(increasing)e(function)1309 2171 y Fs(g)s Fx(\()p Fs(z)t Fx(\))e(:=)e Fy(E)p Fx(\()p Fs(T)1789 2134 y Fr(z)1776 2197 y Fo([0)p Fr(;x)p Fo(])1914 2171 y Fx(\))125 b(for)58 b Fs(z)29 b Fq(2)c Fs(E)236 2355 y Fx(the)g(recurrence)g(theorem)g(of)g(Kac)g(in)f(its)g(familiar)e (v)m(ersion)j(and)f(the)h(Mark)m(o)m(v)h(prop)s(ert)m(y)f(imply)1092 2566 y Fy(P)p Fx(\()p Fs(T)1251 2585 y Fo([0)p Fr(;x)p Fo(])1415 2566 y Fs(<)g Fq(1)p Fx(\))83 b(=)1874 2451 y Fi(Z)1947 2607 y Fp(f)p Fr(X)2040 2616 y Fd(0)2075 2607 y Fp(\024)p Fr(x)p Fp(g)2238 2566 y Fs(T)2291 2585 y Fo([0)p Fr(;x)p Fo(])2444 2566 y Fs(d)p Fy(P)1720 2784 y Fq(\025)1874 2669 y Fi(Z)1947 2820 y Fr(A)2033 2784 y Fs(T)2086 2802 y Fo([0)p Fr(;x)p Fo(])2239 2784 y Fs(d)p Fy(P)1720 3001 y Fx(=)1874 2887 y Fi(Z)1947 3038 y Fr(A)2033 3001 y Fx(\()p Fs(m)20 b Fx(+)g Fs(g)s Fx(\()p Fs(X)2415 3015 y Fr(m)2483 3001 y Fx(\)\))15 b Fs(d)p Fy(P)1720 3172 y Fq(\025)83 b Fy(P)p Fx(\()p Fs(A)p Fx(\))15 b(\()p Fs(m)22 b Fx(+)e Fs(g)s Fx(\()p Fs(x)p Fx(\)\))15 b Fs(:)236 3356 y Fx(Therefore)30 b Fs(g)s Fx(\()p Fs(x)p Fx(\))d Fs(<)e Fq(1)p Fx(,)30 b(as)h(had)f(to)h(b)s(e)f(sho)m(wn.)377 3499 y(\(b\))d(The)e(extended)h(v)m(ersion)g(of)g(the)g(recurrence)g (theorem)h(of)f(Kac)g(needed)g(here)g(concerns)236 3611 y(the)31 b(in\014nite)e(measure)i Fs(\045)c Fx(:=)f Fs(\026)20 b Fq(\012)h Fs(P)33 b Fq(\012)21 b Fs(P)33 b Fq(\012)21 b Fs(:)15 b(:)g(:)47 b Fx(on)2080 3547 y Fi(N)2187 3647 y Fr(n)p Fp(\025)p Fo(0)2354 3611 y Fm(B)p Fx(\()p Fs(E)5 b Fx(\),)32 b(whic)m(h)e(is)g(shift)g(in)m(v)-5 b(arian)m(t)236 3724 y(b)m(y)31 b(the)g(in)m(v)-5 b(ariance)30 b(of)h Fs(\026)p Fx(.)41 b(No)m(w)32 b(the)f(standard)f(pro)s(of)g(in)f(the)i (case)h(of)f(probabilit)m(y)d(measures)236 3837 y(is)i(easily)f(c)m (hec)m(k)m(ed)j(to)g(w)m(ork)e(as)h(w)m(ell)e(for)h Fs(\045)p Fx(,)h(resulting)d(in)637 3934 y Fi(Z)710 4085 y Fr(B)800 4049 y Fy(E)p Fx(\()p Fs(T)970 4004 y Fr(y)957 4076 y(B)1018 4049 y Fx(\))15 b Fs(\026)p Fx(\()p Fs(dy)s Fx(\))26 b(=)1410 3934 y Fi(Z)1484 4085 y Fr(E)1572 4049 y Fy(P)p Fx(\()p Fs(T)1744 4004 y Fr(y)1731 4076 y(B)1818 4049 y Fs(<)f Fq(1)p Fx(\))15 b Fs(\026)p Fx(\()p Fs(dy)s Fx(\))126 b(for)30 b(all)57 b Fs(B)30 b Fq(2)25 b Fm(B)p Fx(\()p Fs(E)5 b Fx(\))15 b Fs(:)236 4265 y Fx(In)42 b(applying)e(this)h(equation)h(to)h Fs(B)49 b Fx(=)c([0)p Fs(;)15 b(x)p Fx(])43 b(it)f(means)g(no)g(restriction)f(to)i(assume)f (again)236 4378 y Fs(\026)p Fx(\([0)p Fs(;)15 b(x)p Fx(]\))27 b Fs(>)e Fx(0,)h(b)s(ecause)e(otherwise)f(an)g(application)f(of)i (\(4.4b\))h(to)g Fs(f)34 b Fx(=)25 b(1)2760 4396 y Fo([0)p Fr(;x)p Fo(])2922 4378 y Fx(and)f(the)g(Mark)m(o)m(v)236 4491 y(prop)s(ert)m(y)36 b(yield)f Fy(P)p Fx(\()p Fs(T)1012 4458 y Fr(x)999 4521 y Fo([0)p Fr(;x)p Fo(])1173 4491 y Fx(=)g Fq(1)p Fx(\))g Fs(>)g Fx(0.)59 b(Under)35 b(this)h (assumption,)g(no)m(w)h(in)e(view)g(of)i(\(4.4a\),)236 4619 y Fy(P)p Fx(\()p Fs(T)408 4574 y Fr(y)395 4650 y Fo([0)p Fr(;x)p Fo(])559 4619 y Fs(<)25 b Fq(1)p Fx(\))h(=)f(1)31 b(for)f(all)f Fs(y)f Fq(2)d Fs(E)36 b Fx(and)29 b(th)m(us)h(b)m(y)h (monotonicit)m(y)970 4861 y Fy(E)p Fx(\()p Fs(T)1140 4823 y Fr(x)1127 4886 y Fo([0)p Fr(;x)p Fo(])1265 4861 y Fx(\))15 b Fs(\026)p Fx(\([0)p Fs(;)g(x)p Fx(]\))86 b Fq(\025)1866 4746 y Fi(Z)1940 4901 y Fo([0)p 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b(Then)30 b(the)g(recursion)636 2576 y Fs(S)692 2590 y Fo(0)756 2576 y Fx(:=)25 b(0)125 b(and)f Fs(S)1374 2591 y Fr(k)r Fo(+1)1532 2576 y Fx(:=)25 b(inf)6 b Fq(f)p Fs(n)25 b(>)g(S)2041 2591 y Fr(k)2109 2576 y Fx(:)g Fs(H)2235 2590 y Fr(n)2302 2576 y Fq(\016)c Fs(:)15 b(:)g(:)21 b Fq(\016)g Fs(H)2636 2590 y Fr(S)2679 2602 y Fe(k)2716 2590 y Fo(+1)2811 2576 y Fx(\()p Fs(y)s Fx(\))k Fq(\024)g Fs(y)s Fq(g)236 2777 y Fx(de\014nes)33 b(a)i(sequence)f(of)g(stopping)f(times)g(with)g(resp)s(ect)h(to)g(\()p Fs(H)2475 2791 y Fr(n)2522 2777 y Fs(;)15 b(n)32 b Fq(2)e Fy(N)p Fx(\),)35 b(whic)m(h)e(b)m(y)h(\(4.4a\))236 2890 y(ma)m(y)d(b)s(e)f(assumed)g(to)h(b)s(e)e(\014nite.)40 b(By)31 b(\(7.1a\))h(it)e(follo)m(ws)g(as)g(in)f(the)i(pro)s(of)f(of)g (\(1.4\))i(that)236 3091 y(\(1\))652 b Fy(E)p Fx(\()p Fs(S)1163 3106 y Fr(k)1225 3091 y Fq(\000)20 b Fs(S)1372 3106 y Fr(k)r Fp(\000)p Fo(1)1505 3091 y Fx(\))26 b(=)f Fy(E)p Fx(\()p Fs(T)1832 3047 y Fr(y)1819 3123 y Fo([0)p Fr(;y)r Fo(])1954 3091 y Fx(\))h 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Fs(T)63 b Fq(\024)1150 4328 y Fi(X)1285 4445 y Fr(k)r Fp(\025)p Fo(0)1478 4328 y Fi(Y)1599 4445 y Fo(0)p Fp(\024)p Fr(i)p Fp(\024)p Fr(k)1840 4408 y Fx(\(1)21 b Fq(\000)f Fx(1)2077 4422 y Fr(A)2130 4432 y Fe(i)2160 4408 y Fx(\))15 b(\()p Fs(S)2301 4423 y Fr(k)r Fo(+1)2455 4408 y Fq(\000)20 b Fs(S)2602 4423 y Fr(k)2644 4408 y Fx(\))36 b(+)f(1)236 4621 y(holds,)27 b(b)s(ecause)g(for)g(\014xed)g Fs(!)j Fx(the)d(righ)m(t-hand)f(side)g(equals)h Fs(S)2375 4636 y Fr(k)2417 4621 y Fx(\()p Fs(!)s Fx(\))14 b(+)g(1,)29 b(if)d Fs(k)k Fx(is)d(the)g(\014rst)g(index)236 4734 y(with)k Fs(!)f Fq(2)d Fs(A)688 4749 y Fr(k)731 4734 y Fx(,)32 b(and)f(is)g(in\014nite,)f(if)h(there)h(is)f(no)g(suc)m(h)h (index.)43 b(By)32 b(cancelling)f(for)g(eac)m(h)i Fs(k)i Fx(the)236 4847 y(factor)c(with)e Fs(i)d Fx(=)f Fs(k)33 b Fx(the)d(b)s(ound)f(for)g Fs(T)43 b Fx(is)30 b(increased,)f(and)h (the)g(summands)f(are)h(comp)s(osed)g(of)236 4960 y(indep)s(enden)m(t)e (factors.)42 b(By)31 b(\(1\))g(and)f(\(2\))h(this)f(yields)1276 5160 y Fy(E)p Fx(\()p Fs(T)13 b Fx(\))26 b Fq(\024)f Fs(#)1657 5123 y Fp(\000)p Fo(1)1766 5160 y Fy(E)p Fx(\()p Fs(T)1936 5116 y Fr(y)1923 5192 y Fo([0)p Fr(;y)r Fo(])2059 5160 y Fx(\))20 b(+)g(1)26 b Fs(<)f Fq(1)15 b Fs(:)970 b Ff(\003)377 5391 y Fx(No)m(w)40 b(the)f(familiar)e(criterion)h(for)g (p)s(ositiv)m(e)g(resp.)66 b(n)m(ull)37 b(recurrence)h(b)m(y)h(mean)g (passage)236 5504 y(times)c(carries)f(o)m(v)m(er)j(from)d(discrete)h (Mark)m(o)m(v)i(c)m(hain)d(theory)h(to)h(the)f(presen)m(t)g(setting)g (in)f(the)1844 5753 y(28)p eop %%Page: 29 29 29 28 bop 236 154 a Fx(follo)m(wing)29 b(form:)377 350 y Fg(\(7.3\))51 b(Theorem)77 b Fk(L)-5 b(et)33 b(the)g(system)h Fx(\()p Fs(E)5 b(;)15 b(\027)6 b Fx(\))26 b Fk(b)-5 b(e)33 b(r)-5 b(e)g(curr)g(ent)34 b(with)f(attr)-5 b(actor)35 b Fs(M)10 b Fk(.)236 504 y Fx(\(a\))91 b Fk(If)33 b Fx(\()p Fs(E)5 b(;)15 b(\027)6 b Fx(\))26 b Fk(is)33 b(p)-5 b(ositive)33 b(r)-5 b(e)g(curr)g(ent)34 b(and)g Fs(x)25 b Fq(2)g Fs(M)10 b Fk(,)33 b(then)969 690 y Fy(E)p Fx(\()p Fs(T)1139 653 y Fr(x)1126 713 y(G)1185 690 y Fx(\))25 b Fs(<)g Fq(1)125 b Fk(for)33 b(al)5 b(l)59 b Fs(G)26 b Fq(2)e Fm(G)p Fx(\()p Fs(E)5 b Fx(\))60 b Fk(with)f Fs(x)25 b Fq(2)g Fs(G)15 b Fx(;)236 876 y(\(b\))90 b Fk(if)33 b Fx(\()p Fs(E)5 b(;)15 b(\027)6 b Fx(\))26 b Fk(is)33 b(nul)5 b(l)32 b(r)-5 b(e)g(curr)g(ent)34 b(and)g Fs(x)25 b Fq(2)g Fs(E)38 b Fk(arbitr)-5 b(ary,)35 b(then)917 1062 y Fy(E)p Fx(\()p Fs(T)1087 1025 y Fr(x)1074 1085 y(G)1134 1062 y Fx(\))25 b(=)g Fq(1)125 b Fk(for)33 b(some)59 b Fs(G)26 b Fq(2)e Fm(G)p Fx(\()p Fs(E)5 b Fx(\))60 b Fk(with)f Fs(x)25 b Fq(2)g Fs(G)15 b(;)236 1248 y Fk(pr)-5 b(ovide)g(d)35 b Fs(E)j Fk(is)33 b(lo)-5 b(c)g(al)5 b(ly)34 b(b)-5 b(ounde)g(d)34 b Fx(\()p Fk(se)-5 b(e)33 b Fx(\(10.2\)\).)377 1444 y Fg(Pr)n(oof.)73 b Fx(\(a\))35 b(Since)e Fs(G)h Fx(b)m(y)g(\(10.4\))i (ma)m(y)f(b)s(e)e(assumed)h(to)g(b)s(e)g(con)m(v)m(ex,)i(this)d(is)g(a) i(sp)s(ecial)236 1557 y(case)d(of)e(\(7.2\).)377 1700 y(\(b\))45 b(By)f(the)g(lo)s(cal)g(b)s(oundedness)d(there)j(exist)g Fs(G)2191 1714 y Fo(0)2279 1700 y Fq(2)k Fm(G)p Fx(\()p Fs(E)5 b Fx(\))45 b(and)f Fs(y)51 b Fq(2)d Fs(E)h Fx(suc)m(h)44 b(that)236 1813 y Fs(x)25 b Fq(2)g Fs(G)470 1827 y Fo(0)535 1813 y Fq(\032)g Fx([0)p Fs(;)15 b(y)s Fx(],)32 b(where)e Fs(y)e Fq(6)p Fx(=)d Fs(x)30 b Fx(ma)m(y)h(b)s(e)f(assumed,)g(b)s (ecause)g(otherwise)g(b)m(y)g(\(7.1b\))1395 1999 y Fy(E)p Fx(\()p Fs(T)1565 1961 y Fr(x)1552 2021 y(G)1607 2030 y Fd(0)1645 1999 y Fx(\))c Fq(\025)f Fy(E)p Fx(\()p Fs(T)1972 1961 y Fr(x)1959 2025 y Fo([0)p Fr(:x)p Fo(])2097 1999 y Fx(\))h(=)f Fq(1)15 b Fs(:)236 2185 y Fx(But)41 b(then)f Fs(G)715 2199 y Fo(0)781 2185 y Fq(\\)27 b Fx([)p Fs(y)s(;)j Fq(\001)15 b Fx(])43 b(=)e Fq(;)g Fx(ma)m(y)g(b)s(e)e(assumed)h(as)g(w) m(ell,)i(b)s(ecause)f(otherwise)e Fs(G)3193 2199 y Fo(0)3273 2185 y Fx(can)i(b)s(e)236 2298 y(decreased)29 b(to)f Fs(G)823 2312 y Fo(0)878 2298 y Fq(n)15 b Fx([)p Fs(y)s(;)31 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y(sen)m(tial)k(for)g(assertion)h(\(b\))f(\(see,)i(ho)m(w)m(ev)m (er,)h(\(9.5\)\).)42 b(Moreo)m(v)m(er,)31 b(assertion)d(\(b\))h (extends)f(again)236 4290 y(to)k(transien)m(t)f(systems,)h(b)s(ecause)f (in)f(this)h(case)h Fy(P)p Fx(\()p Fs(T)2083 4257 y Fr(x)2070 4316 y(G)2157 4290 y Fx(=)26 b Fq(1)p Fx(\))h Fs(>)g Fx(0)32 b(whenev)m(er)f Fs(G)h Fx(is)e(relativ)m(ely)236 4403 y(compact.)377 4670 y Fy(8.)64 b(F)-9 b(urther)33 b(limit)f(theorems.)54 b Fx(F)-8 b(rom)30 b(the)f(results)f(of)h (Section)f(6)i(it)e(is)g(easily)g(deriv)m(ed)236 4783 y(that)34 b(the)f(distributions)c(of)34 b Fs(X)1311 4797 y Fr(n)1358 4783 y Fs(;)c(n)g Fq(\025)f Fx(0)p Fs(;)34 b Fx(con)m(v)m(erge)h(in)d(the)h(p)s(ositiv)m(e)f(recurren)m(t)h(case)h (w)m(eakly)236 4896 y(to)j(the)f(stationary)g(distribution)c(\(and)k (otherwise)f(v)-5 b(aguely)36 b(to)h(0\).)58 b(Actually)-8 b(,)37 b(the)f(class)g(of)236 5009 y(functions,)29 b(for)i(whic)m(h)e (con)m(v)m(ergence)j(holds,)e(is)f(considerably)f(larger:)377 5205 y Fg(\(8.1\))50 b(Pr)n(oposition)75 b Fk(L)-5 b(et)32 b(the)g(system)g Fx(\()p Fs(E)5 b(;)15 b(\027)6 b Fx(\))26 b Fk(b)-5 b(e)32 b(p)-5 b(ositive)32 b(r)-5 b(e)g(curr)g(ent)33 b(with)g(stationary)236 5318 y(distribution)h Fs(\026)p Fk(.)41 b(Then,)33 b(for)g(arbitr)-5 b(ary)35 b(initial)e(law)h Fs(\026)2122 5332 y Fo(0)2194 5318 y Fk(and)f(with)h Fs(\026)2623 5332 y Fr(n)2695 5318 y Fx(:=)25 b Fs(\026)2871 5332 y Fo(0)2925 5318 y Fs(P)2996 5285 y Fr(n)3043 5318 y Fk(,)1261 5504 y Fs(\026)1316 5518 y Fr(n)1363 5504 y Fs(f)34 b Fq(!)25 b Fs(\026f)134 b Fk(for)33 b(al)5 b(l)59 b Fs(f)34 b Fq(2)25 b(R)p Fx(\()p Fs(E)5 b Fx(\))15 b Fs(:)1844 5753 y Fx(29)p eop %%Page: 30 30 30 29 bop 377 142 a Fg(Pr)n(oof.)71 b Fx(Since)33 b Fs(f)43 b Fx(is)32 b(b)s(ounded,)g(application)g(of)i(\(1.6\))h(with)d(initial) f(v)-5 b(ariable)33 b Fs(X)3281 156 y Fo(0)3351 142 y Fx(=)d Fs(x)3504 156 y Fo(0)236 255 y Fx(resp.)40 b Fs(X)539 269 y Fo(0)604 255 y Fx(=)25 b Fs(x)30 b Fx(yields)685 488 y Fs(\026)740 502 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1257 y Fr(n)1746 1243 y Fq(\024)f Fs(y)s Fx(\))k(to)h(the)f(limit)e Fs(F)13 b Fx(\()p Fs(y)s Fx(\))26 b(:=)f Fs(\026)p Fx(\([0)p Fs(;)15 b(y)s Fx(]\).)42 b(T)-8 b(o)29 b(extend)236 1356 y(this)g(result)h(to)h(the)f(general)h (setting,)g(an)f(appropriate)f(metric)h(on)h Fl(M)2678 1370 y Fo(1)2717 1356 y Fx(\()p Fs(E)5 b Fx(\))31 b(is)e(needed:)377 1552 y Fg(\(8.2\))51 b(Pr)n(oposition)78 b Fk(The)32 b(de\014nition)943 1756 y Fs(d)p Fx(\()p Fs(\026)1080 1770 y Fo(1)1119 1756 y Fs(;)15 b(\026)1214 1770 y Fo(2)1254 1756 y Fx(\))26 b(:=)f(sup)14 b Fq(fj)p Fs(\026)1713 1770 y Fo(1)1753 1756 y Fx(\()p Fs(B)5 b Fx(\))20 b Fq(\000)g Fs(\026)2063 1770 y Fo(2)2102 1756 y Fx(\()p Fs(B)5 b Fx(\))p Fq(j)26 b Fx(:)g Fs(B)j Fq(2)c Fm(B)2612 1710 y Fj(#)2649 1756 y Fx(\()p Fs(E)5 b Fx(\))p Fq(g)236 1960 y Fk(yields)33 b(a)g(metric)g(on)g Fl(M)1065 1974 y Fo(1)1104 1960 y Fx(\()p Fs(E)5 b Fx(\))34 b Fk(with)f(the)h(fol)5 b(lowing)33 b(pr)-5 b(op)g(erties:)236 2165 y Fx(\(a\))343 b Fs(d)p Fx(\()p 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Fx(\))25 b Fs(>)g(y)s Fx(\))15 b Fs(dy)s Fx(.)377 2962 y(\(b\))31 b(By)g(\(a\))977 3075 y Fs(\026)1032 3089 y Fr(n)1079 3075 y Fs(f)j Fq(!)25 b Fs(\026f)134 b Fx(for)58 b Fs(f)34 b Fq(2)25 b(B)1903 3029 y Fj(#)1939 3075 y Fx(\()p Fs(E)5 b Fx(\))59 b(with)e(0)25 b Fq(\024)g Fs(f)35 b Fq(\024)25 b Fx(1)15 b Fs(;)236 3241 y Fx(whic)m(h)32 b(implies)e(con)m(v)m(ergence)35 b(for)e Fs(f)38 b Fq(2)29 b(K)1699 3255 y Fo(0)1772 3241 y Fx(and)j(th)m(us)g(for)h Fs(f)39 b Fq(2)29 b(K)q Fx(\()p Fs(E)5 b Fx(\))34 b(according)f(to)g (\(10.6a\).)236 3354 y(Besides)d Fs(\026)614 3368 y Fr(n)686 3354 y Fq(!)698 3379 y Fr(w)793 3354 y Fs(\026)g Fx(this)f(pro)m(v)m (es)i Fs(d)g Fx(to)g(b)s(e)e(indeed)g(a)i(metric.)377 3497 y(\(c\))h(By)e(\(a\))i(this)d(follo)m(ws)h(from)g Fs(\026P)13 b(f)34 b Fx(=)1800 3428 y Fi(R)1845 3538 y Fp(H)p Fo([)p Fr(E)t Fo(])2033 3497 y Fs(\026)p Fx(\()p Fs(f)c Fq(\016)21 b Fs(h)p Fx(\))15 b Fs(\027)6 b Fx(\()p Fs(dh)p Fx(\).)862 b Ff(\003)377 3681 y Fx(An)25 b(application)e(of)i (\(10.3\))i(to)f Fs(E)1518 3648 y Fp(\003)1582 3681 y Fx(yields)d Fs(d)p Fx(\()p Fs(")1956 3695 y Fr(x)2001 3681 y Fs(;)15 b(")2083 3695 y Fr(y)2126 3681 y Fx(\))25 b(=)g(1)g(for)g Fs(x)g Fq(6)p Fx(=)g Fs(y)s Fx(,)h(sho)m(wing)e(metric) h(con-)236 3794 y(v)m(ergence)i(in)e(general)g(to)i(b)s(e)e(m)m(uc)m(h) g(stronger)h(than)f(w)m(eak)i(con)m(v)m(ergence.)41 b(Using)25 b(the)h(notion)f(of)236 3907 y(a)d(\\)15 b(splitting)k(p)s(oin)m(t)c(") 21 b(\(see)h Fs(z)k Fx(in)20 b(the)h(follo)m(wing)f(pro)s(of)7 b(\),)23 b(as)f(in)m(tro)s(duced)d(b)m(y)i(Dubins/F)-8 b(reedman)236 4020 y([13])30 b(for)e Fs(E)j Fx(=)25 b([0)p Fs(;)15 b Fx(1])30 b(and)e(extended)h(b)m(y)f(Bhattac)m(hary)m(a/Ma)5 b(jumdar)31 b([8,)f(9])f(to)g(higher)f(dimen-)236 4133 y(sion,)i(the)g(w)m(eak)i(con)m(v)m(ergence)g(from)e(\(8.1\))j(can)d(b) s(e)g(strengthened)g(to)h(metric)f(con)m(v)m(ergence:)377 4329 y Fg(\(8.3\))54 b(Theorem)81 b Fk(If)35 b(the)g(system)h Fx(\()p Fs(E)5 b(;)15 b(\027)6 b Fx(\))30 b Fk(is)35 b(p)-5 b(ositive)36 b(r)-5 b(e)g(curr)g(ent)36 b(with)g(stationary)h (dis-)236 4442 y(tribution)c Fs(\026)p Fk(,)g(then)1109 4555 y Fs(d)p Fx(\()p Fs(\026)1246 4569 y Fo(0)1302 4555 y Fs(P)1373 4517 y Fr(n)1419 4555 y Fs(;)15 b(\026)p Fx(\))26 b Fq(!)f Fx(0)125 b Fk(for)34 b(al)5 b(l)58 b Fs(\026)2217 4569 y Fo(0)2282 4555 y Fq(2)25 b Fl(M)2448 4569 y Fo(1)2487 4555 y Fx(\()p Fs(E)5 b Fx(\))15 b Fs(:)377 4793 y Fg(Pr)n(oof.)93 b Fx(1.)71 b(Applying)38 b(\(1.3\),)45 b(c)m(ho)s(ose)d Fs(z)j Fx(with)39 b Fs(\026)p Fx(\([0)p Fs(;)15 b(z)t Fx(]\))44 b Fs(>)e Fx(0)e(and,)j(applying)38 b(\(5.3\),)236 4906 y(c)m(ho)s(ose)32 b Fs(\016)d(>)c Fx(0)30 b(suc)m(h)g(that)1423 5019 y Fs(\026)p Fx(\([0)p Fs(;)15 b(z)t Fx(]\))23 b Fq(^)c Fs(\026)p Fx(\([)p Fs(z)t(;)32 b Fq(\001)15 b Fx(]\))26 b Fs(>)f(\016)19 b(:)236 5186 y Fx(Then,)40 b(for)e(giv)m(en)g Fs(")h(>)f Fx(0)h(and)f(with)f Fs(\026)1630 5200 y Fr(n)1715 5186 y Fx(:=)i Fs(\026)1905 5200 y Fo(0)1959 5186 y Fs(P)2030 5153 y Fr(n)2077 5186 y Fx(,)i(there)d(exists)g(a)h(\014nite)e(subset)h Fs(A)g Fx(of)h Fs(E)236 5299 y Fx(satisfying)714 5503 y Fs(\026)p Fx(\()p Fs(E)25 b Fq(n)c Fs(A)1030 5457 y Fj(#)1066 5503 y Fx(\))26 b Fq(\024)f Fs(\016)19 b(")125 b Fx(and)f Fs(\026)1775 5517 y Fr(n)1821 5503 y Fx(\()p Fs(E)26 b Fq(n)21 b Fs(A)2083 5457 y Fj(#)2119 5503 y Fx(\))26 b Fq(\024)f Fs(\016)19 b(")83 b Fx(for)30 b(all)82 b Fs(n)25 b Fq(2)g Fy(N)15 b Fs(;)1844 5753 y Fx(30)p eop %%Page: 31 31 31 30 bop 236 154 a Fx(as)31 b(follo)m(ws)e(from)h(\(1.3\))i(and)e (\(8.1\),)i(applied)c(to)j Fs(f)k Fx(=)25 b(1)2147 182 y Fr(A)2200 157 y Fj(#)2240 154 y Fx(.)41 b(Moreo)m(v)m(er,)32 b(again)f(b)m(y)f(\(8.1\),)i(there)236 267 y(exists)e Fs(k)f Fq(2)c Fy(N)30 b Fx(suc)m(h)g(that)780 471 y Fy(P)p Fx(\()p Fs(X)968 434 y Fr(x)961 494 y(k)1039 471 y Fq(\024)25 b Fs(z)45 b Fx(for)31 b(all)40 b Fs(x)25 b Fq(2)g Fs(A)1730 425 y Fj(#)1766 471 y Fx(\))h Fq(\025)f Fs(\016)128 b Fx(and)c Fy(P)p Fx(\()p Fs(X)2550 434 y Fo(0)2543 494 y 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Fx(satisfying)29 b Fs(A)709 2020 y Fj(#)771 2061 y Fq(\032)c Fs(h)919 2028 y Fp(\000)p Fo(1)1013 2061 y Fx([)p Fs(B)5 b Fx(])26 b Fq(2)e Fm(B)1328 2020 y Fj(#)1365 2061 y Fx(\()p Fs(E)5 b Fx(\))31 b(for)f Fs(h)c Fq(2)f(H)1918 2075 y Fp(\003)1957 2061 y Fx(.)41 b(Therefore)648 2298 y Fq(j)p Fs(\026)728 2312 y Fr(n)775 2298 y Fx(\()p Fs(B)5 b Fx(\))21 b Fq(\000)f Fs(\026)p Fx(\()p Fs(B)5 b Fx(\))p Fq(j)83 b(\024)1492 2183 y Fi(Z)1565 2339 y Fp(H)p Fo([)p Fr(E)t Fo(])1753 2298 y Fq(j)p Fs(\026)1833 2312 y Fr(n)p Fp(\000)p Fo(1)1970 2298 y Fx(\()p Fs(h)2057 2261 y Fp(\000)p Fo(1)2152 2298 y Fx([)p Fs(B)5 b Fx(]\))21 b Fq(\000)e Fs(\026)p Fx(\()p Fs(h)2564 2261 y Fp(\000)p Fo(1)2659 2298 y Fx([)p Fs(B)5 b Fx(]\))p Fq(j)15 b Fs(\027)6 b Fx(\()p Fs(dh)p Fx(\))1338 2469 y Fq(\024)83 b Fx(\(1)21 b Fq(\000)e Fs(\016)s Fx(\))c Fs(d)p Fx(\()p Fs(\026)1913 2483 y Fr(n)p Fp(\000)p Fo(1)2052 2469 y Fs(;)g(\026)p Fx(\))36 b(+)f Fs(\016)19 b(")125 b Fx(for)30 b(all)57 b Fs(n)25 b Fq(2)g Fy(N)15 b Fs(;)236 2674 y Fx(b)s(ecause)30 b(b)m(y)f(the)h(\014rst)e(part)i Fs(\027)6 b Fx(\()p Fq(H)q Fx([)p Fs(E)f Fx(])19 b Fq(n)g(H)1682 2688 y Fp(\003)1721 2674 y Fx(\))25 b Fq(\024)g Fx(1)19 b Fq(\000)f Fs(\016)33 b Fx(and)c(on)h Fq(H)2482 2688 y Fp(\003)2550 2674 y Fx(the)g(in)m(tegrand)f(is)f(b)s(ounded)236 2787 y(b)m(y)i Fs(\016)19 b(")p Fx(.)236 2941 y(\(2\))87 b(In)30 b(the)g(case)i Fs(z)39 b(=)-55 b Fq(2)25 b Fs(B)5 b Fx(,)30 b(i.e.)41 b([)p Fs(z)t(;)31 b Fq(\001)15 b Fx(])26 b Fq(\032)e Fs(E)i Fq(n)21 b Fs(B)5 b Fx(,)30 b(replace)g Fq(H)2330 2955 y Fp(\003)2400 2941 y Fx(b)m(y)1304 3145 y Fq(H)1382 3108 y Fp(\003)1447 3145 y Fx(:=)25 b Fq(f)p Fs(h)h Fq(2)f(H)q Fx([)p Fs(E)5 b Fx(])26 b(:)f Fs(h)p Fx(\(0\))i Fq(\025)e Fs(z)t Fq(g)15 b Fs(;)236 3349 y Fx(satisfying)43 b Fs(h)707 3316 y Fp(\000)p Fo(1)801 3349 y Fx([)p Fs(B)5 b Fx(])48 b(=)f Fq(;)d Fx(for)g Fs(h)k Fq(2)f(H)1619 3316 y Fp(\003)1658 3349 y Fx(.)81 b(Then)42 b(it)i(follo)m(ws)e(similarly)-8 b(,)45 b(in)d(fact)j(somewhat)236 3462 y(simpler,)28 b(that)827 3667 y Fq(j)p Fs(\026)907 3681 y Fr(n)954 3667 y Fx(\()p Fs(B)5 b Fx(\))21 b Fq(\000)f Fs(\026)p Fx(\()p Fs(B)5 b Fx(\))p Fq(j)25 b(\024)g Fx(\(1)c Fq(\000)f Fs(\016)s Fx(\))15 b Fs(d)p Fx(\()p Fs(\026)1977 3681 y Fr(n)p Fp(\000)p Fo(1)2116 3667 y Fs(;)g(\026)p Fx(\))125 b(for)30 b(all)57 b Fs(n)25 b Fq(2)g Fy(N)15 b Fs(:)236 3871 y Fx(Com)m(bining)28 b(b)s(oth)i(cases)h(and)f(putting)f Fs(\015)i Fx(:=)25 b(1)c Fq(\000)e Fs(\016)35 b Fx(this)29 b(yields)1329 4075 y Fs(d)p Fx(\()p Fs(\026)1466 4089 y Fr(n)1514 4075 y Fs(;)15 b(\026)p Fx(\))25 b Fq(\024)g Fs(\015)c(d)p Fx(\()p Fs(\026)1970 4089 y Fr(n)p Fp(\000)p Fo(1)2107 4075 y Fs(;)15 b(\026)p Fx(\))21 b(+)f Fs(\016)f(")236 4279 y Fx(and)30 b(th)m(us)g(b)m(y)g(recursion)1031 4483 y Fs(d)p Fx(\()p Fs(\026)1168 4497 y Fr(n)1216 4483 y Fs(;)15 b(\026)p Fx(\))25 b Fs(<)g(\015)1519 4446 y Fr(n)1582 4483 y Fs(d)p Fx(\()p Fs(\026)1719 4497 y Fo(0)1759 4483 y Fs(;)15 b(\026)p Fx(\))20 b(+)g Fs(")125 b Fx(for)30 b(all)57 b Fs(n)25 b Fq(2)g Fy(N)15 b Fs(:)236 4688 y Fx(In)30 b(view)f(of)i Fs(\015)f(<)25 b Fx(1)31 b(and)f Fs(d)25 b Fq(\024)g Fx(1)31 b(this)e(pro)m(v)m(es)i(the)g(assertion.) 1150 b Ff(\003)377 4872 y Fx(In)26 b(case)h(the)f(state)i(space)f(is)e (b)s(ounded)f(\(or)j(at)g(least)f(of)g(the)h(form)f Fs(E)k Fx(=)25 b Fs(A)2921 4831 y Fj(#)2983 4872 y Fx(with)g Fs(A)i Fx(\014nite\),)236 4985 y(the)33 b(pro)s(of)f(is)f(easily)h(c)m (hec)m(k)m(ed)i(to)g(w)m(ork)e(with)f Fs(")f Fx(=)e(0,)34 b(i.e.)47 b(there)32 b(is)g(geometric)i(con)m(v)m(ergence,)236 5098 y(whic)m(h)29 b(in)g(addition)g(is)g(uniform)g(in)g Fs(\026)1555 5112 y Fo(0)1594 5098 y Fx(.)377 5241 y(T)-8 b(o)32 b(conclude)e(this)g(section)h(b)m(y)g(a)h(la)m(w)f(of)g(large)g (n)m(um)m(b)s(ers,)f(ergo)s(dicit)m(y)h(of)g(the)g(stationary)236 5354 y(v)m(ersion)42 b(of)g(the)h(pro)s(cess)f(\()p Fs(X)1280 5368 y Fr(n)1327 5354 y Fs(;)31 b(n)45 b Fq(\025)f Fx(0\))f(is)f(essen) m(tial.)76 b(Actually)-8 b(,)45 b(b)m(y)d(means)h(of)f(the)h(dual)236 5466 y(pro)s(cess)30 b(from)g(Section)g(6,)h(a)g(stronger)g(result)e (can)h(b)s(e)g(established:)1844 5753 y(31)p eop %%Page: 32 32 32 31 bop 377 154 a Fg(\(8.4\))53 b(Pr)n(oposition)c Fk(L)-5 b(et)35 b(the)g(system)g Fx(\()p Fs(E)5 b(;)15 b(\027)6 b Fx(\))29 b Fk(b)-5 b(e)35 b(p)-5 b(ositive)35 b(r)-5 b(e)g(curr)g(ent)36 b(with)f(stationary)236 267 y(distribution)f Fs(\026)p Fk(.)41 b(Then)33 b(the)g(pr)-5 b(o)g(c)g(ess)35 b Fx(\()p Fs(X)1653 281 y Fr(n)1701 267 y Fs(;)30 b(n)25 b Fq(\025)g Fx(0\))34 b Fk(with)f(initial)g(law)h Fs(\026)e Fk(is)h(mixing.)377 463 y Fg(Pr)n(oof.)60 b Fx(Extending)26 b Fs(H)1262 477 y Fr(n)1308 463 y Fs(;)31 b(n)25 b Fq(2)f Fy(N)p Fs(;)j Fx(let)g Fs(H)1867 477 y Fr(n)1913 463 y Fs(;)15 b(n)26 b Fq(2)e Fy(Z)p Fs(;)j Fx(b)s(e)f(indep)s(enden)m(t)f(random)h(v)-5 b(ariables)236 576 y(with)29 b(distribution)e Fs(\027)6 b Fx(.)40 b(Then)30 b(b)m(y)g(\(6.5\))i(and)e(\(6.6\))1026 731 y Fs(X)1108 693 y Fp(0)1101 753 y Fr(n)1173 731 y Fx(:=)141 b(lim)1295 789 y Fr(n)p Fp(\025)p Fr(m)p Fp(!\0001)1666 731 y Fs(H)1742 745 y Fr(n)1809 731 y Fq(\016)20 b Fs(:)15 b(:)g(:)22 b Fq(\016)e Fs(H)2142 745 y Fr(m)2224 731 y Fx(\(0\))26 b Fq(2)f Fs(E)63 b Fx(a.s.)16 b Fs(:)236 927 y Fx(The)30 b(con)m(tin)m(uit)m(y)g(of)h(the)f(mappings)f Fs(H)1598 941 y Fr(n)1645 927 y Fx(\()p Fs(!)s Fx(\))h(yields)1181 1082 y Fs(X)1263 1044 y Fp(0)1256 1104 y Fr(n)1328 1082 y Fx(=)25 b Fs(H)1500 1096 y Fr(n)1547 1082 y Fx(\()p Fs(X)1664 1044 y Fp(0)1657 1104 y Fr(n)p Fp(\000)p Fo(1)1794 1082 y Fx(\))59 b(a.s.)125 b(for)58 b Fs(n)25 b Fq(2)g Fy(N)15 b Fs(:)236 1237 y Fx(Since)40 b Fs(X)566 1204 y Fp(0)559 1260 y Fo(0)639 1237 y Fx(is)g(indep)s(enden)m(t)e(of)j(\()p Fs(H)1491 1251 y Fr(n)1538 1237 y Fs(;)15 b(n)42 b Fq(2)g Fy(N)p Fx(\))f(and)f Fq(L)p Fx(\()p Fs(X)2303 1204 y Fp(0)2296 1260 y Fo(0)2336 1237 y Fx(\))i(=)h Fs(\026)d Fx(b)m(y)g(\(6.7\),)46 b(the)40 b(pro)s(cesses)236 1350 y(\()p Fs(X)346 1364 y Fr(n)394 1350 y Fs(;)30 b(n)35 b Fq(\025)g Fx(0\))i(and)e(\()p Fs(X)1061 1317 y Fp(0)1054 1372 y Fr(n)1102 1350 y Fs(;)30 b(n)35 b Fq(\025)g Fx(0\))i(ha)m(v)m(e) g(the)f(same)h(distribution,)d(and)i(th)m(us)g(it)g(is)f(su\016cien)m (t)236 1463 y(to)e(pro)m(v)m(e)f(the)g(assertion)g(for)f(\()p Fs(X)1392 1430 y Fp(0)1385 1485 y Fr(n)1433 1463 y Fs(;)f(n)d Fq(\025)h Fx(0\).)45 b(T)-8 b(o)32 b(this)f(end)g(denote)h(b)m(y)g Fs(\033)j Fx(and)c Fs(\033)3048 1430 y Fp(0)3103 1463 y Fx(the)h(shift)f(in)236 1576 y Fs(W)38 b Fx(:=)481 1511 y Fi(Q)575 1618 y Fr(n)p Fp(2)p Fh(Z)752 1576 y Fq(H)q Fx([)p Fs(E)5 b Fx(])31 b(and)e Fs(W)1258 1543 y Fp(0)1306 1576 y Fx(:=)1428 1511 y Fi(Q)1521 1611 y Fr(n)p Fp(\025)p Fo(0)1687 1576 y Fs(E)1759 1543 y Fp(\003)1799 1576 y Fx(,)i(resp)s(ectiv)m(ely)-8 b(,)30 b(and)g(consider)f(the)i (mapping)841 1770 y Fs(\034)36 b Fx(:)25 b(\()p Fs(h)1054 1784 y Fr(n)1102 1770 y Fs(;)15 b(n)25 b Fq(2)g Fy(Z)p Fx(\))g Fq(7!)1548 1676 y Fi(\020)1728 1770 y Fx(lim)1613 1828 y Fr(n)p Fp(\025)p Fr(m)p Fp(!\0001)1985 1770 y Fs(h)2037 1784 y Fr(n)2104 1770 y Fq(\016)c Fs(:)15 b(:)g(:)21 b Fq(\016)g Fs(h)2414 1784 y Fr(m)2496 1770 y Fx(\(0\))p Fs(;)31 b(n)25 b Fq(\025)g Fx(0)2888 1676 y Fi(\021)236 1968 y Fx(from)39 b Fs(W)52 b Fx(to)40 b Fs(W)817 1935 y Fp(0)840 1968 y Fx(,)i(whic)m(h)d(is)f(easily)h(c)m(hec)m(k)m(ed)i (to)g(b)s(e)d(measurable,)k(due)c(to)j(the)e(top)s(ological)236 2081 y(prop)s(erties)34 b(of)h Fs(E)849 2048 y Fp(\003)889 2081 y Fx(.)55 b(Then)34 b(the)h(mappings)f Fs(\033)k Fx(and)c Fs(\033)2113 2048 y Fp(0)2172 2081 y Fx(are)h(conjugate)i (under)c Fs(\034)10 b Fx(,)37 b(i.e.)55 b(satisfy)236 2194 y Fs(\034)27 b Fq(\016)17 b Fs(\033)28 b Fx(=)d Fs(\033)596 2161 y Fp(0)637 2194 y Fq(\016)17 b Fs(\034)10 b Fx(.)40 b(Therefore)28 b(the)h(mixing)e(prop)s(ert)m(y)h(of)g Fs(\033)k Fx(with)27 b(resp)s(ect)i(to)g(the)g(pro)s(duct)e(mea-)236 2307 y(sure)442 2242 y Fi(N)549 2349 y Fr(n)p Fp(2)p Fh(Z)726 2307 y Fs(\027)49 b Fx(carries)28 b(o)m(v)m(er)h(to)g Fs(\033)1463 2274 y Fp(0)1515 2307 y Fx(with)e(resp)s(ect)h(to)h(its)e (image)h(b)m(y)g Fs(\034)10 b Fx(.)40 b(Since)28 b(this)f(ob)m(viously) 236 2431 y(is)j(the)g(distribution)d(of)j(\()p Fs(X)1200 2398 y Fp(0)1193 2453 y Fr(n)1241 2431 y Fs(;)h(n)24 b Fq(\025)h Fx(0\),)32 b(the)e(assertion)g(follo)m(ws.)1034 b Ff(\003)377 2615 y Fx(T)-8 b(o)48 b(b)s(e)e(complete,)52 b(it)47 b(has)g(to)h(b)s(e)e(men)m(tioned)h(that)h(in)e(general)h(the)g (tail)f Fs(\033)s Fq(\000)p Fx(\014eld)g(of)236 2728 y(\()p Fs(X)346 2742 y Fr(n)394 2728 y Fs(;)30 b(n)46 b Fq(\025)f Fx(0\),)i(ev)m(en)c(under)e(stationarit)m(y)-8 b(,)47 b(need)42 b(not)h(b)s(e)f(trivial.)75 b(A)43 b(coun)m (terexample)g(is)236 2841 y(pro)m(vided)38 b(b)m(y)i(the)f(Can)m(tor)h (system,)i(where)d Fs(X)1920 2855 y Fr(n)p Fp(\000)p Fo(1)2097 2841 y Fx(can)g(b)s(e)g(reconstructed)h(from)f Fs(X)3280 2855 y Fr(n)3367 2841 y Fx(with)236 2954 y(probabilit)m(y)22 b(1,)k(and)d(th)m(us)h(the)g(tail)g Fs(\033)s Fq(\000)p Fx(\014eld)f(of)h(\()p Fs(X)1985 2968 y Fr(n)2032 2954 y Fs(;)31 b(n)25 b Fq(\025)g Fx(0\))g(coincides)e(with)g(the)h(full)e Fs(\033)s Fq(\000)p Fx(\014eld)236 3067 y(generated)32 b(b)m(y)e(the)g(pro)s(cess)g(up)g(to)h(sets)g(of)f(probabilit)m(y)e(0.) 377 3210 y(No)m(w)j(a)g(fairly)e(general)h(la)m(w)g(of)h(large)f(n)m (um)m(b)s(ers)f(can)i(b)s(e)f(deriv)m(ed:)377 3406 y Fg(\(8.5\))64 b(Theorem)99 b Fk(L)-5 b(et)44 b(the)g(system)g Fx(\()p Fs(E)5 b(;)15 b(\027)6 b Fx(\))46 b Fk(b)-5 b(e)43 b(p)-5 b(ositive)45 b(r)-5 b(e)g(curr)g(ent)45 b(with)f(stationary)236 3519 y(distribution)34 b Fs(\026)p Fk(.)41 b(Then)33 b(for)h(arbitr)-5 b(ary)34 b(initial)g(law)1273 3645 y Fx(1)p 1268 3686 55 4 v 1268 3769 a Fs(n)1348 3626 y Fi(X)1483 3742 y Fo(0)p Fp(\024)p Fr(m)f Fx(0.)41 b(Then)29 b(the)h(initial)d(la)m(w)j Fs(")2350 620 y Fr(x)2424 606 y Fx(and)f(the)i(function)d Fs(f)35 b Fx(=)25 b(1)3333 620 y Fr(F)3422 606 y Fx(are)236 719 y(suited)k(for)i(a)f(coun)m(terexample.)377 986 y Fy(9.)115 b(Strictly)52 b(order-preserving)h(systems.)99 b Fx(Throughout)44 b(Sections)h(1)g({)h(8)f(the)236 1099 y(system)36 b(\()p Fs(E)5 b(;)15 b(\027)6 b Fx(\))36 b(has)g(alw)m(a)m(ys)h(b)s(een)e(supp)s(osed)f(to)j(b)s(e)e (irreducible.)54 b(T)-8 b(o)37 b(see)g(the)f(crucial)f(role)236 1212 y(of)k(this)f(assumption)f(consider)h(the)h(example)f(follo)m (wing)g(\(3.4\).)67 b(If)39 b(in)e(this)h(case)i(the)f(state)236 1325 y(space)32 b(is)f(enlarged)g(from)g([0)p Fs(;)15 b Fx(1[)33 b(to)g([0)p Fs(;)15 b Fx(1])33 b(\(extending)e(the)h (mappings)e Fs(h)2767 1339 y Fr(i)2827 1325 y Fx(con)m(tin)m(uously\),) h(the)236 1438 y(de\014nition)40 b Fs(\026)k Fx(=)g Fs(")908 1452 y Fo(1)989 1438 y Fx(yields)c(another)i(in)m(v)-5 b(arian)m(t)41 b(measure.)75 b(This)40 b(observ)-5 b(ation)41 b(suggests)i(a)236 1550 y(restriction)30 b(of)g(the)h(original)d(state) k(space:)377 1746 y Fg(\(9.1\))51 b(Definition)81 b Fx(F)-8 b(or)30 b(an)g(arbitrary)f(system)h(\()p Fs(E)5 b(;)15 b(\027)6 b Fx(\))26 b(the)31 b(\\)15 b Fk(r)-5 b(e)g(duc)g(e)g(d)34 b(state)f(sp)-5 b(ac)g(e)17 b Fx(")30 b(is)236 1859 y(giv)m(en)h(b)m(y) f(the)g(subspace)944 2037 y Fi(b)923 2063 y Fs(E)h Fx(:=)25 b Fq(f)p Fs(x)h Fq(2)f Fs(E)30 b Fx(:)c Fy(P)p Fx(\()p Fs(X)1687 2026 y Fo(0)1680 2086 y Fr(n)1753 2063 y Fq(\025)f Fs(x)p Fx(\))h Fs(>)f Fx(0)42 b(for)30 b(some)42 b Fs(n)25 b Fq(2)f Fy(N)p Fq(g)15 b Fs(:)377 2298 y Fx(If)32 b Fs(E)37 b Fx(is)30 b(totally)i(ordered,)1334 2271 y Fi(b)1314 2298 y Fs(E)37 b Fx(is)30 b(ob)m(viously)h(an)g(op)s(en)g(or)h(closed)g (decreasing)f(subset)g(of)h Fs(E)236 2411 y Fx(and)37 b(th)m(us)g(again)h(admissible)c(in)i(the)i(sense)g(of)f(Section)h(0.) 62 b(Moreo)m(v)m(er,)42 b(it)37 b(mak)m(es)h(sense)f(to)236 2523 y(restrict)30 b(the)h(mappings)e Fs(h)h Fx(to)1331 2497 y Fi(b)1310 2523 y Fs(E)6 b Fx(,)30 b(b)s(ecause)236 2728 y(\()p Fq(\003)p Fx(\))757 b Fs(h)p Fx([)1206 2701 y Fi(b)1185 2728 y Fs(E)6 b Fx(])25 b Fq(\032)1425 2701 y Fi(b)1404 2728 y Fs(E)88 b Fx(for)31 b Fs(\027)6 b Fq(\000)p 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Fo(0)1312 3249 y Fx(\)\))42 b Fq(\025)e Fy(P)p Fx(\()p Fs(X)1723 3212 y Fo(0)1716 3272 y Fr(n)1789 3249 y Fq(\025)25 b Fs(x)1937 3263 y Fo(0)1976 3249 y Fx(\))15 b Fy(P)p Fx(\()p Fs(H)2208 3263 y Fr(n)p Fo(+1)2346 3249 y Fx(\()p Fs(x)2433 3263 y Fo(0)2473 3249 y Fx(\))26 b Fq(\025)f Fs(h)2682 3263 y Fo(0)2722 3249 y Fx(\()p Fs(x)2809 3263 y Fo(0)2848 3249 y Fx(\)\))h Fs(>)f Fx(0)15 b Fs(:)236 3453 y Fx(Therefore,)31 b(with)e(the)h(notation)980 3658 y Fq(H)q Fx(\()p Fs(y)s Fx(\))25 b(:=)g Fq(f)p Fs(h)h Fq(2)f(H)q Fx([)p Fs(E)5 b Fx(])26 b(:)g Fs(h)p Fx(\()p Fs(x)1947 3672 y Fo(0)1987 3658 y Fx(\))f Fq(\025)g Fs(y)s Fq(g)125 b Fx(for)58 b Fs(y)28 b Fq(2)d Fs(E)c(;)236 3862 y Fx(eac)m(h)40 b Fs(h)501 3876 y Fo(0)580 3862 y Fq(2)f(H)q Fx([)p Fs(E)5 b Fx(])39 b(with)f Fs(h)1187 3876 y Fo(0)1227 3862 y Fx(\()p Fs(x)1314 3876 y Fo(0)1353 3862 y Fx(\))50 b Fs(=)-55 b Fq(2)1548 3835 y Fi(b)1528 3862 y Fs(E)44 b Fx(is)38 b(con)m(tained)h(in)e(the)i(union)e(of)i(all)e(sets)j Fq(H)q Fx(\()p Fs(y)s 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b(this)f(consideration)g(sho)m(ws,)k(to)d(treat)h(only)e (irreducible)d(systems)k(means)g(no)g(real)236 4539 y(restriction)32 b(under)e(a)j(total)g(ordering.)45 b(This,)32 b(ho)m(w)m(ev)m(er,)i(do) s(es)e(not)h(hold)e(for)h(a)h(general)f(state)236 4652 y(space,)42 b(as)d(the)g(follo)m(wing)e(examples)h(sho)m(w.)66 b(Cho)s(ose)38 b Fs(E)45 b Fx(=)39 b Fy(R)2502 4619 y Fo(2)2502 4675 y(+)2600 4652 y Fx(and)f(let)h Fs(\027)44 b Fx(b)s(e)38 b(supp)s(orted)236 4765 y(b)m(y)c(constan)m(t)g(mappings) e(\(resulting)g(in)g(indep)s(enden)m(t)f(v)-5 b(ariables)33 b Fs(X)2644 4779 y Fr(n)2691 4765 y Fs(;)d(n)g Fq(\025)h Fx(0\),)k(taking)e(their)236 4878 y(v)-5 b(alues)28 b(in)f(the)i (totally)f(disordered)e(subset)i Fs(D)g Fx(:=)d Fq(f)p Fs(x)h Fq(2)f Fs(E)31 b Fx(:)25 b Fs(x)2400 4892 y Fo(1)2456 4878 y Fx(+)16 b Fs(x)2595 4892 y Fo(2)2659 4878 y Fx(=)25 b(1)p Fq(g)30 b Fx(only)-8 b(.)39 b(If)28 b Fs(\027)34 b Fx(assigns)236 4991 y(p)s(ositiv)m(e)h(mass)h(to)h(eac)m(h)g(constan) m(t)g(in)e(a)i(dense)e(subset)h(of)g Fs(D)s Fx(,)h(the)g(reduced)e (state)i(space)3492 4964 y Fi(b)3471 4991 y Fs(E)236 5104 y Fx(is)31 b(easily)g(seen)i(to)f(b)s(e)g(no)g(longer)f(lo)s (cally)g(compact.)47 b(If)31 b(on)h(the)h(other)f(hand)f Fs(\027)38 b Fx(has)31 b(no)h(p)s(oin)m(t)236 5217 y(masses)f(at)g (all,)e(ob)m(viously)1019 5421 y Fs(h)p Fx([)1118 5395 y Fi(b)1096 5421 y Fs(E)6 b Fx(])21 b Fq(\\)1316 5395 y Fi(b)1295 5421 y Fs(E)31 b Fx(=)25 b Fq(;)114 b Fx(for)30 b Fs(\027)6 b Fq(\000)p Fx(almost)30 b(all)57 b Fs(h)26 b Fq(2)e(H)q Fx([)p Fs(E)5 b Fx(])15 b Fs(:)1844 5753 y Fx(33)p eop %%Page: 34 34 34 33 bop 377 154 a Fx(This)40 b(observ)-5 b(ation)41 b(is)g(the)g(motiv)-5 b(ation)42 b(to)g(in)m(v)m(estigate)h(systems)e (\()p Fs(E)5 b(;)15 b(\027)6 b Fx(\))45 b(that)d(are)g(not)236 267 y(necessarily)31 b(irreducible)e(but)i(strictly)g (order-preserving.)43 b(Before)33 b(in)m(tro)s(ducing)d(this)h(notion) 236 380 y(the)g(necessary)g(measurabilit)m(y)d(has)i(to)h(b)s(e)f (settled:)377 576 y Fg(\(9.2\))57 b(Pr)n(oposition)87 b Fk(The)38 b(subsp)-5 b(ac)g(e)39 b Fq(J)16 b Fx([)p Fs(E)5 b Fx(])38 b Fk(c)-5 b(onsisting)39 b(of)e(al)5 b(l)38 b(mappings)i Fs(h)34 b Fq(2)g(H)q Fx([)p Fs(E)5 b Fx(])236 689 y Fk(such)33 b(that)1198 802 y Fs(h)p Fx(\()p Fs(x)1337 816 y Fo(1)1377 802 y Fx(\))26 b Fs(<)f(h)p Fx(\()p Fs(x)1673 816 y Fo(2)1713 802 y Fx(\))83 b Fk(whenever)h Fs(x)2330 816 y Fo(1)2394 802 y Fs(<)25 b(x)2542 816 y Fo(2)236 966 y Fk(is)33 b(of)g(typ)-5 b(e)33 b Fs(G)702 981 y Fr(\016)773 966 y Fk(\(i.e.)41 b(a)33 b(Polish)h(sp)-5 b(ac)g(e\).)377 1162 y Fg(Pr)n(oof.)61 b Fx(Denote)31 b(b)m(y)f Fs(S)k Fx(the)c(op)s(en)f(set)h(of)f(all)g(pairs)f(\()p Fs(x)2332 1176 y Fo(1)2372 1162 y Fs(;)15 b(x)2464 1176 y Fo(2)2503 1162 y Fx(\))30 b(with)e Fs(x)2826 1176 y Fo(1)2891 1162 y Fs(<)d(x)3039 1176 y Fo(2)3078 1162 y Fx(.)41 b(Since)28 b Fs(E)35 b Fx(is)236 1275 y(lo)s(cally)29 b(compact)j(and)e(second)g(coun)m(table,)h(it)f(follo)m(ws)f(that)1048 1474 y Fs(S)h Fx(=)1230 1394 y Fi([)1337 1516 y Fr(l)q Fp(2)p Fh(N)1509 1474 y Fx(\()p Fs(K)1628 1437 y Fr(l)1621 1497 y Fo(1)1681 1474 y Fq(\002)20 b Fs(K)1856 1437 y Fr(l)1849 1497 y Fo(2)1888 1474 y Fx(\))84 b(with)e Fs(K)2351 1437 y Fr(l)2344 1497 y(i)2402 1474 y Fq(2)24 b Fm(K)p Fx(\()p Fs(E)5 b Fx(\))15 b Fs(:)236 1674 y Fx(Therefore)30 b(it)g(is)g(su\016cien)m(t)f(to)j(pro)m(v)m(e)1134 1873 y Fq(H)1211 1888 y Fr(l)1262 1873 y Fx(:=)25 b Fq(f)p Fs(h)h Fq(2)f(H)q Fx([)p Fs(E)5 b Fx(])26 b(:)f Fs(h)p Fx([)p Fs(K)2029 1836 y Fr(l)2022 1896 y Fo(1)2062 1873 y Fx(])c Fq(\002)f Fs(h)p Fx([)p Fs(K)2360 1836 y Fr(l)2353 1896 y Fo(2)2393 1873 y Fx(])25 b Fq(\032)g Fs(S)5 b Fq(g)236 2073 y Fx(to)37 b(b)s(e)e(op)s(en)g(for)g(all)g Fs(l)h Fq(2)d Fy(N)p Fx(.)56 b(In)35 b(view)g(of)h Fs(h)p Fx([)p Fs(K)1907 2040 y Fr(l)1900 2097 y(i)1933 2073 y Fx(])f Fq(2)e Fm(K)p Fx(\()p Fs(E)5 b Fx(\))37 b(a)f(theorem)g(of)g (W)-8 b(allace)36 b(\(see)h(e.g.)236 2186 y([16]\))d(applies,)c(i.e.)45 b Fs(h)28 b Fq(2)f(H)1181 2201 y Fr(l)1238 2186 y Fx(is)k(equiv)-5 b(alen)m(t)31 b(to)i(the)f(existence)g(of)g(sets)g Fs(G)2778 2200 y Fr(i)2834 2186 y Fq(2)27 b Fm(G)p Fx(\()p Fs(E)5 b Fx(\))34 b(satisfying)236 2299 y Fs(h)p Fx([)p Fs(K)397 2266 y Fr(l)390 2323 y(i)424 2299 y Fx(])25 b Fq(\032)g Fs(G)641 2313 y Fr(i)700 2299 y Fx(and)k Fs(G)947 2313 y Fo(1)1007 2299 y Fq(\002)19 b Fs(G)1168 2313 y Fo(2)1233 2299 y Fq(\032)25 b Fs(S)5 b Fx(.)41 b(Since)29 b(the)h(sets)h Fq(f)p Fs(h)26 b Fq(2)f(H)q Fx([)p Fs(E)5 b Fx(])25 b(:)h Fs(h)p Fx([)p Fs(K)2673 2266 y Fr(l)2666 2323 y(i)2699 2299 y Fx(])g Fq(\032)f Fs(G)2917 2313 y Fr(i)2945 2299 y Fq(g)31 b Fx(are)f(op)s(en,)g(the)236 2412 y(assertion)g(is)g (established,)2303 b Ff(\003)377 2596 y Fx(No)m(w)31 b(the)g(cen)m(tral)g(assumption)e(for)h(this)f(section)i(can)f(b)s(e)g (made)g(precise:)377 2792 y Fg(\(9.3\))58 b(Definition)95 b Fx(The)36 b(system)h(\()p Fs(E)5 b(;)15 b(\027)6 b Fx(\))38 b(is)e(called)g(\\)15 b Fk(strictly)40 b(or)-5 b(der-pr)g(eserving)17 b Fx(",)39 b(if)236 2905 y(the)31 b(follo)m(wing)e(t)m(w)m(o)i(conditions)e(are)i(satis\014ed:)236 3104 y(\(a\))1274 b Fs(\027)6 b Fx(\()p Fq(J)16 b Fx([)p Fs(E)5 b Fx(]\))27 b(=)e(1)15 b Fs(;)236 3284 y Fx(\(b\))794 b Fs(\027)1202 3247 y Fr(n)1249 3284 y Fx(\()p Fs(h)p Fx(\(0\))27 b Fs(>)e Fx(0\))h Fs(>)f Fx(0)125 b(for)30 b(some)59 b Fs(n)24 b Fq(2)h Fy(N)15 b Fs(:)377 3514 y Fx(Clearly)-8 b(,)30 b(condition)f(\(a\))i(is)f(of)g(relev)-5 b(ance)31 b(only)f(in)f(conjunction)g(with)g(condition)g(\(b\).)377 3626 y(T)-8 b(o)31 b(discuss)d(brie\015y)g(an)i(imp)s(ortan)m(t)f(sp)s (ecial)g(case,)i(consider)e(generalized)g(autoregressiv)m(e)236 3739 y(mo)s(dels)e(on)g Fs(E)k Fx(=)25 b Fy(R)935 3706 y Fr(d)935 3762 y Fo(+)994 3739 y Fx(,)k(where)e Fs(\027)34 b Fx(is)27 b(supp)s(orted)f(b)m(y)h(a\016ne)h(maps)f Fs(h)f Fx(:)g Fs(x)f Fq(7!)g Fs(Ax)15 b Fx(+)g Fs(b)p Fx(.)40 b(If)27 b Fs(A)h Fx(and)f Fs(b)236 3852 y Fx(are)g(comp)s(osed) f(of)g(the)h(\(nonnegativ)m(e\))g(v)-5 b(ariables)25 b Fs(a)2041 3867 y Fr(ik)2134 3852 y Fx(and)g Fs(b)2345 3866 y Fr(i)2374 3852 y Fx(,)i(resp)s(ectiv)m(ely)-8 b(,)27 b(then)f(conditions)236 3965 y(\(a\))32 b(and)d(\(b\))i(are)g (satis\014ed)e(as)i(so)s(on)f(as)236 4165 y(\(a)316 4132 y Fp(0)340 4165 y Fx(\))652 b Fy(P)p Fx(\()p Fs(a)1181 4179 y Fr(i)p Fo(1)1265 4165 y Fx(+)20 b Fs(:)15 b(:)g(:)21 b Fx(+)f Fs(a)1621 4180 y Fr(id)1711 4165 y Fs(>)25 b Fx(0\))h(=)f(1)125 b(for)58 b(1)26 b Fq(\024)f Fs(i)g Fq(\024)g Fs(d)15 b(;)236 4345 y Fx(\(b)322 4312 y Fp(0)345 4345 y Fx(\))1105 b Fy(P)p Fx(\()p Fs(b)1630 4359 y Fo(1)1685 4345 y Fs(:)15 b(:)g(:)h(b)1845 4360 y Fr(d)1911 4345 y Fs(>)25 b Fx(0\))h Fs(>)f Fx(0)15 b Fs(:)236 4544 y Fx(This)39 b(example)i(should)e(b)s(e)h(compared)h(with)f(the)h(mo)s (del)f(in)f([4].)74 b(While)39 b(the)i(state)i(space)236 4657 y(there)36 b(is)e(enlarged)h(to)h Fs(E)i Fx(=)33 b Fy(R)1342 4624 y Fr(d)1383 4657 y Fx(,)k(only)d(mappings)g Fs(h)f Fx(:)h Fs(x)f Fq(7!)g Fs(ax)24 b Fx(+)f Fs(b)35 b Fx(with)f(strictly)g(p)s(ositiv)m(e)236 4770 y(scalar)25 b(factors)i Fs(a)e Fx(are)g(admitted,)i(imp)s(osing)22 b(in)i(addition)g(strong)h(momen)m(t)h(conditions)e(on)h(the)236 4883 y(v)-5 b(ariables)29 b Fs(a)i Fx(and)e Fs(b)p Fx(.)377 4996 y(The)h(notions)g(in)f(\(9.1\))j(and)e(\(9.3\))i(are)f(related)f (b)m(y)g(the)h(follo)m(wing)e(facts:)377 5192 y Fg(\(9.4\))52 b(Lemma)77 b Fk(If)33 b(the)g(system)h Fx(\()p Fs(E)5 b(;)15 b(\027)6 b Fx(\))27 b Fk(is)33 b(strictly)g(or)-5 b(der-pr)g(eserving,)35 b(then)f(any)f Fs(x)26 b Fq(2)3492 5165 y Fi(b)3471 5192 y Fs(E)236 5305 y Fk(satis\014es)236 5504 y Fx(\(a\))829 b Fy(P)p Fx(\()p Fs(X)1368 5466 y Fo(0)1361 5527 y Fr(n)1434 5504 y Fs(>)25 b(x)p Fx(\))h Fs(>)f Fx(0)125 b Fk(for)33 b(some)59 b Fs(n)25 b Fq(2)f Fy(N)15 b Fs(;)1844 5753 y Fx(34)p eop %%Page: 35 35 35 34 bop 236 131 a Fx(\(b\))1023 b Fs(x)25 b(<)g(y)127 b Fk(for)33 b(some)59 b Fs(y)28 b Fq(2)2308 105 y Fi(b)2287 131 y Fs(E)21 b(;)236 304 y Fx(\(c\))855 b Fs(h)p Fx(\()p Fs(x)p Fx(\))26 b Fq(2)1508 278 y Fi(b)1487 304 y Fs(E)130 b Fk(for)33 b(al)5 b(l)59 b Fs(h)25 b Fq(2)g(N)33 b(\\)20 b(J)d Fx([)p Fs(E)5 b Fx(])15 b Fs(:)377 572 y Fg(Pr)n(oof.)148 b Fx(Since)57 b Fq(J)17 b Fx([)p Fs(E)5 b Fx(])58 b(is)f(stable)h (under)e(comp)s(osition,)64 b Fs(\027)6 b Fx(\()p Fq(J)16 b Fx([)p Fs(E)5 b Fx(]\))73 b(=)e(1)58 b(implies)236 685 y Fs(\027)287 652 y Fr(k)330 685 y Fx(\()p Fq(J)16 b Fx([)p Fs(E)5 b Fx(]\))40 b(=)f(1)f(for)h(all)e Fs(k)42 b Fq(2)c Fy(N)p Fx(.)65 b(In)37 b(the)i(sequel)f(denote)h(the)g(supp)s (ort)d(of)j Fs(\027)3013 652 y Fr(k)3094 685 y Fx(b)m(y)f Fq(N)3316 652 y Fr(k)3397 685 y Fx(and)236 798 y(its)30 b(elemen)m(ts)h(b)m(y)f Fs(h)913 813 y Fr(k)956 798 y Fx(.)377 941 y(\(a\))i(By)e(the)h(assumptions)e(on)h Fs(x)g Fx(and)g(\()p Fs(E)5 b(;)15 b(\027)6 b Fx(\))26 b(there)31 b(are)g Fs(l)c Fq(2)e Fy(N)30 b Fx(and)g Fs(m)25 b Fq(2)g Fy(N)30 b Fx(suc)m(h)g(that)994 1137 y Fs(\027)1045 1099 y Fr(l)1071 1137 y Fx(\()p Fs(h)1158 1152 y Fr(l)1185 1137 y Fx(\(0\))c Fq(\025)f Fs(x)p Fx(\))h Fs(>)f Fx(0)83 b(and)f Fs(\027)2039 1099 y Fr(m)2106 1137 y Fx(\()p Fs(h)2193 1151 y Fr(m)2260 1137 y Fx(\(0\))26 b Fs(>)f Fx(0\))i Fs(>)d Fx(0)15 b Fs(:)236 1333 y Fx(With)30 b Fs(n)25 b Fx(=)g Fs(l)d Fx(+)e Fs(m)30 b Fx(and)1234 1446 y Fq(J)1296 1460 y Fr(a)1362 1446 y Fx(:=)25 b Fq(f)p Fx(\()p Fs(h)1615 1461 y Fr(l)1642 1446 y Fs(;)15 b(h)1734 1460 y Fr(m)1802 1446 y Fx(\))25 b(:)h Fs(h)1965 1461 y Fr(l)2011 1446 y Fq(\016)21 b Fs(h)2129 1460 y Fr(m)2211 1446 y Fx(\(0\))27 b Fs(>)e(x)p Fq(g)236 1609 y Fx(this)k(implies)516 1805 y Fs(\027)567 1768 y Fr(n)614 1805 y Fx(\()p Fs(h)701 1819 y Fr(n)749 1805 y Fx(\(0\))d Fs(>)f(x)p Fx(\))h(=)f Fs(\027)1246 1768 y Fr(l)1291 1805 y Fq(\012)20 b Fs(\027)1433 1768 y Fr(m)1515 1805 y Fx(\()p Fq(J)1612 1819 y Fr(a)1653 1805 y Fx(\))26 b Fq(\025)f Fs(\027)1861 1768 y Fr(l)1907 1805 y Fq(\012)20 b Fs(\027)2049 1768 y Fr(m)2130 1805 y Fx(\()p Fs(h)2217 1820 y Fr(l)2244 1805 y Fx(\(0\))26 b Fq(\025)f Fs(x;)31 b(h)2641 1819 y Fr(m)2708 1805 y Fx(\(0\))26 b Fs(>)f Fx(0\))41 b Fs(>)f Fx(0)15 b Fs(:)377 2002 y Fx(\(b\))43 b(By)f(\(a\))i(there)e(are)h Fs(h)1314 1969 y Fo(0)1314 2024 y Fr(m)1426 2002 y Fq(2)i(N)1620 1969 y Fr(m)1729 2002 y Fx(with)c Fs(h)2000 1969 y Fo(0)2000 2024 y Fr(m)2067 2002 y Fx(\(0\))46 b Fs(>)f Fx(0)e(and)f Fs(h)2673 1969 y Fo(0)2673 2024 y Fr(n)2765 2002 y Fq(2)j(N)2959 1969 y Fr(n)3034 2002 y Fq(\\)28 b(J)16 b Fx([)p Fs(E)5 b Fx(])44 b(with)236 2115 y Fs(y)28 b Fx(:=)e Fs(h)483 2082 y Fo(0)483 2137 y Fr(n)530 2115 y Fx(\(0\))g Fs(>)f(x)p Fx(.)41 b(Then)29 b Fs(h)1174 2082 y Fo(0)1174 2137 y Fr(n)1242 2115 y Fq(\016)20 b Fs(h)1359 2082 y Fo(0)1359 2137 y Fr(m)1442 2115 y Fx(\(0\))26 b Fs(>)f(y)33 b Fx(and)d(th)m(us) 1218 2311 y Fq(J)1280 2326 y Fr(b)1339 2311 y Fx(:=)c Fq(f)p Fx(\()p Fs(h)1593 2325 y Fr(n)1641 2311 y Fs(;)15 b(h)1733 2325 y Fr(m)1800 2311 y Fx(\))26 b(:)f Fs(h)1963 2325 y Fr(n)2031 2311 y Fq(\016)c Fs(h)2149 2325 y Fr(m)2231 2311 y Fx(\(0\))26 b Fs(>)f(y)s Fq(g)236 2507 y Fx(de\014nes)e(an)h(op) s(en)f(subset)g(of)h Fq(H)q Fx([)p Fs(E)5 b Fx(])i Fq(\002)g(H)q Fx([)p Fs(E)e Fx(])25 b(in)m(tersecting)e(the)h(supp)s(ort)e(of)i Fs(\027)2849 2474 y Fr(n)2903 2507 y Fq(\012)7 b Fs(\027)3032 2474 y Fr(m)3098 2507 y Fx(.)38 b(Therefore)1078 2703 y Fs(\027)1129 2666 y Fr(n)p Fo(+)p Fr(m)1293 2703 y Fx(\()p Fs(h)1380 2717 y Fr(n)p Fo(+)p Fr(m)1545 2703 y Fx(\(0\))26 b Fs(>)f(y)s Fx(\))h(=)f Fs(\027)2038 2666 y Fr(n)2105 2703 y Fq(\012)19 b Fs(\027)2246 2666 y Fr(m)2328 2703 y Fx(\()p Fq(J)2425 2718 y Fr(b)2459 2703 y Fx(\))26 b Fs(>)f Fx(0)15 b Fs(;)236 2900 y Fx(hence)31 b(in)e(particular)g Fs(y)f Fq(2)1194 2873 y Fi(b)1173 2900 y Fs(E)6 b Fx(.)377 3042 y(\(c\))34 b(Cho)s(ose)f Fs(h)892 3009 y Fo(0)892 3066 y(1)961 3042 y Fx(=)c Fs(h)k Fx(and)f Fs(h)1377 3009 y Fo(0)1377 3065 y Fr(n)1457 3042 y Fx(as)h(in)f(\(b\).)48 b(Then)32 b Fs(h)2166 3009 y Fo(0)2166 3066 y(1)2228 3042 y Fq(\016)22 b Fs(h)2347 3009 y Fo(0)2347 3065 y Fr(n)2409 3042 y Fx(\(0\))31 b Fs(>)e(h)p Fx(\()p Fs(x)p Fx(\),)35 b(and)d(it)g(follo)m(ws)g(as)236 3155 y(ab)s(o)m(v)m(e,)g (considering)d(no)m(w)1161 3352 y Fq(J)1223 3366 y Fr(c)1283 3352 y Fx(:=)c Fq(f)p Fx(\()p Fs(h)1536 3366 y Fo(1)1577 3352 y Fs(;)15 b(h)1669 3366 y Fr(n)1717 3352 y Fx(\))25 b(:)h Fs(h)1880 3366 y Fo(1)1940 3352 y Fq(\016)20 b Fs(h)2057 3366 y Fr(n)2120 3352 y Fx(\(0\))26 b Fs(>)f(h)p Fx(\()p Fs(x)p Fx(\))p Fq(g)15 b Fs(;)236 3548 y Fx(that)31 b Fs(h)p Fx(\()p Fs(x)p Fx(\))26 b Fq(2)740 3522 y Fi(b)719 3548 y Fs(E)6 b Fx(.)2656 b Ff(\003)377 3732 y Fx(The)30 b(crucial)f(prop)s(erties)g(of)i(the)f(reduced)g(state)i(space)f(follo) m(w)e(readily:)377 3928 y Fg(\(9.5\))51 b(Pr)n(oposition)78 b Fk(If)32 b(the)h(system)g Fx(\()p Fs(E)5 b(;)15 b(\027)6 b Fx(\))27 b Fk(is)32 b(strictly)i(or)-5 b(der-pr)g(eserving,)34 b(then)236 4124 y Fx(\(a\))1487 4098 y Fi(b)1466 4124 y Fs(E)47 b Fk(is)33 b(lo)-5 b(c)g(al)5 b(ly)34 b(c)-5 b(omp)g(act)18 b Fs(;)236 4298 y Fx(\(b\))748 b Fs(h)p Fx([)1204 4271 y Fi(b)1182 4298 y Fs(E)6 b Fx(])26 b Fq(\032)1422 4271 y Fi(b)1402 4298 y Fs(E)88 b Fk(for)33 b Fs(\027)6 b Fq(\000)p Fk(almost)34 b(al)5 b(l)59 b Fs(h)25 b Fq(2)g(H)q Fx([)p Fs(E)5 b Fx(])15 b Fs(;)236 4471 y Fx(\(c\))1489 4445 y Fi(b)1469 4471 y Fs(E)47 b Fk(is)32 b(lo)-5 b(c)g(al)5 b(ly)35 b(b)-5 b(ounde)g(d)16 b Fs(:)377 4739 y Fg(Pr)n(oof.)62 b Fx(\(a\))920 4712 y Fi(b)900 4739 y Fs(E)35 b Fx(is)30 b(a)h(decreasing)f(subset)g(of)h Fs(E)5 b Fx(,)31 b(hence)g(b)m(y)f(\(9.4b\))i(op)s(en)e(in)f Fs(E)36 b Fx(and)30 b(th)m(us)236 4852 y(again)h(lo)s(cally)e(compact.) 377 4995 y(\(b\))f(Since)780 4968 y Fi(b)760 4995 y Fs(E)k Fx(is)26 b(second)h(coun)m(table)g(and)g(b)m(y)g(\(9.4b\))h(co)m(v)m (ered)h(b)m(y)e(the)g(family)e(of)i(op)s(en)g(sets)236 5107 y Fq(f)p Fs(x)33 b Fq(2)479 5081 y Fi(b)458 5107 y Fs(E)k Fx(:)c Fs(x)f(<)f(y)s Fq(g)p Fs(;)g(y)k Fq(2)1148 5081 y Fi(b)1127 5107 y Fs(E)6 b Fx(,)35 b(there)g(are)g Fs(y)1698 5122 y Fr(k)1772 5107 y Fq(2)1885 5081 y Fi(b)1864 5107 y Fs(E)40 b Fx(suc)m(h)34 b(that)2402 5081 y Fi(b)2381 5107 y Fs(E)j Fx(=)2588 5043 y Fi(S)2672 5149 y Fr(k)r Fp(2)p Fh(N)2860 5107 y Fx([0)p Fs(;)15 b(y)3015 5122 y Fr(k)3059 5107 y Fx(].)52 b(Therefore)236 5248 y Fs(h)p Fx([)335 5222 y Fi(b)313 5248 y Fs(E)6 b Fx(])34 b Fq(\032)569 5222 y Fi(b)549 5248 y Fs(E)40 b Fx(if)34 b(and)h(only)f(if)g Fs(h)p Fx(\()p Fs(y)1350 5263 y Fr(k)1393 5248 y Fx(\))g Fq(2)1576 5222 y Fi(b)1555 5248 y Fs(E)41 b Fx(for)35 b(all)f Fs(k)i Fq(2)d Fy(N)p Fx(,)j(as)g(holds)d(indeed)h(b)m(y)h (\(9.4c\))i(for)e(all)236 5361 y Fs(h)26 b Fq(2)f(N)33 b(\\)20 b(J)c Fx([)p Fs(E)5 b Fx(].)377 5504 y(\(c\))32 b(Lo)s(cal)e(b)s(oundedness)e(is)h(immediate)h(from)g(\(9.4b\).)1162 b Ff(\003)1844 5753 y Fx(35)p eop %%Page: 36 36 36 35 bop 377 157 a Fx(Since)35 b(the)h(assumptions)d(\(E2\))k(and)e (\(E3\))h(clearly)f(carry)g(o)m(v)m(er)i(from)e Fs(E)41 b Fx(to)3113 131 y Fi(b)3092 157 y Fs(E)5 b Fx(,)37 b(to)s(o,)h(the)236 270 y(reduced)j(state)h(space)g(is)e(again)h(admissible)d(in)i(the)h (sense)h(of)f(Section)g(0.)73 b(Therefore)41 b(the)236 383 y(reduction)27 b(of)h(a)g(strictly)f(order-preserving)f(system)i (\()p Fs(E)5 b(;)15 b(\027)6 b Fx(\))26 b(can)i(b)s(e)g(summarized)e (as)i(follo)m(ws:)236 496 y(Disregarding)36 b(the)h Fs(\027)6 b Fq(\000)p Fx(n)m(ull)35 b(set)j(of)f(mappings)e(with)h Fs(h)p Fx([)2236 470 y Fi(b)2215 496 y Fs(E)6 b Fx(])37 b Fq(6\032)2478 470 y Fi(b)2457 496 y Fs(E)5 b Fx(,)39 b(let)2736 493 y Fi(b)2731 496 y Fs(\027)k Fx(b)s(e)37 b(the)g(image)g(of)h Fs(\027)236 609 y Fx(under)31 b(the)i(\(con)m(tin) m(uous\))g(mapping)d(that)k(assigns)d(to)i Fs(h)d Fq(2)e(H)q Fx([)p Fs(E)5 b Fx(])33 b(its)f(restriction)3148 582 y Fi(b)3147 609 y Fs(h)d Fq(2)g(H)q Fx([)3441 583 y Fi(b)3421 609 y Fs(E)5 b Fx(].)236 722 y(Then)22 b(the)h(system)g(\()965 695 y Fi(b)944 722 y Fs(E)6 b(;)1062 719 y Fi(b)1057 722 y Fs(\027)g Fx(\))23 b(is)f(irreducible,)f(and)i(an)f(asso)s (ciated)i(pro)s(cess)e(\()2803 695 y Fi(b)2777 722 y Fs(X)2852 736 y Fr(n)2900 722 y Fs(;)30 b(n)25 b Fq(\025)g Fx(0\))f(b)s(eha)m(v)m(es)236 835 y(as)h(\()p Fs(X)452 849 y Fr(n)500 835 y Fs(;)31 b(n)24 b Fq(\025)h Fx(0\),)i(whenev)m(er)e (the)g(initial)d(la)m(w)j(is)f(supp)s(orted)f(b)m(y)2473 808 y Fi(b)2453 835 y Fs(E)5 b Fx(.)39 b(By)25 b(\(9.5c\),)j(moreo)m(v) m(er,)g(the)236 948 y(results)h(\(5.7\))i(and)f(\(7.3\))h(on)f (attractor)i(and)d(mean)h(passage)h(time)e(simplify)d(for)k(the)g (reduced)236 1061 y(system.)377 1328 y Fy(10.)65 b(Order)34 b(and)h(top)s(ology)-9 b(.)57 b Fx(A)30 b(set)h(is)f(called)f(an)h Fk(or)-5 b(der)g(e)g(d)36 b(top)-5 b(olo)g(gic)g(al)36 b(sp)-5 b(ac)g(e)31 b Fx(\(OTS\),)236 1441 y(if)f(its)f(top)s(ology)i (and)f(\(partial\))g(order)g(are)g(compatible,)g(i.e.)1117 1644 y Fs(R)c Fx(:=)g Fq(f)p Fx(\()p Fs(x)1466 1658 y Fo(1)1506 1644 y Fs(;)15 b(x)1598 1658 y Fo(2)1638 1644 y Fx(\))25 b(:)h Fs(x)1801 1658 y Fo(1)1865 1644 y Fq(\024)f Fs(x)2013 1658 y Fo(2)2053 1644 y Fq(g)41 b(2)f Fm(F)p Fx(\()p Fs(E)25 b Fq(\002)20 b Fs(E)5 b Fx(\))15 b Fs(:)236 1847 y Fx(By)22 b(symmetry)f(this)g(holds)f(as)h(w)m(ell)g(for)g(the)h (in)m(v)m(erse)f(ordering,)i(hence)e(the)h(diagonal)f(is)g(a)g(closed) 236 1959 y(subset)28 b(of)i Fs(E)22 b Fq(\002)17 b Fs(E)5 b Fx(,)30 b(and)e(th)m(us)g(eac)m(h)i(OTS)e(is)g(a)h(Hausdor\013)f (space.)41 b(Moreo)m(v)m(er,)32 b(eac)m(h)e(subspace)236 2072 y(with)i(the)g(induced)f(ordering)g(and)h(eac)m(h)i(pro)s(duct)e (space)h(with)e(the)i(pro)s(duct)f(ordering)f(yield)236 2185 y(again)g(an)f(OTS.)377 2298 y(The)40 b(simplest)e(example)i(of)g (an)g(OTS)f(is)g(pro)m(vided)g(b)m(y)h(a)h(totally)f(ordered)f(set)i Fs(E)5 b Fx(,)43 b(the)236 2411 y(top)s(ology)26 b(b)s(eing)e (generated)i(b)m(y)g(the)f(\\)16 b(op)s(en)24 b(in)m(terv)-5 b(als)15 b(")25 b Fs(E)16 b Fq(n)10 b Fx([)p Fs(x;)31 b Fq(\001)15 b Fx(])26 b(and)f Fs(E)16 b Fq(n)10 b Fx([)15 b Fq(\001)g Fs(;)g(x)p Fx(].)41 b(While)24 b(the)236 2524 y(meaning)j(of)g(a)g(strict)g(inequalit)m(y)f(\\)15 b Fs(x)1535 2538 y Fo(1)1600 2524 y Fs(<)25 b(x)1748 2538 y Fo(2)1803 2524 y Fx(")i(is)f(clear)h(in)f(this)g(case,)j(more)f (care)g(is)e(necessary)236 2637 y(in)j(the)i(general)f(case:)377 2833 y Fg(\(10.1\))67 b(Definition)111 b Fx(Let)45 b Fs(E)50 b Fx(b)s(e)44 b(an)h(arbitrary)f(OTS.)g(Then)f(\\)16 b Fs(x)2890 2847 y Fo(1)2979 2833 y Fs(<)49 b(x)3151 2847 y Fo(2)3205 2833 y Fx(")c(means)236 2946 y(existence)31 b(of)g(disjoin)m(t)d(neigh)m(b)s(orho)s(o)s(ds)g Fs(G)1728 2960 y Fr(i)1787 2946 y Fx(of)j Fs(x)1943 2960 y Fr(i)2001 2946 y Fx(suc)m(h)f(that)236 3149 y(\(a\))890 b Fs(G)1312 3163 y Fo(1)1377 3149 y Fq(2)25 b Fm(G)1534 3103 y Fj(#)1570 3149 y Fx(\()p Fs(E)5 b Fx(\))84 b(and)f Fs(G)2097 3163 y Fo(2)2162 3149 y Fq(2)25 b Fm(G)2319 3103 y Fj(")2355 3149 y Fx(\()p Fs(E)5 b Fx(\))15 b Fs(;)236 3335 y Fx(\(b\))1089 b Fs(y)1491 3349 y Fo(1)1555 3335 y Fq(\024)25 b Fs(y)1696 3349 y Fo(2)1818 3335 y Fx(for)83 b Fs(y)2055 3349 y Fr(i)2108 3335 y Fq(2)25 b Fs(G)2265 3349 y Fr(i)2309 3335 y Fs(:)377 3568 y Fx(This)k(implies)f(in)h(particular)1134 3771 y Fs(S)h Fx(:=)c Fq(f)p Fx(\()p Fs(x)1474 3785 y Fo(1)1514 3771 y Fs(;)15 b(x)1606 3785 y Fo(2)1646 3771 y Fx(\))25 b(:)h Fs(x)1809 3785 y Fo(1)1873 3771 y Fs(<)f(x)2021 3785 y Fo(2)2061 3771 y Fq(g)41 b(2)f Fm(G)p Fx(\()p Fs(E)26 b Fq(\002)20 b Fs(E)5 b Fx(\))236 3974 y(and)30 b(the)h(transitivit)m(y)e(la)m(w)810 4176 y Fs(x)862 4190 y Fo(1)927 4176 y Fs(<)c(x)1075 4190 y Fo(3)1197 4176 y Fx(whenev)m(er)83 b Fs(x)1701 4190 y Fo(1)1766 4176 y Fs(<)25 b(x)1914 4190 y Fo(2)1978 4176 y Fq(\024)g Fs(x)2126 4190 y Fo(3)2249 4176 y Fx(or)83 b Fs(x)2465 4190 y Fo(1)2529 4176 y Fq(\024)25 b Fs(x)2677 4190 y Fo(2)2742 4176 y Fs(<)g(x)2890 4190 y Fo(3)2944 4176 y Fs(:)377 4379 y Fx(Another)31 b(notion)e(com)m(bining)g(order)h(and)g (top)s(ology)h(app)s(ears)e(in)g(Sections)h(5)h(and)f(7:)377 4575 y Fg(\(10.2\))50 b(Definition)79 b Fx(Let)29 b Fs(E)34 b Fx(b)s(e)29 b(an)f(arbitrary)g(OTS.)g(Then)g Fs(E)34 b Fx(is)28 b(\\)16 b Fk(lo)-5 b(c)g(al)5 b(ly)33 b(b)-5 b(ounde)g(d)17 b Fx(",)236 4688 y(if)30 b(an)m(y)g Fs(x)25 b Fq(2)g Fs(E)36 b Fx(has)30 b(a)h(b)s(ounded)d(neigh)m(b)s(orho)s(o)s (d.)377 4872 y(While)36 b(this)h(condition)f(is)g(clearly)h (satis\014ed)f(under)g(a)i(total)g(ordering,)g(it)f(ma)m(y)h(w)m(ell)e (b)s(e)236 4985 y(violated)30 b(in)f(the)i(general)f(case:)42 b(consider,)30 b(for)g(instance,)g(the)h(subspace)1188 5188 y Fs(E)g Fx(=)25 b Fq(f)p Fx(\()p Fs(x)1514 5202 y Fo(1)1554 5188 y Fs(;)15 b(x)1646 5202 y Fo(2)1685 5188 y Fx(\))26 b Fq(2)f Fy(R)1910 5151 y Fo(2)1910 5211 y(+)1995 5188 y Fx(:)g Fs(x)2097 5202 y Fo(1)2157 5188 y Fx(+)20 b Fs(x)2300 5202 y Fo(2)2364 5188 y Fq(\024)25 b Fx(1)p Fq(g)15 b Fs(:)377 5391 y Fx(The)33 b(deep)s(est)h(result)e (on)i(order)f(and)g(top)s(ology)h(used)e(here)i(is)e(the)i(analogue)g (of)g(Tietze's)236 5504 y(extension)c(theorem,)h(due)f(to)h(Nac)m(h)m (bin:)1844 5753 y(36)p eop %%Page: 37 37 37 36 bop 377 154 a Fg(\(10.3\))58 b(Pr)n(oposition)88 b Fk(L)-5 b(et)38 b Fs(E)43 b Fk(b)-5 b(e)38 b(a)g(c)-5 b(omp)g(act)40 b(OTS)e(and)h Fs(E)2635 168 y Fo(0)2712 154 y Fk(b)-5 b(e)38 b(a)g(close)-5 b(d)39 b(subsp)-5 b(ac)g(e.)236 267 y(Then)33 b(any)g(function)g Fs(f)1050 281 y Fo(0)1114 267 y Fq(2)25 b(C)1253 226 y Fj(")1289 267 y Fx(\()p Fs(E)1391 281 y Fo(0)1431 267 y Fx(\))33 b Fk(c)-5 b(an)33 b(b)-5 b(e)33 b(extende)-5 b(d)34 b(to)f(a)g (function)g Fs(f)h Fq(2)25 b(C)2912 226 y Fj(")2948 267 y Fx(\()p Fs(E)5 b Fx(\))p Fk(.)377 463 y Fg(Pr)n(oof.)61 b Fx(See)31 b([33,)h(Corollary)d(3.4)i(and)f(Theorem)g(3.6].)1113 b Ff(\003)377 647 y Fx(The)29 b(follo)m(wing)f(fact)i(is)e(an)h (immediate)f(consequence)i(of)g(\(10.3\))h(and)e(therefore)h(supplied) 236 760 y(with)f(its)h(simple)e(pro)s(of:)377 956 y Fg(\(10.4\))49 b(Pr)n(oposition)73 b Fk(L)-5 b(et)31 b Fs(E)36 b Fk(b)-5 b(e)30 b(\(a)h(subsp)-5 b(ac)g(e)32 b(of)9 b(\))31 b(a)g(c)-5 b(omp)g(act)33 b(OTS.)d(Then)h(the)f(class)236 1069 y(of)j(c)-5 b(onvex)33 b(op)-5 b(en)34 b(sets)f(is)f(a)h(b)-5 b(ase)34 b(of)e(the)h(top)-5 b(olo)g(gy.)377 1265 y Fg(Pr)n(oof.)100 b Fx(Assume)43 b Fs(E)48 b Fx(to)43 b(b)s(e)f(compact)i(and)e(apply)f (\(10.3\))k(to)f Fs(E)2808 1279 y Fo(0)2893 1265 y Fx(=)i Fq(f)p Fs(x)3107 1279 y Fo(1)3147 1265 y Fs(;)15 b(x)3239 1279 y Fo(2)3278 1265 y Fq(g)44 b Fx(with)236 1378 y Fs(x)288 1392 y Fo(1)366 1378 y Fq(6)p Fx(=)38 b Fs(x)527 1392 y Fo(2)605 1378 y Fx(to)h(see)g(that)g Fq(C)1137 1336 y Fj(")1173 1378 y Fx(\()p Fs(E)5 b Fx(\))39 b(separates)g(the)f (p)s(oin)m(ts)g(of)g Fs(E)5 b Fx(.)65 b(Therefore)38 b Fq(C)2949 1336 y Fj(")2985 1378 y Fx(\()p Fs(E)5 b Fx(\))39 b(induces)e(a)236 1491 y(Hausdor\013)g(top)s(ology)h(in)e Fs(E)42 b Fx(coarser)c(than)f(the)h(underlying)c(compact)39 b(top)s(ology)-8 b(,)40 b(and)d(th)m(us)236 1604 y(b)s(oth)32 b(top)s(ologies)g(agree.)47 b(Since)32 b(sets)g Fq(f)p Fs(x)d Fq(2)f Fs(E)34 b Fx(:)28 b Fs(a)h(<)f(f)10 b Fx(\()p Fs(x)p Fx(\))28 b Fs(<)g(b)p Fq(g)33 b Fx(are)g(con)m(v)m(ex)g(for)f Fs(f)38 b Fq(2)28 b(C)3364 1562 y Fj(")3400 1604 y Fx(\()p Fs(E)5 b Fx(\))236 1716 y(and)30 b Fs(a;)15 b(b)26 b Fq(2)f Fy(R)30 b Fx(and)g(con)m(v)m(exit)m(y)i(is)e(stable)g(under)e (in)m(tersection,)j(the)f(assertion)g(follo)m(ws.)176 b Ff(\003)377 1901 y Fx(The)38 b(next)g(result)e(is)h(crucial)g(for)g (the)h(in)m(tro)s(duction)e(of)i(admissible)d(state)k(spaces)f Fs(E)44 b Fx(in)236 2014 y(Section)30 b(0:)377 2210 y Fg(\(10.5\))54 b(Pr)n(oposition)82 b Fk(L)-5 b(et)35 b Fs(E)40 b Fk(b)-5 b(e)34 b(a)i(lo)-5 b(c)g(al)5 b(ly)36 b(c)-5 b(omp)g(act)37 b(OTS)e(and)g Fs(E)2886 2177 y Fp(\003)2955 2210 y Fx(=)29 b Fs(E)e Fq([)22 b(f1g)35 b Fk(its)236 2323 y(A)n(lexandr)-5 b(ov)37 b(c)-5 b(omp)g(acti\014c)g (ation.)53 b(Then,)36 b(de\014ning)f Fs(x)30 b Fq(\024)f(1)35 b Fk(for)h(al)5 b(l)35 b Fs(x)30 b Fq(2)g Fs(E)40 b Fk(makes)c Fs(E)3255 2290 y Fp(\003)3330 2323 y Fk(again)236 2435 y(an)d(OTS)g(if)f(and)i(only)f(if)236 2597 y Fx(\()p Fq(\003)p 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Fs(R)2110 3181 y Fp(\003)2164 3219 y Fx(])26 b Fq(2)f Fm(K)p Fx(\()p Fs(E)2469 3181 y Fp(\003)2509 3219 y Fx(\))15 b Fs(;)236 3380 y Fx(where)30 b Fq(1)35 b Fs(=)-55 b Fq(2)25 b Fs(K)785 3339 y Fj(#)851 3380 y Fx(and)30 b(th)m(us)g(indeed)f Fs(K)1600 3339 y Fj(#)1661 3380 y Fq(2)c Fm(K)p Fx(\()p Fs(E)5 b Fx(\).)377 3523 y(2.)50 b(Let)34 b(con)m(v)m(ersely)g(condition)d(\()p Fq(\003)p Fx(\))k(b)s(e)d(satis\014ed)h(and)f Fs(R)2380 3490 y Fp(\003)2453 3523 y Fx(b)s(e)g(de\014ned)g(as)i(ab)s(o)m(v)m(e.) 50 b(Then,)236 3636 y(to)31 b(pro)m(v)m(e)h(\()p Fs(E)700 3603 y Fp(\003)760 3636 y Fq(\002)20 b Fs(E)923 3603 y Fp(\003)963 3636 y Fx(\))g Fq(n)h Fs(R)1154 3603 y Fp(\003)1223 3636 y Fx(to)31 b(b)s(e)f(op)s(en,)g(c)m(ho)s(ose)h(an)m (y)g(\()p Fs(x)2250 3650 y Fo(1)2290 3636 y Fs(;)15 b(x)2382 3650 y Fo(2)2422 3636 y Fx(\))30 b(in)f(this)h(set.)236 3791 y(\(1\))87 b(If)30 b Fq(f)p Fs(x)626 3805 y Fo(1)666 3791 y Fs(;)15 b(x)758 3805 y Fo(2)797 3791 y Fq(g)26 b(\032)f Fs(E)5 b Fx(,)31 b(then)f(there)h(are)f Fs(G)1754 3805 y Fr(i)1808 3791 y Fq(2)25 b Fm(G)p Fx(\()p Fs(E)5 b Fx(\))27 b Fq(\032)e Fm(G)p Fx(\()p Fs(E)2408 3758 y Fp(\003)2448 3791 y Fx(\))31 b(suc)m(h)f(that)667 3953 y Fs(x)719 3967 y Fr(i)772 3953 y Fq(2)25 b Fs(G)929 3967 y Fr(i)1083 3953 y Fx(and)124 b Fs(G)1425 3967 y Fo(1)1485 3953 y Fq(\002)20 b Fs(G)1647 3967 y Fo(2)1727 3953 y Fq(\032)40 b Fx(\()p Fs(E)26 b Fq(\002)20 b Fs(E)5 b Fx(\))21 b Fq(n)f Fs(R)41 b Fq(\032)g Fx(\()p Fs(E)2579 3915 y Fp(\003)2639 3953 y Fq(\002)20 b Fs(E)2802 3915 y Fp(\003)2842 3953 y Fx(\))g Fq(n)h Fs(R)3033 3915 y Fp(\003)3087 3953 y Fs(:)236 4115 y Fx(\(2\))75 b(If)24 b Fq(f)p Fs(x)608 4129 y Fo(1)647 4115 y Fs(;)15 b(x)739 4129 y Fo(2)779 4115 y Fq(g)26 b(6\032)f Fs(E)5 b Fx(,)26 b(then)e Fs(x)1322 4129 y Fo(1)1387 4115 y Fx(=)h Fq(1)f Fx(and)f Fs(x)1820 4129 y Fo(2)1885 4115 y Fq(2)i Fs(E)5 b Fx(.)39 b(Since)23 b Fs(E)30 b Fx(is)23 b(lo)s(cally)g(compact,)k (there)e(is)e(a)236 4227 y(neigh)m(b)s(orho)s(o)s(d)29 b Fs(G)879 4241 y Fo(2)945 4227 y Fq(2)c Fm(G)p Fx(\()p Fs(E)5 b Fx(\))28 b Fq(\032)e Fm(G)p Fx(\()p Fs(E)1547 4194 y Fp(\003)1587 4227 y Fx(\))32 b(of)f Fs(x)1810 4241 y Fo(2)1880 4227 y Fx(with)e(closure)i Fs(K)h Fq(2)26 b Fm(K)p Fx(\()p Fs(E)5 b Fx(\).)43 b(By)31 b(condition)f(\()p Fq(\003)p Fx(\),)236 4340 y(therefore,)35 b Fs(K)731 4299 y Fj(#)797 4340 y Fq(2)29 b Fm(K)p Fx(\()p Fs(E)5 b Fx(\))34 b(as)g(w)m(ell,)f(and)f(th)m(us)h Fs(G)1905 4354 y Fo(1)1975 4340 y Fx(:=)d Fs(E)2173 4307 y Fp(\003)2235 4340 y Fq(n)22 b Fs(K)2386 4299 y Fj(#)2452 4340 y Fq(2)30 b Fm(G)p Fx(\()p Fs(E)2721 4307 y Fp(\003)2761 4340 y Fx(\))k(is)e(a)i(neigh)m(b)s(orho)s(o)s(d)236 4453 y(of)d Fs(x)392 4467 y Fo(1)431 4453 y Fx(.)41 b(Then)29 b(it)h(is)g(easily)f (c)m(hec)m(k)m(ed)j(that)f(\()p Fs(G)1803 4467 y Fo(1)1864 4453 y Fq(\002)20 b Fs(G)2026 4467 y Fo(2)2066 4453 y Fx(\))g Fq(\\)g Fs(R)2272 4420 y Fp(\003)2336 4453 y Fx(=)25 b Fq(;)p Fx(.)971 b Ff(\003)377 4638 y Fx(No)m(w)31 b(an)g(appropriate)e(v)m(ersion)h(of)g(Stone's)h(theorem)g(can)g(b)s(e) e(established:)377 4833 y Fg(\(10.6\))53 b(Pr)n(oposition)79 b Fk(L)-5 b(et)34 b Fs(E)39 b Fk(b)-5 b(e)33 b(a)h(lo)-5 b(c)g(al)5 b(ly)36 b(c)-5 b(omp)g(act)35 b(OTS)f(satisfying)g(c)-5 b(ondition)35 b Fx(\()p Fq(\003)p Fx(\))236 4946 y Fk(of)e Fx(\(10.5\))p Fk(.)44 b(Then)236 5108 y Fx(\(a\))888 b Fq(K)1308 5122 y Fo(0)1373 5108 y Fx(:=)25 b Fq(f)p Fs(f)1584 5122 y Fo(1)1644 5108 y Fq(\000)20 b Fs(f)1780 5122 y Fo(2)1844 5108 y Fx(:)26 b(0)f Fq(\024)g Fs(f)2106 5122 y Fr(i)2159 5108 y Fq(2)g(K)2315 5063 y Fj(#)2352 5108 y Fx(\()p Fs(E)5 b Fx(\))p Fq(g)236 5270 y Fk(is)33 b(a)g(dense)g(subsp)-5 b(ac)g(e)33 b(of)g Fq(K)q Fx(\()p Fs(E)5 b Fx(\))34 b Fk(with)g(r)-5 b(esp)g(e)g(ct)34 b(to)f(the)g(uniform)h(norm,)236 5432 y Fx(\(b\))1235 b Fq(K)q Fx(\()p Fs(E)5 b Fx(\))27 b Fq(\032)e(R)p Fx(\()p Fs(E)5 b Fx(\))15 b Fs(:)1844 5753 y Fx(37)p eop %%Page: 38 38 38 37 bop 377 154 a Fg(Pr)n(oof.)85 b Fx(Since)37 b Fs(E)1093 121 y Fp(\003)1171 154 y Fx(according)h(to)g(\(10.5\))i(is)d(a)i (compact)g(OTS,)e(it)g(follo)m(ws)g(as)h(in)f(the)236 267 y(pro)s(of)30 b(of)g(\(10.4\))j(that)1216 380 y Fq(C)1269 342 y Fp(\003)1264 402 y Fo(0)1334 380 y Fx(:=)25 b Fq(f)p Fs(f)1555 342 y Fp(\003)1545 402 y Fo(1)1614 380 y Fq(\000)20 b Fs(f)1760 342 y Fp(\003)1750 402 y Fo(2)1824 380 y Fx(:)26 b(0)g Fq(\024)f Fs(f)2097 342 y Fp(\003)2087 402 y Fr(i)2161 380 y Fq(2)f(C)2299 334 y Fj(#)2336 380 y Fx(\()p Fs(E)2443 342 y Fp(\003)2483 380 y Fx(\))p Fq(g)236 536 y Fx(separates)32 b(the)g(p)s(oin)m(ts)e(of)i Fs(E)1241 503 y Fp(\003)1281 536 y Fx(.)43 b(All)30 b(further)g (conditions)g(in)g(Stone's)i(theorem)g(are)g(ob)m(viously)236 649 y(satis\014ed,)g(and)g(th)m(us)g Fq(C)1047 616 y Fp(\003)1042 672 y Fo(0)1119 649 y Fx(is)f(dense)h(in)f Fq(C)5 b Fx(\()p Fs(E)1730 616 y Fp(\003)1771 649 y Fx(\).)46 b(Let)33 b(no)m(w)g Fs(f)k Fq(2)29 b(K)q Fx(\()p Fs(E)5 b Fx(\))33 b(b)s(e)f(giv)m(en)g(and)g Fs(f)3250 616 y Fp(\003)3321 649 y Fx(b)s(e)g(its)236 762 y(trivial)d(extension)h(to)h Fs(E)1092 729 y Fp(\003)1132 762 y Fx(.)40 b(Then)30 b(there)g(are)h(nonnegativ)m(e)g(functions)e Fs(f)2771 729 y Fp(\003)2761 788 y Fr(ik)2852 762 y Fq(2)c(C)2991 720 y Fj(#)3027 762 y Fx(\()p Fs(E)3134 729 y Fp(\003)3175 762 y Fx(\))30 b(with)1104 944 y Fq(k)p Fs(f)1204 907 y Fp(\003)1194 967 y Fr(k)1264 944 y Fq(\000)20 b Fs(f)1410 907 y Fp(\003)1448 944 y Fq(k)26 b(!)f Fx(0)125 b(for)g Fs(f)2094 907 y Fp(\003)2084 967 y Fr(k)2158 944 y Fx(:=)25 b Fs(f)2334 907 y Fp(\003)2324 967 y Fo(1)p Fr(k)2421 944 y Fq(\000)20 b Fs(f)2567 907 y Fp(\003)2557 967 y Fo(2)p Fr(k)2650 944 y Fs(;)236 1127 y Fx(where)31 b(b)m(y)h(adding)e (suitable)g(constan)m(ts)j Fs(f)1726 1094 y Fp(\003)1716 1154 y Fr(ik)1783 1127 y Fx(\()p Fq(1)p Fx(\))28 b(=)f(0)32 b(can)g(b)s(e)f(ac)m(hiev)m(ed.)45 b(The)31 b(restriction)g(of)236 1240 y(\()p Fs(f)326 1207 y Fp(\003)316 1266 y Fr(ik)403 1240 y Fq(\000)20 b Fx(1)p Fs(=k)s Fx(\))669 1207 y Fo(+)760 1240 y 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Fr(\033)363 5287 y Fq(\025)25 b Fx(0)p Fs(;)31 b(\033)d Fq(2)d(f\000)p Fs(;)15 b Fx(+)p Fq(g)p Fs(;)31 b Fx(de\014ned)e(b)m(y)586 5485 y Fs(f)641 5448 y Fr(\033)687 5485 y Fx(\()p Fs(x)p Fx(\))d(:=)g(sup)1124 5391 y Fi(n)1209 5404 y(X)1344 5527 y Fr(k)r Fp(2)p Fh(N)1532 5485 y Fx(\()p Fs(f)10 b Fx(\()p Fs(x)1709 5500 y Fr(k)r Fo(+1)1842 5485 y Fx(\))21 b Fq(\000)f Fs(f)10 b Fx(\()p Fs(x)2131 5500 y Fr(k)2173 5485 y Fx(\)\))2255 5448 y Fr(\033)2327 5485 y Fx(:)26 b Fs(x)2430 5499 y Fo(1)2495 5485 y Fq(\024)f Fs(x)2643 5499 y Fo(2)2707 5485 y Fq(\024)g Fs(:)15 b(:)g(:)26 b Fq(\024)f Fs(x)3097 5391 y Fi(o)3168 5485 y Fs(:)1844 5753 y Fx(38)p eop %%Page: 39 39 39 38 bop 236 154 a Fx(T)-8 b(o)31 b(v)m(erify)f Fs(f)678 121 y Fr(\033)749 154 y Fq(2)25 b(U)9 b Fx(\()p Fs(E)c Fx(\),)32 b(consider)d(the)h(Borel)h(measurable)e(function)668 345 y Fs(g)714 307 y Fr(\033)762 345 y Fx(\()p Fs(x)849 359 y Fo(1)889 345 y Fs(;)15 b(x)981 359 y Fo(2)1020 345 y Fs(;)g(:)g(:)g(:)i Fx(;)e Fs(x)p Fx(\))26 b(:=)f(1)1501 363 y Fp(f)p Fr(x)1576 372 y Fd(1)1611 363 y Fp(\024)p Fr(x)1706 372 y Fd(2)1740 363 y Fp(\024)p Fr(:::)o Fp(\024)p Fr(x)p Fp(g)2018 264 y Fi(X)2154 387 y Fr(k)r Fp(2)p Fh(N)2341 345 y Fx(\()p Fs(f)10 b Fx(\()p Fs(x)2518 360 y Fr(k)r Fo(+1)2651 345 y Fx(\))21 b Fq(\000)f Fs(f)10 b Fx(\()p Fs(x)2940 360 y Fr(k)2982 345 y Fx(\)\))3064 307 y Fr(\033)236 561 y Fx(on)46 b(the)g(Suslin)d(space)j Fs(E)1164 528 y Fh(N)1269 561 y Fq(\002)30 b Fs(E)5 b Fx(.)88 b(Then)45 b(the)h(set)g(of)g Fs(x)51 b Fq(2)g Fs(E)g Fx(satisfying)45 b(an)h(inequalit)m(y)236 674 y Fs(f)291 641 y Fr(\033)337 674 y Fx(\()p Fs(x)p Fx(\))h Fs(>)e(\015)j Fx(equals)42 b(the)g(pro)5 b(jection)43 b(of)f(the)h(set)g(of)g(\()p Fs(x)2259 688 y Fo(1)2299 674 y Fs(;)15 b(x)2391 688 y Fo(2)2431 674 y Fs(;)g(:)g(:)g(:)h Fx(;)f Fs(x)p Fx(\))46 b Fq(2)g Fs(E)2944 641 y Fh(N)3046 674 y Fq(\002)28 b Fs(E)47 b Fx(satisfy-)236 787 y(ing)40 b Fs(g)444 754 y Fr(\033)492 787 y Fx(\()p Fs(x)579 801 y Fo(1)619 787 y Fs(;)15 b(x)711 801 y Fo(2)750 787 y Fs(;)g(:)g(:)g(:)i Fx(;)e Fs(x)p Fx(\))44 b Fs(>)e(\015)k Fx(on)m(to)c(the)f(second)g(factor)h(and)f(is)f(therefore)h(a)g (Suslin,)g(hence)236 900 y(univ)m(ersally)28 b(measurable,)i(set)h(for) f(all)f Fs(\015)h Fq(\025)25 b Fx(0.)377 1043 y(\(b\))34 b(In)f(the)h(sp)s(ecial)e(case)j Fs(f)40 b Fx(=)31 b(1)1536 1057 y Fr(B)1631 1043 y Fx(the)i(functions)g Fs(f)2241 1010 y Fr(\033)2321 1043 y Fx(are)h(in)m(teger-v)-5 b(alued.)50 b(This)32 b(yields)236 1156 y(the)f(represen)m(tation)1171 1269 y Fs(B)e Fx(=)1365 1188 y Fi([)1473 1311 y Fr(k)r Fp(2)p Fh(N)1661 1269 y Fx(\()p Fq(f)p Fs(f)1796 1231 y Fo(+)1880 1269 y Fq(\025)c Fs(k)s Fq(g)c(n)f(f)p Fs(f)2257 1231 y Fp(\000)2342 1269 y Fq(\025)k Fs(k)s Fq(g)p Fx(\))15 b Fs(;)236 1429 y Fx(where)30 b(the)h(righ)m(t-hand)e(side)g(is)h(in)f (fact)i(a)g(\014nite)e(union.)1254 b Ff(\003)377 1613 y Fx(It)25 b(has)f(to)h(b)s(e)e(men)m(tioned)h(that)h(no)f(n)m(ull)e (sets)j(in)m(terv)m(ene)f(under)f(a)i(total)g(ordering,)f(b)s(ecause) 236 1726 y(in)29 b(this)h(case)h(increasing)e(functions)g(are)i(Borel)f (measurable.)1577 2036 y(REFERENCES)269 2219 y Fw([1])62 b(Alpuim,)42 b(M.,)g(A)n(tha)n(yde,)f(E.)e(\(1990\).)69 b(On)38 b(the)h(stationary)e(distribution)i(of)g(some)f(extremal)419 2319 y(Mark)n(o)n(vian)25 b(sequences.)36 b(J.)27 b(Appl.)38 b(Prob.)e Fc(27)p Fw(,)27 b(291{302.)269 2437 y([2])62 b(Arnold,)27 b(L.)h(\(1998\).)35 b(Random)28 b(dynamical)f(systems.)36 b(Springer,)27 b(New)g(Y)-7 b(ork.)269 2555 y([3])62 b(Asm)n(ussen,)27 b(S.)h(\(1987\).)36 b(Applied)28 b(probabilit)n(y)e (and)i(queues.)36 b(Wiley)-7 b(,)28 b(New)g(Y)-7 b(ork.)269 2673 y([4])62 b(Babillot,)42 b(M.,)h(Bougerol,)e(P)-7 b(.,)42 b(Elie,)h(L.)c(\(1997\).)72 b(The)40 b(random)f(di\013erence)g (equation)g Fv(X)3415 2685 y Fu(n)3479 2673 y Fw(=)419 2773 y Fv(A)481 2785 y Fu(n)526 2773 y Fv(X)595 2785 y Fu(n)p Fb(\000)p Fa(1)735 2773 y Fw(+)10 b Fv(B)873 2785 y Fu(n)946 2773 y Fw(in)28 b(the)g(critical)f(case.)36 b(Ann.)h(Probab.)f Fc(25)p Fw(,)27 b(478{493.)269 2891 y([5])62 b(Barnsley)-7 b(,)26 b(M.)i(\(1993\).)36 b(F)-7 b(ractals)26 b(ev)n(erywhere,)g(2nd)h(edition.)37 b(Academic)28 b(Press,)e(London.)269 3009 y([6])62 b(Barnsley)-7 b(,)31 b(M.,)h(Elton,)g(J.)g(\(1988\).)46 b(A)32 b(new)g(class)e(of)h(Mark)n (o)n(v)e(pro)r(cesses)h(for)h(image)g(enco)r(ding.)419 3109 y(Adv.)37 b(Appl.)h(Prob.)d Fc(20)p Fw(,)27 b(14{32.)269 3227 y([7])62 b(Benda,)20 b(M.)f(\(1998\).)32 b(Sc)n(h)n(w)n(ac)n(h)17 b(k)n(on)n(traktiv)n(e)g(sto)r(c)n(hastisc)n(he)g(dynamisc)n(he)h (Systeme.)34 b(PhD)18 b(thesis,)419 3326 y(Univ)n(ersit)n(y)27 b(of)g(Munic)n(h.)269 3445 y([8])62 b(Bhattac)n(hary)n(a,)38 b(R.,)j(Ma)5 b(jumdar,)40 b(M.)f(\(1999\).)67 b(On)38 b(a)g(theorem)g(of)g(Dubins)h(and)f(F)-7 b(reedman.)419 3544 y(J.)27 b(Theor.)36 b(Prob.)g(1067{1087.)269 3662 y([9])62 b(Bhattac)n(hary)n(a,)30 b(R.,)k(Ma)5 b(jumdar,)32 b(M.)g(\(1999\).)49 b(Con)n(v)n(ergence)30 b(to)i(equilibrium)g(of)g (random)f(dy-)419 3762 y(namical)e(systems)g(generated)f(b)n(y)i(i.)14 b(i.)g(d.)43 b(monotone)28 b(maps,)i(with)g(applications)f(to)g (economics.)419 3862 y(Statist.)37 b(T)-7 b(extb)r(o)r(oks)27 b(Monogr.)35 b Fc(158)p Fw(,)27 b(713{741.)34 b(Dekk)n(er,)27 b(New)g(Y)-7 b(ork.)228 3980 y([10])61 b(Bougerol,)31 b(P)-7 b(.,)32 b(Picard,)g(N.)g(\(1992\).)48 b(Stationarit)n(y)30 b(of)i(GAR)n(CH)g(pro)r(cesses)f(and)g(of)h(some)f(non-)419 4079 y(negativ)n(e)26 b(time)i(series.)36 b(J.)28 b(Econ.)36 b Fc(52)p Fw(,)27 b(115{127.)228 4197 y([11])61 b(Brandt,)31 b(A.,)h(F)-7 b(rank)n(en,)31 b(P)-7 b(.,)32 b(Lisek,)f(B.)g(\(1990\).) 46 b(Stationary)30 b(sto)r(c)n(hastic)g(mo)r(dels.)47 b(Wiley)-7 b(,)32 b(New)419 4297 y(Y)-7 b(ork.)228 4415 y([12])61 b(Diaconis,)27 b(P)-7 b(.,)27 b(F)-7 b(reedman,)27 b(D.)h(\(1999\).)36 b(Iterated)26 b(random)h(functions.)37 b(SIAM)28 b(Rev.)37 b Fc(41)p Fw(,)27 b(45{76.)228 4533 y([13])61 b(Dubins,)27 b(L.,)f(F)-7 b(reedman,)26 b(D.)h(\(1966\).)35 b(In)n(v)-5 b(arian)n(t)25 b(probabilities)g(for)h(certain)f(Mark)n(o)n (v)f(pro)r(cesses.)419 4633 y(Ann.)37 b(Math.)g(Stat.)g Fc(37)p Fw(,)28 b(837{848.)228 4751 y([14])61 b(Elton,)24 b(J.)g(\(1987\).)34 b(An)24 b(ergo)r(dic)f(theorem)g(for)g(iterated)h (maps.)35 b(Ergo)r(dic)22 b(Theory)h(Dyn.)36 b(Syst.)g Fc(7)p Fw(,)419 4851 y(481{488.)228 4969 y([15])61 b(Em)n(brec)n(h)n (ts,)37 b(P)-7 b(.,)39 b(Goldie,)g(C.)e(\(1994\).)64 b(P)n(erp)r(etuities)36 b(and)h(random)e(equations.)64 b(Asymptotic)419 5069 y(statistics)27 b(\(Prague)f(1993\),)g(75{86.)35 b(Ph)n(ysica,)26 b(Heidelb)r(erg.)228 5187 y([16])61 b(Engelking,)42 b(R.)f(\(1989\).)75 b(General)39 b(top)r(ology)-7 b(,)43 b(2nd)d(edition.)76 b(P)n(olish)40 b(Scien)n(ti\014c)g (Publishers,)419 5286 y(W)-7 b(arza)n(w)n(a.)228 5404 y([17])61 b(F)-7 b(oguel,)34 b(S.)g(\(1973\).)53 b(The)33 b(ergo)r(dic)f(theory)h(of)g(p)r(ositiv)n(e)g(op)r(erators)f(on)h(con)n (tin)n(uous)f(functions.)419 5504 y(Ann.)37 b(Sc.)g(Norm.)g(Sup)r(er.)g (Pisa)26 b Fc(27)p Fw(,)i(19{51.)1844 5753 y Fx(39)p eop %%Page: 40 40 40 39 bop 228 154 a Fw([18])61 b(Glasserman,)25 b(P)-7 b(.,)26 b(Y)-7 b(ao,)26 b(D.)h(\(1995\).)36 b(Sto)r(c)n(hastic)25 b(v)n(ector)g(di\013erence)i(equations)e(with)i(stationary)419 254 y(co)r(e\016cien)n(ts.)36 b(J.)28 b(Appl.)37 b(Prob.)f Fc(32)p Fw(,)27 b(851{866.)228 374 y([19])61 b(Goldie,)29 b(C.)f(\(1991\).)38 b(Implicit)30 b(renew)n(al)d(theory)g(and)i(tails)f (of)g(solutions)g(of)h(random)e(equations.)419 474 y(Ann.)37 b(Appl.)h(Probab.)d Fc(1)p Fw(,)28 b(126{166.)228 594 y([20])61 b(Goldie,)27 b(C.,)h(Maller,)f(R.)h(\(2000\).)35 b(Stabilit)n(y)28 b(of)g(p)r(erp)r(etuities.)37 b(Ann.)h(Probab.)d Fc(28)p Fw(,)27 b(1195{1218.)228 714 y([21])61 b(Helland,)49 b(I.,)g(Nilsen,)h(T.)45 b(\(1976\).)87 b(On)44 b(a)h(general)e(random)h (exc)n(hange)f(mo)r(del.)89 b(J.)45 b(Appl.)419 814 y(Prob.)35 b Fc(13)p Fw(,)28 b(781{790.)228 934 y([22])61 b(H\177)-42 b(ogn\177)g(as,)21 b(G.,)h(Muhk)n(erjea,)g(A.)f(\(1995\).)33 b(Probabilit)n(y)19 b(measures)h(on)g(semigroups.)33 b(Plen)n(um)21 b(Press,)419 1034 y(London.)228 1154 y([23])61 b(Jarner,)28 b(S.,)i(Tw)n(eedie,)g(R.)f(\(2001\).)41 b(Lo)r(cally)29 b(con)n(tracting)e(iterated)i(functions)h(and)f (stabilit)n(y)g(of)419 1254 y(Mark)n(o)n(v)c(c)n(hains.)36 b(J.)28 b(Appl.)37 b(Prob.)f Fc(38)p Fw(,)27 b(494{507.)228 1374 y([24])61 b(Kellerer,)25 b(H.)i(\(1992\).)36 b(Ergo)r(dic)25 b(b)r(eha)n(viour)g(of)i(a\016ne)f(recursions)f(I,)i(I)r(I,)g(I)r(I)r (I.)g(Preprin)n(ts,)e(Univ)n(er-)419 1474 y(sit)n(y)i(of)h(Munic)n(h)g (\(h)n(ttp://www.mathematik.uni-m)n(uenc)n(hen.de/)p Ft(\030)p Fw(k)n(ellerer\).)228 1594 y([25])61 b(Kellerer,)23 b(H.)g(\(1995\).)34 b(Order-preserving)20 b(random)j(dynamical)f (systems.)35 b(Preprin)n(t,)23 b(Univ)n(ersit)n(y)419 1694 y(of)k(Munic)n(h)h(\(h)n(ttp://www.mathematik.uni-m)n(uenc)n (hen.de/)p Ft(\030)p Fw(k)n(ellerer\).)228 1814 y([26])61 b(Kifer,)27 b(Y.)h(\(1986\).)36 b(Ergo)r(dic)26 b(theory)h(of)g(random) g(transformations.)35 b(Birkh\177)-42 b(auser,)26 b(Basel.)228 1935 y([27])61 b(Letac,)20 b(G.)g(\(1986\).)33 b(A)20 b(con)n(traction)d(principle)j(for)e(certain)h(Mark)n(o)n(v)e(c)n (hains)i(and)g(its)g(applications.)419 2034 y(Con)n(temp.)37 b(Math.)g Fc(50)p Fw(,)27 b(263{273.)228 2155 y([28])61 b(Lindley)-7 b(,)31 b(D.)g(\(1952\).)43 b(The)31 b(theory)e(of)i(a)e (queue)i(with)f(a)g(single)g(serv)n(er.)43 b(Pro)r(c.)h(Cam)n(b.)g (Philos.)419 2254 y(So)r(c.)37 b Fc(48)p Fw(,)27 b(277{289.)228 2375 y([29])61 b(Lo)n(ynes,)37 b(R.)f(\(1962\).)62 b(The)36 b(stabilit)n(y)g(of)g(a)g(queue)g(with)g(non-indep)r(enden)n(t)h(in)n (ter-arriv)-5 b(al)34 b(and)419 2474 y(service)26 b(times.)38 b(Pro)r(c.)d(Cam)n(b.)i(Philos.)f(So)r(c.)g Fc(58)p Fw(,)28 b(497{520.)228 2595 y([30])61 b(Lund,)35 b(B.,)f(Meyn,)h(P)-7 b(.,)34 b(Tw)n(eedie,)g(R.)f(\(1996\).)52 b(Computable)33 b(exp)r(onen)n(tial)g(con)n(v)n(ergence)d(rates)419 2694 y(for)d(sto)r(c)n(hastically)f(ordered)g(Mark)n(o)n(v)g(pro)r(cesses.) 35 b(Ann.)j(Appl.)f(Probab.)e Fc(6)p Fw(,)28 b(218{237.)228 2815 y([31])61 b(Mairesse,)28 b(J.)h(\(1997\).)40 b(Pro)r(ducts)28 b(of)h(irreducible)f(random)g(matrices)h(in)g(the)g(\(max,+\))g (algebra.)419 2914 y(Adv.)37 b(Appl.)h(Prob.)d Fc(29)p Fw(,)27 b(444{477.)228 3035 y([32])61 b(Meyn,)30 b(P)-7 b(.,)30 b(Tw)n(eedie,)g(R.)g(\(1993\).)42 b(Mark)n(o)n(v)27 b(c)n(hains)i(and)h(sto)r(c)n(hastic)f(stabilit)n(y)-7 b(.)43 b(Springer,)29 b(New)419 3134 y(Y)-7 b(ork.)228 3255 y([33])61 b(Nac)n(h)n(bin,)27 b(L.)h(\(1965\).)35 b(T)-7 b(op)r(ology)26 b(and)i(order.)35 b(V)-7 b(an)28 b(Nostrand,)f(Princeton.)228 3375 y([34])61 b(Rac)n(hev,)25 b(S.,)h(Samoro)r(dnitsky)-7 b(,)25 b(G.)h(\(1995\).)35 b(Limit)26 b(la)n(ws)f(for)g(a)h(sto)r(c)n(hastic)e(pro)r(cess)h(and)g (random)419 3475 y(recursion)h(arising)g(in)i(probabilit)n(y)f(mo)r (delling.)36 b(Adv.)i(Appl.)f(Prob.)f Fc(27)p Fw(,)27 b(185{202.)228 3595 y([35])61 b(Rac)n(hev,)28 b(S.,)i(T)-7 b(o)r(doro)n(vic,)27 b(P)-7 b(.)29 b(\(1990\).)39 b(On)29 b(the)g(rate)f(of)h(con)n(v)n(ergence)d(of)j(some)f(functionals)h(of)g (a)419 3695 y(sto)r(c)n(hastic)e(pro)r(cess.)35 b(J.)28 b(Appl.)37 b(Prob.)f Fc(27)p Fw(,)27 b(805{814.)228 3815 y([36])61 b(Revuz,)27 b(D.)i(\(1984\).)35 b(Mark)n(o)n(v)26 b(c)n(hains,)g(2nd)i(edition.)37 b(North-Holland,)27 b(Amsterdam.)228 3936 y([37])61 b(Tw)n(eedie,)47 b(R.)c(\(1976\).)83 b(Criteria)42 b(for)g(classifying)h(general)f(Mark)n(o)n(v)f(c)n (hains.)83 b(Adv.)h(Appl.)419 4035 y(Prob.)35 b Fc(8)p Fw(,)28 b(737{771.)228 4156 y([38])61 b(V)-7 b(erv)i(aat,)30 b(W.)h(\(1979\).)43 b(On)30 b(a)g(sto)r(c)n(hastic)f(di\013erence)h (equation)g(and)g(a)f(represen)n(tation)g(of)h(non-)419 4255 y(negativ)n(e)c(in\014nitely)j(divisible)e(random)g(v)-5 b(ariables.)35 b(Adv.)j(Appl.)f(Prob.)f Fc(11)p Fw(,)27 b(750{783.)228 4376 y([39])61 b(v)n(on)36 b(W)-7 b(eizs\177)-42 b(ac)n(k)n(er,)38 b(H.)f(\(1974\).)64 b(Zur)36 b(Gleic)n(h)n(w)n (ertigk)n(eit)g(zw)n(eier)g(Arten)h(der)g(Randomisierung.)419 4475 y(Man)n(uscr.)f(Math.)h Fc(11)p Fw(,)27 b(91{94.)228 4596 y([40])61 b(Y)-7 b(aha)n(v,)36 b(J.)f(\(1975\).)59 b(On)36 b(a)e(\014xed)i(p)r(oin)n(t)f(theorem)g(and)g(its)h(sto)r(c)n (hastic)f(equiv)-5 b(alen)n(t.)60 b(J.)35 b(Appl.)419 4695 y(Prob.)g Fc(12)p Fw(,)28 b(605{611.)1844 5753 y Fx(40)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF