%!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: II.dvi %%Pages: 25 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips II %DVIPSParameters: dpi=300, compressed, comments removed %DVIPSSource: TeX output 1999.04.22:1611 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 0 0 0]N /nn 0 N /IE 0 N 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%%Feature: *Resolution 300dpi TeXDict begin %%PaperSize: a4 %%EndSetup %%Page: 0 1 0 0 bop 118 253 a Fp(Ergo)r(dic)23 b(Beha)n(viour)g(of)h(A\016ne)e (Recursions)g(I)r(I)118 372 y(In)n(v)l(arian)n(t)i(Measures)35 b(and)g(Ergo)r(dic)23 b(Theorems)118 548 y Fo(Hans)17 b(G.)f(Kellerer)118 637 y(Mathematisc)o(hes)e(Institut)i(der)g(Univ)o (ersit\177)-24 b(at)15 b(M)q(\177)-26 b(unc)o(hen)118 727 y(Theresienstra\031e)16 b(39,)25 b(D-8000)18 b(M)q(\177)-26 b(unc)o(hen)17 b(2,)24 b(German)o(y)118 903 y(Octob)q(er)16 b(30,)h(1992)118 1079 y Fn(Summary)-5 b(.)22 b Fo(This)f(pap)q(er)g(is) f(concerned)g(with)g(the)g(discrete{time)d(Mark)o(o)o(v)j(pro)q(cess) 118 1140 y(\()p Fm(X)177 1147 y Fl(n)201 1140 y Fo(\))220 1147 y Fl(n)p Fk(\025)p Fj(0)320 1140 y Fo(solving)i(the)h(sto)q(c)o (hastic)g(di\013erence)f(equation)31 b Fm(X)1292 1147 y Fl(n)1341 1140 y Fo(=)25 b Fm(Y)1432 1147 y Fl(n)1456 1140 y Fm(X)1496 1147 y Fl(n)p Fk(\000)p Fj(1)1580 1140 y Fo(+)16 b Fm(Z)1667 1147 y Fl(n)1713 1140 y Fo(for)118 1200 y Fm(n)e Fi(2)g Fn(N)p Fo(,)h(where)g(\()p Fm(Y)468 1207 y Fl(n)491 1200 y Fm(;)8 b(Z)546 1207 y Fl(n)570 1200 y Fo(\))589 1208 y Fl(n)p Fk(2)p Fh(N)686 1200 y Fo(is)15 b(a)g(sequence)f(of)j(i.i.d.)8 b(random)k(v)m(ariables)k (indep)q(enden)o(t)118 1260 y(of)g(the)g(initial)f(v)m(ariable)g Fm(X)620 1267 y Fj(0)656 1260 y Fo(and,)i(in)e(accordance)h(with)g (most)f(applications,)h(the)f(state)118 1320 y(space)j(is)g(restricted) e(to)i Fn(R)622 1327 y Fj(+)652 1320 y Fo(.)26 b(In)17 b(this)h(part)g(the)f(emphasis)g(is)h(on)g(the)g(recurren)o(t)e(case,) 118 1380 y(where)i(existence)e(and)i(uniqueness)g(of)g(an)g(in)o(v)m (arian)o(t)f(measure)g(as)h(w)o(ell)f(as)h(mean)f(and)118 1441 y(p)q(oin)o(t)o(wise)f(ergo)q(dic)g(theorems)f(can)i(b)q(e)f (established.)118 2682 y(1991)21 b(Mathematics)c(Sub)s(ject)h (Classi\014cation.)30 b(Primary)18 b(47A35,)i(60J05;)i(Secondary)118 2742 y(54H20,)17 b(60G30.)p eop %%Page: 1 2 1 1 bop 118 77 a Fn(In)n(tro)r(duction)118 195 y Fo(This)20 b(is)f(the)g(con)o(tin)o(uation)g(of)g(w)o(ork)h(b)q(egun)g(in)f Fm(:)8 b(:)g(:)38 b Fo(and)20 b(to)f(b)q(e)h(\014nished)f(in)g Fm(:)8 b(:)g(:)19 b Fo(.)30 b(It)118 255 y(studies)16 b(a\016ne)g(recursions)g(on)h Fn(R)749 262 y Fj(+)779 255 y Fo(,)e(i.e.)20 b(sequences)c(\()p Fm(X)1173 262 y Fl(n)1197 255 y Fo(\))1216 262 y Fl(n)p Fk(\025)p Fj(0)1300 255 y Fo(de\014ned)g(b)o(y)268 355 y Fm(X)308 362 y Fl(n)345 355 y Fo(=)e Fm(Y)425 362 y Fl(n)449 355 y Fm(X)489 362 y Fl(n)p Fk(\000)p Fj(1)569 355 y Fo(+)d Fm(Z)651 362 y Fl(n)762 355 y Fo(for)25 b Fm(n)14 b Fi(2)g Fn(N)p Fm(:)118 455 y Fo(Here,)22 b(\()p Fm(Y)301 462 y Fl(n)325 455 y Fm(;)8 b(Z)380 462 y Fl(n)404 455 y Fo(\))423 464 y Fl(n)p Fk(2)p Fh(N)526 455 y Fo(is)22 b(a)g(sequence)f(of)i(indep)q (enden)o(t)e(iden)o(tically)f(distributed)h Fn(R)1718 437 y Fj(2)1718 468 y(+)1747 455 y Fo({)118 516 y(v)m(alued)d(random)g (v)m(ariables)g(whic)o(h)g(is)g(indep)q(enden)o(t)f(of)i(the)f(initial) e(v)m(ariable)i Fm(X)1640 523 y Fj(0)1678 516 y Fi(\025)f Fo(0.)118 576 y(Without)e(loss)h(of)f(generalit)o(y)l(,)f(the)g(common) f(distribution)i Fm(\027)j Fo(of)e(\()p Fm(X)1390 583 y Fl(n)1414 576 y Fm(;)8 b(Z)1469 583 y Fl(n)1492 576 y Fo(\))p Fm(;)g(n)14 b Fi(2)g Fn(N)p Fo(,)h(will)118 636 y(b)q(e)h(assumed)g(to)h(b)q(elong)g(to)f(the)g(class)h Fi(N)23 b Fo(de\014ned)16 b(in)g(Section)g(0.)191 696 y(A)21 b(cen)o(tral)g(result)g(of)h(P)o(art)f(I)g(then)h(states)g(that) g(the)f(lo)o(w)o(er)g(and)h(upp)q(er)g(limit)c(of)118 756 y(\()p Fm(X)177 763 y Fl(n)201 756 y Fo(\))220 763 y Fl(n)p Fk(\025)p Fj(0)304 756 y Fo(are)e(constan)o(ts)h Fm(x)p 601 763 28 2 v 15 w Fo(and)p 738 730 V 16 w Fm(x)p Fo(,)f(indep)q(enden)o(t)e(of)j(the)e(initial)f(la)o(w)i Fi(L)p Fo(\()p Fm(X)1531 763 y Fl(o)1551 756 y Fo(\).)21 b(Accord-)118 816 y(ingly)c(the)g(sequence)g(is)g(called)g(recurren)o (t)f(in)h(the)h(case)f Fm(x)p 1178 823 V 16 w(<)f Fi(1)h Fo(and)i(transien)o(t)e(in)g(the)118 877 y(case)j Fm(x)p 225 884 V 19 w Fo(=)g Fi(1)p Fo(.)32 b(The)20 b(aim)e(of)i(P)o(art)g(I) q(I)f(is)h(a)g(more)e(sp)q(eci\014c)h(study)h(of)g(the)g(asymptotic)118 937 y(b)q(eha)o(viour)c(in)g(the)g(recurren)o(t)f(case;)h(its)g(con)o (ten)o(ts)g(are)g(summarized)d(b)q(elo)o(w.)191 1026 y Fn(Section)23 b(4.)34 b Fo(F)l(or)21 b(information)e(ab)q(out)j (limit)c(p)q(oin)o(ts)j(of)f(the)h(sequence)e(\()p Fm(X)1660 1033 y Fl(n)1684 1026 y Fo(\))1703 1033 y Fl(n)p Fk(\025)p Fj(0)118 1086 y Fo(di\013eren)o(t)d(from)f Fm(x)p 426 1093 V 16 w Fo(and)p 566 1060 V 18 w Fm(x)h Fo(existence)f(and)i (uniqueness)f(of)h(in)o(v)m(arian)o(t)f(measures)f(for)i(the)118 1146 y(corresp)q(onding)23 b(transition)g(k)o(ernel)d(are)i(essen)o (tial.)38 b(Here,)22 b(in)g(accordance)g(with)g(the)118 1207 y(top)q(ological)17 b(structure)g(of)g(the)f(state)h(space,)f (only)h(lo)q(cally)f(\014nite)g(measures)f(are)i(of)g(in-)118 1267 y(terest.)30 b(The)19 b(supp)q(ort)i Fm(M)k Fo(of)19 b(suc)o(h)g(an)h(in)o(v)m(arian)o(t)f(measure)f Fm(\026)h Fo(satis\014es)h(a)g(functional)118 1327 y(equation)d(\(4.1\),)g(b)o(y) f(whic)o(h)h(lo)o(w)o(er)f(and)h(upp)q(er)g(limit)e(can)i(b)q(e)g(iden) o(ti\014ed)e(as)j(minim)n(um)118 1387 y(and)e(maxim)o(um)c(of)k Fm(M)21 b Fo(\(4.3\).)g(Moreo)o(v)o(er,)14 b Fm(M)21 b Fo(inherits)15 b(connectedness)h(from)e(the)i(sup-)118 1447 y(p)q(ort)e(of)f(the)g(join)o(t)f(la)o(w)h Fi(L)p Fo(\()p Fm(Y)631 1454 y Fl(n)655 1447 y Fm(;)8 b(Z)710 1454 y Fl(n)734 1447 y Fo(\))k(\(4.4\).)21 b(An)12 b(imp)q(ortan)o(t)g (prop)q(ert)o(y)h(of)g Fm(\026)g Fo(itself)f(consists)118 1508 y(in)k(the)h(fact)g(that)g(the)f(measure)g Fm(\026)p Fo(\([0)p Fm(;)8 b(t)p Fo(]\))16 b(gro)o(ws)h(only)g(p)q(olynomially)d (for)j Fm(t)e Fi(!)f(1)p Fo(,)i(un-)118 1568 y(less)g(the)g(underlying) g(a\016ne)g(maps)f(are)i(expansiv)o(e)e(almost)g(surely)h(\(4.5\).)191 1657 y Fn(Section)h(5.)22 b Fo(The)15 b(main)g(result)g(of)h(this)f (section)h(is)f(basic)h(for)g(all)f(that)h(follo)o(ws.)21 b(T)l(o)118 1717 y(eac)o(h)c(recurren)o(t)f(sequence)g(\()p Fm(X)702 1724 y Fl(n)726 1717 y Fo(\))745 1724 y Fl(n)p Fk(\025)p Fj(0)831 1717 y Fo(it)h(assigns)h(an)g(essen)o(tially)d (unique)i(lo)q(cally)f(\014nite)118 1777 y(in)o(v)m(arian)o(t)j (measure)f(\(5.6\).)31 b(While)18 b(its)h(existence)f(is)h(settled)g(b) o(y)g(usual)h(compactness)118 1838 y(argumen)o(ts)c(\(5.3\),)h(the)g (uniqueness)g(requires)f(an)h(elab)q(orate)h(lo)q(calization.)23 b(It)17 b(is)g(com-)118 1898 y(plicated)c(b)o(y)h(the)f(fact)h(that)h (the)e(hitting)h(k)o(ernel)e(of)j(an)f(in)o(terv)m(al)f([0)p Fm(;)8 b(t)p Fo(])13 b(with)h Fm(x)p 1532 1905 V 13 w(<)g(t)g(<)f Fi(1)p Fo(,)118 1958 y(though)18 b(b)q(eing)f(still)f(sto)q(c)o (hastic,)h(need)f(no)h(longer)g(b)q(e)g(a)g(F)l(eller)e(k)o(ernel.)22 b(T)l(o)17 b(determine)118 2018 y(the)d(in)o(v)m(arian)o(t)g(measure)f (explicitly)l(,)e(an)k(in)o(tegral)e(equation)h(for)h(its)f(Laplace)g (transform)118 2078 y(is)g(a)o(v)m(ailable)g(\(5.7\),)h(whic)o(h,)f(ho) o(w)o(ev)o(er,)f(is)h(more)g(or)h(less)f(of)h(theoretical)e(in)o (terest.)20 b(In)14 b(the)118 2139 y(example)118 2258 y(\(E\))79 b Fm(X)308 2265 y Fl(n)346 2258 y Fo(=)14 b Fm(Y)426 2265 y Fl(n)449 2258 y Fm(X)489 2265 y Fl(n)p Fk(\000)p Fj(1)570 2258 y Fo(+)d(1)87 b(with)g Fn(P)p Fo(\()p Fm(Y)997 2265 y Fl(n)1035 2258 y Fo(=)1092 2224 y(1)p 1092 2246 25 2 v 1092 2292 a(2)1121 2258 y(\))14 b(=)1211 2224 y(1)p 1211 2246 V 1211 2292 a(2)1254 2258 y(=)g Fn(P)p Fo(\()p Fm(Y)1391 2265 y Fl(n)1429 2258 y Fo(=)f(2\))118 2371 y(it)j(leads)g(to)h(the)f(functional)g(equation) 268 2490 y Fm( )r Fo(\()p Fm(u)p Fo(\))d(=)g Fm(e)455 2469 y Fk(\000)p Fl(u)518 2456 y Fo(1)p 518 2478 V 518 2524 a(2)556 2490 y(\()p Fm( )r Fo(\()633 2456 y Fm(u)p 633 2478 28 2 v 635 2524 a Fo(2)665 2490 y(\))e(+)g Fm( )r Fo(\(2)p Fm(u)p Fo(\)\))87 b(for)25 b Fm(u)14 b(>)f Fo(0)p Fm(;)118 2603 y Fo(whic)o(h)j(can)g(hardly)g(b)q(e)h(solv)o(ed.)191 2692 y Fn(Section)e(6.)20 b Fo(In)13 b(view)g(of)h(the)f(computational) g(problems)f(it)h(is)g(imp)q(ortan)o(t)g(to)h(study)118 2752 y(the)i(in)o(v)m(arian)o(t)g(measure)g Fm(\026)h Fo(at)g(least)f(qualitativ)o(ely)l(.)k(The)c(\014rst)h(assertion)g (concerns)g(its)933 2877 y(1)p eop %%Page: 2 3 2 2 bop 118 77 a Fo(supp)q(ort)15 b Fm(M)5 b Fo(,)14 b(whic)o(h,)f(whenev)o(er)g(un)o(b)q(ounded,)h(can)g(b)q(e)g(sho)o(wn)g (to)g(b)q(e)g(an)g(in)o(terv)m(al)f(\(6.1\).)118 137 y(More)20 b(precisely)l(,)e(here)h(as)i(in)e(some)g(of)h(the)g(follo)o (wing)f(results,)h Fn(P)p Fo(\()p Fm(Y)1434 144 y Fl(n)1478 137 y Fo(=)g(0\))g(=)g(0)g(has)118 197 y(to)d(b)q(e)f(assumed.)21 b(It)16 b(is)g(not)h(hard)g(to)f(sho)o(w)h Fm(\026)g Fo(to)f(b)q(e)h(either)e(absolutely)h(con)o(tin)o(uous)g(or)118 258 y(singular)h(with)g(resp)q(ect)g(to)g(Leb)q(esgue)g(measure)f (\(6.2\).)23 b(The)17 b(\014rst)g(statemen)o(t)f(applies,)118 318 y(unless)j(the)g(join)o(t)g(la)o(w)g Fi(L)p Fo(\()p Fm(Y)641 325 y Fl(n)665 318 y Fm(;)8 b(Z)720 325 y Fl(n)744 318 y Fo(\))19 b(is)h(singular)f(\(6.3\).)31 b(It)19 b(is)g(more)f(in)o(v)o(olv)o(ed)f(to)i(pro)o(v)o(e)118 378 y(that)f Fm(\026)p Fo(,)g(apart)h(from)d(a)i(trivial)f(exception,)f (is)h(nonatomic)g(\(6.4\).)26 b(The)18 b(section)f(closes)118 438 y(with)f(a)g(stabilit)o(y)f(result)g(that)h(ma)o(y)f(b)q(e)h(used)g (for)g(appro)o(ximating)f(the)g(in)o(v)m(arian)o(t)h(mea-)118 498 y(sure)g(\(6.5\).)191 588 y Fn(Section)26 b(7.)41 b Fo(Here)22 b(the)g(main)g(ergo)q(dic)h(theorems)e(for)i(ratios)h(are) f(established.)118 648 y(While)d(the)g(v)o(ersion)g(for)g(means)g (follo)o(ws)g(easily)g(from)f(earlier)h(results)g(\(7.1\),)h(for)g(the) 118 708 y(p)q(oin)o(t)o(wise)f(v)o(ersion)g(a)h(lo)q(calization)f(as)h (in)g(Section)f(5)h(is)f(necessary)h(\(7.4\).)34 b(It)20 b(mak)o(es)118 768 y(essen)o(tial)c(use)i(of)f(the)g(fact)g(that)h(the) f(hitting)g(k)o(ernel)f(from)g(Section)h(5)g(enjo)o(ys)g(at)h(least)118 828 y(some)h(w)o(eak)o(er)g(F)l(eller)f(prop)q(ert)o(y)l(.)33 b(As)20 b(a)g(consequence)f(of)h(the)g(ergo)q(dic)g(theorems)f(the)118 889 y(supp)q(ort)14 b Fm(M)k Fo(and)c(its)f(complem)o(en)o(t)c(can)k(b) q(e)g(iden)o(ti\014ed)f(with)h(the)f(conserv)m(ativ)o(e)g(and)i(dis-) 118 949 y(sipativ)o(e)g(part)h(of)f(the)h(pro)q(cess)g(\(7.5\).)21 b(If)14 b(sp)q(ecialized)f(to)i(the)f(example)f(\(E\))h(ab)q(o)o(v)o (e,)h(this)118 1009 y(means)g(that)i(the)f(set)g(of)h(limit)c(p)q(oin)o (ts)k(of)f(the)g(sequence)f(\()p Fm(X)1252 1016 y Fl(n)1276 1009 y Fo(\))1295 1016 y Fl(n)p Fk(\025)p Fj(0)1380 1009 y Fo(equals)h(the)g(in)o(terv)m(al)118 1069 y([2)p Fm(;)8 b Fi(1)p Fo(])16 b(almost)g(surely)l(.)23 b(Rather)17 b(general)g(results)g(on)g(irreducibilit)o(y)d(and)k(ap)q(erio)q(dicit) o(y)118 1129 y(conclude)e(the)g(section)f(\(7.6\).)118 1306 y Fn(4.)25 b(Excessiv)n(e)16 b(and)k(in)n(v)m(arian)n(t)e (measures)118 1424 y Fo(Basic)23 b(for)g(the)h(ergo)q(dic)f(theorems)f (in)h(Sections)g(5)h(and)g(7)f(are)h(some)e(prop)q(erties)h(of)118 1484 y(measures)17 b Fm(\026)9 b Fi(2)g(M)p Fo(\()p Fn(R)530 1491 y Fj(+)559 1484 y Fo(\))18 b(that)h(are)f(excessiv)o(e)e(or)i(in)o (v)m(arian)o(t)g(with)g(resp)q(ect)f(to)i Fm(\027)s Fo(.)27 b(Here)118 1544 y(the)17 b(reference)e(to)i Fm(\027)s Fo(,)g(suppressed)g(in)g(general,)f(actually)g(refers)h(to)g(the)g (corresp)q(onding)118 1605 y(k)o(ernel)i Fm(P)7 b Fo(.)35 b(Moreo)o(v)o(er,)20 b(it)g(should)h(b)q(e)g(noted)g(that)g(for)g(this) f(section)h(it)f(is)g(irrelev)m(an)o(t)118 1665 y(whether)g(the)g (underlying)g(distribution)g Fm(\027)j Fo(is)e(recurren)o(t)e(or)h (transien)o(t.)34 b(The)20 b(results)118 1725 y(concern)g(mainly)e(the) i(supp)q(ort)h(of)g Fm(\026)p Fo(,)g(where)e(in)h(the)g(\014rst)g (assertion)p 1467 1686 37 2 v 21 w Fm(A)g Fo(denotes)g(the)118 1785 y(closure)c(of)h(a)f(subset)h Fm(A)f Fo(of)g Fn(R)677 1792 y Fj(+)706 1785 y Fo(:)118 1874 y Fn(\(4.1\))28 b(Prop)r(osition.)c Fg(With)17 b(the)h(mapping)268 1976 y Fm(h)13 b Fo(:)39 b Fn(R)404 1983 y Fj(+)444 1976 y Fi(\002)11 b Fn(R)536 1956 y Fj(2)536 1988 y(+)579 1976 y Fi(3)j Fo(\()p Fm(x)p Fo(;)8 b Fm(y)r(;)g(z)r Fo(\))13 b Fi(!)g Fm(xy)g Fo(+)e Fm(z)16 b Fi(2)e Fn(R)1105 1983 y Fj(+)118 2078 y Fg(the)k(supp)n(ort)f(M)g(of)g(a)g(me)n(asur)n(e)g Fm(\026)d Fi(2)g(M)p Fo(\()p Fn(R)932 2085 y Fj(+)961 2078 y Fo(\))k Fg(satis\014es:)118 2180 y Fo(\(a\))88 b Fm(M)19 b Fi(\033)p 386 2137 213 2 v 13 w Fm(h)p Fo([)p Fm(M)d Fi(\002)11 b Fm(N)5 b Fo(])p Fm(;)58 b(if)22 b(\026)c(is)f (excessiv)r(e;)118 2294 y Fo(\(b\))87 b Fm(M)20 b Fo(=)p 388 2251 V 13 w Fm(h)p Fo([)p Fm(M)c Fi(\002)11 b Fm(N)5 b Fo(])p Fm(;)58 b(if)22 b(\026)c(is)f(inv)r(ar)q(iant:)118 2408 y Fg(Pr)n(o)n(of.)23 b Fo(The)18 b(measure)e Fm(\026P)24 b Fo(is)17 b(the)g(image)f(of)i Fm(\026)12 b Fi(\012)g Fm(\027)20 b Fo(under)d(the)g(con)o(tin)o(uous)g(mapping)118 2468 y Fm(h)p Fo(.)38 b(Therefore,)23 b(due)f(to)g(a)g(general)g (result)f(from)g(top)q(ological)i(measure)d(theory)l(,)j(the)118 2528 y(supp)q(ort)18 b(of)e Fm(\026P)24 b Fo(is)16 b(the)g(closure)g (of)g(the)g(image)f(of)268 2630 y(supp)25 b(\()p Fm(\026)11 b Fi(\012)g Fm(\027)s Fo(\))j(=)g Fm(M)i Fi(\002)11 b Fm(N)118 2732 y Fo(under)16 b Fm(h)p Fo(,)g(whic)o(h)g(clearly)f(pro)o (v)o(es)g(\(a\))i(and)g(\(b\))p Fm(:)49 b Ff(2)933 2877 y Fo(2)p eop %%Page: 3 4 3 3 bop 191 77 a Fo(Equally)16 b(simple)e(is)i(the)g(follo)o(wing)g (auxiliary)f(result:)118 166 y Fn(\(4.2\))28 b(Lemma.)22 b Fg(If)15 b Fm(\026)g Fi(2)f(M)p Fo(\()p Fn(R)744 173 y Fj(+)773 166 y Fo(\))i Fg(is)g(a)g(nontrivial)h(exc)n(essive)h(me)n (asur)n(e)d(with)i(supp)n(ort)118 226 y Fm(M)5 b Fg(,)18 b(then)316 303 y Fm(z)p 273 325 111 2 v 273 371 a Fo(1)11 b Fi(\000)g Fm(y)402 337 y Fi(2)j Fm(M)93 b Fg(for)24 b Fo(\()p Fm(y)r(;)8 b(z)r Fo(\))13 b Fi(2)h Fm(N)30 b(w)q(ith)17 b(y)f(<)e Fo(1)p Fm(:)118 464 y Fg(Pr)n(o)n(of.)23 b Fo(With)16 b(an)g(arbitrary)g Fm(x)699 471 y Fj(0)732 464 y Fi(2)f Fm(M)k Fi(6)p Fo(=)14 b Fi(;)h Fo(and)i(\()p Fm(y)r(;)8 b(z)r Fo(\))15 b(as)h(ab)q(o)o(v)o(e,)g(\(4.1a\))g(yields)f (recur-)118 524 y(siv)o(ely)268 626 y Fm(x)296 633 y Fl(n)333 626 y Fo(:=)e Fm(y)d(x)460 633 y Fl(n)p Fk(\000)p Fj(1)539 626 y Fo(+)h Fm(z)16 b Fi(2)e Fm(M)92 b Fo(for)26 b Fm(n)14 b Fi(2)g Fn(N)p Fm(:)118 727 y Fo(Since)h Fm(M)22 b Fo(is)16 b(closed,)g(this)g(implies)316 804 y Fm(z)p 273 826 V 273 872 a Fo(1)11 b Fi(\000)g Fm(y)402 838 y Fo(=)j(lim)530 845 y Fl(n)p Fk(!1)632 838 y Fo(\()p Fm(y)677 817 y Fl(n)700 838 y Fm(x)728 845 y Fj(0)759 838 y Fo(+)d Fm(y)834 817 y Fl(n)p Fk(\000)p Fj(1)902 838 y Fm(z)i Fo(+)e Fm(:)d(:)g(:)i Fo(+)h Fm(z)r Fo(\))j(=)g(lim)1290 845 y Fl(n)p Fk(!1)1392 838 y Fm(x)1420 845 y Fl(n)1457 838 y Fi(2)g Fm(M)r(:)50 b Ff(2)191 962 y Fo(An)17 b(excessiv)o(e)e (measure)g Fm(\026)i Fo(ob)o(viously)g(preserv)o(es)e(this)i(prop)q (ert)o(y)l(,)f(if)h Fm(P)24 b Fo(is)17 b(replaced)118 1022 y(b)o(y)i(some)f(p)q(o)o(w)o(er)i Fm(P)498 1004 y Fl(n)522 1022 y Fo(.)30 b(Th)o(us)20 b(it)f(is)g(clear)g(that)h(for)g (\()p Fm(y)1147 1029 y Fl(m)1180 1022 y Fm(;)8 b(z)1225 1029 y Fl(m)1258 1022 y Fo(\))19 b Fi(2)g Fm(N)25 b Fo(with)19 b Fm(y)1550 1029 y Fl(m)1603 1022 y Fm(<)g Fo(1)h(the)118 1082 y(supp)q(ort)g Fm(M)j Fo(con)o(tains)c(not)f(only)g(the)g(\014xed) g(p)q(oin)o(ts)h Fm(z)1140 1089 y Fl(m)1181 1082 y Fm(=)8 b Fo(\(1)k Fi(\000)f Fm(y)1342 1089 y Fl(m)1375 1082 y Fo(\))18 b(of)h(the)f(asso)q(ciated)118 1143 y(a\016ne)c(maps)f Fm(g)394 1150 y Fl(m)441 1143 y Fo(:)h Fm(x)f Fi(!)h Fm(y)598 1150 y Fl(m)631 1143 y Fm(x)6 b Fo(+)g Fm(z)732 1150 y Fl(m)779 1143 y Fo(but)14 b(as)g(w)o(ell)f(those)h(of)g(the)g (comp)q(osition)f Fm(g)1573 1150 y Fj(1)1599 1143 y Fi(\016)6 b Fm(:)i(:)g(:)e Fi(\016)g Fm(g)1748 1150 y Fl(n)118 1203 y Fo(for)17 b(an)o(y)f Fm(n)e Fi(2)g Fn(N)p Fo(.)191 1263 y(The)j(next)e(result)h(is)g(closely)f(related)h(to)g(\(1.3\):)118 1352 y Fn(\(4.3\))28 b(Theorem.)23 b Fg(If)18 b Fm(\026)d Fi(2)h(M)p Fo(\()p Fn(R)788 1359 y Fj(+)817 1352 y Fo(\))i Fg(is)h(a)f(nontrivial)h(invariant)g(me)n(asur)n(e)f(with)g(sup-)118 1412 y(p)n(ort)f Fm(M)5 b Fg(,)17 b(then,)i(with)f(the)f(notation)i(of) 24 b Fo(\(1.3\))p Fg(,)118 1526 y Fo(\(a\))96 b(inf)11 b Fm(M)28 b Fo(=)14 b(inf)s Fi(f)601 1492 y Fm(z)p 559 1514 V 559 1560 a Fo(1)d Fi(\000)g Fm(y)688 1526 y Fo(:)i(\()p Fm(y)r(;)8 b(z)r Fo(\))14 b Fi(2)g Fm(N)926 1533 y Fl(c)943 1526 y Fi(g)p Fm(;)118 1683 y Fo(\(b\))96 b(sup)8 b Fm(M)j Fo(=)j(sup)p Fi(f)616 1649 y Fm(z)p 573 1671 V 573 1717 a Fo(1)e Fi(\000)f Fm(y)703 1683 y Fo(:)i(\()p Fm(y)r(;)8 b(z)r Fo(\))13 b Fi(2)i Fm(N)941 1690 y Fl(c)958 1683 y Fi(g)87 b Fm(if)22 b(N)1172 1690 y Fl(e)1205 1683 y Fo(=)14 b Fi(;)p Fm(;)420 1778 y Fo(=)f Fi(1)549 b Fg(otherwise.)118 1867 y(Pr)n(o)n(of.)23 b Fo(1.)36 b(If)20 b(inf)h Fm(M)26 b Fo(and)c(sup)f Fm(M)26 b Fo(are)21 b(denoted)g(b)o(y)g Fm(m)p 1163 1874 43 2 v 20 w Fo(and)p 1326 1841 V 22 w Fm(m)p Fo(,)g(resp)q(ectiv)o(ely)l(,)e(it)i(is)118 1927 y(immedi)o(ate)14 b(from)h(\(4.2\))h(that)268 2029 y Fm(m)p 268 2036 V 13 w Fi(\024)e Fo(inf)g Fi(f)p Fm(:)8 b(:)g(:)o Fi(g)87 b Fo(and)p 807 2003 V 88 w Fm(m)13 b Fi(\025)h Fo(sup)d Fi(f)p Fm(:)d(:)g(:)o Fi(g)p Fm(:)118 2131 y Fo(Moreo)o(v)o(er,)p 346 2104 V 15 w Fm(m)13 b Fo(=)h Fi(1)i Fo(holds)h(in)f(the)g(case)g Fm(N)931 2138 y Fl(e)963 2131 y Fi(6)p Fo(=)e Fi(;)p Fo(.)21 b(Indeed,)15 b(\(4.1a\))i(implies)118 2232 y(\(1\))88 b Fm(M)13 b Fi(\\)f Fo(]0)p Fm(;)c Fi(1)p Fo([)g Fi(6)p Fo(=)13 b Fi(;)p Fm(;)118 2334 y Fo(and)k(0)d Fi(6)p Fo(=)g Fm(x)331 2341 y Fj(0)364 2334 y Fi(2)g Fm(M)22 b Fo(com)o(bined)14 b(with)i(\()p Fm(y)852 2341 y Fj(0)872 2334 y Fm(;)8 b(z)917 2341 y Fj(0)936 2334 y Fo(\))14 b Fi(2)g Fm(N)1055 2341 y Fl(e)1089 2334 y Fo(yields,)h(again)i(b)o(y)f(\(4.1a\),)268 2436 y Fm(M)j Fi(3)14 b Fm(y)407 2415 y Fl(n)405 2448 y Fj(0)430 2436 y Fm(x)458 2443 y Fj(0)488 2436 y Fo(+)d Fm(y)563 2415 y Fl(n)p Fk(\000)p Fj(1)561 2448 y(0)632 2436 y Fm(z)655 2443 y Fj(0)685 2436 y Fo(+)g Fm(:)d(:)g(:)i Fo(+)h Fm(z)874 2443 y Fj(0)908 2436 y Fi(!)i(1)p Fm(:)191 2537 y Fo(2.)22 b(T)l(o)17 b(pro)o(v)o(e)e(the)h(in)o(v)o(erse)f (inequalit)o(y)f(for)i Fm(m)p 1000 2544 V Fo(,)g(abbreviate)268 2648 y Fm(\015)g Fo(:=)e(inf)g Fi(f)518 2614 y Fm(z)p 475 2636 111 2 v 475 2682 a Fo(1)d Fi(\000)g Fm(y)604 2648 y Fo(:)j(\()p Fm(y)r(;)8 b(z)r Fo(\))13 b Fi(2)h Fm(N)842 2655 y Fl(c)859 2648 y Fi(g)p Fm(;)933 2877 y Fo(3)p eop %%Page: 4 5 4 4 bop 118 77 a Fo(whic)o(h)16 b(so)h(far)f(ma)o(y)f(b)q(e)h (in\014nite,)f(and)i(c)o(ho)q(ose)g Fm(\016)e Fi(2)f Fo([0)p Fm(;)8 b(\015)s Fo([.)20 b(Then)268 175 y Fm(y)9 b(\016)k Fo(+)e Fm(z)16 b Fi(\025)d Fm(\016)89 b Fo(for)16 b(all)25 b(\()p Fm(y)r(;)8 b(z)r Fo(\))13 b Fi(2)h Fm(N)r(;)118 273 y Fo(b)q(ecause)j(this)f(inequalit)o(y)e(is)i(trivial)f(for)h Fm(y)g Fi(\025)d Fo(1.)22 b(Therefore)268 371 y Fi(f)p Fm(x)13 b Fo(:)g Fm(y)r(x)e Fo(+)g Fm(z)k(<)f(\016)r Fi(g)f(\032)h(f)p Fm(x)f Fo(:)h Fm(x)f(<)h(\016)r Fi(g)87 b Fo(for)16 b(all)25 b(\()p Fm(y)r(;)8 b(z)r Fo(\))13 b Fi(2)h Fm(N)r(;)118 469 y Fo(while)h(b)o(y)h(the)g(in)o(v)m(ariance)g (of)g 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Fo(for)16 b Fm(\027)s Fo(-almost)g(all)25 b(\()p Fm(y)r(;)8 b(z)r Fo(\))13 b Fi(2)h Fn(R)1356 939 y Fj(2)1356 972 y(+)1385 959 y Fm(:)118 1057 y Fo(Since)g(b)q(oth)i(sides)e(of)h(this)g (equation)f(are)h(con)o(tin)o(uous)g(functions)g(of)g Fm(x)f Fo(and)h(\()p Fm(y)r(;)8 b(z)r Fo(\),)14 b(they)118 1117 y(agree)i(therefore)g(on)h(the)f(supp)q(ort)h(of)g Fm(\026)11 b Fi(\012)g Fm(\027)s Fo(,)16 b(hence)268 1216 y(\()p Fm(y)r(x)10 b Fo(+)h Fm(z)r Fo(\))g Fi(^)g Fm(\015)17 b Fo(=)d Fm(x)c Fi(^)i Fm(\015)90 b Fo(for)25 b Fm(x)14 b Fi(2)g Fm(M)30 b Fo(and)25 b(\()p Fm(y)r(;)8 b(z)r Fo(\))14 b Fi(2)g Fm(N)r(:)118 1314 y Fo(By)i(c)o(ho)q(osing)h (\()p Fm(y)436 1321 y Fj(0)455 1314 y Fm(;)8 b(z)500 1321 y Fj(0)519 1314 y Fo(\))14 b Fi(2)g Fm(N)22 b Fo(with)16 b Fm(z)794 1321 y Fj(0)827 1314 y Fm(>)e Fo(0)j(th)o(us)268 1412 y(\()p Fm(y)311 1419 y Fj(0)330 1412 y Fm(x)11 b Fo(+)g Fm(z)441 1419 y Fj(0)460 1412 y Fo(\))g Fi(^)h Fm(\015)k Fo(=)e Fm(x)d Fi(^)g Fm(\015)90 b Fo(for)17 b(all)24 b Fm(x)14 b Fi(2)g Fm(M)r(:)118 1510 y Fo(This)k(pro)o(v)o(es) f(indeed)g(the)h(inequalit)o(y)e Fm(m)p 853 1517 43 2 v 16 w Fi(\025)g Fm(\015)s Fo(,)i(b)q(ecause)g(an)o(y)g Fm(x)1331 1517 y Fj(0)1367 1510 y Fi(2)f Fm(M)g Fi(\\)12 b Fo([0)p Fm(;)c(\015)s Fo([)18 b(w)o(ould)118 1570 y(lead)e(to)h Fm(y)305 1577 y Fj(0)324 1570 y Fm(x)352 1577 y Fj(0)383 1570 y Fo(+)11 b Fm(z)455 1577 y Fj(0)488 1570 y Fo(=)j Fm(x)568 1577 y Fj(0)587 1570 y Fo(,)i(hence)f Fm(y)776 1577 y Fj(0)810 1570 y Fm(<)f Fo(1,)i(and)h(th)o(us)f(to)g(the)g(con)o (tradiction)268 1677 y Fm(x)296 1684 y Fj(0)329 1677 y Fo(=)429 1643 y Fm(z)452 1650 y Fj(0)p 386 1665 129 2 v 386 1711 a Fo(1)11 b Fi(\000)g Fm(y)495 1718 y Fj(0)533 1677 y Fi(\025)j Fm(\015)s(:)191 1800 y Fo(3.)21 b(The)14 b(corresp)q(onding)h(pro)q(of)g(for)p 856 1773 43 2 v 14 w Fm(m)p Fo(,)f(under)f(the)h(additional)g(h)o(yp)q(othesis)g Fm(N)1649 1807 y Fl(e)1681 1800 y Fo(=)g Fi(;)p Fo(,)118 1860 y(di\013ers)i(only)g(at)h(the)f(b)q(eginning.)22 b(De\014ne)16 b(here)268 1967 y Fm(\015)g Fo(:=)e(sup)d Fi(f)532 1933 y Fm(z)p 489 1955 111 2 v 489 2001 a Fo(1)h Fi(\000)e Fm(y)619 1967 y Fo(:)j(\()p Fm(y)r(;)8 b(z)r Fo(\))13 b Fi(2)h Fm(N)856 1974 y Fl(c)874 1967 y Fi(g)p Fm(;)118 2090 y Fo(whic)o(h)i(no)o(w)g(ma)o(y)f(b)q(e)h(assumed)g(to)h (b)q(e)f(\014nite,)f(and)i(c)o(ho)q(ose)g Fm(\016)e Fi(2)f Fo([)p Fm(\015)s(;)8 b Fi(1)p Fo([.)20 b(Then)268 2188 y Fm(y)9 b(\016)k Fo(+)e Fm(z)16 b Fi(\024)d Fm(\016)89 b Fo(for)16 b(all)25 b(\()p Fm(y)r(;)8 b(z)r Fo(\))13 b Fi(2)h Fm(N)r(;)118 2286 y Fo(b)q(ecause)j Fm(N)f Fi(n)11 b Fm(N)429 2293 y Fl(c)463 2286 y Fo(ma)o(y)j(con)o(tain)i(\()p Fm(y)r(;)8 b(z)r Fo(\))13 b(=)h(\(1)p Fm(;)8 b Fo(0\))17 b(only)l(.)k(Therefore)268 2384 y Fi(f)p Fm(x)13 b Fo(:)g Fm(y)r(x)e Fo(+)g Fm(z)k Fi(\024)f Fm(\016)r Fi(g)f(\033)h(f)p Fm(x)f Fo(:)h Fm(x)f Fi(\024)h Fm(\016)r Fi(g)86 b Fo(for)17 b(all)24 b(\()p Fm(y)r(;)8 b(z)r Fo(\))13 b Fi(2)h Fm(N)r(;)118 2482 y Fo(and)j(the)f(pro)q(of)h(con)o(tin)o(ues)f(in)g(complete)d (analogy)18 b(to)e(part)h(2)p Fm(:)50 b Ff(2)191 2571 y Fo(While)21 b(it)g(is)h(easily)f(seen)g(that)h(\(b\))g(extends)f(to)h (excessiv)o(e)e(measures,)h(this)h(fails)118 2632 y(for)16 b(\(a\).)21 b(Indeed,)15 b(if)g Fm(\027)j Fo(is)e(transien)o(t,)e(the)i (excessiv)o(e)d(measure)h Fm(")1313 2639 y Fj(0)1341 2632 y Fo(\()1360 2598 y Fe(P)1412 2639 y Fl(n)p Fk(\025)p Fj(0)1489 2632 y Fm(P)1527 2614 y Fl(n)1551 2632 y Fo(\))h(is)g(lo)q (cally)118 2692 y(\014nite)e(according)h(to)f(\(2.2b\))h(with)g(inf)d Fm(M)19 b Fo(=)14 b(0,)g(while)e(the)i(in\014m)o(um)c(on)k(the)f(righ)o (t{hand)118 2752 y(side)j(ma)o(y)f(b)q(e)h(arbitrarily)f(large.)933 2877 y(4)p eop %%Page: 5 6 5 5 bop 191 77 a Fo(A)16 b(simple)e(consequence)h(of)i(\(4.3\))f(is)h (the)f(follo)o(wing)f(result:)118 166 y Fn(\(4.4\))28 b(Prop)r(osition.)c Fg(The)38 b(supp)n(ort)g(M)g(of)g(a)g(nontrivial)h (invariant)g(me)n(asur)n(e)118 226 y Fm(\026)14 b Fi(2)g(M)p Fo(\()p Fn(R)329 233 y Fj(+)358 226 y Fo(\))k Fg(is)f(an)h(interval)h (whenever)g(N)f(is)f(c)n(onne)n(cte)n(d.)118 316 y(Pr)n(o)n(of.)23 b Fo(With)16 b(0)f Fi(6)p Fo(=)e Fm(x)515 323 y Fj(0)549 316 y Fi(2)h Fm(M)21 b Fo(\(see)16 b(\(1\))h(in)f(the)g(pro)q(of)h(of)f (\(4.3\)\))h(the)f(set)268 417 y Fm(M)315 424 y Fl(n)352 417 y Fo(:=)d Fi(f)p Fm(y)468 397 y Fl(n)491 417 y Fm(x)519 424 y Fj(0)550 417 y Fo(+)e Fm(y)625 397 y Fl(n)p Fk(\000)p Fj(1)693 417 y Fm(z)i Fo(+)e Fm(:)d(:)g(:)i Fo(+)h Fm(z)16 b Fo(:)d(\()p Fm(y)r(;)8 b(z)r Fo(\))13 b Fi(2)h Fm(N)5 b Fi(g)118 519 y Fo(is)16 b(the)g(con)o(tin)o(uous)g(image)f(of)i(a)f (connected)g(set,)g(hence)118 621 y(\(1\))88 b(])p Fm(m)p 282 628 43 2 v 324 631 a Fl(n)347 621 y Fm(;)p 369 594 V 8 w(m)412 628 y Fl(n)435 621 y Fo([)14 b Fi(\032)f Fm(M)562 628 y Fl(n)600 621 y Fi(\032)g Fo([)p Fm(m)p 666 628 V 10 x Fl(n)732 621 y Fm(;)p 754 594 V 8 w(m)797 628 y Fl(n)820 621 y Fo(])87 b(for)25 b Fm(n)14 b Fi(2)g Fn(N)118 722 y Fo(with)h(appropriate)h(b)q(ounds)g Fm(m)p 658 729 V 11 x Fl(n)740 722 y Fo(and)p 833 696 V 15 w Fm(m)876 729 y Fl(n)899 722 y Fo(.)21 b(Moreo)o(v)o(er,)14 b(according)h(to)h(\(4.1a\))g(and)f(\(4.3\))118 824 y(\(2\))88 b Fm(M)315 831 y Fl(n)352 824 y Fi(\032)14 b Fm(M)92 b Fo(for)25 b Fm(n)14 b Fi(2)g Fn(N)p Fm(;)118 938 y Fo(\(3\))88 b Fm(m)p 268 945 V 310 949 a Fl(n)348 938 y Fi(!)13 b Fo(inf)f Fm(M)92 b Fo(and)p 784 912 V 88 w Fm(m)827 945 y Fl(n)864 938 y Fi(!)14 b Fo(sup)8 b Fm(M)r(;)118 1053 y Fo(where)19 b(in)f(the)h(case)g Fm(N)553 1060 y Fl(e)590 1053 y Fi(6)p Fo(=)f Fi(;)h Fo(the)g(c)o(hoice)f Fm(x)953 1060 y Fj(0)990 1053 y Fi(6)p Fo(=)h(0)g(is)g(essen)o(tial.)28 b(T)l(ogether,)20 b(\(1\))f({)g(\(3\))118 1113 y(pro)o(v)o(e)d(the)g (assertion)p Fm(:)49 b Ff(2)191 1202 y Fo(As)16 b(will)e(b)q(e)i(seen)f (in)g(\(6.1\),)h(the)f(con)o(v)o(erse)g(of)h(\(4.4\))f(fails)h(in)f(a)h (surprisingly)f(general)118 1262 y(sense.)191 1322 y(The)k(\014nal)g (part)g(of)f(this)h(section)f(concerns)g(prop)q(erties)h(not)g(only)f (of)h(the)f(supp)q(ort)118 1383 y(but)i(of)g(excessiv)o(e)e(measures)g (themselv)o(es.)29 b(The)20 b(follo)o(wing)g(tec)o(hnical)e(result)h (will)g(b)q(e)118 1443 y(imp)q(ortan)o(t:)118 1532 y Fn(\(4.5\))28 b(Prop)r(osition.)c Fg(L)n(et)18 b Fo(\()p Fm(y)715 1539 y Fj(0)735 1532 y Fm(;)8 b(z)780 1539 y Fj(0)799 1532 y Fo(\))16 b Fi(2)g Fn(R)925 1514 y Fj(2)925 1544 y(+)972 1532 y Fg(with)j Fm(y)1103 1539 y Fj(0)1139 1532 y Fm(<)c Fo(1)k Fg(b)n(e)g(given.)27 b(Then)19 b(for)f Fm(p)f(>)e Fo(0)118 1592 y Fg(and)j Fm(s)c(>)f(z)324 1599 y Fj(0)352 1592 y Fm(=)8 b Fo(\(1)k Fi(\000)f Fm(y)513 1599 y Fj(0)532 1592 y Fo(\))18 b Fg(ther)n(e)f(exist)i(\014nite)f(c)n (onstants)268 1694 y Fm(\013)c Fo(=)f Fm(\013)p Fo(\()p Fm(p)p Fo(\))88 b Fg(and)g Fm(\015)17 b Fo(=)c Fm(\015)s Fo(\()p Fm(p;)8 b(s)p Fo(\))118 1796 y Fg(such)18 b(that)g(under)f(the) h(hyp)n(othesis)268 1897 y Fm(\027)s Fo(\([0)p Fm(;)8 b(y)398 1904 y Fj(0)417 1897 y Fo([)g Fi(\002)g Fo([0)p Fm(;)g(z)569 1904 y Fj(0)588 1897 y Fo([\))13 b Fi(\025)h Fm(p)118 1999 y Fg(e)n(ach)k(exc)n(essive)h(me)n(asur)n(e)e Fm(\026)d Fi(2)g(M)p Fo(\()p Fn(R)832 2006 y Fj(+)861 1999 y Fo(\))j Fg(satis\014es)268 2101 y Fm(\026)p Fo(\([0)p Fm(;)8 b(t)p Fo(]\))13 b Fi(\024)g Fm(\015)f(\026)p Fo(\([0)p Fm(;)c(s)p Fo(]\))13 b Fm(t)724 2080 y Fl(\013)835 2101 y Fg(for)25 b Fm(t)13 b Fi(\025)h Fm(s:)118 2202 y Fg(Pr)n(o)n(of.)23 b Fo(If)16 b Fm(\026)h Fo(is)f(excessiv)o(e)e(with)i(resp)q(ect)g(to)g Fm(\027)s Fo(,)h(then)268 2304 y Fm(\026)p Fo(\([0)p Fm(;)8 b(t)p Fo(]\))13 b Fm(\027)s Fo(\([0)p Fm(;)8 b(y)570 2311 y Fj(0)589 2304 y Fo([)g Fi(\002)g Fo([0)p Fm(;)g(z)741 2311 y Fj(0)760 2304 y Fo([\))41 b Fi(\024)g Fo(\()p Fm(\026)12 b Fi(\012)f Fm(\027)s Fo(\)\()p Fi(f)p Fo(\()p Fm(x)p Fo(;)d Fm(y)r(;)g(z)r Fo(\))k(:)i Fm(y)r(x)c Fo(+)h Fm(z)16 b Fi(\024)d Fm(y)1543 2311 y Fj(0)1563 2304 y Fm(t)e Fo(+)g Fm(z)1664 2311 y Fj(0)1683 2304 y Fi(g)p Fo(\))834 2406 y Fi(\024)41 b Fm(\026)p Fo(\([0)8 b Fm(;)17 b(y)1063 2413 y Fj(0)1083 2406 y Fm(t)10 b Fo(+)h Fm(z)1183 2413 y Fj(0)1203 2406 y Fo(]\))86 b(for)17 b(all)24 b Fm(t)14 b Fi(\025)f Fo(0)p Fm(:)118 2507 y Fo(This)j(yields)g(the)g (estimate)268 2609 y Fm(\026)p Fo(\([0)p Fm(;)8 b(t)p Fo(]\))40 b Fi(\024)i Fm(p)572 2589 y Fk(\000)p Fj(1)619 2609 y Fm(\026)p Fo(\([0)8 b Fm(;)17 b(y)768 2616 y Fj(0)787 2609 y Fm(t)11 b Fo(+)g Fm(z)888 2616 y Fj(0)907 2609 y Fo(]\))467 2711 y Fi(\024)42 b Fm(p)572 2690 y Fk(\000)p Fj(2)619 2711 y Fm(\026)p Fo(\([0)8 b Fm(;)17 b(y)768 2718 y Fj(0)787 2711 y Fo(\()p Fm(y)830 2718 y Fj(0)850 2711 y Fm(t)11 b Fo(+)g Fm(z)951 2718 y Fj(0)970 2711 y Fo(\))g(+)g Fm(z)1072 2718 y Fj(0)1091 2711 y Fo(]\))p Fm(;)933 2877 y Fo(5)p eop %%Page: 6 7 6 6 bop 118 77 a Fo(hence)16 b(b)o(y)f(iteration)268 179 y Fm(\026)p Fo(\([0)p Fm(;)8 b(t)p Fo(]\))40 b Fi(\024)i Fm(p)572 158 y Fk(\000)p Fl(k)621 179 y Fm(\026)p Fo(\([0)8 b Fm(;)17 b(y)772 158 y Fl(k)770 191 y Fj(0)801 179 y Fm(t)10 b Fo(+)h Fm(y)904 158 y Fl(k)q Fk(\000)p Fj(1)902 191 y(0)970 179 y Fm(z)993 186 y Fj(0)1024 179 y Fo(+)g Fm(:)d(:)g(:)i Fo(+)h Fm(z)1213 186 y Fj(0)1233 179 y Fo(]\))467 292 y Fi(\024)42 b Fm(p)572 271 y Fk(\000)p Fl(k)621 292 y Fm(\026)p Fo(\([0)8 b Fm(;)17 b(y)772 271 y Fl(k)770 304 y Fj(0)801 292 y Fm(t)10 b Fo(+)927 258 y Fm(z)950 265 y Fj(0)p 883 280 129 2 v 883 326 a Fo(1)i Fi(\000)f Fm(y)993 333 y Fj(0)1017 292 y Fo(]\))87 b(for)25 b Fm(k)16 b Fi(2)e Fn(N)p Fm(:)118 416 y Fo(Therefore)268 518 y Fm(\026)p Fo(\([0)p Fm(;)8 b(t)p Fo(]\))13 b Fi(\024)g Fm(p)516 497 y Fk(\000)p Fl(k)565 518 y Fm(\026)p Fo(\([0)p Fm(;)8 b(s)p Fo(]\))87 b(whenev)o(er)f Fm(y)1127 497 y Fl(k)1125 530 y Fj(0)1156 518 y Fm(t)11 b Fo(+)1282 484 y Fm(z)1305 491 y Fj(0)p 1239 506 V 1239 552 a Fo(1)h Fi(\000)e Fm(y)1348 559 y Fj(0)1387 518 y Fi(\024)j Fm(s:)118 642 y Fo(This)j(condition)h(holds)f(for)h Fm(k)f Fi(2)e Fn(N)i Fo(satisfying)268 752 y Fm(k)f Fi(\025)f Fo([\(log)9 b(\()p Fm(s)i Fi(\000)616 719 y Fm(z)639 726 y Fj(0)p 573 741 V 573 787 a Fo(1)g Fi(\000)g Fm(y)682 794 y Fj(0)706 752 y Fo(\))g Fi(\000)g Fo(log)f Fm(t)p Fo(\))e Fm(=)g Fo(log)h Fm(y)1030 759 y Fj(0)1050 752 y Fo(])k Fi(\025)h Fo(0)p Fm(;)118 876 y Fo(with)20 b(the)f(con)o(v)o(en)o(tion)g(log)9 b Fm(y)663 883 y Fj(0)702 876 y Fo(=)20 b Fi(\0001)g Fo(for)g Fm(y)971 883 y Fj(0)1010 876 y Fo(=)g(0.)32 b(Since)19 b(for)h Fm(t)g Fi(\025)f Fm(s)h Fo(this)g(b)q(ound)h(is)118 937 y(nonnegativ)o(e,)16 b(\014nally)268 1038 y Fm(\026)p Fo(\([0)p Fm(;)8 b(t)p Fo(]\))13 b Fi(\024)g Fm(p)516 1018 y Fk(\000)p Fj(\([)p Fl(:)6 b(:)g(:)p Fj(])f(+)i(1\))700 1038 y Fm(\026)p Fo(\([0)p Fm(;)h(s)p Fo(]\))86 b(for)25 b Fm(t)14 b Fi(\025)g Fm(s;)118 1140 y Fo(and)j(a)g(simple)d (computation)h(pro)o(vides)h(the)g(constan)o(ts)268 1261 y Fm(\013)e Fo(:=)f(log)c Fm(p)f(=)g Fo(log)j Fm(y)610 1268 y Fj(0)717 1261 y Fo(and)88 b Fm(\015)16 b Fo(:=)995 1228 y(1)p 995 1250 25 2 v 995 1295 a Fm(p)1035 1261 y Fo(\()p Fm(s)11 b Fi(\000)1186 1228 y Fm(z)1209 1235 y Fj(0)p 1143 1250 129 2 v 1143 1295 a Fo(1)g Fi(\000)g Fm(y)1252 1302 y Fj(0)1277 1261 y Fo(\))1296 1241 y Fk(\000)p Fl(\013)1348 1261 y Fm(:)49 b Ff(2)191 1385 y Fo(The)20 b(essen)o(tial)e(con)o(ten)o(t)h(of)g(this)h(result)f(lies)f(in)h(the)g (fact)g(that)h(the)f(gro)o(wth)h(of)g(an)118 1446 y(excessiv)o(e)14 b(measure)h(is)h(only)g(p)q(olynomial)f(whenev)o(er)g Fn(P)p Fo(\()p Fm(Y)26 b(<)13 b Fo(1\))i Fm(>)e Fo(0.)191 1506 y(The)k(last)f(result)g(of)g(this)g(section)g(will)f(b)q(e)i (required)e(for)h(stabilit)o(y)f(theorems:)118 1595 y Fn(\(4.6\))28 b(Lemma.)22 b Fg(L)n(et)c Fi(N)24 b(3)16 b Fm(\027)706 1602 y Fl(k)756 1568 y Fj(w)744 1595 y Fi(!)g Fm(\027)22 b Fg(and)d Fm(\026)981 1602 y Fl(k)1019 1595 y Fi(2)e(M)p Fo(\()p Fn(R)1190 1602 y Fj(+)1219 1595 y Fo(\))h Fg(b)n(e)i(exc)n(essive)g(with)f(r)n(esp)n(e)n(ct)118 1655 y(to)f Fm(\027)201 1662 y Fl(k)222 1655 y Fg(.)k(Then)c Fm(\026)415 1662 y Fl(k)466 1629 y Fj(v)451 1655 y Fi(!)c Fm(\026)g Fi(2)g(M)p Fo(\()p Fn(R)726 1662 y Fj(+)755 1655 y Fo(\))j Fg(implies)h(that)g Fm(\026)g Fg(is)f(exc)n(essive)i (with)f(r)n(esp)n(e)n(ct)f(to)g Fm(\027)s Fg(.)118 1744 y(Pr)n(o)n(of.)23 b Fo(V)l(ague)13 b(con)o(v)o(ergence)d(is)i(implied)e (b)o(y)h(w)o(eak)h(con)o(v)o(ergence)f(and)i(compatible)d(with)118 1805 y(forming)15 b(pro)q(duct)i(measures,)e(hence)268 1906 y Fm(\026)297 1913 y Fl(k)329 1906 y Fi(\012)c Fm(\027)403 1913 y Fl(k)454 1880 y Fj(v)438 1906 y Fi(!)j Fm(\026)d Fi(\012)g Fm(\027:)118 2008 y Fo(By)16 b(monotone)g(appro)o(ximation)f (this)h(yields)268 2074 y Fe(R)304 2110 y Fm(g)10 b(d)p Fo(\()p Fm(\026)i Fi(\012)f Fm(\027)s Fo(\))j Fi(\024)f Fo(lim)8 b(inf)728 2117 y Fl(k)q Fk(!1)828 2074 y Fe(R)864 2110 y Fm(g)i(d)p Fo(\()p Fm(\026)970 2117 y Fl(k)1003 2110 y Fi(\012)h Fm(\027)1077 2117 y Fl(k)1098 2110 y Fo(\))88 b(for)25 b(0)14 b Fi(\024)g Fm(g)i Fi(2)e(C)s Fo(\()p Fn(R)1555 2117 y Fj(+)1587 2110 y Fi(\002)s Fn(R)1671 2089 y Fj(2)1671 2122 y(+)1700 2110 y Fo(\))p Fm(:)118 2211 y Fo(Therefore,)i(with)g Fm(P)495 2218 y Fl(k)533 2211 y Fo(denoting)g(the)g(k)o(ernel)f(corresp)q(onding)i(to)g Fm(\027)1358 2218 y Fl(k)1379 2211 y Fo(,)268 2313 y Fm(\026)8 b(P)f(f)48 b Fo(=)495 2278 y Fe(R)531 2313 y Fm(f)5 b Fo(\()p Fm(y)r(x)10 b Fo(+)h Fm(z)r Fo(\))d Fm(\026)p Fo(\()p Fm(dx)p Fo(\))p Fm(\027)s Fo(\()p Fm(dy)r(;)g(dz)r Fo(\))414 2415 y Fi(\024)42 b Fo(lim)8 b(inf)638 2422 y Fl(k)q Fk(!1)738 2379 y Fe(R)774 2415 y Fm(f)d Fo(\()p Fm(y)r(x)11 b Fo(+)g Fm(z)r Fo(\))d Fm(\026)1017 2422 y Fl(k)1039 2415 y Fo(\()p Fm(dx)p Fo(\))p Fm(\027)1154 2422 y Fl(k)1175 2415 y Fo(\()p Fm(dy)r(;)g(dz)r Fo(\))415 2517 y(=)42 b(lim)8 b(inf)638 2524 y Fl(k)q Fk(!1)738 2517 y Fm(\026)767 2524 y Fl(k)789 2517 y Fm(P)820 2524 y Fl(k)842 2517 y Fm(f)414 2618 y Fi(\024)42 b Fo(lim)8 b(inf)638 2625 y Fl(k)q Fk(!1)738 2618 y Fm(\026)767 2625 y Fl(k)789 2618 y Fm(f)415 2720 y Fo(=)42 b Fm(\026f)92 b Fo(for)26 b(0)14 b Fi(\024)f Fm(f)20 b Fi(2)14 b(K)q Fo(\()p Fn(R)1005 2727 y Fj(+)1034 2720 y Fo(\))p Fm(;)933 2877 y Fo(6)p eop %%Page: 7 8 7 7 bop 118 77 a Fo(whic)o(h)16 b(pro)o(v)o(es)f Fm(\026P)22 b Fi(\024)13 b Fm(\026:)50 b Ff(2)191 166 y Fo(It)21 b(should)g(b)q(e)h(men)o(tioned)d(that)i(this)g(result)g(do)q(es)g(not) h(carry)f(o)o(v)o(er)f(to)h(in)o(v)m(arian)o(t)118 226 y(measures,)15 b(as)i(can)f(b)q(e)h(sho)o(wn)f(b)o(y)g(somewhat)g(in)o (v)o(olv)o(ed)e(examples.)118 403 y Fn(5.)25 b(Existence)16 b(and)j(uniqueness)f(of)h(in)n(v)m(arian)n(t)g(measures)118 521 y Fo(T)l(o)f(deriv)o(e)e(ergo)q(dic)h(theorems)f(in)h(the)g (recurren)o(t)f(case,)h(the)g(follo)o(wing)g(\\lo)q(calization")118 581 y(is)f(essen)o(tial:)118 671 y Fn(\(5.1\))28 b(De\014nition.)c Fg(L)n(et)17 b Fm(\027)j Fg(b)n(e)e(r)n(e)n(curr)n(ent)f(and)h Fm(x)p 1038 678 28 2 v 13 w(<)c(t)f(<)h Fi(1)p Fg(.)22 b(Then:)118 760 y Fo(\(a\))268 742 y Fl(t)274 760 y Fm(P)j Fg(denotes)18 b(the)g Fo(\\)p Fg(hitting)h(kernel")g(b)n(elonging)h(to) d Fm(P)25 b Fg(and)18 b Fo([0)p Fm(;)8 b(t)p Fo(])p Fg(,)16 b(i.e.)268 834 y Fl(t)274 855 y Fm(P)7 b Fo(\()p Fm(x)p Fo(;)h Fm(B)s Fo(\))13 b(:=)h Fn(P)557 834 y Fl(x)579 855 y Fo(\()p Fm(X)638 862 y Fl(T)680 855 y Fi(2)g Fm(B)s Fo(\))86 b Fg(for)25 b Fm(x)13 b Fi(2)h Fo([0)p Fm(;)8 b(t)p Fo(])24 b Fg(and)h Fm(B)17 b Fi(2)d(B)r Fo(\([0)p Fm(;)8 b(t)p Fo(]\))p Fm(;)118 949 y Fg(wher)n(e)268 1044 y Fm(T)20 b Fo(:=)13 b(inf)s Fi(f)p Fm(n)h Fi(2)g Fn(N)g Fo(:)f Fm(X)681 1051 y Fl(n)719 1044 y Fi(2)h Fo([0)p Fm(;)8 b(t)p Fo(])p Fi(g)p Fo(;)118 1139 y(\(b\))87 b(\()289 1121 y Fl(t)296 1139 y Fm(X)336 1146 y Fl(n)360 1139 y Fo(\))379 1146 y Fl(n)p Fk(\025)p Fj(0)465 1139 y Fg(denotes)19 b(the)f Fo(\\)p Fg(sojourn)f(pr)n(o)n(c)n(ess")f(b)n (elonging)k(to)e Fo(\()p Fm(X)1442 1146 y Fl(n)1466 1139 y Fo(\))1485 1146 y Fl(n)p Fk(\025)p Fj(0)1571 1139 y Fg(and)g Fo([0)p Fm(;)8 b(t)p Fo(])p Fg(,)118 1199 y(i.e.)268 1273 y Fl(t)274 1293 y Fm(X)314 1300 y Fl(n)352 1293 y Fo(:=)13 b Fm(X)457 1300 y Fl(T)478 1304 y Fd(n)589 1293 y Fg(for)24 b Fm(n)14 b Fi(\025)g Fo(0)p Fm(;)118 1388 y Fg(wher)n(e)20 b Fm(T)287 1395 y Fj(0)325 1388 y Fm(<)f(T)411 1395 y Fj(1)449 1388 y Fm(<)f(:)8 b(:)g(:)20 b Fg(ar)n(e)f(the)i(r)n(andom)e(times)h(when)h Fo(\()p Fm(X)1251 1395 y Fl(n)1275 1388 y Fo(\))1294 1395 y Fl(n)p Fk(\025)p Fj(0)1383 1388 y Fg(is)f(in)g Fo([0)p Fm(;)8 b(t)p Fo(])19 b Fg(and)h(the)118 1448 y(notation)312 1430 y Fl(t)319 1448 y Fm(X)363 1430 y Fl(x)359 1461 y(n)403 1448 y Fg(is)d(use)n(d)g(in)h(the)g(c)n(ase)g Fm(X)849 1455 y Fj(0)883 1448 y Fo(=)13 b Fm(x)p Fg(.)191 1538 y Fo(F)l(or)18 b(easy)g(reference)f(a)h(simple)e(conclusion)h (from)g(probabilistic)g(p)q(oten)o(tial)h(theory)118 1598 y(is)e(stated)h(explicitly:)118 1687 y Fn(\(5.2\))28 b(Lemma.)22 b Fg(L)n(et)13 b Fm(\027)j Fg(b)n(e)d(r)n(e)n(curr)n(ent)g (and)g Fm(\026)h Fi(2)g(M)p Fo(\()p Fn(R)1164 1694 y Fj(+)1193 1687 y Fo(\))g Fg(b)n(e)f(exc)n(essive.)23 b(If)1564 1669 y Fl(t)1571 1687 y Fm(\026)13 b Fg(denotes)118 1747 y(the)18 b(r)n(estriction)f(of)h Fm(\026)f Fg(to)h Fo([0)p Fm(;)8 b(t)p Fo(])p Fm(;)24 b(x)p 727 1754 V 14 w(<)14 b(t)f(<)h Fi(1)p Fg(,)j(then)118 1836 y Fo(\(a\))268 1818 y Fl(t)274 1836 y Fm(\026)h Fg(is)f(invariant)h(with)g(r)n(esp)n (e)n(ct)f(to)907 1818 y Fl(t)914 1836 y Fm(P)7 b Fg(,)118 1926 y Fo(\(b\))87 b Fm(\026)18 b Fg(is)f(invariant)h(with)g(r)n(esp)n (e)n(ct)f(to)h Fm(P)7 b Fg(.)118 2015 y(Pr)n(o)n(of.)23 b Fo(1.)f(If)16 b Fm(I)402 2022 y Fl(A)446 2015 y Fo(for)g Fm(A)e Fi(2)g(B)r Fo(\()p Fn(R)714 2022 y Fj(+)742 2015 y Fo(\))j(denotes)f(the)g(k)o(ernel)268 2110 y Fm(I)290 2117 y Fl(A)318 2110 y Fo(\()p Fm(x)p Fo(;)8 b Fi(\001)p Fo(\))13 b(:=)g(1)522 2117 y Fl(A)551 2110 y Fo(\()p Fm(x)p Fo(\))8 b Fm(")648 2117 y Fl(x)757 2110 y Fo(for)25 b Fm(x)14 b Fi(2)g Fn(R)971 2117 y Fj(+)1000 2110 y Fm(;)118 2204 y Fo(the)i(crucial)f(p)q(oin)o(t)i(is)f(the)g(inequalit)o(y)118 2299 y(\(1\))88 b Fm(\026)p Fo(\()p Fm(I)338 2306 y Fl(A)374 2266 y Fe(P)426 2306 y Fl(n)p Fk(\025)p Fj(0)503 2299 y Fo(\()p Fm(P)7 b(I)582 2308 y Fh(R)615 2312 y Fc(+)635 2308 y Fk(n)p Fl(A)681 2299 y Fo(\))700 2278 y Fl(n)723 2299 y Fo(\))14 b Fi(\024)g Fm(\026)118 2394 y Fo(\(see)i(e.g.)21 b(IX,)15 b(\(62.4\))i(and)f(\(31.6\),)h(in)f([8]\).)191 2454 y(2.)21 b(Multiplied)12 b(b)o(y)h Fm(P)7 b(I)610 2461 y Fl(A)653 2454 y Fo(from)13 b(the)g(righ)o(t)h(and)g(sp)q (ecialized)f(to)h Fm(A)g Fo(=)g([0)p Fm(;)8 b(t)p Fo(],)k(\(1\))j (yields)268 2549 y(\()287 2528 y Fl(t)293 2549 y Fm(\026)330 2528 y Fl(t)337 2549 y Fm(P)7 b Fo(\)\()p Fm(B)s Fo(\))41 b Fi(\024)h Fo(\()p Fm(\026P)7 b Fo(\)\()p Fm(B)s Fo(\))513 2650 y Fi(\024)42 b Fm(\026)p Fo(\()p Fm(B)s Fo(\))514 2752 y(=)594 2731 y Fl(t)600 2752 y Fm(\026)p Fo(\()p Fm(B)s Fo(\))87 b(for)25 b Fm(B)17 b Fi(2)d(B)r Fo(\([0)p Fm(;)8 b(t)p Fo(]\))p Fm(:)933 2877 y Fo(7)p eop %%Page: 8 9 8 8 bop 118 77 a Fo(This)16 b(pro)o(v)o(es)g(\(a\),)g(b)q(ecause)653 59 y Fl(t)659 77 y Fm(\026)h Fo(is)f(\014nite)g(and)973 59 y Fl(t)980 77 y Fm(P)23 b Fo(is)16 b(a)h(sto)q(c)o(hastic)g(k)o (ernel.)191 137 y(3.)28 b(F)l(or)18 b(0)f Fi(\024)g Fm(f)22 b Fi(2)c(K)q Fo(\()p Fn(R)640 144 y Fj(+)669 137 y Fo(\))g(c)o(ho)q (ose)g(no)o(w)h Fm(t)d(>)h(x)p 1055 144 28 2 v 18 w Fo(with)h(supp)8 b Fm(f)23 b Fi(\032)17 b Fo([0)p Fm(;)8 b(t)p Fo(])17 b(and)i(denote)118 197 y(the)d(restriction)f(of)i Fm(f)22 b Fo(to)16 b([0)p Fm(;)8 b(t)p Fo(])15 b(b)o(y)771 179 y Fl(t)778 197 y Fm(f)5 b Fo(.)21 b(Then)c(\(a\))f(implies)268 297 y Fm(\026f)47 b Fo(=)448 277 y Fl(t)455 297 y Fm(\026)492 277 y Fl(t)499 297 y Fm(f)368 399 y Fo(=)42 b(\()467 379 y Fl(t)474 399 y Fm(\026)511 379 y Fl(t)518 399 y Fm(P)7 b Fo(\))583 379 y Fl(t)590 399 y Fm(f)368 501 y Fi(\024)41 b Fo(\()p Fm(\026P)7 b Fo(\))p Fm(f)93 b Fo(for)25 b(0)14 b Fi(\024)g Fm(f)19 b Fi(2)14 b(K)q Fo(\()p Fn(R)1034 508 y Fj(+)1063 501 y Fo(\))p Fm(;)118 601 y Fo(once)i(more)f(b)o(y)h(\(1\),)g(and)h(this)f(yields)f(the)h (inequalit)o(y)e Fm(\026)h Fi(\024)e Fm(\026P)24 b Fo(needed)16 b(for)g(\(b\))p Fm(:)49 b Ff(2)191 690 y Fo(No)o(w)24 b(the)g(existence)e(of)j(an)f(in)o(v)m(arian)o(t)f(measure)g(can)h(b)q (e)h(pro)o(v)o(ed)e(in)h(the)f(usual)118 750 y(w)o(a)o(y)17 b(\(see)g(e.g.)25 b([14]\),)18 b(under)f(some)g(simpli\014cation)e(due) i(to)h(the)g(monotonicit)o(y)l(.)23 b(More)118 810 y(generally)l(,)15 b(the)h(follo)o(wing)g(v)o(ersion)f(is)h(required)f(in)h(Section)g(7:) 118 900 y Fn(\(5.3\))28 b(Prop)r(osition.)c Fg(If)17 b Fm(\027)k Fg(is)c(r)n(e)n(curr)n(ent)g(and)g Fm(x)p 1038 907 V 14 w(<)d(t)f(<)h Fi(1)p Fg(,)j(the)h(me)n(asur)n(es)268 1000 y Fm(\045)293 1007 y Fl(n)316 1000 y Fo(\()p Fm(B)s Fo(\))c(:=)473 967 y Fe(P)525 1007 y Fj(0)p Fk(\024)p Fl(m)h Fo(0)p Fm(;)118 1628 y Fo(whic)o(h)j(is)g(p)q(ossible)g(in)g(view)f(of)i Fm(t)d(>)g(x)p 807 1635 V Fo(.)24 b(With)17 b Fm(\026)1029 1635 y Fl(m)1078 1628 y Fo(:=)d Fi(L)p Fo(\()p Fm(X)1237 1635 y Fl(m)1272 1628 y Fo(\))j(this)g(implies)d(b)o(y)j(mono-)118 1688 y(tonicit)o(y)268 1788 y Fn(P)p Fo(\()p Fm(X)365 1795 y Fl(m)p Fj(+)p Fl(l)451 1788 y Fi(\024)c Fm(t)p 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b(K)q Fo(\()p Fn(R)558 1018 y Fj(+)587 1011 y Fo(\))i(suc)o(h)g(that)268 1113 y(1)292 1120 y Fj([0)p Fl(;r)q Fk(\000)p Fl(\016)q Fj(])416 1113 y Fi(\024)e Fm(f)19 b Fi(\024)13 b Fo(1)588 1120 y Fj([0)p Fl(;r)q Fj(])655 1113 y Fm(:)118 1214 y Fo(Then)j(according)h(to)g(\(2.6\))f (the)g(set)268 1316 y Fi(f)p Fm(m)d Fi(\025)h Fo(0)g(:)6 b(1)484 1324 y Fj([0)p Fl(;r)q Fj(])549 1316 y Fo(\()p Fm(X)612 1296 y Fl(x)608 1328 y(m)642 1316 y Fo(\))14 b Fm(<)g Fo(1)751 1324 y Fj([0)p Fl(;r)q Fk(\000)p Fl(\016)q Fj(])861 1316 y Fo(\()p Fm(X)924 1296 y Fj(0)920 1328 y Fl(m)954 1316 y Fo(\))p Fi(g)g(\032)f(f)p Fm(m)h Fi(\025)f Fo(0)i(:)6 b Fm(f)f Fo(\()p Fm(X)1349 1296 y Fl(x)1345 1328 y(m)1378 1316 y Fo(\))14 b(=)f(0)p Fm(;)8 b(f)d Fo(\()p Fm(X)1600 1296 y Fj(0)1596 1328 y Fl(m)1631 1316 y Fo(\))13 b(=)h(1)p Fi(g)118 1418 y Fo(is)i(almost)g(surely)f (\014nite,)g(hence)h(with)g(probabilit)o(y)f(1)118 1519 y(\(3\))88 b(1)292 1527 y Fj([0)p Fl(;r)q Fj(])358 1519 y Fo(\()p Fm(X)421 1499 y Fl(x)417 1532 y(m)451 1519 y Fo(\))14 b Fi(\025)f Fo(1)560 1527 y Fj([0)p Fl(;r)q Fk(\000)p Fl(\016)q Fj(])671 1519 y Fo(\()p Fm(X)734 1499 y Fj(0)730 1532 y Fl(m)764 1519 y Fo(\))87 b(for)16 b(almost)g(all)24 b Fm(m)14 b Fi(\025)f Fo(0)p Fm(:)118 1621 y Fo(Since)k(for)h Fm(r)g Fo(=)f Fm(t)h Fo(the)f(terms)g(on)h(the) g(righ)o(t{hand)h(side)e(of)i(b)q(oth)f(\(2\))h(and)f(\(3\))h(sum)d(up) 118 1681 y(to)h Fi(1)f Fo(almost)f(surely)l(,)g(this)h(implies)268 1783 y Fm(Q)307 1763 y Fj(0)307 1795 y Fl(s)p Fk(\000)p Fl(\016)7 b(;)e(t)417 1783 y Fi(\024)14 b Fm(Q)509 1763 y Fl(x)509 1795 y(s;t)563 1783 y Fi(\024)g Fm(Q)655 1763 y Fj(0)655 1795 y Fl(s)5 b(;)h(t)p Fk(\000)p Fl(\016)776 1783 y Fo(a.s.)118 1885 y(No)o(w)14 b(assertion)h(\(b\))f(follo)o(ws)g (for)g Fm(\016)h Fo(=)f(1)p Fm(=k)j Fi(!)c Fo(0)i(from)e(the)g(h)o(yp)q (othesis)i(\()p Fm(s;)8 b(t)p Fo(\))13 b Fi(2)h Fm(C)1652 1867 y Fj(0)1671 1885 y Fm(:)49 b Ff(2)191 1974 y Fo(A)o(t)16 b(the)g(next)f(step)i(the)f(p)q(oin)o(t)o(wise)f(ergo)q(dic)i(theorem)d (en)o(ters:)118 2063 y Fn(\(5.5\))28 b(Lemma.)22 b Fg(L)n(et)17 b Fm(\027)k Fg(b)n(e)d(r)n(e)n(curr)n(ent)e(and)i Fm(\026)c Fi(2)g(M)p Fo(\()p Fn(R)1186 2070 y Fj(+)1216 2063 y Fo(\))j Fg(b)n(e)h(a)f(nontrivial)i(invariant)118 2123 y(me)n(asur)n(e.)j(Then)c(for)f Fo(\()p Fm(s;)8 b(t)p Fo(\))13 b Fi(2)h Fm(C)732 2105 y Fj(0)751 2123 y Fg(,)j(with)h(the)g (notation)g(of)g Fo(\(5.4\))p Fg(:)118 2225 y Fo(\(a\))88 b Fn(E)p Fo(\()p Fm(Q)p 324 2242 39 2 v 362 2204 a Fl(x)362 2245 y(s;t)403 2225 y Fo(\))14 b(=)g Fm(\026)p Fo(\([0)p Fm(;)8 b(s)p Fo(]\))g Fm(=)g(\026)p Fo(\([0)p Fm(;)g(t)p Fo(]\))13 b(=)h Fn(E)p Fo(\()p 972 2186 V Fm(Q)1010 2196 y Fl(x)1010 2238 y(s;t)1051 2225 y Fo(\))87 b Fg(for)17 b(al)r(l)26 b Fm(x)14 b Fi(2)g Fo([0)p Fm(;)8 b(t)p Fo(])p Fm(;)118 2376 y Fo(\(b\))278 2342 y(1)p 275 2364 30 2 v 275 2410 a Fm(n)309 2342 y Fe(P)362 2383 y Fj(0)p Fk(\024)p Fl(m)f(r)18 b Fo(suc)o(h)e(that)118 2125 y(\(1\))88 b Fm(\015)16 b Fo(:=)e Fm(\026)p Fo(\([0)p Fm(;)8 b(t)p Fo(]\))13 b Fm(>)h Fo(0)87 b(and)h Fm(\015)904 2105 y Fk(0)929 2125 y Fo(:=)14 b Fm(\026)1024 2105 y Fk(0)1036 2125 y Fo(\([0)p Fm(;)8 b(t)p Fo(]\))k Fm(>)i Fo(0)p Fm(;)118 2239 y Fo(\(2\))88 b Fm(\025)p Fo(\(\()p Fm(C)15 b Fi(n)c Fm(C)459 2219 y Fj(0)478 2239 y Fo(\))497 2246 y Fl(t)511 2239 y Fo(\))j(=)g(0)p Fm(;)118 2354 y Fo(whic)o(h)19 b(is)g(p)q(ossible)h(according)g(to)g(\(5.4a\).)32 b(If)1008 2335 y Fl(t)1014 2354 y Fm(\026)20 b Fo(and)1162 2335 y Fl(t)1168 2354 y Fm(\026)1197 2335 y Fk(0)1229 2354 y Fo(denote)f(the)g(restrictions)g(of)118 2414 y Fm(\026)e Fo(and)h Fm(\026)289 2396 y Fk(0)317 2414 y Fo(to)g([0)p Fm(;)8 b(t)p Fo(],)15 b(the)i(measures)e Fm(\015)821 2396 y Fk(\000)p Fj(1)c Fl(t)883 2414 y Fm(\026)17 b Fo(and)h(\()p Fm(\015)1072 2396 y Fk(0)1083 2414 y Fo(\))1102 2396 y Fk(\000)p Fj(1)11 b Fl(t)1164 2414 y Fm(\026)1193 2396 y Fk(0)1222 2414 y Fo(are)17 b(con)o(tained)f(in)h Fi(M)1643 2421 y Fj(1)1662 2414 y Fo(\()p Fn(R)1723 2421 y Fj(+)1753 2414 y Fo(\))118 2474 y(b)o(y)h(\(1\))h(and)g(according)f (to)h(\(5.5a\))g(agree)f(for)h(sets)g([0)p Fm(;)8 b(s)p Fo(])17 b(with)h(\()p Fm(s;)8 b(t)p Fo(\))17 b Fi(2)h Fm(C)1531 2456 y Fj(0)1550 2474 y Fo(,)h(hence)e(on)118 2534 y Fi(B)r Fo(\([0)p Fm(;)8 b(t)p Fo(]\),)14 b(b)o(y)i(\(2\).)21 b(Therefore)697 2516 y Fl(t)704 2534 y Fm(\026)16 b Fo(and)844 2516 y Fl(t)850 2534 y Fm(\026)879 2516 y Fk(0)908 2534 y Fo(are)g(linearly)f(dep)q(enden)o(t,)g(whic)o(h)g(for)i Fm(t)c Fi(!)h(1)118 2594 y Fo(extends)i(to)g Fm(\026)h Fo(and)g Fm(\026)525 2576 y Fk(0)537 2594 y Fm(:)49 b Ff(2)191 2684 y Fo(Tw)o(o)17 b(commen)o(ts)c(on)k(this)f(result)g(are)g (in)g(order:)118 2744 y(|)25 b(Uniqueness)14 b(holds)i(within)e(the)h (class)h(of)f(lo)q(cally)g(\014nite)f(measures)g(only)l(,)h(as)h(is)f (seen)920 2877 y(11)p eop %%Page: 12 13 12 12 bop 118 77 a Fo(already)22 b(b)o(y)h(a)f(deterministic)e (example:)31 b(If)22 b Fm(Y)36 b Fo(=)25 b(1)p Fm(=)p Fo(2)g(=)g Fm(Z)t Fo(,)e(then)g Fm(\027)j Fo(is)c(recurren)o(t)118 137 y(with)f(in)o(v)m(arian)o(t)f(measure)f Fm(\026)i Fo(=)h Fm(")770 144 y Fj(1)789 137 y Fo(.)34 b(Since)20 b Fm(x)h Fi(!)g Fm(x=)p Fo(2)i(+)f(1)p Fm(=)p Fo(2)g(is)e(a)h (bijection)f(of)h(the)118 197 y(set)e Fm(A)f Fo(:=)g Fn(Q)8 b Fi(\\)g Fo(]1)p Fm(;)g Fi(1)p Fo([,)19 b(ho)o(w)o(ev)o(er,)f (the)h(de\014nition)f Fm(\026)1109 179 y Fk(0)1121 197 y Fo(\()p Fm(B)s Fo(\))g(:=)g Fi(j)p Fm(A)13 b Fi(\\)g Fm(B)s Fi(j)18 b Fo(yields)g(another)118 258 y(\()p Fm(\033)r Fo({\014nite\))e(in)o(v)m(arian)o(t)f(measure.)118 318 y(|)25 b(In)d(the)g(transien)o(t)g(case)g(non)o(trivial)g(in)o(v)m (arian)o(t)f(measures)h Fm(\026)16 b Fi(2)g(M)p Fo(\()p Fn(R)1539 325 y Fj(+)1568 318 y Fo(\))23 b(ma)o(y)e(b)q(e)118 378 y(absen)o(t,)c(as)h(follo)o(ws)f(from)f(\(4.3a\))i(in)f(the)g(case) g Fm(Y)26 b Fi(\025)15 b Fo(1,)j(or)f(presen)o(t,)g(as)g(can)h(b)q(e)f (sho)o(wn)118 438 y(in)f(the)g(case)g Fm(Y)25 b Fo(=)14 b Fm(\015)j Fi(2)8 b Fo(]0)p Fm(;)g Fo(1[)16 b(b)o(y)g(a)h(limiting)c (pro)q(cedure.)191 498 y(It)19 b(is)h(another)g(question)f(ho)o(w)h(to) g(get)f(the)g(in)o(v)m(arian)o(t)g(measure)f(from)h(\(5.6\).)31 b(It)19 b(is)118 559 y(not)h(hard)f(to)h(translate)f(the)g(equation)f Fm(\026P)27 b Fo(=)18 b Fm(\026)h Fo(in)o(to)g(an)g(in)o(tegral)g (equation)g(for)g(the)118 619 y(function)c Fm(F)7 b Fo(\()p Fm(t)p Fo(\))13 b(:=)g Fm(\026)p Fo(\([0)p Fm(;)8 b(t)p Fo(]\),)14 b(but)h(in)g(general)f(it)h(is)f(imp)q(ossible)g(to)h(solv)o (e)f(it.)21 b(P)o(assing)15 b(to)118 679 y(Laplace)i(transforms)f (simpli\014es)e(at)i(least)g(the)g(in)o(tegral)g(equation:)118 768 y Fn(\(5.7\))28 b(Prop)r(osition.)c Fg(L)n(et)17 b Fm(\027)k Fg(and)c Fm(\026)h Fg(b)n(e)g(given)h(ac)n(c)n(or)n(ding)e (to)g Fo(\(5.6\))p Fg(.)23 b(Then:)118 868 y Fo(\(a\))88 b Fm( )r Fo(\()p Fm(u)p Fo(\))13 b(:=)446 833 y Fe(R)482 868 y Fm(e)505 848 y Fk(\000)p Fl(ux)574 868 y Fm(\026)p Fo(\()p Fm(dx)p Fo(\))i Fm(<)e Fi(1)87 b Fg(for)25 b Fm(u)13 b(>)h Fo(0)p Fm(;)118 969 y Fo(\(b\))87 b Fg(up)18 b(to)f(a)h(sc)n(alar,)f Fm( )i Fg(is)e(uniquely)i(determine)n(d)f(by)f (the)h(e)n(quation)268 1069 y Fm( )r Fo(\()p Fm(u)p Fo(\))13 b(=)432 1033 y Fe(R)468 1069 y Fm( )r Fo(\()p Fm(uy)r Fo(\))p Fm(e)617 1048 y Fk(\000)p Fl(uz)692 1069 y Fm(\027)s Fo(\()p Fm(dy)r(;)8 b(dz)r Fo(\))87 b Fg(for)24 b Fm(u)14 b(>)g Fo(0)p Fm(:)118 1169 y Fg(Pr)n(o)n(of.)23 b Fo(While)16 b(\(a\))g(is)g(a)h(simple)d(consequence)h(of)i(\(4.5\),)f(\(b\))g (follo)o(ws)g(from)268 1234 y Fe(R)304 1269 y Fm(e)327 1249 y Fk(\000)p Fl(ux)404 1269 y Fo(\()p Fm(\026P)7 b Fo(\)\()p Fm(dx)p Fo(\))15 b(=)666 1234 y Fe(R)h(R)738 1269 y Fm(e)761 1249 y Fk(\000)p Fl(u)p Fj(\()p Fl(xy)q Fj(+)p Fl(z)q Fj(\))930 1269 y Fm(\026)p Fo(\()p Fm(dx)p Fo(\))8 b Fm(\027)s Fo(\()p Fm(dy)r(;)g(dz)r Fo(\))118 1370 y(and)17 b(\(5.6\),)f(b)q(ecause)g(the)g(Laplace)h(transform)f (determines)e Fm(\026:)49 b Ff(2)191 1459 y Fo(If)19 b Fm(Y)30 b Fo(and)20 b Fm(Z)j Fo(are)d(indep)q(enden)o(t)e(\(as)i(in)e 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(determination)f(is)i(in)g(general)g(out)h(of)g(reac)o(h.)118 2255 y(Th)o(us)h(it)e(is)h(imp)q(ortan)o(t)g(to)g(obtain)h(at)g(least)f (a)h(qualitativ)o(e)d(description.)191 2315 y(A)21 b(\014rst)g(prop)q (ert)o(y)f(of)h(the)g(supp)q(ort)h Fm(M)k Fo(of)21 b Fm(\026)g Fo(en)o(tered)f(already:)30 b(\(1.3\))21 b(and)g(\(4.3\))118 2376 y(com)o(bine)14 b(to)j(the)f(equations)270 2485 y(inf)11 b Fm(M)47 b Fo(=)41 b Fm(x)p 510 2492 28 2 v 28 w Fo(=)27 b(inf)17 b Fi(f)777 2451 y Fm(z)p 734 2473 111 2 v 734 2519 a Fo(1)11 b Fi(\000)g Fm(y)864 2485 y Fo(:)i(\()p Fm(y)r(;)8 b(z)r Fo(\))13 b Fi(2)h Fm(N)1101 2492 y Fl(c)1119 2485 y Fi(g)p Fm(;)268 2629 y Fo(sup)8 b Fm(M)35 b Fo(=)p 510 2603 28 2 v 41 w Fm(x)28 b Fo(=)f(sup)q Fi(f)777 2596 y Fm(z)p 735 2618 111 2 v 735 2664 a Fo(1)11 b Fi(\000)g Fm(y)864 2629 y Fo(:)j(\()p Fm(y)r(;)8 b(z)r Fo(\))13 b Fi(2)h Fm(N)1102 2636 y Fl(c)1119 2629 y Fi(g)87 b Fo(if)25 b Fm(N)1324 2636 y Fl(e)1356 2629 y Fo(=)14 b Fi(;)p Fm(;)562 2752 y Fo(=)22 b Fi(1)561 b Fo(otherwise.)920 2877 y(12)p eop %%Page: 13 14 13 13 bop 118 77 a Fo(It)21 b(is)g(a)h(natural)f(question)g(to)h(ask)f (under)h(whic)o(h)e(conditions,)i(less)f(restrictiv)o(e)e(than)118 137 y(in)g(\(4.4\),)h(the)f(whole)h(in)o(terv)m(al)e([)p Fm(x)p 734 144 28 2 v -1 w(;)p 783 111 V 8 w(x)p Fo(])h(is)g(exhausted) g(b)o(y)g Fm(M)5 b Fo(.)31 b(A)19 b(surprisingly)g(general)118 197 y(answ)o(er)e(is)f(giv)o(en)f(b)o(y)h(the)g(follo)o(wing)g(result)f (\(for)i(a)g(sp)q(ecial)e(case)i(see)e([2]\):)118 287 y Fn(\(6.1\))28 b(Theorem.)23 b Fg(L)n(et)c Fm(\027)k Fg(b)n(e)d(r)n(e)n(curr)n(ent)f(with)h Fn(P)p Fo(\()p Fm(Y)30 b Fo(=)18 b(0\))h(=)f(0)i Fg(and)p 1466 260 V 20 w Fm(x)e Fo(=)g Fi(1)p Fg(.)29 b(Then)118 347 y(the)18 b(invariant)g(me)n(asur)n(e)f Fm(\026)h Fg(has)f(the)h(supp)n(ort)268 449 y Fm(M)h Fo(=)14 b([)p Fm(x)p 400 456 V -1 w(;)8 b Fi(1)p Fo([)p Fm(:)118 550 y Fg(Pr)n(o)n(of.)23 b Fo(1.)e(By)14 b(the)f(h)o(yp)q(othesis)i Fn(P)p Fo(\()p Fm(Y)h Fo(=)6 b(0\))14 b(=)g(0)g(p)q(oin)o(ts)h(\(0)p Fm(;)8 b(z)r Fo(\))e Fi(2)g Fm(N)18 b Fo(cannot)d(b)q(e)f(isolated,)118 610 y(hence)268 712 y Fm(x)p 268 719 V 13 w Fo(=)g(inf)d Fi(f)501 678 y Fm(z)p 458 700 111 2 v 458 746 a Fo(1)h Fi(\000)e Fm(y)588 712 y Fo(:)j(\()p Fm(y)r(;)8 b(z)r Fo(\))13 b Fi(2)h Fm(N)30 b Fo(with)25 b(0)14 b Fm(<)g(y)i(<)d Fo(1)p Fi(g)p Fm(:)118 836 y Fo(Th)o(us)k(it)e(su\016ces)h(to)h(pro)o (v)o(e)e(that)i Fm(x)d Fi(2)g Fm(M)21 b Fo(if)268 947 y Fm(x)296 954 y Fj(0)329 947 y Fo(=)429 913 y Fm(z)452 920 y Fj(0)p 386 935 129 2 v 386 981 a Fo(1)11 b Fi(\000)g Fm(y)495 988 y Fj(0)533 947 y Fm(<)j(x)f(<)h Fi(1)87 b Fo(with)g(\()p Fm(y)1040 954 y Fj(0)1060 947 y Fm(;)8 b(z)1105 954 y Fj(0)1124 947 y Fo(\))14 b Fi(2)g Fm(N)55 b Fo(and)50 b(0)15 b Fm(<)e(y)1540 954 y Fj(0)1574 947 y Fm(<)g Fo(1)p Fm(:)118 1074 y Fo(T)l(o)18 b(this)f(end)h(p)q(oin)o (ts)f(\()p Fm(y)r(;)8 b(z)r Fo(\))17 b(and)h(the)f(asso)q(ciated)i (mappings)e Fm(g)g Fo(:)f Fm(x)f Fi(!)h Fm(y)r(x)11 b Fo(+)h Fm(z)19 b Fo(will)e(b)q(e)118 1134 y(iden)o(ti\014ed)f (throughout)j(this)e(pro)q(of.)25 b(It)17 b(will)f(b)q(e)h (accomplished)f(b)o(y)g(constructing)i(t)o(w)o(o)118 1194 y(sequences)d Fm(n)368 1201 y Fj(1)388 1194 y Fm(;)8 b(n)439 1201 y Fj(2)459 1194 y Fm(;)g(:)g(:)g(:)13 b Fi(2)h Fn(N)i Fo(and)h Fm(g)777 1201 y Fj(1)797 1194 y Fm(;)8 b(g)842 1201 y Fj(2)862 1194 y Fm(;)g(:)g(:)g(:)13 b Fi(2)h Fm(N)21 b Fo(suc)o(h)16 b(that)268 1296 y(0)e(=)g Fm(n)387 1303 y Fj(0)420 1296 y Fi(\024)g Fm(n)502 1303 y Fj(1)535 1296 y Fi(\024)g Fm(:)8 b(:)g(:)16 b(;)268 1410 y(g)291 1417 y Fl(i)319 1410 y Fo(=)e(\()p Fm(y)414 1417 y Fl(i)427 1410 y Fm(;)8 b(z)472 1417 y Fl(i)486 1410 y Fo(\))87 b(with)25 b Fm(y)736 1417 y Fl(i)764 1410 y Fm(>)13 b Fo(0)p Fm(;)268 1524 y Fo(0)h Fi(\024)f Fm(x)e Fi(\000)g Fm(x)475 1531 y Fl(k)510 1524 y Fi(\024)i Fo(\(1)f Fi(\000)f Fm(y)691 1531 y Fj(0)710 1524 y Fo(\))729 1503 y Fl(k)750 1524 y Fo(\()p Fm(x)g Fi(\000)g Fm(x)886 1531 y Fj(0)905 1524 y Fo(\))88 b(for)25 b Fm(x)1123 1531 y Fl(k)1158 1524 y Fo(=)13 b Fm(g)1232 1531 y Fj(1)1264 1524 y Fi(\016)e Fm(:)d(:)g(:)i Fi(\016)h Fm(g)1427 1531 y Fl(n)1448 1537 y Fd(k)1481 1524 y Fo(\()p Fm(x)1528 1531 y Fj(0)1547 1524 y Fo(\))p Fm(:)118 1638 y Fo(Clearly)l(,)k(this)h (implies)e Fm(x)f Fi(2)h Fm(M)22 b Fo(b)o(y)16 b(\(4.1a\))h(and)f (\(4.2\).)191 1698 y(2.)21 b(It)13 b(su\016ces)h(to)g(construct)g Fm(n)762 1705 y Fl(k)q Fj(+1)842 1698 y Fo(and)g Fm(g)957 1705 y Fl(n)978 1711 y Fd(k)1004 1705 y Fj(+)6 b(1)1057 1698 y Fm(;)i(:)g(:)g(:)f(;)h(g)1189 1705 y Fl(n)1210 1711 y Fd(k)q Fc(+1)1284 1698 y Fo(from)13 b Fm(n)1426 1705 y Fl(k)1461 1698 y Fo(and)h Fm(g)1576 1705 y Fj(1)1597 1698 y Fm(;)8 b(:)g(:)g(:)f(;)h(g)1729 1705 y Fl(n)1750 1711 y Fd(k)118 1759 y Fo(under)24 b(the)g(additional)g(assumption)f Fm(x)887 1766 y Fl(k)935 1759 y Fm(<)k(x)p Fo(,)e(b)q(ecause)f (otherwise)g(the)f(de\014nition)118 1819 y Fm(n)147 1826 y Fl(k)q Fj(+1)227 1819 y Fo(=)14 b Fm(n)308 1826 y Fl(k)346 1819 y Fo(w)o(orks.)21 b(No)o(w)16 b(the)g(h)o(yp)q(othesis)p 939 1792 28 2 v 17 w Fm(x)d Fo(=)h Fi(1)i Fo(en)o(ters,)f(pro)o(viding) 268 1920 y Fm(g)293 1900 y Fk(0)291 1933 y Fl(i)319 1920 y Fo(=)f(\()p Fm(y)416 1900 y Fk(0)414 1933 y Fl(i)427 1920 y Fm(;)8 b(z)474 1900 y Fk(0)472 1933 y Fl(i)486 1920 y Fo(\))14 b Fi(2)g Fm(N)55 b Fo(with)25 b Fm(y)806 1900 y Fk(0)804 1933 y Fl(i)831 1920 y Fm(>)14 b Fo(0)88 b(for)25 b(1)14 b Fi(\024)g Fm(i)f Fi(\024)h Fm(m)118 2022 y Fo(suc)o(h)i(that)268 2124 y Fm(\016)290 2131 y Fl(k)324 2124 y Fo(:=)e Fm(y)414 2131 y Fj(1)441 2124 y Fm(:)8 b(:)g(:)g(y)531 2131 y Fl(n)552 2137 y Fd(k)573 2124 y Fo(\()p Fm(g)617 2103 y Fk(0)615 2136 y Fj(1)646 2124 y Fi(\016)j Fm(:)d(:)g(:)j Fi(\016)g Fm(g)812 2103 y Fk(0)810 2136 y Fl(m)854 2124 y Fo(\()p Fm(x)901 2131 y Fj(0)921 2124 y Fo(\))g Fi(\000)g Fm(x)1029 2131 y Fj(0)1048 2124 y Fo(\))j Fm(>)g(x)c Fi(\000)h Fm(x)1249 2131 y Fl(k)1270 2124 y Fm(:)118 2226 y Fo(Indeed,)17 b(this)g(follo)o(ws)g(from)g(lim)8 b(sup)823 2233 y Fl(n)p Fk(!1)926 2226 y Fm(X)970 2207 y Fl(x)990 2212 y Fc(0)966 2238 y Fl(n)1025 2226 y Fo(=)16 b Fi(1)p Fo(,)h(b)q(ecause)h(the)f (relations)g Fm(Y)1654 2233 y Fl(n)1694 2226 y Fm(>)e Fo(0)118 2286 y(and)i(\()p Fm(Y)260 2293 y Fl(n)284 2286 y Fm(;)8 b(Z)339 2293 y Fl(n)362 2286 y Fo(\))14 b Fi(2)g Fm(N)22 b Fo(hold)16 b(almost)g(surely)l(.)191 2346 y(3.)22 b(Let)16 b(no)o(w)h Fm(l)d Fi(2)g Fn(N)i Fo(b)q(e)h(de\014ned)f(b)o(y) 118 2448 y(\(1\))88 b Fm(y)294 2427 y Fl(l)p Fk(\000)p Fj(1)292 2460 y(0)351 2448 y Fm(\016)373 2455 y Fl(k)408 2448 y Fm(>)14 b(x)d Fi(\000)f Fm(x)576 2455 y Fl(k)611 2448 y Fi(\025)k Fm(y)690 2427 y Fl(l)688 2460 y Fj(0)716 2448 y Fm(\016)738 2455 y Fl(k)759 2448 y Fm(:)118 2549 y Fo(Then)i(the)g(construction)h(can)f(b)q(e)h(con)o(tin)o(ued)e(b)o(y) h Fm(n)1084 2556 y Fl(k)q Fj(+1)1164 2549 y Fo(=)e Fm(n)1245 2556 y Fl(k)1277 2549 y Fo(+)d Fm(l)h Fo(+)f Fm(m)16 b Fo(and)268 2674 y Fm(g)291 2681 y Fl(i)319 2674 y Fo(:=)384 2626 y Fe(n)429 2640 y Fm(g)452 2647 y Fj(0)700 2640 y Fo(for)26 b Fm(n)813 2647 y Fl(k)848 2640 y Fm(<)14 b(i)f Fi(\024)h Fm(n)1012 2647 y Fl(k)1044 2640 y Fo(+)d Fm(l)q(;)428 2708 y(g)453 2691 y Fk(0)451 2723 y Fl(i)p Fk(\000)p Fj(\()p Fl(n)525 2729 y Fd(k)551 2723 y Fj(+)6 b Fl(l)p Fj(\))698 2708 y Fo(for)25 b Fm(n)810 2715 y Fl(k)843 2708 y Fo(+)11 b Fm(l)j(<)g(i)f Fi(\024)h Fm(n)1085 2715 y Fl(k)q Fj(+1)1151 2708 y Fm(:)2415 2626 y Fe(o)920 2877 y Fo(13)p eop %%Page: 14 15 14 14 bop 118 77 a Fo(Indeed,)15 b(since)h Fm(x)438 84 y Fj(0)473 77 y Fo(is)g(a)h(\014xed)f(p)q(oin)o(t)g(of)h Fm(g)887 84 y Fj(0)921 77 y Fo(=)d(\()p Fm(y)1016 84 y Fj(0)1035 77 y Fm(;)8 b(z)1080 84 y Fj(0)1099 77 y Fo(\),)16 b(b)o(y)g(a\016nit)o(y)268 179 y Fm(x)296 186 y Fl(k)q Fj(+1)373 179 y Fi(\000)11 b Fm(x)451 186 y Fl(k)514 179 y Fo(=)41 b Fm(g)616 186 y Fj(1)648 179 y Fi(\016)11 b Fm(:)d(:)g(:)i Fi(\016)h Fm(g)811 186 y Fl(n)832 192 y Fd(k)865 179 y Fi(\016)g Fm(g)926 158 y Fl(l)924 191 y Fj(0)955 179 y Fo(\()p Fm(g)999 158 y Fk(0)997 191 y Fj(1)1028 179 y Fi(\016)g Fm(:)d(:)g(:)i Fi(\016)h Fm(g)1193 158 y Fk(0)1191 191 y Fl(m)1236 179 y Fo(\()p Fm(x)1283 186 y Fj(0)1302 179 y Fo(\)\))513 280 y Fi(\000)41 b Fm(g)616 287 y Fj(1)648 280 y Fi(\016)11 b Fm(:)d(:)g(:)i Fi(\016)h Fm(g)811 287 y Fl(n)832 293 y Fd(k)865 280 y Fi(\016)g Fm(g)926 260 y Fl(l)924 293 y Fj(0)955 280 y Fo(\()p Fm(x)1002 287 y Fj(0)1021 280 y Fo(\))514 382 y(=)41 b Fm(y)617 389 y Fj(1)645 382 y Fm(:)8 b(:)g(:)g(y)735 389 y Fl(n)756 395 y Fd(k)777 382 y Fm(y)803 362 y Fl(l)801 394 y Fj(0)832 382 y Fo(\()p Fm(g)876 362 y Fk(0)874 394 y Fj(1)905 382 y Fi(\016)j Fm(:)d(:)g(:)i Fi(\016)h Fm(g)1070 362 y Fk(0)1068 394 y Fl(m)1113 382 y Fo(\()p Fm(x)1160 389 y Fj(0)1179 382 y Fo(\))g Fi(\000)g Fm(x)1287 389 y Fj(0)1306 382 y Fo(\))514 484 y(=)41 b Fm(y)619 463 y Fl(l)617 496 y Fj(0)645 484 y Fm(\016)667 491 y Fl(k)688 484 y Fm(:)118 586 y Fo(By)16 b(\(1\))g(this)g(implies)e(the)i(required)f(inequalities)268 687 y Fm(x)296 694 y Fl(k)q Fj(+1)376 687 y Fo(=)e Fm(x)455 694 y Fl(k)487 687 y Fo(+)e Fm(y)562 667 y Fl(l)560 700 y Fj(0)588 687 y Fm(\016)610 694 y Fl(k)645 687 y Fi(\024)j Fm(x;)268 801 y(x)c Fi(\000)h Fm(x)384 808 y Fl(k)q Fj(+1)464 801 y Fo(=)j(\()p Fm(x)d Fi(\000)f Fm(x)651 808 y Fl(k)673 801 y Fo(\))h Fi(\000)f Fm(y)776 808 y Fj(0)804 801 y Fm(y)830 781 y Fl(l)p Fk(\000)p Fj(1)828 814 y(0)888 801 y Fm(\016)910 808 y Fl(k)945 801 y Fm(<)j Fo(\(1)f Fi(\000)f Fm(y)1125 808 y Fj(0)1144 801 y Fo(\)\()p Fm(x)g Fi(\000)g Fm(x)1299 808 y Fl(k)1320 801 y Fo(\))p Fm(:)49 b Ff(2)191 916 y Fo(Clearly)l(,)17 b(the)h(condition)g Fn(P)p Fo(\()p Fm(Y)28 b Fo(=)16 b(0\))i(=)e(0)i(is)g(essen)o(tial)f (for)h(this)g(result,)g(as)g(is)g(seen)118 976 y(from)f(the)g(trivial)g (case)h Fm(Y)27 b Fo(=)17 b(0,)h(where)f Fm(\026)h Fo(is)g(\(a)g(m)o (ultiple)d(of)s(\))j(the)g(distribution)f(of)h Fm(Z)t Fo(.)191 1036 y(The)13 b(argumen)o(t)f(used)h(in)g(the)f(follo)o(wing)h (pro)q(of)h(can)f(b)q(e)g(traced)g(bac)o(k)g(to)g(Karlin)f([22];)118 1096 y(it)k(applies)g(to)g(lo)q(cally)g(\014nite)f(measures)h(as)h(w)o (ell:)118 1185 y Fn(\(6.2\))28 b(Theorem.)23 b Fg(L)n(et)18 b Fm(\027)j Fg(b)n(e)d(r)n(e)n(curr)n(ent)g(with)g Fn(P)p Fo(\()p Fm(Y)27 b Fo(=)15 b(0\))h(=)f(0)p Fg(.)25 b(Then)18 b(the)h(invariant)118 1246 y(me)n(asur)n(e)e Fm(\026)h Fg(is)g(either)g(absolutely)h(c)n(ontinuous)f(or)g(singular)g(with)g(r) n(esp)n(e)n(ct)f(to)h(L)n(eb)n(esgue)118 1306 y(me)n(asur)n(e.)118 1395 y(Pr)n(o)n(of.)23 b Fo(F)l(rom)15 b(the)h(equation)268 1518 y(\()p Fm(\026P)7 b Fo(\)\()p Fm(B)s Fo(\))13 b(=)516 1483 y Fe(R)552 1525 y Fl(y)q(>)p Fj(0)629 1518 y Fm(\026)8 b Fo(\()690 1484 y Fm(B)14 b Fi(\000)d Fm(z)p 690 1506 126 2 v 740 1552 a(y)821 1518 y Fo(\))g Fm(d\027)91 b Fo(for)25 b Fm(B)16 b Fi(2)e(B)r Fo(\()p Fn(R)1270 1525 y Fj(+)1299 1518 y Fo(\))118 1642 y(it)j(is)f(clear)h(that)g Fm(\026)f Fi(\034)f Fm(\025)i Fo(implies)e Fm(\026P)22 b Fi(\034)15 b Fm(\025)j Fo(as)f(w)o(ell.)23 b(If,)16 b(therefore,)g Fm(\026)h Fo(is)g(decomp)q(osed)118 1702 y(in)o(to)d(the)g(absolutely)f(con)o(tin)o(uous)h(part)h Fm(\026)901 1709 y Fl(c)932 1702 y Fo(and)g(the)f(singular)g(part)h Fm(\026)1422 1709 y Fl(s)1440 1702 y Fo(,)f(it)g(follo)o(ws)g(from)268 1804 y Fm(\026)297 1811 y Fl(c)314 1804 y Fm(P)19 b Fo(+)11 b Fm(\026)442 1811 y Fl(s)460 1804 y Fm(P)21 b Fo(=)14 b Fm(\026P)21 b Fo(=)14 b Fm(\026)g Fo(=)g Fm(\026)821 1811 y Fl(c)850 1804 y Fo(+)d Fm(\026)928 1811 y Fl(s)118 1906 y Fo(that)19 b Fm(\026)255 1913 y Fl(c)272 1906 y Fm(P)24 b Fi(\024)17 b Fm(\026)412 1913 y Fl(c)430 1906 y Fo(.)26 b(Th)o(us)19 b(b)o(y)e(\(5.6\))i(there)e(exists)h(a)g (constan)o(t)h Fm(\015)1313 1913 y Fl(c)1348 1906 y Fo(suc)o(h)f(that)h Fm(\026)1597 1913 y Fl(c)1631 1906 y Fo(=)e Fm(\015)1711 1913 y Fl(c)1729 1906 y Fm(\026)p Fo(,)118 1966 y(hence)11 b(also)i(a)f(constan)o(t)h Fm(\015)597 1973 y Fl(s)627 1966 y Fo(suc)o(h)f(that)h Fm(\026)864 1973 y Fl(s)896 1966 y Fo(=)h Fm(\015)973 1973 y Fl(s)992 1966 y Fm(\026)p Fo(.)20 b(No)o(w)12 b Fm(\026)1191 1973 y Fl(c)1211 1966 y Fi(^)r Fm(\026)1275 1973 y Fl(s)1308 1966 y Fo(=)i(0)e(implies)d Fm(\015)1582 1973 y Fl(c)1602 1966 y Fi(^)r Fm(\015)1662 1973 y Fl(s)1695 1966 y Fo(=)14 b(0)118 2026 y(and)j(v)o(eri\014es)e (the)h(assertion)p Fm(:)50 b Ff(2)191 2115 y Fo(The)24 b(remark)e(follo)o(wing)i(\(6.1\))g(sho)o(ws)g(that)h(the)e(condition)h Fn(P)p Fo(\()p Fm(Y)30 b Fo(=)18 b(0\))27 b(=)g(0)d(is)118 2176 y(essen)o(tial)15 b(for)i(\(6.2\))f(as)h(w)o(ell.)191 2236 y(It)c(is)g(another)g(question)g(ho)o(w)g(to)g(decide)f(on)i(the)e (alternativ)o(e)g(in)g(\(6.2\).)21 b(While)12 b(some)118 2296 y(sp)q(ecial)17 b(cases)h(are)g(considered)f(in)h(Section)f(10,)h (the)g(only)f(general)h(result)f(is)g(pro)o(vided)118 2356 y(b)o(y)f(the)g(follo)o(wing)g(necessary)g(conditions)g(for)h (singularit)o(y)e(of)i Fm(\026)p Fo(:)118 2445 y Fn(\(6.3\))28 b(Prop)r(osition.)c Fg(L)n(et)c Fm(\027)j Fg(b)n(e)d(r)n(e)n(curr)n (ent)f(and)i(the)f(invariant)h(me)n(asur)n(e)e Fm(\026)i Fg(b)n(e)f(sin-)118 2506 y(gular)e(with)g(r)n(esp)n(e)n(ct)f(to)g(L)n (eb)n(esgue)h(me)n(asur)n(e.)k(Then:)118 2595 y Fo(\(a\))88 b Fm(\027)20 b Fg(is)e(singular)g(with)g(r)n(esp)n(e)n(ct)e(to)i Fm(\025)902 2577 y Fj(2)922 2595 y Fg(,)118 2684 y Fo(\(b\))87 b Fm(\027)294 2691 y Fl(y)333 2684 y Fg(and)17 b Fm(\027)451 2691 y Fl(z)489 2684 y Fg(ar)n(e)f(singular)j(with)e(r)n(esp)n(e)n(ct)g (to)h Fm(\025)p Fg(,)g(if)31 b Fm(\027)17 b Fo(=)d Fm(\027)1320 2691 y Fl(y)1352 2684 y Fi(\012)c Fm(\027)1425 2691 y Fl(z)1445 2684 y Fg(.)920 2877 y Fo(14)p eop %%Page: 15 16 15 15 bop 118 77 a Fg(Pr)n(o)n(of.)23 b Fo(\(a\))17 b(Let)f Fm(B)h Fi(2)d(B)r Fo(\()p Fn(R)634 84 y Fj(+)662 77 y Fo(\))i(satisfy)268 170 y Fm(\025)p Fo(\()p Fm(B)s Fo(\))d(=)h(0)88 b(and)f Fm(\026)p Fo(\()p Fn(R)806 177 y Fj(+)847 170 y Fi(n)11 b Fm(B)s Fo(\))i(=)h(0)p Fm(:)118 264 y Fo(Then)i(on)h(the)f (one)h(hand)118 357 y(\(1\))88 b Fm(\025)296 337 y Fj(2)316 357 y Fo(\()p Fi(f)p Fo(\()p Fm(y)r(;)8 b(z)r Fo(\))13 b(:)g Fm(y)r(x)d Fo(+)h Fm(z)16 b Fi(2)e Fm(B)s Fi(g)p Fo(\))g(=)859 322 y Fe(R)904 357 y Fm(\025)p Fo(\()p Fm(B)g Fi(\000)c Fm(y)r(x)p Fo(\))e Fm(\025)p Fo(\()p Fm(dy)r Fo(\))14 b(=)g(0)87 b(for)17 b(all)24 b Fm(x)14 b Fi(2)g Fn(R)1708 364 y Fj(+)1737 357 y Fm(;)118 451 y Fo(while)h(on)i(the)f(other)h(hand)268 544 y(0)41 b(=)h Fm(\026)p Fo(\()p Fi(f)p Fm(x)14 b Fo(:)f Fm(x)19 b(=)-29 b Fi(2)14 b Fm(B)s Fi(g)p Fo(\))333 646 y(=)42 b(\()p Fm(\026)11 b Fi(\012)g Fm(\027)s Fo(\)\()p Fi(f)p Fo(\()p Fm(x)p Fo(;)d Fm(y)r(;)g(z)r Fo(\))13 b(:)g Fm(y)r(x)e Fo(+)g Fm(z)21 b(=)-30 b Fi(2)14 b Fm(B)s Fi(g)p Fo(\))333 747 y(=)413 712 y Fe(R)457 747 y Fm(\027)s Fo(\()p Fi(f)p Fo(\()p Fm(y)r(;)8 b(z)r Fo(\))13 b(:)h Fm(y)r(x)c Fo(+)h Fm(z)22 b(=)-30 b Fi(2)14 b Fm(B)s Fi(g)p Fo(\))8 b Fm(\026)p Fo(\()p Fm(dx)p Fo(\))p Fm(:)118 841 y Fo(In)16 b(view)g(of)g Fm(\026)e Fi(6)p Fo(=)g(0)j(this)f(ensures)g(the)g(existence)f(of)h Fm(x)1124 848 y Fj(0)1157 841 y Fi(2)e Fn(R)1246 848 y Fj(+)1292 841 y Fo(suc)o(h)i(that)118 934 y(\(2\))88 b Fm(\027)s Fo(\()p Fi(f)p Fo(\()p Fm(y)r(;)8 b(z)r Fo(\))13 b(:)g Fm(y)r(x)544 941 y Fj(0)574 934 y Fo(+)e Fm(z)22 b(=)-30 b Fi(2)14 b Fm(B)s Fi(g)f Fo(=)h(0)p Fm(:)118 1027 y Fo(By)i(\(1\))g(and)h(\(2\),)f(therefore,)f Fm(\025)706 1009 y Fj(2)743 1027 y Fo(and)i Fm(\027)i Fo(are)d(supp)q(orted)i(b)o (y)d(disjoin)o(t)h(sets.)191 1088 y(\(b\))h(With)f Fm(B)i Fo(as)f(ab)q(o)o(v)o(e)f(it)g(follo)o(ws)g(no)o(w)h(that)268 1181 y(0)41 b(=)h(\()p Fm(\026)11 b Fi(\012)g Fm(\027)546 1188 y Fl(y)578 1181 y Fi(\012)g Fm(\027)652 1188 y Fl(z)672 1181 y Fo(\)\()p Fi(f)p Fo(\()p Fm(x)p Fo(;)d Fm(y)r(;)g(z)r Fo(\))k(:)i Fm(y)r(x)c Fo(+)h Fm(z)22 b(=)-30 b Fi(2)14 b Fm(B)s Fi(g)p Fo(\))333 1283 y(=)413 1247 y Fe(R)457 1283 y Fm(\027)481 1290 y Fl(y)502 1283 y Fo(\()p Fi(f)p Fm(y)h Fo(:)f Fm(y)r(x)c Fo(+)h Fm(z)22 b(=)-30 b Fi(2)14 b Fm(B)s Fi(g)p Fo(\))8 b Fm(\026)p Fo(\()p Fm(dx)p Fo(\))g Fm(\027)1056 1290 y Fl(z)1076 1283 y Fo(\()p Fm(dz)r Fo(\))p Fm(:)118 1376 y Fo(Since)14 b(supp)7 b Fm(\026)15 b Fi(6)p Fo(=)f Fi(f)p Fo(0)p Fi(g)h Fo(\(see)g(\(1\))h(in)f(the)f(pro) q(of)j(of)e(\(4.3\)\),)h(this)f(ensures)g(the)g(existence)e(of)118 1436 y Fm(x)146 1443 y Fj(0)179 1436 y Fm(>)h Fo(0)j(and)g Fm(z)390 1443 y Fj(0)423 1436 y Fi(\025)c Fo(0)k(suc)o(h)f(that)268 1549 y Fm(\025)8 b Fo(\()328 1515 y Fm(B)14 b Fi(\000)d Fm(z)452 1522 y Fj(0)p 328 1537 143 2 v 376 1583 a Fm(x)404 1590 y Fj(0)476 1549 y Fo(\))j(=)g(0)87 b(and)h Fm(\027)862 1556 y Fl(y)891 1549 y Fo(\()p Fn(R)952 1556 y Fj(+)992 1549 y Fi(n)1033 1515 y Fm(B)14 b Fi(\000)d Fm(z)1157 1522 y Fj(0)p 1033 1537 V 1081 1583 a Fm(x)1109 1590 y Fj(0)1181 1549 y Fo(\))j(=)g(0)p Fm(:)118 1662 y Fo(The)i(corresp)q (onding)i(argumen)o(t)d(for)h Fm(\027)848 1669 y Fl(z)885 1662 y Fo(is)g(ev)o(en)f(simpler)p Fm(:)47 b Ff(2)191 1752 y Fo(It)21 b(is)g(an)h(easy)g(consequence)e(of)i(\(1.4\))g(that)g (the)f(indep)q(endence)f(of)i Fm(Y)32 b Fo(and)22 b Fm(Z)k Fo(is)118 1812 y(essen)o(tial)18 b(in)g(\(b\).)30 b(It)18 b(applies,)h(ho)o(w)o(ev)o(er,)e(in)i(the)f(additiv)o(e)g(or)h(m)o (ultiplicati)o(v)o(e)c(mo)q(del.)118 1872 y(Moreo)o(v)o(er,)g(it)h (should)h(b)q(e)g(men)o(tioned)d(that)j(\(a\))g(and)g(\(b\))g(are)f(b)o (y)g(no)h(means)f(su\016cien)o(t)118 1932 y(for)h(singularit)o(y)e(of)i Fm(\026)f Fo(\(in)g(this)g(con)o(text)g(see)f(Section)h(10\).)191 1992 y(The)d(follo)o(wing)g(result)f(is)h(easily)f(established)g(for)h (measures)f Fm(\026)h Fo(con)o(taining)g(an)g(atom)118 2053 y(with)20 b(maximal)c(mass)k(\(see)f([9]\).)31 b(Since)19 b(this)h(ma)o(y)e(fail)h(for)h(lo)q(cally)f(\014nite)g(measures,)118 2113 y(the)d(general)g(pro)q(of)h(gets)g(more)e(in)o(v)o(olv)o(ed:)118 2202 y Fn(\(6.4\))28 b(Theorem.)23 b Fg(L)n(et)16 b Fm(\027)j Fg(b)n(e)e(r)n(e)n(curr)n(ent)e(with)i Fn(P)p Fo(\()p Fm(Y)25 b Fo(=)14 b(0\))g(=)g(0)j Fg(and)f Fm(x)p 1424 2209 28 2 v 14 w(<)p 1518 2176 V 14 w(x)o Fg(.)22 b(Then)17 b(the)118 2262 y(invariant)h(me)n(asur)n(e)f Fm(\026)h Fg(is)f(nonatomic.)118 2351 y(Pr)n(o)n(of.)23 b Fo(1.)f(It)15 b(follo)o(ws)g(as)i(in)e(\(6.2\))h(that)g Fm(\026)g Fo(is)f(either)g (nonatomic)g(or)h(purely)e(atomic.)20 b(It)118 2412 y(su\016ces,)15 b(therefore,)h(to)g(sho)o(w)h(that)g(in)f(the)g(second)g(case)118 2505 y(\(1\))88 b Fm(Z)17 b Fo(=)d Fm(x)p 370 2512 V 11 w Fo(\(1)d Fi(\000)g Fm(Y)g Fo(\))p Fm(;)118 2598 y Fo(th)o(us)20 b(in)f(view)f(of)i(\(1.4\))g(obtaining)g(a)f(con)o (tradiction)g(to)h(the)f(h)o(yp)q(othesis)h Fm(x)p 1540 2605 V 19 w(<)p 1644 2572 V 19 w(x)o Fo(.)31 b(T)l(o)118 2659 y(this)16 b(end)g(consider)g(the)g(coun)o(table)g(set)268 2752 y Fm(R)e Fo(:=)f Fi(f)p Fm(x)h Fi(2)g Fn(R)540 2759 y Fj(+)583 2752 y Fo(:)f Fm(\026)p Fo(\()p Fi(f)p Fm(x)p Fi(g)p Fo(\))h Fm(>)g Fo(0)p Fi(g)p Fm(:)920 2877 y Fo(15)p eop %%Page: 16 17 16 16 bop 118 77 a Fo(By)16 b(the)g(in)o(v)m(ariance)f(of)i Fm(\026)f Fo(it)g(satis\014es)268 173 y(0)e(=)g Fm(\026)p Fo(\()p Fn(R)448 180 y Fj(+)480 173 y Fi(n)s Fm(R)p Fo(\))g(=)630 140 y Fe(P)682 180 y Fl(x)p Fk(2)p Fl(R)765 173 y Fm(\026)p Fo(\()p Fi(f)p Fm(x)p Fi(g)p Fo(\))8 b Fm(P)f Fo(\()p Fm(x)p Fo(;)h Fn(R)1067 180 y Fj(+)1099 173 y Fi(n)s Fm(R)p Fo(\))p Fm(;)118 269 y Fo(hence)16 b(the)g(de\014nition)279 348 y Fe(b)268 364 y Fm(P)7 b Fo(\()p Fm(x;)h(x)403 344 y Fk(0)414 364 y Fo(\))13 b(:=)h Fm(P)7 b Fo(\()p Fm(x)p Fo(;)h Fi(f)p Fm(x)672 344 y Fk(0)683 364 y Fi(g)p Fo(\))87 b(for)25 b Fm(x;)8 b(x)975 344 y Fk(0)1000 364 y Fi(2)14 b Fm(R)118 460 y Fo(yields)h(a)i(Mark)o(o)o(v)e(c)o(hain)h(on)h(the)f (state)g(space)h Fm(R)f Fo(with)h(in)o(v)m(arian)o(t)e(measure)272 552 y Fe(b)268 556 y Fm(\026)p Fo(\()p Fm(x)p Fo(\))e(:=)h Fm(\026)p Fo(\()p Fi(f)p Fm(x)p Fi(g)p Fo(\))87 b(for)25 b Fm(x)13 b Fi(2)h Fm(R:)191 652 y Fo(2.)29 b(F)l(or)19 b Fm(x)p 348 659 28 2 v 17 w(<)f(t)f(<)h Fi(1)g Fo(de\014ne)751 634 y Fl(t)778 636 y Fe(b)766 652 y Fm(P)26 b Fo(and)920 634 y Fl(t)940 648 y Fe(b)935 652 y Fm(\026)19 b Fo(in)f(analogy)i(to) 1287 634 y Fl(t)1293 652 y Fm(P)26 b Fo(and)1447 634 y Fl(t)1454 652 y Fm(\026)18 b Fo(in)h(Section)f(5.)118 712 y(Then)246 694 y Fl(t)272 696 y Fe(b)260 712 y Fm(P)24 b Fo(is)16 b(again)h(a)g(sto)q(c)o(hastic)g(k)o(ernel)e(with)h (strictly)f(p)q(ositiv)o(e)h(and)h(\014nite)e(in)o(v)m(arian)o(t)118 772 y(measure)316 754 y Fl(t)335 768 y Fe(b)331 772 y Fm(\026)p Fo(.)46 b(By)23 b(classical)h(Mark)o(o)o(v)f(c)o(hain)h (theory)g(this)g(implies)e(that)i(all)g(states)118 832 y Fm(x)15 b Fi(2)f Fm(R)e Fi(\\)g Fo([0)p Fm(;)c(t)p Fo(])16 b(are)h(\(p)q(ositiv)o(e\))e(recurren)o(t)h(with)h(resp)q(ect)f (to)1260 814 y Fl(t)1287 816 y Fe(b)1275 832 y Fm(P)7 b Fo(,)17 b(hence)f(also)h(recurren)o(t)118 893 y(with)f(resp)q(ect)g (to)467 877 y Fe(b)456 893 y Fm(P)7 b Fo(.)21 b(F)l(or)16 b Fm(t)e Fi(!)g(1)i Fo(this)g(extends)f(to)i(all)f(states)g Fm(x)e Fi(2)g Fm(R)p Fo(.)22 b(The)16 b(recurren)o(t)118 953 y(Mark)o(o)o(v)h(k)o(ernel)453 937 y Fe(b)442 953 y Fm(P)24 b Fo(is)18 b(in)f(addition)h(irreducible.)23 b(Indeed,)16 b(eac)o(h)h(restriction)g(of)1644 949 y Fe(b)1640 953 y Fm(\026)g Fo(to)h(a)118 1013 y(single)c(class)h(yields) e(again)j(an)f(in)o(v)m(arian)o(t)f(measure)f(for)1170 997 y Fe(b)1159 1013 y Fm(P)22 b Fo(and)15 b(its)f(trivial)f(extension) h(to)118 1073 y Fn(R)160 1080 y Fj(+)207 1073 y Fo(an)19 b(in)o(v)m(arian)o(t)e(measure)g(for)h Fm(P)7 b Fo(.)27 b(Th)o(us)18 b(b)o(y)g(the)f(uniqueness)h(prop)q(ert)o(y)g(of)g Fm(\026)g Fo(there)118 1133 y(can)i(exist)f(only)h(one)g(class.)31 b(Therefore,)20 b(again)h(b)o(y)e(classical)g(Mark)o(o)o(v)g(c)o(hain)g (theory)l(,)118 1194 y(ev)o(ery)g Fm(\033)r Fo({\014nite)h(\(not)i (necessarily)d(lo)q(cally)h(\014nite)8 b(!\))34 b(excessiv)o(e)19 b(measure)g(for)1630 1177 y Fe(b)1618 1194 y Fm(P)28 b Fo(is)21 b(in)118 1254 y(fact)16 b(a)h(m)o(ultiple)c(of)506 1250 y Fe(b)502 1254 y Fm(\026)p Fo(.)191 1314 y(3.)22 b(No)o(w)16 b(b)o(y)g(the)g(h)o(yp)q(othesis)g Fn(P)p Fo(\()p Fm(Y)25 b Fo(=)14 b(0\))g(=)g(0)j(the)f(sets)268 1410 y Fm(A)p Fo(\()p Fm(x;)8 b(x)402 1389 y Fk(0)412 1410 y Fo(\))14 b(:=)f Fi(f)p Fo(\()p Fm(y)r(;)8 b(z)r Fo(\))13 b(:)h Fm(y)r(x)c Fo(+)h Fm(z)16 b Fo(=)d Fm(x)918 1389 y Fk(0)930 1410 y Fi(g)p Fm(;)21 b(x)14 b Fi(2)g Fm(R;)118 1506 y Fo(are)i Fm(\027)s Fo({almost)g(disjoin)o(t)g(for)h (\014xed)f Fm(x)803 1488 y Fk(0)814 1506 y Fo(,)g(hence)268 1568 y Fe(P)320 1608 y Fl(x)p Fk(2)p Fl(R)412 1585 y Fe(b)400 1601 y Fm(P)7 b Fo(\()p Fm(x;)h(x)535 1581 y Fk(0)546 1601 y Fo(\))14 b(=)631 1568 y Fe(P)683 1608 y Fl(x)p Fk(2)p Fl(R)766 1601 y Fm(\027)s Fo(\()p Fm(A)p Fo(\()p Fm(x;)8 b(x)946 1581 y Fk(0)957 1601 y Fo(\)\))14 b Fi(\024)f Fo(1)88 b(for)16 b(all)25 b Fm(x)1352 1581 y Fk(0)1377 1601 y Fi(2)14 b Fm(R;)118 1697 y Fo(i.e.)21 b(the)16 b(equidistribution)g(on)h Fm(R)g Fo(is)f(excessiv)o(e)f(for) 1105 1681 y Fe(b)1093 1697 y Fm(P)8 b Fo(.)22 b(Therefore,)16 b(according)g(to)h(part)118 1757 y(2)g(of)f(the)g(pro)q(of,)118 1853 y(\(2\))272 1849 y Fe(b)268 1853 y Fm(\026)p Fo(\()p Fm(x)p Fo(\))d(=)h(1)88 b(for)16 b(all)25 b Fm(x)13 b Fi(2)h Fm(R)118 1949 y Fo(ma)o(y)h(b)q(e)h(assumed)g(in)g(the)g (sequel.)191 2009 y(4.)22 b(Since)16 b Fm(\026)h Fo(is)f(lo)q(cally)g (\014nite,)f(the)h(supp)q(ort)i Fm(M)k Fo(of)16 b Fm(\026)h Fo(in)f(view)g(of)h(\(2\))f(m)o(ust)f(consist)118 2070 y(of)i(isolated)f(p)q(oin)o(ts,)g(hence)f Fm(R)i Fo(m)o(ust)e(con)o (tain)h Fm(x)p 995 2077 V 14 w Fo(=)d(min)7 b Fm(M)e Fo(.)22 b(No)o(w)16 b(b)o(y)f(\(1.2a\))268 2165 y Fm(y)r(x)p 294 2172 V 10 w Fo(+)c Fm(z)16 b Fi(\025)d Fm(x)p 472 2172 V 50 w Fo(for)j Fm(\027)s Fo({almost)g(all)25 b(\()p Fm(y)r(;)8 b(z)r Fo(\))p Fm(;)118 2261 y Fo(whic)o(h)16 b(in)g(view)f(of)i Fn(P)p Fo(\()p Fm(Y)25 b Fo(=)14 b(0\))g(=)g(0)i (implies)268 2357 y Fm(y)r(x)10 b Fo(+)h Fm(z)16 b(>)d(x)p 471 2364 V 50 w Fo(for)j Fm(\027)s Fo({almost)h(all)24 b(\()p Fm(y)r(;)8 b(z)r Fo(\))86 b(whenev)o(er)24 b Fm(x)14 b(>)f(x)p 1422 2364 V(:)118 2453 y Fo(By)j(the)g(in)o(v)m(ariance)f(of) 569 2449 y Fe(b)565 2453 y Fm(\026)h Fo(this)g(yields)268 2549 y(1)41 b(=)413 2515 y Fe(P)465 2556 y Fl(x)p Fk(2)p Fl(R)557 2532 y Fe(b)546 2549 y Fm(P)7 b Fo(\()p Fm(x;)h(x)p 653 2556 V -1 w Fo(\))333 2650 y(=)413 2617 y Fe(P)465 2657 y Fl(x)p Fk(2)p Fl(R)548 2650 y Fm(\027)s Fo(\()p Fi(f)p Fo(\()p Fm(y)r(;)g(z)r Fo(\))14 b(:)f Fm(y)r(x)d Fo(+)h Fm(z)16 b Fo(=)e Fm(x)p 975 2657 V -1 w Fi(g)p Fo(\))333 2752 y(=)42 b Fm(\027)s Fo(\()p Fi(f)p Fo(\()p Fm(y)r(;)8 b(z)r Fo(\))13 b(:)g Fm(y)r(x)p 661 2759 V 11 w Fo(+)e Fm(z)16 b Fo(=)d Fm(x)p 839 2759 V Fi(g)p Fo(\))p Fm(;)920 2877 y Fo(16)p eop %%Page: 17 18 17 17 bop 118 77 a Fo(and)17 b(\(1\))g(is)f(established)p Fm(:)49 b Ff(2)191 166 y Fo(The)18 b(remark)e(follo)o(wing)h(\(6.1\))h (sho)o(ws)h(again)f(that)g(the)f(condition)h Fn(P)p Fo(\()p Fm(Y)27 b Fo(=)17 b(0\))f(=)g(0)118 226 y(is)g(essen)o(tial)f(for)i (\(6.4\).)191 287 y(The)h(\014nal)f(result)g(of)h(this)f(section)g(is)g (a)h(strong)g(stabilit)o(y)e(statemen)o(t,)g(v)m(alid)h(under)118 347 y(an)g(appropriate)g(normalization:)118 436 y 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Fl(T)735 189 y Fc(0)758 172 y Fo(\))e(=)f(1)p Fm(:)118 266 y Fo(Since)i Fm(f)22 b Fo(is)16 b(b)q(ounded,)h(\(2\))f (and)h(\(3\))g(com)o(bine)d(to)268 361 y Fn(E)p Fo(\()p Fm(f)5 b Fo(\()p Fm(X)416 338 y Fl(x)436 344 y Fd(k)412 373 y Fl(T)433 379 y Fd(k)457 361 y Fo(\)\))14 b Fi(!)f Fn(E)p Fo(\()p Fm(f)5 b Fo(\()p Fm(X)720 339 y Fl(x)740 344 y Fc(0)716 373 y Fl(T)737 378 y Fc(0)760 361 y Fo(\)\))p Fm(;)118 456 y Fo(and)17 b(\(1\))g(is)f(established)p Fm(:)49 b Ff(2)191 545 y Fo(A)o(t)16 b(the)g(next)f(step)i(\(5.5\))f (is)g(essen)o(tial:)118 634 y Fn(\(7.3\))28 b(Lemma.)22 b Fg(L)n(et)e Fm(\027)j Fg(b)n(e)d(r)n(e)n(curr)n(ent)g(with)g (invariant)h(me)n(asur)n(e)e Fm(\026)p Fg(.)31 b(Supp)n(ose,)21 b(with)118 695 y(the)d(notations)g(of)26 b Fo(\(7.2\))18 b Fg(and)25 b Fo(\(5.4\))p Fg(,)118 789 y Fo(\()p Fi(\003)p Fo(\))87 b Fm(t)14 b Fi(2)g Fm(D)388 769 y Fj(0)495 789 y Fg(and)88 b Fm(C)699 769 y Fj(0)695 802 y Fl(t)743 789 y Fg(dense)18 b(in)26 b Fo([0)p Fm(;)8 b(t)p Fo(])118 884 y Fg(and,)19 b(in)f(ac)n(c)n(or)n(danc)n(e)g(with)h Fo(\(4.3\))p Fg(,)775 866 y Fl(t)782 884 y Fm(\026)f Fg(to)h(b)n(e)f(normalize)n(d.)25 b(Then)19 b(the)g(pr)n(o)n(c)n(ess)e Fo(\()1614 866 y Fl(t)1620 884 y Fm(X)1660 891 y Fl(n)1684 884 y Fo(\))1703 891 y Fl(n)p Fk(\025)p Fj(0)118 944 y Fg(is)d(stationary)g(and)h(er)n(go)n(dic,)f(if)h Fm(X)749 951 y Fj(0)783 944 y Fg(is)f(distribute)n(d)h(ac)n(c)n(or)n(ding)f(to)g (\(the)h(trivial)g(extension)118 1005 y(of)5 b(\))200 986 y Fl(t)207 1005 y Fm(\026)p Fg(.)118 1094 y(Pr)n(o)n(of.)23 b Fo(1.)f(The)16 b(follo)o(wing)g(fact)g(will)f(b)q(e)i(used:)k(the)16 b(measures)268 1208 y Fm(\045)293 1187 y Fl(x)293 1220 y(n)330 1208 y Fo(:=)403 1174 y(1)p 400 1196 30 2 v 400 1242 a Fm(n)434 1175 y Fe(P)487 1215 y Fj(0)p Fk(\024)p Fl(m)h(x)379 1101 y Fj(0)485 1094 y Fo(and)88 b(supp)9 b Fm(f)784 1101 y Fl(i)812 1094 y Fi(\032)k Fo([0)p Fm(;)8 b(t)p Fo(])49 b(for)25 b Fm(i)14 b Fo(=)g(1)p Fm(;)19 b Fo(2)p Fm(;)118 1208 y Fo(\(2\))88 b Fm(t)24 b 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V 13 w Fi(\024)14 b Fm(x)g(<)f(s:)118 1702 y Fo(Indeed,)i(\014x)h Fm(x)g Fo(and)h(c)o(ho)q(ose)g Fm(t)c(>)h(s)i Fo(and)h Fm(l)e Fi(\025)e Fo(0)k(suc)o(h)f(that)268 1804 y Fn(P)p Fo(\()p Fm(X)369 1783 y Fl(s)365 1816 y Fj(1)401 1804 y Fi(\024)e Fm(t)p Fo(\))f Fm(>)h Fo(0)88 b(and)g Fn(P)p Fo(\()p Fm(X)935 1783 y Fl(t)931 1816 y(l)964 1804 y Fm(<)14 b(s)p Fo(\))f Fm(>)h Fo(0)p Fm(:)118 1906 y Fo(Then)i(b)o(y)g(\(1.2a\)) h(and)g(monotonicit)o(y)268 2007 y Fn(P)p Fo(\()p Fm(x)p 325 2014 V 13 w Fi(\024)d Fm(X)463 1987 y Fl(x)459 2020 y(l)499 2007 y Fm(<)g(s)p Fo(\))g(=)f Fn(P)p Fo(\()p Fm(X)759 1987 y Fl(x)755 2020 y(l)796 2007 y Fm(<)h(s)p Fo(\))f Fi(\025)h Fn(P)p Fo(\()p Fm(X)1057 1987 y Fl(t)1053 2020 y(l)1086 2007 y Fm(<)g(s)p Fo(\))g Fm(>)g Fo(0)p Fm(:)118 2109 y Fo(With)i Fm(\026)273 2116 y Fj(1)307 2109 y Fo(:=)d Fi(L)p Fo(\()p Fm(X)469 2091 y Fl(x)465 2121 y Fj(1)492 2109 y Fo(\))j(moreo)o(v)o(er)268 2211 y Fn(P)p Fo(\()p Fm(x)p 325 2218 V 13 w Fi(\024)e Fm(X)463 2190 y Fl(x)459 2223 y(l)p Fj(+1)531 2211 y Fm(<)g(s)p Fo(\))g Fi(\025)691 2175 y Fe(R)727 2218 y Fj([)p Fl(x)p 737 2225 20 2 v(;t)p Fj(])800 2211 y Fn(P)p Fo(\()p Fm(x)p 857 2218 28 2 v 14 w Fi(\024)f Fm(X)995 2189 y Fl(x)1015 2194 y Fc(1)991 2223 y Fl(l)1049 2211 y Fm(<)g(s)p Fo(\))e Fm(\026)1182 2218 y Fj(1)1203 2211 y Fo(\()p Fm(dx)1275 2218 y Fj(1)1294 2211 y Fo(\))p Fm(;)118 2312 y Fo(where)16 b(as)h(ab)q(o)o(v)o(e)268 2414 y Fn(P)p Fo(\()p Fm(x)p 325 2421 V 13 w Fi(\024)d Fm(X)463 2392 y Fl(x)483 2397 y Fc(1)459 2427 y Fl(l)516 2414 y Fm(<)g(s)p Fo(\))g Fm(>)g Fo(0)87 b(for)25 b Fm(x)898 2421 y Fj(1)932 2414 y Fi(2)14 b Fo([)p Fm(x)p 993 2421 V -1 w(;)8 b(t)p Fo(])118 2516 y(and)268 2617 y Fm(\026)297 2624 y Fj(1)317 2617 y Fo(\([)p Fm(x)p 350 2624 V -1 w(;)g(t)p Fo(]\))13 b(=)g Fn(P)p Fo(\()p Fm(x)p 571 2624 V 14 w Fi(\024)h Fm(X)710 2597 y Fl(x)706 2630 y Fj(1)746 2617 y Fi(\024)g Fm(t)p Fo(\))f Fi(\025)h Fn(P)p Fo(\()p Fm(X)1003 2597 y Fl(s)999 2630 y Fj(1)1035 2617 y Fi(\024)g Fm(t)p Fo(\))f Fm(>)h Fo(0)p Fm(:)920 2877 y Fo(23)p eop %%Page: 24 25 24 24 bop 191 77 a Fo(4.)22 b(By)15 b(com)o(bining)f Fm(k)j Fo(steps)f(according)g(to)g(\(2\),)g Fm(i)g Fo(times)d Fm(l)k Fo(steps)f(and)g Fm(j)j Fo(times)14 b Fm(l)c Fo(+)g(1)118 137 y(steps)22 b(according)g(to)g(\(3\),)h(and)f Fm(m)g Fo(steps)f(according)h(to)g(\(1\),)h(the)f(Mark)o(o)o(v)f(prop)q(ert)o (y)118 197 y(yields)268 299 y Fn(P)306 279 y Fl(x)326 284 y Fc(0)345 299 y Fo(\()p Fm(X)404 306 y Fl(n)442 299 y Fi(2)14 b Fm(G)527 306 y Fj(0)547 299 y Fo(\))g Fm(>)g Fo(0)87 b(for)25 b Fm(n)14 b Fo(=)g Fm(k)f Fo(+)e Fm(il)h Fo(+)f Fm(j)s Fo(\()p Fm(l)g Fo(+)g(1\))h(+)f Fm(m:)118 401 y Fo(Since)k Fm(i;)8 b(j)17 b Fi(\025)c Fo(0)k(are)f(arbitrary)l(,)g(the)g(requiremen)o(t)d(is)j(met)f(b)o(y) 268 502 y Fm(n)297 509 y Fj(0)330 502 y Fo(:=)f Fm(k)f Fo(+)e(\()p Fm(l)g Fi(\000)g Fo(1\))p Fm(l)h Fo(+)f Fm(m:)49 b Ff(2)191 604 y Fo(In)16 b(view)g(of)g(\(7.6\))h(the)f(follo)o(wing)g (op)q(en)h(problem)e(can)h(b)q(e)h(p)q(osed:)22 b(is)16 b(it)g(p)q(ossible)g(to)118 664 y(strengthen)21 b(\(7.1\))g(to)f(a)h (strong)h(ratio)f(limit)d(theorem)h(as)i(v)m(alid)f(for)h(irreducible)d (and)118 725 y(ap)q(erio)q(dic)e(recurren)o(t)f(random)h(w)o(alk?)118 899 y Fn(References)141 1010 y Fb(2.)24 b(Barnsley)l(,)19 b(M.,)e(Elton,)h(J.:)25 b(A)18 b(new)f(class)h(of)g(Mark)o(o)o(v)e(pro) q(cesses)i(for)f(image)h(enco)q(ding.)201 1066 y(Adv.)i(Appl.)h(Prob.)f Fa(20)p Fb(,)15 b(14{32)f(\(1988\))141 1125 y(8.)24 b(Dellac)o(herie,) 17 b(C.,)d(Mey)o(er,)h(P)l(.A.:)k(Probabilit)o(\023)-21 b(es)16 b(et)f(p)q(oten)o(tiel,)h(Chapitres)g(IX{XI.)f(P)o(aris:)201 1182 y(Hermann)g(1983)141 1240 y(9.)24 b(Dubins,)15 b(L.,)g(F)l (reedman,)f(D.:)19 b(In)o(v)m(arian)o(t)14 b(probabilities)j(for)d (certain)h(Mark)o(o)o(v)e(pro)q(cesses.)201 1297 y(Ann.)21 b(Math.)e(Stat.)g Fa(37)p Fb(,)c(837{848)e(\(1966\))119 1355 y(14.)23 b(F)l(oguel,)d(S.:)27 b(The)20 b(ergo)q(dic)g(theory)e (of)h(p)q(ositiv)o(e)h(op)q(erators)e(on)h(con)o(tin)o(uous)g (functions.)201 1412 y(Ann.)i(Sc.)f(Norm.)f(Sup)q(er.)i(Pisa)15 b Fa(27)p Fb(,)g(19{51)f(\(1973\))119 1471 y(22.)23 b(Karlin,)15 b(S.:)k(Random)14 b(w)o(alks)f(arising)h(in)g(learning)h(mo)q(dels.)20 b(P)o(ac.)f(J.)13 b(Math.)19 b Fa(3)p Fb(,)14 b(725{756)201 1528 y(\(1953\))920 2877 y Fo(24)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF