; TeX output 2002.07.01:1413썠:-iӍ-;(7"Vff cmbx10Order-preservingRandomDynamicalSystems"N+XQ cmr12HansG.KellererbhMathematiscrhesInstitutderUniversit atMSvunchencTheresienstrasse39,D{80333MSvuncrhen,GermanyJJune30,1995/s0N cmbx12Summary.xThisppapSerisconcernedwithstocrhasticrecursions,g cmmi12X2cmmi8n=URH̹nP(X̹nK cmsy8 |{Ycmr81)sfor~nQ-!", cmsy102N,where(H̹nP) n2t : cmbx9N#isasequenceofi.i.d.randomtransformationsinsR̸+ ZindepSendenrt=oftheinitialvXariableX̸0.0IfthetransformationsaresupposedstobSeconrtinuousandorder-preserving,Mthisresultsinquitenaturalnotionsofs(topSological)ܷrecurrenceandtransience. IntherecurrenrtcaseexistenceandsuniquenessofaninrvXariantmeasureaswrellasmeanandpSointwiseergoSdicstheoremscareestablished.<Moreorver,RthecassoSciatedattractorisinrvestigated.sTheS$classi cationiscompletedbrydistinguishingpSositiveandnullrecurrencescorrespSonding,Q"respectivrelyV,to*thecaseofa niteorin niteinrvXariant*measure.sEquivXalenrtlyV,thisamountsto niteorin nitemeanpassagetimes.sMathematicsFSubjectClassi cations(1991):Primary60J05,60H25;fSecondarys58F11,54H20.*썠:-iӍ{-sIntro`duction sIn}thispapSerrandomdynamicalsystemsarestudiedunderthefollorwingas-ssumptionsbonspaceandtime:theevrolutionissuppSosedtodevelopinthespacesofXrealsandtoproSceedindiscretetimestartingatzero.Thrustheprocessissgivrenbyasequence(X̹nP)̹n0otsatisfyingtherecursion8X̹n=URH̹nP(X̹n1)for%n2N;swhereDX̸0 isareal-vXaluedrandomvariableandH̹nP;n2NDarerandomtrans-sformations*|fromRtoR.TVoobtaintheMarkrov*|propSertythevXariablesX̸0;sH̸1;H̸2;:::"ZareWsuppSosedtobeindependenrt,stoobtainastationarytransitionskrernelthevXariablesH̸1;H̸2;:::X*aresuppSosedtobeidenrticallydistributed.!Ifstheircommondistributionn~iscarriedbrya nitefamilyoftransformations,sthis6resultsinan\iteratedfunctionsystem",madepSopularduringthepastsdecade8JbryBarnsleyV,Eltonandothers(seee.g. ![2,3,8,9]).On8Jtheothershand,=Iit,isshorwn,forinstance,bryKiferinTheorem1.1of[18]that,withoutsrestrictingbthesuppSortof,pthemodelaborvebprovidesjustanotherpSossibilitystoinrtroSduceanyMarkovchainwithstatespaceR..Intapplications,Xsucrharepresentationarisesoftenquitenaturallyasissexhibitedbrythefollowingexamples(seealso[1,24,25]):s(1)ThesimplestqueuingproScessisgivrenby2X̹n=UR(X̹n1/t+U̹nP) +xfor-n2Nswithѫi.i.d. vXariablesU̹nP;nF2N.Here,KlX̹n1Vwdenotesѫthewraitingtimeofscustomermn1andU̹n=URS̹n10(T̹nT T̹n1)isthebalancebSetrweenhisservicestimeandthearrivXaltimesofcustomersn1andn.s(2)A\sarvingsproScess"isde nedbyz@X̹n=URU̹nPX̹n1/t+V̹nPfor+n2Nswith3]i.i.d. vXariables(U̹nP;V̹n);n2N. Here,X̹n1)denotes3]thebalanceofassarvings0accountattimenf1, U̹n theinrterestW/in ationrateduringpSeriodnsandRFV̹nthedepSositmadeattimen(thisandasurvreyofotheranerecursionssinbiologyV,economics,phrysicsetc.8canbSefoundinVervXaat[29]).s(3)An\excrhangeproScess"isde nedby$X̹n=UR(X̹n1/tU̹nP)_V̹nPfor+n2Nswith3si.i.d.BvXariables(U̹nP;V̹n);n:2N.BHere,EX̹n1?is3stheutilitryofsomeequip-smenrtyinuseattimen :1,U̹n "FitsylossinutilityduringpSeriodyn,andV̹n "Fthesutilitry4ofanewequipmentavXailableattimen(forthisandrelatedexamplesssee,forinstance,HellandandNilsen[13])..TheO rstexampledi ersfromthesecondandthethirdonebryaparticularscrharacteristic:4theQstate0isaregenerationpSoint,4andthusthequeuingproScesss tsinrtotheDoSeblin-HarristheoryfornotnecessarilydiscreteMarkovchains.@1썠:-iӍ{-sSincenthedominatingmeasure,) whoseexistenceispSostulatedinthistheoryV,sfails toexistingeneral,Qthereareattempts,forinstancebryRosenblatt[27]sorijTwreedie[28],toclassifyMarkovchainsbystressingthetopSologicalstruc-stureofthestatespace.0Evrenundercontinuityassumptionsontheunderlyingskrernel,Έhowever,thisresultsinavXarietryofnotionsoftransienceandnullorspSositivrerecurrence,beingthruslessconvincingthantheclassicalnotionsforsdiscrete֍Markrovchains.ThepresentpapSer,therefore,emphasizestheordersstructurea^ofthestatespace,|motivXatedbrytwoobservXationsholdingforvarioussapplications. First,the transformationshUR:x!(xF+u)2+x,hUR:x!uxF+vn9,andsh$:x!(x/u)_vappSearing'intheexamplesaborve'areallorder-preservings(observingkWU̹n ث0[0in(2)).Second,intheseexamplesthepropSerstatespacesisR̸+ r(observingV̹n %S}0in(2)and(3)).~Asitturnsout,acceptingthesetrwosrestrictionsTpresenrtsanappropriatecompromiseinordertoobtainasatisfyingstheoryaswrellassubstantialapplications..Whilethereexistsanextensivreliteraturebasedonthemetricstructureofsthestatespacebryrequiring,Bsforinstance,an\arveragecontractivity"ofthesunderlying transformations,]thereareonlyafewpapSerswithspecialemphasisson7theorderstructure.˺TheearliestonedatesbacrktoDubinsandFVreedman[7],swho/limit,horwever,their/investigationstothecompactstatespace[0,1].#ThissrestrictionisgivrenupinYVahav[30], whostudiesconcaveincreasingmappingssfrom+R̸+ HgtoR̸+x.ExtendingthestatespacefurthertoR,Bhattacrharaya+andsWVarymireinSectionISI.14of[4]takeupa\splitting"conditionfrom[7].>Insall̯thistreatmenrtsthemaininterestconcernstheexistenceanduniquenessofsstationaryfdistributions.ThisholdsaswrellforBrandtetal., whoconsiderinsSection1.3X@ofN[5]order-preservingmappingsinpartiallyorderedProlishspacessrequiring,horwever,appropriatecompactnessandcontractionpropSerties..TVo$see uctuationaspSectstobeasinrterestingasequilibriumresults,3con-ssiderWanexcrhangeproScesswithdeterministiclossofutilityV,)sayU̹n=UR1.pThenitsisPueasilyestablishedthata(unique)stationarydistributionexistsifandonlyifsthe7'utilitryV̹nwofthesubstitutehasa niteexpSectation. Thisfails,[ forinstance,sifV̹n hasthedensitry鍑Pf̸1(x)UR=(x+1) 2\|or)$resp._thepSotentialkernel,`bymeansofhittingprobabilitiesresp._meanspassagetimes,EHorbrymeansofauniqueinvXariantmeasureall ndtheircoun-sterpartPinthepresenrtpapSer(fora rstorientationseetheremarksfollowings(6.5),(9.8),(10.1)andpreceding(11.4))..FinallyV,aUhistoricalremarkisinorder.*Thiswrorkoriginatedfromathree-spartpapSerdevrotedtothespecialcaseofrecursionsڍ\X̹n=URU̹nPX̹n1/t+V̹nPwith0$U̹n;V̹nUR0for%n2N:sThisanemoSdel,)whicrhforconstantU̹n#containsinparticular rst-orderauto-sregressivre+proScesses,isofspecialimportance,becauseitmaryservetoapprox-simate"morecomplexsituationsbrylinearization.(DuringtherefereeingproScesssofm[16],Fhorwever,itmbSecameclearthatmostresultsrelyonthetopologicalandsorder9structureofthestatespaceonlyandmakrenouseofthelinearstruc-sture.tMoreorver, fitturnedoutthatwithinamoregeneralframewrorknotonlyssevreralproSofscouldbesimpli edbutalsosevreraltheoremscouldbestrength-sened.ThisRledtothedecisiontodevrelop rstthegeneraltheoryinthepresentspapSerandtodealwiththespecialfeaturesofanerecursionsinasubsequenrtspapSer.Asitistobeexpected,Gthereare,forinstance,strongercriteriaforspSositivreW/nullBrecurrenceortransience,%)iftheunderlyingmappingsarecom-spatible,withthelinearstructure./aTVostatejustoneofthemainresultsof[17],slet(S̹nP)̹n0EbSetherandomwralkwithincrementslogfU̹nN(UR1).+Then,lundersaZwreakbSoundednessconditiononV̹n (andexcludingthedegeneratecaseofsaNcommon xedpSoinrtoftheunderlyingmappings),gthefollowingtrichotomysholds:ժthe$=proScess(X̹nP)̹n0 ispositivrerecurrentresp.nullrecurrentresp.tran-ssienrt,ify6theassoSciatedrandomwalk(S̹nP)̹n0divergesto1resp.oscillatessbSetrween1and+1resp.8divrergesto+1.sPreliminaries sThroughoutLthepapSerthestatespaceEisasubinrtervXalofRsatisfyingsmin2 -E]=0andendorwedwithitsBoreln9-algebraB]m(E).Asusual,C5(E)de-snotesthespaceofbSoundedconrtinuousfunctionsf%:ݘE!RandK,`(E)thessubspace4consistingoffunctionswithcompactsuppSort.LEmploryingtheordersstructure,R(E)denotesthespaceof(\regular")functionsf:[E!RharvingslimitswithinRfromtherighrtandfromthelefteverywhere(includingsup^E)sandV(E)thesubspaceconsistingoffunctionsofbSoundedvXariation..Let|thespaceofconrtinuous|mappingsfromE0toEbSeendorwed|withthescompactRopSentopologyV.ThentheclosedsubspaceH[E]oforder-preservingsmappingsrinheritsaProlishtopSologyV,]whichisalsotheinitialtopSologywithsrespSecttotheevXaluationmapshs!h(x);x2E.1SinceH[E]isstablewithsrespSecttocompositionandcompositionisconrtinuous,H[E]isatopologicalssemigroup.R SinceHbEycanbSeemrbeddedinrtoH[E],_themapping(x;h)!h(x)sisconrtinuous,toSo..M(E)denotestheclassofloScally nitemeasuresonEandM̸1(E)thessubSclassPconsistingofprobabilitrymeasures. LIff9Odenotestheintegral@3(+썠:-iӍ{-sof'afunctionfG,&thenM(E)isendorwed'withthevXague(wreak2)topSologyV,si.e.,the#linitialtopSologywithrespecttothemappingsi!f;f2K,`(E);sthecorrespSondingconrvergenceisdenotedbry %!Rvu.]OnthesubspaceM̸1(E)sthis2inducesthewreak(narrow)topSologyV,Ui.e. ~theinitialtopologywithre-sspSectntothemappings!f;fҷ2C5(E);Qthencorrespondingconrvergencenissdenotedbry?!Rw=..The$7mainobjectofthispapSeristhespaceNİ[E]ofdistributionsonH[E].sThe֏semigroupstructureofH[E]inducesaconrvolution֏inNİ[E]andmakressthis9JspaceatopSologicalsemigroupitself. $Correspondingporwers9Jaresimplysdenotedbryǟ2nj,i.e.%ȍCoCu cmex10CZOnfG(h(x))ǟ nj(dh)UR=CZUQ:::CZ+f(h̸1j:::h̹nP(x))(dh̸1):::ʜ(dh̹nP)SsforxpX2E;fW2C5(E)andn2N,whileǟ20 |Sistheunitmeasure"̹h `ywithhbSeingstheRidenrtitymap.DSinceH[E]isagainaPolishspace,q1inparticularthesuppSortsN+iswrell-de nedfor2URNİ[E].m.NorwUthestoSchasticmoSdelcanbepreciselyinrtroduced.Letbegivren,Aonssomeprobabilitryspace( ;A;P),s(1)aUsequenceofindepSendenrtrandomvXariablesH̹n:UR !H[E]Uwithidenticalsdistribution2URNİ[E],s(2)arandomvXariableX̸0V:UR !EthatisindepSendenrtof(H̹nP) n2N.sThisde nesan\order-preservingrandomdynamicalsystem"bry3q.X̹n=UR n9(X̹n1;H̹nP)with4 (x;h)UR=h(x);swhicrhinthesequelwillbSebrie ywrittenashX̹n=URH̹nP(X̹n1)for%n2N:sThrusX^thedistributionof(X̹nP)̹n0*iscompletelydeterminedby2Nİ[E]andstheMinitiallarw̸0V=URL(X̸0). Here,theprimarycompSonentis,andallnotionssto^bSede nedwilldependonthatdistribution. MThisdependencewill,zhorwever,sbSesuppressedintherelatednotations,becauseoissupposedtobe xed..Asusual,theinitiallarwislargelyofsecondaryimpSortanceonlyV. %IfinsparticularX̸0V=URx,thiswillbSeexpressedbrythenotation(X2xRAn:j)̹n0,i.e.3cÏX xڍn =URH̹nR:::H̸1(x)for%x2E and'Cn0:sThrusforgeneral̸0conditionalprobabilitiesaregivenbyr!P xH((X̹nP;nUR0)2B)=P((X xڍn:j;n0)2B)swithananologousequationforconditionalexpSectations.m.ClearlyV,(X̹nP)̹n0issahomogeneousMarkrovsproScess. Itstransitionkrernel,salwraysdenotedbryP,transformsafunctionf2onEintoPf2givenby%ȍyK1PfG(x)UR=CZ㌟zHl[Erظ]Mf(h(x))(dh)for%x2E@49%썠:-iӍ{-sandameasureonEinrtoPngivenbyge wP(B)UR=CZ㌟zE(h(x)2B)(dx)for%BX2B]m(E);swhicrhforan9 nitemeasurebyFVubiniequals_P(B)UR=CZ㌟zHl[Erظ]M(h(x)2B)(dh)for%BX2B]m(E):.ThekrernelPbSelongstotheclassP[E]ofMarkovkernelsfromEtoEssatisfyingthefollorwingtwoconditions:s(1)PntransformsC5(E)inrtoitself,s(2)P)transformscbSoundedincreasingfunctionsinrtofunctionsofthesametypSe.sItk3hastobSemenrtionedherethatthemapping!0P fromNİ[E]toP[E]issneitherinjectivrenorsurjective.Z.FinallyV,%it hastobSepoinrtedoutthat,%incontrasttotopSologyandorder,sthe}=algebraicstructureofthestatespaceisnottakrenintoaccount. Thussdistributions2; Nİ[E]andǟ20 2Nİ[E20P]arecalled\conjugate"andharvetosbSe6classi edinthesamewrayV,Iif6thereisanorder-preservinghomeomorphismsgË:URE20ע!Esucrhthatǟ20uistheimageofunderthemappinghUR!gn921khgn9..FVor_anexampleinthecaseE=cR̸+ G=E20 8let_h̸+2H[E]_bSestrictlysincreasingwithh̸+x(x)UR>xforallxUR2Eâandde neh̺ q2URH[E]bryh̺(x)UR=0forsxURh̸+x(0)andh̺(x)UR=h21RA+ \|(x)otherwise.6Thenadistribution2URNİ[E]withssuppSortxNo=Gfh̺x;h̸+gisconjugatetothedistributionǟ20׋2GNİ[E20P]bSelongingstothequeuingproScessv12X̹n=UR(X̹n1/t+U̹nP) +xfor-n2NswithindepSendenrtvXariablesU̹nP;nUR2N;satisfyingV:uP(U̹n=UR1)=(fh̺xg) and&,P(U̹n=+1)=(fh̸+xg):sIndeed,itiseasilycrheckedthatanrystrictlyincreasingfunctiong2iC5([0;1])swithSgn9(0)=0andgn9(1)=h̸+x(0)canbSeextendedtoanappropriatehomeo-smorphismbrythede nitionv1^1Wgn9(x+n)UR=h nڍ+x(g(x))for%0x1 and&,n2N:s1.Lowerandupp`erlimit sAsDindiscreteMarkrovDchaintheoryV,the rstquestiontobSesettledconcernssthexappropriatedecompSositionofthestatespaceE.jFVorsometGT2E,itxmaryssplitf4inrtotwointervXalsE̸1='[0;t[andE̸2='EnE̸1 &8sucrhthath[E̹id]E̹iforsalmost}allhO2H[E]}andiO=1;2.'With}0=minE1asreferencestatethescorrespSonding\class"canbecrharacterizedexplitelyaswellasimplicitly:@5I썠:-iӍ{-s(1.1)Prop`osition 35.@ cmti12For352URNİ[E]theset_QE̸0V:=YyC[ {URn2NzfxUR2E i:P(X 0ڍn x)>0g#sis35thesmallestsubintervalI$ofELsatisfyingthecffonditionss(a)Ө0UR2I;s(b))(h[I]URI)=1:sPrffoof.1. ToSinceanryopSenorclosedintervXalIinERcontaining0cansbSeLrepresenrtedasacountableunionofintervXals[0;x̹k#],IthesubsetofH[E]soSccurringin(b)isapparenrtlyoftypSeG̹T(orevenclosed)andthusinparticularsmeasurable.0Since'mthisrepresenrtationappliestoIE=E̸0qitself,6inestablishings(b)mitsucestoshorwthat,۞xp=x̹k bSeingm xed,h(x)2E̸0 kqformalmostallshUR2H[E]..2.8TVothisendcrhoSosenUR2NwithP(X20RAn URx)>0andde newJH̹t:=URfh2H[E]:h(x)tgfor%t2E:sThentheindepSendenceofX20RAn /andH̹n+1otyields$fhUR2H[E]:h(x)2E̸0gCfhUR2H[E]:P(X 0ڍn+1h(x))>0gCfhUR2H[E]:P(X 0ڍn x;H̹n+1(x)h(x))>0g\=fhUR2H[E]:P(H̹n+1(x))h(x))>0g\=fhUR2H[E]:(H߹h(x)P)>0g:sDuetohUR2H߹h(x)thisimplies-֞fhUR2H[E]:h(x)=2E̸0ggfhUR2H[E]:(H߹h(x)P)=0g㍍gߟC[٥fH߹h(x) :URh2H[E]UPwith!t(H߹h(x)P)=0g:sSinceethesetsH̹t/decreaseforincreasingt,0thisunioncanbSereplacedbryascounrtablemoneandisthusitselfanullsetwithrespSectto,ͧashadtobesshorwn..3.{ FinallyV,ېletathesubinrtervXalIofE_xsatisfy(a)and(b).Theniterationsyieldstheequationǟ2nj(h[I]URI)=1andthrusinparticularg P(X 0ڍn 2URI)=ǟ nj(h(0)2I)=1forall6E>n2N:sFVorxx2E]/satisfyingtheconditionP(X20RAn Jx)>0thisimpliesx2I,شthrussprorvingE̸0tobSeminimal.83Tq lasy102 .TheҥinrtervXalE̸0 canbSeanopenorclosedsubsetofEasitmaryhappensalreadyinthefollorwingtwotrivialexamples:@6W썠:-iӍ{-s(1)the\deterministicsystem"(E;h),Hwhere ^="̹h jforsomemappingshG2H[E],v(yieldingZBdeterministicvXariablesX̹nP;nG2N;ZBwhenevrerX̸0 Fisacon-sstanrt;s(2)the\indepSendenrtsystem"(E;),>whereZistheimageofsomemeasures_2M̸1(E)withrespSecttothecanonicalinjectionjofEinrtoH[E],zyieldingsindepSendenrtvXariablesX̹nP;nUR2N;withdistribution.f.IfqtheinrtervXalE̸0WuisapropSersubsetofE,itsupperendpoinrtiseasilyiden-sti ed: s(1.2)Prop`ositionhThehsuprffemum&feR]ڍx ~ofthesetE̸0(de nedby(1.1)isgivenby͓eW&feR]ڍxo=URminfxUR2E i:h(x)URxgwheneverLQ&feR]ڍxV2E:sPrffoof.FVordeacrhx>f2ET{withdh(x)URxtheinrtervXalI/=[0;x]satis esthesconditionin(1.1)andthruscontainsE̸0,whichimplies&feR]ڍx u x.3Ontheothershand,thede nitionofE̸0 qyieldsh(x)5UR&feR]ڍx forx5<&feR]ڍx U,whicrhbythecontinu-sitryofhUR2H[E]impliesh(&feR]ڍxR)URURߟ&feR]ڍx.wheneverthecondition&feR]ڍx 2UREissatis ed.R2 .Theqassumption&feR]ڍxP2EÈisnorealrestriction,bSecauseEbSeingreplacedbrysfe f fbE*e:="Eڬ[&f&feR]ڍxRgmappingsh2H[E]harveuniqueextensionsw}feÊ hX2H[fe f fbE f],-wherespSossibly1hastobeadjoinedtoR̸+x.d,Moreorver,if&feR]ڍxL2mE.andthesupportNsofois nite,&feR]ڍxclearlyisa xedpSoinrtofsomehUR2N@.f.TheimplicitdescriptionofE̸0suggeststhefollorwingnotion:s(1.3)De nition The distribution2URNİ[E]iscffalled \irreducible",{ifthesetsE̸09de neffd35by(1.1)coincideswiththestatespaceE..IrreducibilitryiapparentlycanalwaysbSeachieved,lreplacingmappingssh52H[E]qbrytheirrestrictionsh̸0 25H[E̸0]. K0:sThenbrymonotonicityandindepSendence)P(X̹nURxin nitelyoftenP)Gϝ*P(limsup 7/ak6!1&bfH߸(k6+1)l:::H̸1(0)URxg)7Gϝ*P(limsup 7/ak6!1&bfH߸(k6+1)l:::H̹k6lK+1(0)URxg)I򍍍`=ϝ*1:sSince\xct2E=sisarbitraryV, limsup*kn!1E??X̹n &feR]ڍx"holdsalmostsurelyV, whilethesinrverseinequalityisobvious.82.Thesecondresultislessimmediate:s(1.6)Theorem tIft2Nİ[E]isirrffeducible,CtheretisacffonstantxfeR&feR]ڍxsatis-sfyingƀliminf(n!1P,X̹n=URxURfeRa.s.(;Tsrffegardless35oftheinitiallaw.sPrffoof.1.8FVorxUR2Eandn0de nelXlƟfe xxnW:=URliminf8㍑Yk6!1'H̹n+k:::H̹n+1(x):7sThentheinequalitryqWXqWfe x|eF0|eF0>2=zliminf8㍒k6!1MH̹n+k:::H̹n+1(X 0ڍn)i>%Jzliminf8㍒k6!1MH̹n+k:::H̹n+1(0)UR=Xfe xA0Ani>scomrbinedwiththeequationL(Xfe x 0 0q)UR=L(Xfe x 0 nZ?)yields$X$fe x00i=URXURfe xA0Ana.s.=foralld]nUR0;shenceXϟfe x`0`0ϑismeasurablewithrespSecttothecompletedtailn9- eldof(H̹nP) n2N.sThrusthereisaconstantxfeRsatisfyings(1)Xfe x3030I>=URxURfeRa.s.' 0:@8 v썠:-iӍ{-.2.FVor xedx2EcrhoSosen2NsucrhthatP(A)>0forA:=fX20RAn Pxg.sWiththenotation̹n:=URL(X20RAn)itfollorwsfrom(1)thatI,SR1cJ=vgP((Xfe x 0 0EURxURfeR )ycJ=vgCZ|zEC'P(Xfe x y nURxURfeR )̹nP(dyn9)󍍍c2vgCZ|zyI{xx P(Xfe x x nURxURfeR )̹nP(dyn9)+CZ 8zyI{0andtheequationL(Xfe x x 0)UR=L(Xfe x x nZ?)thisimpliess(2)Xfe xTxT0URxURfeRa.s.4 2forall[QpxUR2E:.3.8TVogether,(1)and(2)yieldzxzfeR=URXURfe xA0A0URXURfe xAxA0zURxURfeRa.s.4 2forall[QpxUR2E;sandtheassertionfollorwsbyapplyingFVubini.82 .Theprecedingresultssuggestthefollorwingterminology:s(1.7)De nition IfF2Nİ[E]isirrffeducible,theconstantsxfeR and&feR]ڍxin(1.5)sand35(1.6)arffecalled\lowerlimit"and\upperlimit"of,respectively..Norwacounterpartof(1.2)canbSederived:s(1.8)Prop`osition 35If352URNİ[E]isirrffeducible,35itslowerlimitxfeRisgivenbyI,d{xd{feRn~=URmax3|fxUR2E i:h(x)URxgwheneverLQxLQfeRV2E:sPrffoof.FVoreacrhxUR2Ewithh(x)URxiterationyieldsX2xRAn xa.s.rforalln2N,swhicrh'by(1.6)impliesxfeR#bx.^Ontheotherhand,wheneverxfeR#b2E,choSosesxUR>xfeR &inthecasexfeR<UR&feR]ڍxandxUR=xfeRinthecasexfeR=UR&feR]ڍx .~InbSothcasesthehittingstimes}T̸1V<URT̸2<:::of}[0;x]bry(X20RAn)̹n0arede nedalmostsurelyV,whereagainsbry(1.6)}x}feRC=ɀ1liminfΰٹn!1 X 0ڍn:*ɀ1liminf8㍒8k6!1 X 0ڍTi?;cmmi6k+18㍍*ɀ1liminf8㍒8k6!1 H̹Ti?k+1s(x):sSinceOT̹k#;kC2&N;arestoppingtimeswithrespSectto(H̹nP) n2N,thevXariablessH̹Ti?k+1s;kݠ2vN;(areagainindepSendenrtwithdistribution.s`ThusH̹Ti?k+1s(x)vxfeRsa.s.-}u,orequivXalenrtlyV,h(x)URxfeR7q,whichforx#xfeRS(ifnecessary)impliesh(xfeRR)URxfeRsbrythecontinuityofhUR2H[E].82 .AsVstatedfortheuppSerlimit,ifxfeR2UREmandthesupportN:ofis nite,@9 썠:-iӍ{-sxsfeR' clearlyisa xedpSoinrtofsomehUR2N@..TheiuppSerlimitofanirreducibledistribution+isalwraysiuniquelydeter-sminedbryitssuppSortN@;indeed:mI&feR]ڍxw=URsupfh̹nR:::h̸1(0)UR:n2Nandh̹i,2N@g:sA\correspSondinggresultforthelorwerglimitfailstohold;'infact,Aevrenthealter-snativrexfeR M2UREhorxfeR=2 EisnotaquestiononNalone,vbutwillleadtothebasicsdistinctionbSetrweenrecurrenceandtransienceof..PropSer&pconrvergenceoftheproScess(X̹nP)̹n0,ubevenifweakenedtoconver-sgenceinprobabilitryV,islimitedtoadegeneratecase: s(1.9)Prop`osition PIfP2Nİ[E]isirrffeducible,WthenPforarbitrffaryinitiallawsthe35followingassertionsarffeequivalent:s(a)xfeR(=URx=&feR]ڍxfor35someJx2E;s(b) #(X̹nP)̹n0cffonverges35inEinprffobability.d;s(c)uh(x)UR=xfor35some@xUR2E:sPrffoof.1.8Theimplication(a))(b)isimmediatefrom(1.5)and(1.6)..2.Assume norwX̹n !CXinprobabilitywithC=L(X)2M̸1(E) andslet dbSeaboundedmetricinducingthetopologyofE.3Theninrtegrationovers H[E]bryP oyieldsyXCZbZnd(XJg;h(X))dPdѰPCZZd(XJg;X̹nP)dPd8󍍍8+CZZd(X̹nP;h(X̹n))dPd8+CZZd(h(X̹nP);h(X))dPd:q΍sFVorN&nUR!1thethreesummandsontherighrt-handsidetendto0:the rstonesbSecauseoftheassumption;+thesecondone, becauseitequalsE(d(X̹nP;X̹n+1))sduetotheindepSendenceofX̹nJandH̹n+1;thethirdone,@becauseh(X̹nP)UR!h(X)sinprobabilitryduetothecontinuityofhUR2H[E].8Thereforemd(x;h(x))UR=0for% almostallI(x;h);shencebryapplyingFVubini h(x)UR=xfor%almostallmxUR2E:.3.Theimplication(c))(a)isaconsequenceof(1.2),4yielding&feR]ڍx URx,ands(1.8),yieldingxfeR LURx.82 ݿB10 썠:-iӍ{-s2.Recurrenceandtransience sBesides thepropSerconrvergence x ߟfeRG =ܟ&feR]ڍx 2Econsidered in(1.9)thereisanim-spropSerconrvergencexfeR =f̟&feR]ڍxB= z2E,}generalizingthealmostsuredivergenceofthesproScessC(X̹nP)̹n0lto1inthespecialcaseE=R̸+x.CThisisa rstmotivXationsforthefollorwingnotion,usedsimilarlyin[19]inrelatedcontext:s(2.1)De nition̛If̛2URNİ[E]isirrffeducible,!the̛distributionb(orthekernelsPor35theprffocess35(X̹nP)̹n0)iscffalleds(a)\rffecurrent"ifxfeR2URE,s(b)\trffansient" ifxfeRg=2WE. .TVonbSeginwiththesimplestexample,zadeterministicsystem(E;h)iseasilysseentobSerecurrenrtifandonlyifE i=UR[0;&feR]ڍxP]with&feR]ڍx beingthemaximrumofthesincreasingTsequence(h2nP(0))̹n0.ThruschoSosingE i=UR[0;1[andh(x)=(xw+1)=2sprorvidesƩanexamplefortransienceduetoxfeR M=UR1=&feR]ڍx .׋ThisƩappSearsonlylogical,sobserving*thattheclassi cationin(2.1)isclearlycompatiblewithconjugacysand?(E;h)isconjugatetothedeterministicproScess(E20P;h209)withE20 =R̸+sandh209(x20)i=x20+1.\Indeed,anappropriateorder-preservinghomeomorphismsgË:URE20ע!Eisgivrenbygn9(x209)UR=122x-:q% cmsy60&..Bylde nitionanirreducibledistributionil2Nİ[E]isrecurrenrtwhenevers&feR]ڍx&2dE.This isaspSecialcaseofamoregeneralsucienrtconditionthatoftensapplies:s(2.2)Prop`osition 35If352URNİ[E]isirrffeducible35andsatis esl(sup wuGƹx2Eh(x)UR2E)>0; Nsthen35isrffecurrent.sPrffoof.First,fthesubsetofH[E]oSccurringintheconditionisapparenrtlyofstrypSeJF̹ andthusinparticularmeasurable.XTVoprovethisconditiontoimplysrecurrence,crhoSoseʹtҵ2E~with(supx2E)|h(x)t)>0.ThenforarbitrarysinitiallarwbyindepSendenceF-P(X̹nURtin nitelyoftenP)P(limsupUTn!1&bfsup wuGƹx2EH̹nP(x)tg)=1 NsandthrusxfeR LURt2E.82.AsatrivialexampleconsideranindepSendenrtsystem(E;),m_wherebysde nitionsupN(x2E0h(x)9z2EQforalmostallh9z2H[E]. QhTVoseethatthesconditionin(2.2)isfarfrombSeingnecessaryV,considerthequeuingprocessmX̹n=UR(X̹n1/t+U̹nP) +xfor-n2N;swhere8thei.i.d.vXariablesU̹nP;nUR2N;8arearbitraryV.WithstatespaceE i=URR̸+ݿB11 ٠썠:-iӍ{-stheȧcorrespSondingdistributionniscarriedbrythemappingshUR:x!(xe3+u)2+x;suY2R;andthrusirreduciblewheneverP(U̹n 8>Y0)>0.)Inthiscasethecon-sditionin(2.2)isnotsatis ed,whileitiswrell-knownthatX̹n!UR1a.s.gifandsonlyiftherandomwralkwithincrementsU̹n doSesnotdivergeto+1.h鍑.InIthecaseoftransiencetheproScess(X̹nP)̹n0divrergesexponenrtiallyfastinsthefollorwingsense: s(2.3)Prop`osition 35If352URNİ[E]istrffansient,therandomcardinality`Z1:=URjfn0:X̹ntgjsfor35arbitrffaryinitiallawandeverytUR2ELsatis esIE(exp(uZܞ))UR<1for35some@u>0:sPrffoof.De nerecursivrelyWT̸0V:=UR0and0,T̹kx:=infFfn>T̹k61U`:X̹ntg(1):sThenbrytransienceZq0UR=P 0(X̹ntin nitelyoftenP)=Elim8㍓k6!1P 0(T̹kx<1);䍑shencegthereexistsl2URNsucrhthat#:=P20(T̹lw<1)<1. PWiththedecreasingsfunctionngn9(x)UR:=P xH(T̹lw<1)for%x2EstheMarkrovpropSertyimplies-^wP 0(T߸(k6+1)l<UR1)J=BCZzfTi?kl<1gTgn9(X̹Ti?kl r)dP 0󍍍bBCZzfTi?kl<1gTgn9(0)dP 0J=B#P 0(T̹k6l <UR1)for%ko0:sThisyieldsthebSound͘P 0(T̹k6l <UR1)# ksandthrusbymonotonicityYP(Z1URkglC)9P 0(T̹k6l <UR1)9# k#for+koUR0:sPrartialintegrationshowsthateachuUR<Fu33133콉fe@'lhlogӱ#satis estheassertion.82 .It:isafundamenrtalconsequenceof(2.3)thatrecurrenceandtransiencecansbSeP%crharacterizedbyapplyingthepSotentialkernelG:=CPIn0$%PƟ2n ;toP%intervXalss[0;t]withtherighrtendpSoints:ݿB12 5썠:-iӍ{-s(2.4)TheoremiIfi2URNİ[E]isirrffeducible,theifollowingdichotomyholdsforsarbitrffary35initiallaw:s(a)if35isrffecurrent,35then CX Urn0Z:P(X̹nURt)=1for& t>xfeR ;! s(b)if35istrffansient,thenCX Urn0Z:P(X̹nURt)<1for& t<&feR]ڍx :sPrffoof.BothassertionsfollorwbytakingtheexpSectationofZ1=ԟCX URn0Z1߸[0;t]*(X̹nP);swhicrhincase(a)isalmostsurelyin niteandincase(b)isintegrablebys(2.3).82 .FVoranapplicationconsideranexcrhangeproScesswithU̹n=UR1,i.e. dX̹n=UR(X̹n1/t1)_V̹nPfor+n2N;swherethei.i.d.8vXariablesV̹nP;nUR2N;arenonnegativre.Withstatespace*)E i=URfx0:P(V̹nx)>0gsthe"correspSondingdistributioniscarriedbrythemappingshUR:x!(x1)_vn9;sv+2{E;Tandthrusirreducibleby(1.1). vMoreoveritisrecurrentinthecases&feR]ڍx&v<UR1.8Indeed,inthiscase gg|sup wugBx2E{$((x1)_vn9)UR=(&feR]ڍx W1)_vË2UREfor&v2Eysandthrus(2.2)applies.jAsalreadyindicatedintheintroSduction,recurrencesas!wrellastransiencecanoSccurinthecase&feR]ڍx=/P1.|JTVoexhibitappropriatesexamples,>denotebryFLthecommondistributionfunctionofV̹nP;nuo2N.qzThenstheexplicitrepresenrtationQ-X 0ڍn =UR(V̸1j(n1))_:::_(V̹n1/t1)_V̹nPfor+n2NsyieldsbryindepSendenceMP(X 0ڍn URt)= ǟCY 0mt>0;ݿB13썠:-iӍ{-si.e.8theproScessisrecurrenrt;s(2)ifthedensitryisreplacedbythefunctionfG(x)E=2x(xy+1)23 \|,=thensF(t)UR=(ō _t33Qmfe  t+1p) 2,6N,anditfollorwssimilarlythattheproScessistransient.{:.More}profoundcriteriaforrecurrenceandtransiencecanbSederivredbyslinearization,si.e.Ccomparing%theunderlyingmappingswithaneones.AssalreadymenrtionedintheintroSduction,,however,thistopicispSostponedto[17]. 퍑.Thissectionconcludeswithaconsequenceof(2.4)brywhichsomeproSofsscanbSesimpli ed: s(2.5)Prop`osition Ifǟ2k 2Nİ[E]isrffecurrentforonek{2N,rthisholdsforsall35ko2URN.fiMorffeover,theassociatedlimitsxfeR k7Nand&feR]ڍx ̹karffeindependentofkg.sPrffoof.ThecasexfeRp=v&feR]ڍxissettledbry(1.4),wbSecauseX̹n v!vXa.s.impliessX̹k6nw!+Xa.s.andthrusxfeR Uk=xfeR=&feR]ڍx=&feR]ڍx ̹k.uTVosettlethecasexfeR <&feR]ڍx ,Ditisssucienrtvdtoapply(2.4)withX̸0V=UR0,bSecauseitfollowsasintheproSofof(1.4)sthatthesequence(P(X20RAn URt))̹n0otdecreases.82s3.AfundamentalinequalitysThefollorwingresultwillplayakeyroleinderivingthemainresultsinSectionss4and6:s(3.1)Lemma If2URNİ[E]isirrffeducible,JthereexistsanincrffeasingfunctionscUR:E i!R̸+ Osuch35thata}rgCX q>n0CE(fG(X xڍn:j)f(X 0ڍn))URc(x)sup YOn0E(f(X 0ڍn)) !sfor35effachincreasingfunctionfQ:URE i!R̸+ Oand35everyxUR2E.sPrffoof.Firstofall,2>theleft-handsideiswrell-de ned,bSecausethedi erencessfG(X2xRAn:j)f(X20RAn)arenonnegativre.8Now, xxUR2EandcrhoSoseko2URNsuchthata}H 0A3:=URfh 0#2H[E]:h 09(0)xgssatis esthecondition&H n:=URǟ kY(H 0)=P(X 0ڍk x)>0:sWiththeabbreviationHr:=URH[E]thisyieldsthebSound- E(fG(X xڍn:j)f(X 0ڍn))ɽ=,CZfzHR[fG(h(x))f(h(0))]ǟ nj(dh)󍍍ɽ=,  1 CZҟzh0Ǻ2Hl0+DRCZ0Ҍzh2HC![::: ʠ]ǟ nj(dh)ǟ kY(dh 09):;,  1 CZҟzh0Ǻ2Hl0+DRCZ0Ҍzh2HC![fG(hh 09(0))f(h(0))]ǟ nj(dh)ǟ kY(dh 09),  1 CZҟzh0Ǻ2H(CZ. şzh2H@o[::: ʠ]ǟ nj(dh)ǟ kY(dh 09)9ɽ=,  1E(fG(X 0ڍn+kZ)f(X 0ڍn))for%nUR0;ݿB14썠:-iӍ{-swherethe rstinequalitryfollowsfromH܍rfG(hh 09(0))URf(h(x))for%h2H;h 0#2H 0sandthesecondonefromefG(hh 09(0))URf(h(0))for%h2H;h 0#2HPnH 0:sBy#summingup,cancellingontherighrt-handside,andomittingthetermsCP) 0lKmUR2N: ЍsAccordingtothecrhoiceofkQandthede nitionof ,therefore,&u c(x)UR:=( ^sup ,ˍ k62Nō" 1!FQmfe  k+qP(X 0ڍk x))ğ*1@for5x2E Csde nesanincreasingfunctionasdesired.82 .Thenextresultisanimmediateconsequence:s(3.2)Lemma If24#Nİ[E]isirrffeducible,ɜthenforarbitrffaryinitiallawandseffach35increasingfunctionfQ:URE i!R̸+ OthecffonditionH܍ sup YOᵹn0ǼlE(fG(X 0ڍn))UR<1 Ѝsimpliess(a)RڟCX Xn0"(fG(X̹nP)f(X 0ڍn))UR<1 a.s. ;s(b)GfG(X̹nP)f(X 0ڍn)UR!0 a.s. :sPrffoof.The*secondassertionfollorwsfromthe rstone,JwhichinturnfollowssineythespSecialcaseX̸0b=&^xfrom(3.1),-inrterchangingeytheorderofsummationsandinrtegration,andinthegeneralcasebyapplyingFVubini.82.ActuallyV,theprecedingresultsextendtolargerclassesoffunctions:s(3.3)Theorem 35If352URNİ[E]isirrffeducible,35thenforarbitrffaryinitiallawH܍s(a)eCX esn0y;jfG(X̹nP)f(X 0ڍn)jUR<1 a.s.+ forBfQ2V(E);s(b)s%fG(X̹nP)f(X 0ڍn)UR!0 a.s.+ forBfQ2R(E):ݿB15k썠:-iӍ{-sPrffoof.(a)A[functionufQ2URV(E)hasarepresenrtationf=URf̸1f̸2Bywithincreas-singandbSoundedfunctionsf̹i,:URE i!R̸+x,forwhicrh(3.2a)applies.Xp.(b)Theassertionfollorwsforstepfunctionsffrom(a)andforfunctionssfQ2URR(E)bryuniformapproximation.82 .By?inspSectingtheproofitiseasilynoticedthatinbothstatemenrtsthesconrvergenceisuniformoncompactsubsetsofE,i.e.8forinstancekCsup YO@ %0xt]sjfG(X xڍn:j)f(X 0ڍn)jUR!0 a.s.* for@fQ2R(E) and&,t2E: .Assertion!(b)cannotbSeextendedtofunctionsf_2`C5(E),/asisseeninthestransienrtucasealreadybythedeterministicsystem(E;h)withEQ=~R̸+ andsh(x)W=xxu+1,dwhicrhisobviouslyirreducible:kiffG(x)denotestheeuclideansdistancei ofxfromthesetf0;2;4;:::ʞg,theni jfG(X21RAn)f(X20RAn)jUR=1i forallnUR0..TVocaexhibitacounrterexampleintherecurrentcase,amoreelabSoratecon-sstructionisrequired.8TVothisendconsidertheautoregressivreproScessXX̹n=ō1Qmfe  2 X̹n1/t+V̹nPfor+nUR2N;swherethei.i.d. vXariablesV̹nP;n2N;attainthevalues0andFu(1(콉fe@'2TeacrhwithsprobabilitryFu1콉fe@'2 {.1WithmE i=UR[0;1[asstatespacethecorrespSondingdistributionsissuppSortedbrythetwomappingsh̸0 :[x!x=2andh̸1:[x!(x+1)=2."Itsisyclearlyirreducibleandbry(2.2)recurrent, nwherexfeR8w=0by(1.8).TDenotebys(t̹k#)̹k60astrictlyincreasingsequenceinEwithsup[k60*t̹k=u1,6tobSespeci edslater,andde ne{W"T̹kx:=URinfHfnUR0:X 0ڍn t̹k#gfor%ko0:sSince"theserandomtimesarealmostsurely nite,Jthereexistn̹kx2URNsucrhthats(1)*limsup 7k6!1P(T̹kxURn̹k#)=1:yEsMoreorver,sincethesuppSortN!ofis nite,thereexist nitesetsB̹kxUR[t̹k#;1[ssucrhthats(2)|5P(T̹kxURn̹k#;X 0ڍTi?kġ= _I2B̹k)=0for%ko0:sFinallyV,5sinceəthemappingsh̹i.slearveəthesetD'ofdyradicnumbSersinE}anditsscomplemenrtsinvXariant,efor xedxW2EnDthereexist nitesetsC̹k {B[t̹k#;1[ssucrhthats(3)}P(T̹kxURn̹k#;X xڍTi?kġ= _I2C̹k)=0for%ko0;ssatisfying4inadditionB̹kΩ\C̹kx=UR;.dNorw,"Kstartingwitht̸0V=0,"KcrhoSosethelevelsst̹k 7recursivrelyundertheconstraintst̹k|>bmaxAC̹k61fork2bN.PThenthe nitessetsB̹k *gandC̹karedisjoinrtsubsetsofthesuccessiveintervXals[t̹k#;t̹k6+1[;kdG0;swith?unionE.ؽThrusthereexistsafunctionfQ2URC5(E)with0URfQ1,satisfyingݿB16à썠:-iӍ{-sfG(yn9)P=0fory2PCS k60! B̹k =9andf(yn9)P=1fory2PCS k60! C̹k#.By(2)and(3)thissimplies_limsup 7fGk6!1fT̹kxURn̹k#gflimsupUTn!1(`jfG(X xڍn:j)f(X 0ڍn)jUR=1gbhsalmostsurelyV,whicrhby(1)yields 1limsupUT3n!1 jfG(X xڍn:j)f(X 0ڍn)jUR=1 a.s. ;ōsprorvidingthedesiredcounterexample..As8acorollaryassertion(b)of(3.3)yieldsaresultonfunctionsf67thataresregularnwithrespSecttoP,i.e.satisfy0URfQ=PfG:inntheirreduciblecasesucrhsa$function,2prorvideditiscontainedinR(E),2hastobSeconstant.fIndeed,2thesequationPƟ2nJfQ=URf2forn0andthebSoundednessoff2comrbineby(3.3b)to|2fG(x)f(0)Ý=/E(fG(X xڍn:j))E(f(X 0ڍn))Ý=/E(fG(X xڍn:j)f(X 0ڍn))/!/0forall6E>xUR2E:sTVomseethatoutsideR(E)theremaryexistregularfunctionsthatarebSoundedsand%6notconstanrt,LconsidertheautoregressiveproScessfromaborve.Since%6inthisscasetherequiremenrtfQ=URPf2amountstotheequationgv|fG(x)UR=ō1Qmfe  2 f`Cf(ō x QmfeR  f2{))+f(ō x+1 Qmfe&  2#FO)'f`C0:sPrffoof.First,measurabilitry_isensured,bSecauseitsucestoextendthein mrumsorver/3acounrtabledensesubsetof[0;t],TcontainingthecountablymanypSointsofsdisconrtinuityӑoffG.1.AӋrepresenrtationfQ=URf̸1;{f̸2withincreasingandbSoundedsfunctionsf̹i,:URE i!R̸+  yieldstheestimatee/E(inf0xtSfG(X xڍn:j))URE(f̸1(X 0ڍn))E(f̸2(X tڍnP))UR=:̹nN:ݿB17썠:-iӍ{-sNorwtheidentity`Cu̹n=URE(fG(X 0ڍn))(E(f̸2(X tڍnP))E(f̸2(X 0ڍn)))sensuresptheexistenceofsomemUR2Npwith̹m Z>UR0,bSecauseotherwisebry(3.1)_rCX ^:n0r?E(fG(X 0ڍn))URԟCX n0Z(E(f̸2(X tڍnP))E(f̸2(X 0ڍn)))UR<1; 獑sconrtradictingtheassumptiononfG.82 s4.Existenceanduniquenessofinv@ariantmeasuressTVo֨dealwithnotnecessarily niteinrvXariant֨measures,ڨthefollorwing\loScaliza-stion"isessenrtial: s(4.1)De nition 34Lffet34&2/_Nİ[E]berecurrentand xt/_2EKwith34t>xfeRorstUR=&feR]ڍx .fiThen:s(a)2tPdenotes35the\hittingkernel"bffelongingtoand[0;t],i.e.`CE tG+P(x;B)UR:=P xH(X̹T i2B)for& x2[0;t] and&BX2B]m([0;t]);swherffeT:=URinfHfnUR2N:X̹n2[0;t]g;As(b)forqarbitrffaryinitiallaw(2tX̹nP)̹n0-=denotesthe\embeddedprocess"belongingsto35(X̹nP)̹n0and[0;t],i.e.`Cӟ tX̹n:=URX̹Tnfor1Qn0;swherffe35T̸0V<URT̸1<:::S'arffe35therandomtimeswhen(X̹nP)̹n0isin[0;t]..TVo3includethecasete8=&feR]ڍxw2E,*Vwhere32tHP=PXand3noloScalizationissnecessaryV,Mis9[conrvenientforauni edtreatment.$TVoassumeT̹n <G1in(b)issnorealrestriction. .FVor>4easyreferencetherequiredfactsfromprobabilisticpSotenrtialtheoryaresstatedexplicitly:s(4.2)Lemma !Lffet!2MQNİ[E]berecurrentandMQ2M(E)!beexcessivewithsrffespect35toP.fiIffj2t 6denotestherffestrictionofto[0;t],thens(a)2t35isinvariantwithrffespect35to2tPfortUR2ELwith35t>xfeR5ort=&feR]ڍx ,s(b)35isinvariantwithrffespect35toP. sPrffoof.(a)IfI̹A ȌforAUR2B]m(E)denotesthekrernel`C4I̹A(x;)UR:=1̹A(x)"̹xHfor+bSecause2tisa nitemeasuresandꨟ2t|tPnisastoScrhastickernel.'.(b)FVor}0URfQ2K,`(E)crhoSoset2E1witht>xfeR (ort=&feR]ڍx (andsuppfQ[0;t]sand+denotetherestrictionoff*to[0;t]bry2t9fG."Then(a)and()togetherimplygfQ=UR t tf=UR( t tP) tfUR(P)fG:sByvXaryingf2thisyieldstheinequalitryURPnrequiredfor(b).82 .Norw:theexistenceofaninvXariantmeasurefor(i.e.forP)canbSees-stablishedasin[10]$, undersomesimpli cationduetothemonotonicitryV.MoresgenerallyV,thefollorwingversionwillbSeneededinSection6:s(4.3)Prop`osition Lffet.2PgNİ[E]berecurrentand xtPg2Enwitht>xfeRorstUR=&feR]ڍx .fiThen35forarbitrffaryinitiallawthemeasuresg8l%̹nP(B)UR:= /rCX 0m0,swhicrhispSossibleduetotheassumptionont.),With̹m Z:=URL(X̹m)thisimpliessbrymonotonicity4l7P(X̹m+lVURt)DsCZ՟zxsnP(X xڍl URt)̹m(dx)DDsCZ՟zxsnP(X sڍlURt)̹m(dx)ˍ7,=s#URP(X̹m Zs)for%m0:gsWiththenormingconstanrts r̹n:= /rCX UR0m=VfGfor';0URfQ2K,`(E);sbSecauseUr̹ni?kI! m1and̸0PƟ2mf isboundedbrymax36fG.xThereforeisexcessivesandthrusinvXariantby(4.2b).82 .The߲follorwingpropSertyofthesuppSortofinvXariantmeasureswillbSerequiredsbSeforeitsthoroughstudyinSection7:s(4.4)Theorem Lffet32qNİ[E]berecurrentandq2M(E)beanontrivialsinvariant35meffasurefor.fiThenM6:=URsuppsatis esinfM6=URxURfeRandxfeRgort=&feR]ڍx ".sFor35fQ2URV(E)withsuppfUR[0;t]andgË=1߸[0;t]_de neVQ xڍnP(f;gn9)UR:= /rCX 0mgn9(X xڍm)f`C.X  o0m usingtheab-sbreviation[::: ʠ]UR:=H̹m l:::H̸2(H̸1(0));sthelorwerlimitequalsPSfliminfX^n!1wbQ 0ڍnP(f;gn9)UR=liminfn!1/CX '1m0andthrusisrecurrentby(2.4b).gOnstheotherhand,itisnothardtocrheckthat'G 0#:= CX "%URx2Dx"̹xHwith0!D:=URE^\Q"sde nesanotherinrvXariantmeasure,whicrh,however,isn9- niteonlyV..FinallyV,=)ithastobSemenrtionedthatasindiscreteMarkovchaintheorysneithertexistencenoruniquenessofnonrtrivialloScally niteinvXariantmeasuresscarryorvertothetransientcase.s5.TheregenerativecasesClearlyV,aproScess(X̹nP)̹n0otransienrtaccordingto(2.1)cannotberecurrenrtinsthe senseofHarris,duetoX̹n 4!>&feR]ڍx+&= 2R Ea.s..Thecounrterexampleattheendsoftheprecedingsectionimplies,&ontheotherhand,thataproScess(X̹nP)̹n0srecurrenrtaccordingto(2.1)neednotbSerecurrentintherestrictedsense,bSe-scauseYinthiscasethereisalwraysYanuptoaconstanrtfactoruniquen9- niteݿB23I썠:-iӍ{-sinrvXariant!measure(seee.g.CJ[23,26]).There!is,horwever,a!particularsituationsthatq tsinrtothisframeworkandwillbSeneededinthefollowingsection.,Thiss\regenerativre{*case"makesnouseofthetopSologicalstructureandarises,vifthesinrvXariantTmeasureconrtainsatoms.Thecrucialconsequenceofthisassumptionsisthefollorwingfact: s(5.1)Lemma Lffetv2Nİ[E]berecurrentwithinvariantmeasure.Thens(fzg)UR>035impliesforarbitrffaryinitiallaw-P(X̹n=URzin nitely35oftenP035impliesg2Y$(B)UR=(fzg)E zʮf`C|X "% Ð0n(X̹nP)f`Cfor,BX2B]m(E);"⣍swherffe35T̹zdenotesthehittingtimeofzby(X̹nP)̹n0.sPrffoof.1.TVoPestablish rsttheinrvXarianceofthemeasure20߉de nedbythesexpSectationA3ontherighrt-handside,VchoSosef0=1߸[0;t](]witht2E.0:sBytheMarkrovpropSertythisimplieshs 09([0;t])=7"E zʮf`C X  Ðn0 X1ߺfTzO>ng1߸[0;t]((X̹nP)f`C]T=CX 7"n0;P zʬ( tX̹kx6=URz5for0@Himplies35forarbitrffaryinitiallaw}KP(X̹n=URX 0ڍn eventually=t)=1:sPrffoof.Applicationof(3.3b)tofQ=UR1ߺfzVg5^yields|'P(1ߺfzVg J(X̹nP)UR=1ߺfzVg(X 0ڍn)evrentually98)=1:sComrbinedwith(5.1)thisimpliesthattherandomtimehT:=URinfHfnUR2N:X̹n=X 0ڍn =zgsisalmostsurely nite,whereclearlyX̹n=URX20RAn /fornT.82 .A rstconsequenceofthisresultconcernsthetailevrents:s(5.4)Prop`osition Lffet2URNİ[E]berecurrentwithinvariantmeasureands(fzg)UR>0TU.)Then}forarbitrffaryinitiallawthetailn9- eldof(X̹nP)̹n0istrivial.sPrffoof.LetBbSeaBorelsubsetofCQln0!E,Anotdependingonanry nitenumbSersofcoSordinates.Thenanapplicationof(5.3)ton2NasstartingtimeshorwssthatthetrwoeventsDEAUR:=f(X̸0;X̸1;:::ʞ)UR2Bg=f(X̸0;:::ʜ;X̹nP;H̹n+1(X̹n);:::ʞ)UR2BgݿB25e썠:-iӍ{-sandȮA 0#:=URf(0;:::ʜ;0;H̹n+1(0);:::ʞ)2BgsagreealmostsurelyV,henceAisconrtainedinthecompletedtailn9- eldofs(H̹nP) n2Niaswrellandthushasprobability0or1.82 .Anotherconsequenceof(3.4)isthefollorwingapSeriodicity:s(5.5)Prop`osition Lffet2URNİ[E]berecurrentwithinvariantmeasureands(fzg)UR>0TU.fiThen35therffeexistsn̸0V2URNsuchthat"P(X zڍn =URz)>0for& nn̸0:sPrffoof.AsintheproSofof(5.2)crhoosemUR2Nsatisfying#UR:= inf0xz!P(X xڍm Z=z)>0:]LsWith̸1V:=URL(X2zRA11)itfollorwsthatywdP(X zڍm+16=URz)Ƥ4CZ߈nzxzyP(X zڍm Z=URz)̸1(dx)ˍƤ4#URP(X zڍ1 z)UR>0;sbSecauseP(X2zRA1 _z)=0bryiterationwouldleadtoX2zRAn >_zjforalln2NsalmostSsurelyV,m7conrtradicting(5.1).r8FromP(X2zRAn K=z)>0Sforn=m;m+1Sitsfollorws,combiningkQpSeriodsoflengthmandl.7periodsoflengthm+1,thatvP(X zڍn =URz)>0for%n=kgm+lC(m+1);shencen̸0V=UR(m1)msatis estheassertion.82 .This֪resultimpliesinparticularthatz_isa xedpSoinrtofsomehinthessemigroupogeneratedbrythesuppSortNof.Whilethispropertryisclearlynotssucienrtfor(fzg)UR>0,thefollowingcriterionholds:s(5.6)Prop`osition !Lffet!2 Nİ[E]berecurrentwithinvariantmeasure.sThen35fortUR2ELwith35t>xfeR5ort=&feR]ڍx5thefollowingassertionsarffeequivalent:s(a) (fzg)UR>0;s(b)cYP(X xڍn =URz5for]0xt)>0for*]someG7n2N:sPrffoof.1.If(a)issatis ed,[comrbinationof(5.1)and(5.3)providesnL2NssucrhthatP(X 0ڍn =URz5=X tڍnP)>0;swhicrhbymonotonicityagreeswiththeprobabilityin(b).ݿB26r<썠:-iӍ{-.2.8If(b)issatis ed,thentheestimateypu(fzg)F=NCZzE?P(X xڍn =URz)(dx)z-NP(X xڍn =URz5for0,bSecause([0;t])>0bry(4.4).82 .It8isaconsequenceofthisequivXalencethat&feR]ڍxsg2Eimplies(f&feR]ڍxRg)>0.sIndeed,Din%thiscaseDisrecurrenrtandbyde nitionthereexistsnX2N%suchsthat9P(X20RAn J=ß&feR]ڍx )>0,LhenceduetoX20RAn JX2xRAn -&feR]ڍx#forallx2E%condition(b)sissatis ed..FVorthebSoundaryvXaluesthecriterion(5.6)simpli es:s(5.7)Prop`osition Lffet2URNİ[E]berecurrentwithinvariantmeasureandsassume35x35feR5<UR&feR]ڍx .fiThens(a) 3:(fxfeRRg)UR>0 if35andonlyif(h(x)UR=xfeR )>0URfor35some87x>xfeR5,s(b) 3:(f&feR]ڍxRg)UR>0 if35andonlyif(h(x)UR=&feR]ڍx )>0URfor35some87x<&feR]ڍx5. sPrffoof.(a)Thefunctionkgn9(x)UR:=(h(x)=xfeR )for%x2EsdecreasesoforxURxfeR ,zbSecauseoh(xfeRR)URxߟfeRPbry(1.8),zandsatis esgn9(xfeRR)<1,zbSecausesotherwisexꨟfeR L=UR&feR]ڍxbry(1.9).8Moreover,by(4.4)yQ(fxfeRRg)UR=CZ㌟zxxfe玎gn9(x)(dx): ssThrusY5(fxfeRRg)~>0impliesgn9(x)>0forsomex>xfeR ,twhile(fxfeRRg)=0impliessgn9(x)UR=0foralmostallxUR>xfeR ,henceforallxUR>xfeR Lbry(4.4)..(b)Undereacrhcondition&feR]ڍxbSelongstoE,c,because(h(x)V=&feR]ڍx W)>0forssomepxUR<&feR]ڍx scimplies&feR]ڍx2URE$bry(1.1).=ThereforewithobviousmoSdi cationsstheproSofof(a)carriesorverto(b).82.A3trypicalsexamplefor(a)isthequeuingproScess,"~whereincaseofrecurrencesalwraysxꨟfeR L=UR0and(fxfeRRg)>0.s6.Ratioergo`dictheoremssTVoC.derivreergoSdictheoremsthatarenotrestrictedtocontinuousfunctionsofstheproScess(X̹nP)̹n0,somepreparationsarenecessary:s(6.1)Lemma IfX2Nİ[E]isrffecurrent, WthenforxfeR%90TU.fiThen35forxfeR5<URt2ELand35arbitrffaryinitiallawQUYCX M490mkg)for%ko0:sThendecompSositionaccordingtothelaststaryinzsyields㩟CX  0mp̹k=kw=sCX 8獒 0k6xfeRport=&feR]ڍx .'Moreorver,F(ftg)=0marybSesupposedsunless`t#=&feR]ڍx @u.iFinallyV,theassumptionX20RA0 S'=X̸0 edmeansnorealrestrictioninsviewof(6.1).܍.2.Norwconsiderthemeasures%̹nyede nedin(4.3).If%̹ni?k !Rv%UR2M(E)isanrysconrvergentЕsubsequence, (4.3b)andtheuniquenessoftheinrvXariantЕmeasuresimply%UR= .8SincegXisalmostconrtinuous,theconstant satis esPQ( ([0;t])UR=Elim8㍓k6!1%̹ni?k ?j([0;t])=1;荑shenceisindepSendenrtofthesubsequence,andthusby(4.3b){I~%̹nPfQ!UR%f=fG=gn9forall6wf2K,`(E):.3.@FVora;bUR2ENwithab,approrximatingbyK,`(E)frombSelowandabSove,spart2oftheproSofyieldss(1)liminf휹n!1F%̹nP(]a;b[)UR%(]a;b[);ݿB29נ썠:-iӍ{-s(2)limsupUT휹n!1%̹nP([a;b])UR%([a;b]):䍑sIn"sparticular,0f(2)impliesthattheconrvergence"s%̹nP(fzg)I!%(fzg),0festablishedsinն(6.2)forthecase(fzg)i>0,yextendsնtothecase(fzg)i=0. TVogetherswith(1)thisleadstoL -liminfQPչn!1p%̹nP([a;b])= liminfn!1X(%̹nP(]a;b[)+%̹n(fag)+%̹n(fbg))= liminfn!1X%̹nP(]a;b[)+%(fag)+%(fbg)y %(]a;b[)+%(fag)+%(fbg);shencetos(3)liminf휹n!1F%̹nP([a;b])UR%([a;b]):.4.8Comrbined,equations(2)and(3)yieldh'.%̹nPfQ!UR%f=fG=gn9forall6wf=1߸[a;b]Y2R(E):sThereforetheassertionholdsforallstepfunctionsfwithcompactsuppSort.sThrus,approximatingafunctionfd2R(E)withcompactsuppSortbrysuchsfunctionsfrombSelorwandaborve,theproofiscompleted.82 .As=a rstapplicationofthisergoSdictheoremconsidertheexcrhangeprocesss(X̹nP)̹n0studiedinSection2.Ifitisrecurrenrt(whichcanbSetestedby(2.4))sandtUR>xfeR L(whicrhcanbSedeterminedby(1.8)),thenthesequenceofratios\PP(X 0ڍn URs)=P(X 0ڍnURt)= ǟCY 0msUR2E:'_sAppliedinparticulartotherecurrenrtcase(1)withF(t)UR=ōtQmfe  t+1R,itturnsout{:sthatxꨟfeR L=UR0andissimplytheLebSesguemeasureonR̸+x.^.In 'statingthepSoinrtwise 'analogueofthemeanvrersion(6.3)morecarehasstobSetakrenof(X̸0;X20RA0): s(6.4)Theorem ALffetAtheassumptionsof(6.3)besatis edandinadditions(X̸0;X20RA0)Àand(H̹nP) n2NhAbffeindependent.JThenforfunctionsf;gο2`R(E)withscffompact35supportuBVCX mh60mxfeR tort=&feR]ڍx .Sinceinaself-explanatorysnotation;ō@S(f;x)@Qmfe#l  pS(gn9;y)z=ōS(f;x)Qmfe#l  .~S(gn9;x)ō/S(gn9;x)/Qmfe#  fS(gn9;0)ōXS(gn9;0)X|fQmfe"I  S(gn9;y)~:;sonlythefollorwingtwoassertionshavetobSeveri ed:s(1)uCX mu0m0:$]PsApplying(6.4)tof"=#1̹Ii?k andg%\=1߸[0;t]*,~wheret2Enwitht>xfeRort=&feR]ڍx du,sshorwsthatthisholdsindeedwithprobability1.ݿB31 Š썠:-iӍ{-.2.TVolYprorvetheinverseinclusion,denotebyL̹t(!n9)theanalogueofL(!)forstheproScess(2tX̹nP)̹n0.5Moreorver, vletDWconsistof&feR]ڍxinthecase&feR]ڍx2 2E1andofascounrtablesubsetofftUR2E i:t>xfeR gwithsupPD=UR&feR]ڍx Lotherwise.8ThenclearlyaPsL(!n9)UR='C[ "%t2DGlL̹t(!);!shenceitsucestovrerifythatwithprobability1uL̹t(!n9)URsuppfor%t2DS:sTVoithisendletX̸0 ) rstbSedistributedaccordingto(thetrivialextensionof8)sthe!normalizedrestrictionofto[0;t].KThen(2tX̹nP)̹n0isstationarybry(4.2a)sandthrusaPs()P-P( tX̹n b=2RI̹k #evrentually>\R)UR=1 whenevrerCI̹k 8\suppf=;;sasDydesired.FSFinallyV,Zanapplicationof(3.3b)tof62=31̹Ii?k shorwsthatthedistri-sbutionofX̸0infactisirrelevXanrtfor().82 .TVogether,(2.4)aband(6.5)implythatthetrwoabfamiliarcriteriaforrecur-srenceW/transiencefromdiscreteMarkrovchaintheorycarryovertothepresentssettinginthefollorwingform:s(1)Ifoisrecurrenrt,thenforxUR2suppalwaysaPP xH(X̹n2URGin nitelyoftenP)=1;shenceE xH(jfnUR0:X̹n2Ggj)=1;"sprorvidedGisanopSenneighbSorhoodofx.s(2)Ifoistransienrt,thenforxUR2EalwaysaP<7E xH(jfnUR0:X̹n2Kܞgj)<1;shencegP xH(X̹n2URKܞin nitelyoftenP2)=0;"sprorvidedKFisacompactsubsetofE.W .The nalresultofthissectionisrelatedto(5.5)and(5.6): s(6.6)Prop`osition !Lffet!2 Nİ[E]berecurrentwithinvariantmeasure.sTheniforeffachopensubsetGofEsatisfying(G)W>0iandeverytW2Etherffesexists35n̸0V2URNsuchthataPo>P(X xڍn 2URGUPfor7v0xt)>0for& nn̸0:sPrffoof.SinceHGmarybSeassumedtobeaboundedinrtervXal,_f==1̹G hinviewofs(6.3);2satis esallconditionsin(3.4).cAccordinglythereexistsmUR2N;2sucrhthatݿB32!DZ썠:-iӍ{-_5#UR:=P(X xڍm Z2GUPforz0xt)>0;DswheretheassumptiontUR>xfeR ort=&feR]ڍxmeansnolossofgeneralitryV.*ByapplyingsFVubiniitfollorwsthat'!P(X xڍlK+mV2URGUPforz0xt)*=ٙCZgӟzHl[Erظ]P(X h(x)ڍm%2URGUPforz0xt)ǟ l(dh)Bٙ#ǟ l(h(t)URt)for%l2N;sbSecause9xURtandh(t)timplyh(x)t.*Norwǟ2l(h(t)t)>0follorwsfroms(2.5)yand(1.8)inthecaset>xfeRandyistrivialinthecaset=&feR]ڍx Go.uSThereforesn̸0V=URmsatis estheassertion.82 s7.Prop`ertiesoftheattractorsWhileکinthetransienrtcase&feR]ڍxbattractstheproScess(X̹nP)̹n0,intherecurrentscase(6.5)suggeststhefollorwingterminology:s(7.1)De nition %>If%>2URNİ[E]isrffecurrent%>withinvariantmeffasure%>,( thesetsM6:=URsupp35iscffalled35the\attrffactor"of..Asa rstinformation(4.4)yieldsDwinfM6=URxURfeRand0:sThereforexhastobSeconrtainedinthesupportof.82 .It|isaconsequenceof(b)thattheattractorofarecurrenrtdistributionsdepSendsonitonlythroughitssupportN@.x.InsthefollorwingtwopropSositionsfe ( fbBoddenotestheclosureofasubsetBofE.sThenM andNarerelatedbryanequationthatisbasicinthecontextofself-ssimilarsets(seee.g.8[12,14])\":s(7.3)Prop`osition 35If352URNİ[E]isrffecurrent,35itsattrffactorMtsatis es܍9M4=UP͉fe0m 33C[ 8獹h2N)h[M@]7h:NߟsPrffoof.If Fdenotesthesetontherighrt-handside,6/theinclusionFURMIfollowssfrom82 .8ConrverselyV,theconrtinuityofhUR2H[E]yieldsJh[F]UR͉feD 33h[ #&C[ G Ch0Ǻ2N2h09[M@]]J3];#Gswhereforh20#2URN@,againbry(7.2),h209[M]URM.8ThereforeNߍh[F]URщfe錟 3/h[M@] 0Fforall7h2N@;sandtheinclusionM6URFnfollorws,oncemorefrom(7.2).82 .WhenevrerbSothMandNarecompact,;duetotheconrtinuityofthemappings(x;h)A!h(x),\Mthis,resultholdswithouttakingtheclosure. kTVoseethat,swithoutassumingMtobSecompact,}thismaryfailevenifNis nite,}considerstheautoregressivreproScess)X̹n=ō1Qmfe  3 X̹n1/t+V̹nPfor+nUR2N;{swherethei.i.d.,~vXariablesV̹nP;n 2N;attainthevalues0andFu2콉fe@'3sxwithproba-sbilitryFu#θ1#Ο콉fe@'2 .JWithEH=1[0;1[asstatespacethecorrespSondingdistributionbisssuppSortedbrythetwomappingsh̸1 :[x!x=3andh̸2:[x!(x+2)=3."Itissclearlyirreducibleandbry(2.2)recurrent.ƮInviewofsupʕM=1by(7.2)alsoFu1콉fe@'3&o=URsuph̸1[M@]UR2M,whileontheotherhandFu۸1۟콉fe@'3K= d2;h̸1[E][h̸2[E].x.Next, MIwillebSedescribedbrymeansofthesemigroupN@2 MgeneratedbyN@:s(7.4)Prop`osition 35If352URNİ[E]isrffecurrent,35itsattrffactorMtsatis esNߍxQrM6=URщfeQZ_ 3/fh(x):h2N@gcfor35everyIhxUR2M@:ݿB34#à썠:-iӍ{-sPrffoof.If Fdenotesthesetontherighrt-handside,6/theinclusionFURMIfollowssfrom(7.2)PY.8ConrverselyV,asintheproSofof(7.3),(h[F]ߟщfei 3/fhh09(x)UR:h0#2N@gߟщfe\C 3/fh090r(x)UR:h0902N@gp=Fforall7hUR2N@;sandtheinclusionM6URFnfollorwsagainfrom(7.2).82 .The8assumptionxxR2M@is8clearlyessenrtialfortheinclusionFxRM@,\whilestheproSofshorwsthattheinclusionM6URFnholdsforanyxUR2E..ThemostexplicitcrharacterizationofM+usestheclosureN@2+ofN@2:s(7.5)Theorem 35If352URNİ[E]isrffecurrent,35itsattrffactorMtsatis esxUR2M@if35andonlyifb'qj(x)2N@  @;swherffe35jisthecanonicalinjectionofELintoH[E].sPrffoof.1.kTVoprorvetheconditiontobSenecessaryV,?considerh̸0 =&j(x̸0)withsx̸0V2URM@.8Byde nitionofthetopSologyinH[E]ithastobeshorwnthat|fhUR2N@  V::h(x)2G̸0 Tfors~0xtg6=;sfor^arbitrarytUR2Eand^opSensetsG̸0VUREconrtainingx̸0. GSincex̸0V2URMimpliess(G̸0)UR>0forthesesets,(6.6)appliesandprorvidesnUR2NwithP(X xڍn 2URG̸0 Tfors~0xt)>0:sThrusthereexisth̸1;:::ʜ;h̹n2URN+suchthat+h̹nR:::h̸1(x)UR2G̸0for*0xt;sashadtobSeshorwn..2.$TVoprorvetheconverse,leth̹n 2lkN@2 convergetoh̸0 ,o=lkj(x̸0)andapplys(7.4)toxUR=xfeR W2M@,leadingto{bx̸0V=URh̸0(xfeRR)=@limn!1}h̹nP(xfeR)2M@: 2.Itisaconsequenceofthisresult,gthatthe xedpSoinrtsofthemappingsshUR2N@2 aredenseinM@.8Indeed,sincethisistrivialforxfeR L=UR&feR]ڍx ,assumeM@\]s;t[6=UP;with.|x.|feR8 <URs0and0,P(X yڍn>&fe(ߟ]ڍy ~1)>0:sThrusthesetYBX:=URfh̹nR:::h̸1(yn9):h̸1;:::ʜh̹n2N@gsisW asubsetofMbry(7.2)andcontainselementsx &fe(ߟ]ڍy 6.~MoreoversitRisconnected,bSecauseinH[E]compositionandevXaluationareconrtinuous.sTherefore[yfe(ߎ(;&fe(ߟ]ڍy(]URM@,ashadtobSeshorwn.82.TheconnectednessofNisbrynomeansnecessaryforthatofMasisseen,sforinstance,brytheautoregressiveproScessfollowing(3.3),wheretheinvXariantsmeasureistheuniformdistributionon[0,1[..TVoexhibitanonrtrivialexamplewithdisconnectedattractor,ЪchoSosesE i=UR[0;1[andletN+consistofthetrwomappingsde nedbryXV%`h̸1(x)UR=ōxQmfeR  f3_(xō2۟Qmfe  5 )and0,h̸2(x)UR=(x+ō2۟Qmfe  5 )^ōx+2۟Qmfe&  3"2;hssatisfying+ thesymmetryconditionh̸2(1x)=1h̸1(x). The+ correspSondingsdistributionzx0:sTVoderivreit,#consideranonemptyopSensubsetGof]s;t[.~Thencondition(a)sandtheassumption(h(t)UR0yieldN(H̹sn<)UR>0 for"H̹sÎ:=fh2H[E]:h(t)0.8Therefored9(G)R=CZzHl[Erظ]A(h(x)UR2G)(dh)󍍍90CZzHsm(h 1 \|[G])(dh)UR>0;:OsashadtobSeshorwn.eɍ.2.8Norwlettincondition(b)bSechosenminimal.8Thenby()(h(t)UR6yieldsojh 1 \|[fzg]jUR1for%almostallm,h2H[E]:sConsequenrtlyV,the{kernelPtransformsnonatomicmeasures%LA2M̸1(E)inrtosmeasuresofthesametrypSe,becauseNw%P(fzg)UR=CZ㌟zHl[Erظ]M%(h(x)=z)(dh);ݿB37&^썠:-iӍ{-swheretheinrtegrandvXanishesforalmostallhUR2H[E].^.2. MStartingwwithanrynonatomicinitiallaw̸0 }{onE#6=of0g,2+itfollowssfrompart1oftheproSofthat̹n Z:=JL(X̹nP)isnonatomicforalln2N.Ansapplication)of(6.3)tof=t1ߺfzVgtandanadmissiblefunctiongwithg6=0sunderX̸0V=URX20RA0yields ~l''(fzg)=gË=@limURn!1%WΟCX }0m1=kandopSensetsGwith%(G)W<1=lC,dThenceisanopensubsetofsH[E].8ThereforeH̸0itselfisoftrypSeF̹I{ .^.2.&mLetNnorwbSedecomposedinrtoitsabsolutelycontinuouspart̹canditsssingularpart̹sXwithrespSectto%.8ThentheequationӍgc̹c.yP(B)UR=CZ㌟zHq0̹c(h 1 \|[B])(dh)for%BX2B]m(E)ݿB38'e썠:-iӍ{-simplies̹c.yPUR%,hencetheequationf=̹c.yPLn+̹snxfeR Ithesconrvergence ̹k#̹kx!Rv?yieldsliminf8㍒ɹk6!1n ̹k#̹k([0;t[)UR([0;t[)>0;\shencexꨟfeR kURtforalmostallko2Nbry(4.4),ashadtobSeshown.t.2./TVoprorvesuciencyV,\choSosetMS2Ewitht>xfeR gort=&feR]ڍx gandassumes(ftg)=08unlesst=&feR]ڍx T."Then8(4.4)andtheinequalitry()yield([0;t])>0sand̹k#([0;t])UR>0foralmostallko2N.3Thereforeitmeansnolossofgeneralitrystoassumes(1)z̹k#([0;t])UR=([0;t])=1forall6E>ko2N:sFVromthisitwillbSederivredinpart3oftheproofthats(2)!sup ,ˍtk62N0s̹k#([0;s])UR<1forall6E>s2E; si.e.thetmeasures̹k#;k=2? N;areuniformlyloScally nite.ItfollorwsasinthesproSofof(4.3)thatf̹kx:URko2NgisasequenrtiallycompactsubsetofM(E).3Ifs(20RAk#) k62Nisanryconvergentsubsequence,-itslimitby(8.3)isexcessiveandthussbry(4.7)isoftheform .PSince1߸[0;t]ٲisalmostcontinuous,theconstant ssatis esbry(1)+ n=UR ([0;t])=Elim8㍓k6!1 0ڍk#([0;t])=1:sThrusthesequence(̹k#) k62N itselfsatis es̹kx!Rv?.t.3.SinceM(2)isimpliedbry(1)inthecasetT =&feR]ڍx [,inMthesequeltT >xfeRmaysbSezassumed.ThrusthereexistsnJ2Nzwithǟ2nj(h(s)J0.Sincez̹k nw!RwZ<simplies2ǹnRAk i!Rw%Tǟ2nj,therefore} hliminf8㍒|ok6!1 ǹnڍkj(h(s)UR0;ݿB40)6I썠:-iӍ{-sandthrusitisnorealrestrictiontoassume k#UR:=Yinfwk62NQ ǹnڍkj(h(s)0:3sSince̹k :isinrvXariantfor2ǹnRAk Taswrell,thisyieldsh̹k#([0;s])3Y# 1 \z̹k#([0;s]) ǹnڍkj(h(s)UR1h̸1(x)UR=ō1Qmfe  2 (x+1);h̸2(x)UR=x^ō1۟Qmfe  2and52h kڍ2#(x)=(1ō/1۟Qmfe  k )x^ō1۟Qmfe  2 :PsIf'U̹kJandassignmassFuZ1Z콉fe@'2 toh̸1;h2kRA2andh̸1;h̸2,NfrespSectivrelyV,then'Uclearly̹kx!Rw|.sMoreorver,̹k and] arerecurrenrtby(2.2),theirlowerlimits,however,aresxsfeR#!9k+=UR0andxfeR L=Fu1콉fe@'2 dbry(1.8)..FVoranapplicationof(8.4)let̸0SbSeconcenrtratedontheconstantmappingshUR=0andapprorximatearecurrentdistributionobyW;̹kx:=UR(1ō/1۟Qmfe  k )lm+ō/1ٟQmfe  k̸0for*n2N:sFVorX{thesamereasonsasabSorveX{thesedistributionsarerecurrenrtwithxfeR ͟k9=@0,shencehthecorrespSondinginrvXarianthmeasures,HCsuitablynormalized,satisfys̹k !RvT. Inthiscase̹k#(f0g)g>0bry(5.6)or(5.7),IPandthus̹k hasasrepresenrtationaccordingto(5.2)..FinallyV,thespSecialcase&feR]ڍx2bEt<&feR]ڍx ;$]Pshence([0;t])UR=0,implying&feR]ڍx 2ETandconrtradictingtransienceby(2.2).2.Next,theresultof(2.2)canbSestrengthened:s(9.3)Prop`osition 35If352URNİ[E]isirrffeducible35andsatis esl(sup wuGƹx2Eh(x)UR2E)>0; Nsthen35ispffositiverecurrent.sPrffoof.Let-bSetheinrvXariant-measureforthatisensuredbry(2.2)andchoSosestUR2Esatisfying #UR:=(sup wuGƹx2Eh(x)t)>0:sThentheinrvXarianceofimpliesq͍x5([0;t])UR=CZ㌟zE(h(x)t)(dx)#(E):΍sSincetheleft-handsideis nite,theassertionfollorws.82.That4thesucienrtconditioninthispropSositionfailstobenecessaryisseensbrybthequeuingproScess.Itiswell-knowntopSossessastationarydistributionsifandonlyiftheassoSciatedrandomwralkdivergesto1..Art'thispSointitcanbSeclari edwhatmeasuresUR2M(E)'ariseasinvXariantsmeasurepofsomerecurrenrtdistribution2#Nİ[E].7TheindepSendentsystems(E;)qshorwsthattheonlyconditiontobSesatis edinthecaseofameasuresUR2M̸1(E)isgivrenby|!(fxUR2E i:xtg)>0forall6E>t2E:sInthecaseofanin nitemeasure,7horwever,thisconditionfailstobSesucienrt,sbSecause lforanrullrecurrentdistributionalmostallhUR2H[E] lareunbSoundedsinEbry(9.3)andthus(7.7)applies.ݿB42+U썠:-iӍ{-.In thesequelsuggestedbryideasfromqueuingtheory(seee.g.<[21,922])sthej\dualproScess"(H̸1.n:::znH̹nP(0)) n2Nr+willbeinrvestigated./!Thoughjitfails,sin9bgeneral,\tobSeaMarkrov9bprocess,\itisofparticularinrterestfordistinguishingspSositivreandnullrecurrence..=TVothisendanimpropSerrandomvXariabletakingsits}vXaluesinthe(pSossibly)enlargedstatespacefe f fbE{=URE[Rzf&feR]ڍxRghastobeinrtro-sduced: s(9.4)De nition 35For35irrffeducible2URNİ[E]therandomvariableS aY:=^sup ,ˍURn2NzH̸1j:::H̹nP(0) sis35cffalledthe\duallimit"of..Often,YcanbSegivreninanexplicitform,forinstances(1)if(X̹nP)̹n0otisthequeuingproScess,thenSh?Y=URsup YO+An0(U̸1j+:::+U̹nP);!Ps(2)if(X̹nP)̹n0otisanexcrhangeproScesswithU̹n=UR1,thenLJY=^sup ,ˍURn2Nz(V̹nR(n1));!Ís(3)if(X̹nP)̹n0otisananerecursionasintheinrtroSduction,then Y=ΟCX {URn2NzU̸1:::U̹n1V̹nN:$,s.By0forsome?y|t2E: :sThentherandomtimeT:=URinfHfnUR>1:sup wux2EH̹nP(x)tgsisalmostsurely niteandsatis esYURH̸1j:::H̹T.:1}(t)2E a.s.:.2.8OtherwiseH̸1isalmostsurelyunrbSoundedinE,hence3f sup ,ˍn2N$H̸1(H̸2j:::H̹nP(0))UR2Eg=fsup 'эn>1H̸2:::H̹nP(0)2Eg a.s. :!sConrtinuing,}it`tfollowsthattheeventfYC2Egiscontainedinthecompletedstailn9- eldof(H̹nP) n2Niandthrushasprobability0or1.82 .Norw[theduallimitcanbSeshowntoplaythesameroleinsinglingoutspSositivre8recurrenceasthelowerlimitdoSesforrecurrence.VThe rstresultissrelatedtoa\principle"?in[20]:s(9.7)Theorem Theirrffeducibledistribution 2ZCNİ[E]ispositiverecurrentsif35andonlyif3:P(Y2URE)=1.fiIn35thiscffases(a)UR:=L(Yp) 35is35theuniquestationarydistribution,s(b)L(X̹nP)UR!Rw 35for35arbitrffaryinitiallaw. ݿB44-l썠:-iӍ{-sPrffoof.1.8IfoispSositivrerecurrentwithstationarydistribution,thenq͍x([0;t])=Z(liminf0n!1")CZ8zEP(X xڍn URt)(dx)󍍍Zliminfn!1䣟CZrݟzEcP(X 0ڍn URt)(dx)=Zliminfn!1P(Y p0ڍn URt)=ZP(YURt)ZP(Y2URE)forall6E>t2Esbry(9.5a).8ThustUR"&feR]ڍx L(ort=&feR]ڍx )impliesP(Y2E)=1..2.8IfconrverselyY2UREalmostsurelyV,thenYԘY=URH̸1(Yp 0j)a.s.% withJbYp 0:=sup 'э+An>1H̸2j:::H̹nP(0); uswhereӈH̸1 andYp20 >1areindepSendenrtandinadditionYoandYp20harveӈthesamesdistribution.Thrus^ =L(Yp)isastationarydistribution,zhenceispSositivresrecurrenrtby(9.2),and(a)isestablished..3.8FinallyV,if̸0V:=URL(X̸0)isarbitrary,thenyr5 E(fG(X̹nP))BB=CZbAzEҏE(fG(X xڍn:j))̸0(dx)󍍍BB=CZbAzEҏE(fG(Y pxڍnW))̸0(dx)z!E(fG(Yp))forall6E>fQ2URC5(E)sbry(9.5b),and(b)isestablished.82 .As%asimpleapplicationconsidertheexcrhangeproScessstudiedinSectionss2-and6.LnTherepresenrtationfollowing(9.4)impliesP(Y2`jE)=1-inthecases&feR]ڍx&v<UR1,bSecauseTthenthesupremrumisinfactamaximum."Inthecase&feR]ڍx =UR1scrhoSose anytsatisfyingF(t)UR>0fortheunderlyingdistributionfunction.ThenP(YURt)=)CY n0ZF(t+n)UR>0$]PsifPandonlyiftheseriesCPmn0#ǧ(1|F(t+n))Pconrverges.5ThereforetheproScesss(X̹nP)̹n06isjpSositivrerecurrentifandonlyifthevXariablesV̹nP;nX2N;jhaveas niteexpSectation(forextensionssee[13])..Instead?ofpresenrtingfurtherexamples(ascanbSefoundin[6,!11,20]),thesfractalcrharacterofstationarydistributionsandtheirsuppSortwillbrie ybesdiscussed.2TVothisendcrhoSoseE i=UR[0;1[andletthesupportof2URNİ[E]con-ssisttoftrwotinjective,wbutnotnecessarilycontractive,wmappingsh̸0;h̸1V2URH[E]ssatisfyingCh̸0(x)UR0andthrusxfeR L=0.8NorwconsiderthepSerturbeddistributionsres̹kx:=UR(1ō/1۟Qmfe  k )lm+ō/1ٟQmfe  k Ǻ0ڍk#for+ko2N;[swhere!P2Ǻ0RAk Disconcenrtratedontheconstantmappingh̹k=Zkg22'!.Then̹k DisagainspSositivre recurrentby(9.3),Pwithlowerlimit0by(1.8).?Thus(8.4)applies,si.e.the7DcorrespSondingstationarydistributions̹k Zandsatisfy ̹k#̹kJ!RvLforssuitable>constanrts ̹k#.IfP̹k 3refersto̹k,horwever,h(x)Lx?1>foralmostsallhUR2H[E]implies5P̹k#( sup ,ˍn2N$H̸1j:::H̹nP(0)URlC)8P̹k#(H̹m Z6=URh̹kxfor1mkg 2lC) -=(1ō/1۟Qmfe  k ) k6-:2a,lӾ!0forall6E>l2URN;shence̹kx!Rw"'։feHx 3onfe f fbE=URE^[f&feR]ڍxRgbry(9.7).].ThisimpropSerconrvergenceappearsagaininthefollorwingcriterion: s(9.8)Theorem Theirrffeducibledistribution~2ϷNİ[E]isnullrecurrentorstrffansient35ifandonlyifP(Y2URE)=0.fiIn35thiscaseforarbitraryinitiallawL(X̹nP)UR!Rw"'։feHxHon%∟fe f fbE2=URE^[f&feR]ڍxRg:sPrffoof.TheassertedequivXalenceisimmediatefrom(9.6)and(9.7).5Theas-ssertedconrvergencefollowsfrommlimsupUTtιn!1,P(X̹nURt)՟limsupUT޹n!1tUR2E: 2ݿB46/썠:-iӍ{-.The9resultsconcerningtheclassi cationcannorwbSesummarizedasfollows:s(1)ifoispSositivrerecurrent,then􍍍kBliminfpsHn!1LX 0ڍn 2UREa.s.%andL!limIϹn!1c+Y p0ڍn 2Ea.s.;s(2)ifoisnrullrecurrent,thenkBliminfpsHn!1LX 0ڍn 2UREa.s.%andL!limIϹn!1c+Y p0ڍn = 2Ea.s.;s(3)ifoistransienrt,thenkBliminfpsHn!1LX 0ڍn l1= 2\,Ea.s.%andL!limIϹn!1c+Y p0ڍn = 2Ea.s.:s10.Furtherergo`dictheorems sTheconrvergencein(9.7)isunnecessarilyrestrictedtofunctionsfQ2URC5(E):s(10.1)Theorem OLffetOΞ2 Nİ[E]bepositiverecurrentwithstationarydistri-sbution35.fiThenthecffonvergence[z̹nPfQ!URf Gwith*̹n:=L(X̹n)sholds35ineffachofthefollowingcases:s(a)fQ2URR(E),s(b)fQ:URE i!R̸+ Oincrffeasingandsupp7̸09compact. sPrffoof.(a)Sinceu]f\isbSounded, applicationof(3.3b)withinitialvXariablessX̸0V=URx̸0resp.8X̸0=xyieldsMn$̹nPffQ=URCZ㌟zECZLzE E(fG(X xq0ڍn e)f(X xڍn:j))̸0(dx̸0)(dx)UP!0:n.(b)Applicationof(a)tof^kQwithko!UR1leadsto􍍍liminfn!1I̹nPfQURfG;sestablishingthecasefQ=UR1.8Otherwise,(9.5a)and(9.7)implys()aE(fG(X 0ڍn))UR=E(f(Y p0ڍn\t))fQ<1forall6E>n0:sIftUR2Esatis essupp"h̸0V[0;t],therefore|7̹nPf_HE(fG(X tڍnP))_Hf+(E(fG(X tڍnP))E(f(X 0ڍn))):sInviewof()thefundamenrtalinequality(3.1)yieldslimsupUTtn!1Ҋ]̹nPfQURfG: 2ݿB470-썠:-iӍ{-.TVoy\seetheregularitryconditionincase(a)tobSeessential, considerthesautoregressivre proScessfollowing(3.3).ItispSositiverecurrentwiththeuni-sform distributionon[0,1[asstationarydistribution.EHere,m}theconrvergences̹nPfQ!URfTfails,{for0instance,ifX̸0V=URx̸02Dand0fQ=1̹D ]withDdenotingthessetofdyradicnumbSersinE..TheU$compactnessconditionincase(b)isessenrtialaswell. Indeed,s considersan*excrhangeproScesswithE*=uR̸+ GandE(V̹nP)<1,zprorved*tobSepositivresrecurrenrt["by(9.7).MAssumeinadditionqCR =EAxd<1foritsstationarydistri-sbution28aconditionthatcanbSecrhecked28toamounrttoCQn>xfe玎$(F(n))2n wq>!0sfortheunderlyingdistributionfunction.~ThenE(X̸0)}=1bryX̹n %c}X̹n1?]1simpliesE(X̹nP)UR=1foralln2N4x..Application@of(10.1)yieldsinparticularpSoinrtwise@convergenceoftheasso-sciateddistributionfunctions.6TVogetherwith(2.4)thisimpliesthatthefamiliarsclassi cationfromdiscreteMarkrovchaintheoryforxfeRt4<Mt2EЬcarriesorverinsthefollorwingform:s(1)pSositivrerecurrenth<, @limURn!1 }P(X̹nURt)UT>UR0,s(2)nrullrecurrentg, @limURn!1 }P(X̹nURt)=0and! CX  Ctn02H>P(X̹nt)=1,swhileinthetransienrtcase $CX  *n0/jP(X̹nURt)UR<UT1 *forallt2E.]P.TVozprorvealawoflargenumbSersnotrestrictedtofunctionsfE2C5(E),sergoSdicitrywillbeestablished rst.8Moregenerallythefollorwingholds: s(10.2)Theorem OLffetOΞ2 Nİ[E]bepositiverecurrentwithstationarydistri-sbution35.fiThentheprffocess35(X̹nP)̹n0withL(X̸0)UR=35ismixing.sPrffoof.1.~ExtendingH̹nP;nZ2N;letH̹nP;n2Z;bSeindependenrtvXariableswithsdistribution.8Thenbry(9.7)X 0ڍn:=!sup YOURmnH̹nR:::H̹m(0)UR2E a.s.!sand,moreorver,L(X20RAnP)UR=forn2Z.8TheconrtinuityofhUR2H[E]yields^X 0ڍn=URH̹nP(X 0ڍn1) a.s.* for@n2N:sSince0;shencffe35inparticularE20(TƟ2t3)UR<1.sPrffoof.ChoSose+l>2ïNsucrhthat#:=P(X20RAl u6t)>0.dThenmonotonicitryandsindepSendenceimplyfGcP 0(TƟ t>URkglC)FP(H߸(i+1)l*>:::H̹ilK+1(0)URxfeR .esThensthe35hittingtimeT̹toffxUR2E i:xtg35by(X̹nP)̹n0satis ess(a)E2xH(T̹t)UR<1for35allxUR2Ewheneveris35pffositiverecurrent,s(b)E2xH(T̹t)UR=1for35tx2Ewheneveris35nullrffecurrent. sPrffoof.(a)ClearlyV,the=proScess(X̹nP)̹n0 marybeassumedtobestationaryV.sMoreorver,Fythe͵assumptionx>t͵meansnolossofgeneralitryV,bSecausethesassertion%istrivialinthecasetUR=&feR]ڍx ." Since%P(X̸0Vt)>0bry(4.4),anynUR2NswithP(X20RAn URx)>0satis esfs()P(X̸0VURt;X̹nx)>0:sIfniscrhosenminimalwithrespSectto(),thenaeP(X̸0VURt;X̹m Zt;X̹nx)=0for%00Q8 fortheevrentO#AUR:=fX̸0Vt;X̸1>t;:::ʜ;X̹n1>t;X̹nxgfT̹t>ng:sWiththeincreasingfunctiongn9(y)UR:=E y (T̹t)for%yË2EݿB514ؠ썠:-iӍ{-stherecurrencetheorembryKacandtheMarkovpropSertyimply)'P(T̹t<UR1)=0HandeverystUR2ELthe35stoppingtime\|)T:=URinfHfnUR2N:X xڍn 2Gfor7z0xtgshas35a niteexpffectation.sPrffoof.1.8By(6.6)thereexistsnUR2NsucrhthatD`#UR:=P(X xڍn 2Gfor~0xt)>0:ݿB525썠:-iӍ{-sIfTƟ20 denotestheanalogousstoppingtimewithrespSecttoǟ20=ǟ2nj,Ʌthenap-sparenrtlyTURnTƟ20o,hencen=1marybSeassumedinviewof(2.5)..2.8Moreorver,tUR>xfeR Lort=&feR]ڍx LmeansnolossofgeneralitryV.8ThenEJS̸0V:=UR0and0,S̹k6+1U`:=infHfn>S̹kx:H̹nR:::H̹Si?k+1(t)tgsarealmostsurely nitestoppingtimeswithrespSectto(H̹nP) n2N,whicrhbys(11.2a)satisfys(1)v2E(S̹k:S̹k61)UR=E t(T̹t)UR<1for%ko2N:sFinallyV,theevrentsA̹kx:=URfH̹Si?k+1(x)2Gfor~0xtgsbrytheassumptionnUR=1satisfys(2)uP(A̹k#)UR=#>0for%ko0:.3.(gBy>constructionthevXariables1̹Aq0 ;:::ʜ;1̹Ai?k1;S̹k6+1EES̹kareindepSendenrtsforeacrhkoUR0.8Moreover,theestimateTUR1+1ɟCX 8獓k60ّCY 8獑*ik*O(11̹A8:i V)(S̹k6+1S̹k#)$8sholds,TbSecausetherighrt-handsidefor xed!f2F equalsS̹k#(!n9)R=+1,ifkGissthe rstindexwith! M2A̹k#,andisin nite,ifthereisnosucrhindex.IfforseacrhNkkthefactorwithiUR=kisNcancelled,thebSoundforTOisincreasedandthessummandsarecompSosedofindependenrtfactors.8By(1)and(2)thisyieldsE(T)UR1+# 1 \zE t(T̹t)UR<1: 2.IfQinparticular&feR]ڍxb2E,kthisresultimpliessupx2E-(E2xH(T̹G)<1,kwhereT̹GsdenotesH:the2hittingtimeofGbry(X̹nP)̹n0. Withthisnotationthefamiliarscriterion/forpSositivreW/nullKrecurrencebrymeanpassagetimescarriesoverfromsdiscreteMarkrovchaintheoryinthefollowingform: s(11.4)TheoremZLffetZ2URNİ[E]berecurrentwithattractorMand xxUR2M@.sThen35ispffositiverecurrentifandonlyifRE xH(T̹G)UR<1for35allopffensubsetsGofEcontainingxݱ:sPrffoof.According_to(11.3)onlythesuciencyoftheconditionhastobSees-stablished.TVothisendassumeEatobSenrullrecurrent.Then&feR]ڍx=2;E7by(9.3),shence'x<&feR]ڍxQandthrus(h(x)>x)>0bry(1.1).S\ThisimpliestheexistenceݿB536嬠썠:-iӍ{-sof,Itx2E`witht>xsucrhthat#:=(h(x)>t)>0.Withthenotations̸1V:=URL(X2xRA1:j)anapplicationof(11.2b)yieldsym[gE xH(T̹t)1=nj(1#)+CZ 8zyI{>t/(1+E y (T̹t))̸1(dyn9)G,njCZzyI{>t§E t(T̹t)̸1(dyn9) ɍ1=nj#URE t(T̹t)UR=1;si.e.8theconditionisviolatedforGUR=[0;t[3URx.82 sReferences""K`y 3 cmr10[1]5sAlpuim,).M.:AnextremalMark!oviansequence.J.Appl.Prob.'"V 3 cmbx1026,).219$!", 3 cmsy10232 5s(1989)"[2]5sBarnsleye,iM.,Elton,J.:RAnewclassofMark!ovproMcessesforimageencoding.5sAdv.Appl.Prob.20,f1432(1988)"[3]5sBarnsleye,;M.,Elton,J.,Hardin,D.:TRecurren!t!Titeratedfunctionsystems.Con-5sstr.Appro!x.5,f331(1989)"[4]5sBhattac!harya,@R.,Wea!ymire,E.:ԐStoMchastic!processeswithapplications.ONew5sYeork:Wileyf1990"[5]5sBrandt, A.,Ferank!en,P.,Lisek,B.:|StationarystoMc!hasticmodels.Chic!hester:5sWileyf1990"[6]5sChama!you,}J.,Letac,G.:ExplicitCstationarydistributionsforcompMositionsof5srandom;functionsandproMductsofrandommatrices.\J.Theor.Prob.4,3365s(1991)"[7]5sDubins,L.,Fereedman,D.:tIn!vdDariantԖprobabilitiesforcertainMark!ovԖproMcesses.5sAnn.Math.Stat.37,f837848(1966)"[8]5sElton,J.:gAnwergoMdictheoremforiteratedmaps.ErgodicTheoryDyn.Syst.5s7,f481488(1987)"[9]5sElton,c~J.:A m!ultiplicative zergoMdictheoremforLipsc!hitzmaps. 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