; TeX output 1999.04.22:1611썠:-iӍ-s7"Vff cmbx10ErgoudicBeha=viourofAneRecursionsIIsIn=v{ariantMeasuresGandErgoudicTheorems*s+XQ cmr12HansG.KellerersMathematiscrhesInstitutderUniversit atMSvunchensTheresienstrae39,D-8000MSvuncrhen2,GermanrysOctobSer30,1992s0N cmbx12Summary.ThispapSerisconcernedwiththediscrete{timeMarkrovprocesss(,g cmmi12X2cmmi8nP)̹nK cmsy8 |{Ycmr804solvingthestoScrhasticdi erenceequationwhX̹n = Y̹nX̹n1D+ Z̹n *aforsnUR-!", cmsy102N,where(Y̹nP;Z̹n) n2t : cmbx9NJisasequenceofi.i.d.WrandomvXariablesindepSendenrtsoftheinitialvXariableX̸0and,inaccordancewithmostapplications,thestatesspaceIisrestrictedtoR̸+x.VqInthisparttheemphasisisontherecurrenrtcase,swhereMexistenceanduniquenessofaninrvXariantMmeasureaswrellasmeanandspSoinrtwiseergodictheoremscanbeestablished.s1991!MathematicsSubjectClassi cation.8KPrimary47A35,60J05;^Secondarys54H20,60G30.*썠:-iӍ{-sIntro`ductionsThisRistheconrtinuationRofwrorkbSegunin:::Ɩandtobe nishedin:::D.tItsstudiesanerecursionsonR̸+x,i.e.8sequences(X̹nP)̹n0otde nedbry@sX̹n=URY̹nPX̹n1/t+Z̹nPfor.*n2N:sHere,$(Y̹nP;Z̹n) n2N3isQrasequenceofindepSendenrtidenticallydistributedR22RA+x{svXalueddrandomvariableswhicrhisindepSendentoftheinitialvXariableX̸0O%K0.sWithout$lossofgeneralitryV,thecommondistributionbof(X̹nP;Z̹n);nUR2N,willsbSeassumedtobelongtotheclassNXde nedinSection0..A.cenrtral/JresultofPartI.thenstatesthattheloweranduppSerlimitofs(X̹nP)̹n0Q0;swhicrhcanhardlybSesolved..Section6.In?6viewofthecomputationalproblemsitisimpSortanrttostudysthezinrvXariantmeasureatleastqualitativelyV.zUThe rstassertionconcernsits@1썠:-iӍ{-ssuppSortL=M@,kwhicrh,wheneverL=unbSounded,canL=beshorwntobeaninrtervXal(6.1).sMoreppreciselyV,"hereasinsomeofthefollorwingresults,P(Y̹n r=ɷ0)=0phasstobSeassumed.AItisnothardtoshorwtobeeitherabsolutelyconrtinuousorssingular`withrespSecttoLebesguemeasure(6.2).The rststatemenrtapplies,sunlessthejoinrtlawL(Y̹nP;Z̹n)issingular(6.3).mwItismoreinvolvedtoprovesthatA,Vapartfromatrivialexception,isnonatomic(6.4).<ThesectionclosesswithastabilitryresultthatmaybSeusedforapproximatingtheinvXariantmea-ssure(6.5)..SectionT7. $HerethemainergoSdictheoremsforratiosareestablished.sWhilethevrersionformeansfollowseasilyfromearlierresults(7.1),0forthespSoinrtwiseversionaloScalizationasinSection5isnecessary(7.4)._Itmakessessenrtial) useofthefactthatthehittingkernelfromSection5enjoysatleastssome'wreakerFVellerpropSertyV.]AsaconsequenceoftheergoSdictheoremsthessuppSortMZanditscomplemenrtcanbeidenrti edwiththeconservXativeanddis-ssipativre0partoftheproScess(7.5).cIfspecializedtotheexample(E)aborve,thissmeansJthatthesetoflimitpSoinrtsofthesequence(X̹nP)̹n0mequalstheintervXals[2;1]BalmostsurelyV.ǭRathergeneralresultsonirreducibilitryandapSeriodicitysconcludethesection(7.6).*s4.Excessiveandinv@ariantmeasuressBasicufortheergoSdictheoremsinSections5and7aresomepropertiesofsmeasures[ 2M(R̸+x)thatareexcessivreorinvXariantwithrespSectto.Heresthereferenceto,:suppressedingeneral,actuallyreferstothecorrespSondingskrernelP.eMoreover,=jitshouldbSenotedthatforthissectionitisirrelevXantswhethertheunderlyingdistributionisrecurrenrtortransient.0"TheresultssconcernmainlythesuppSortof,whereinthe rstassertionfe fbA denotesthesclosureofasubsetAofR̸+x:s(4.1)Prop`osition..@ cmti12With35themapping@shUR: URR̸+ R 2ڍ+ q3(x;yn9;z)UR!xy+z52R̸+sthe35suppffortMofameasureUR2M(R̸+x)35satis es:s(a)@wkM6URщfe3=ɟ 3/h[MN@]6; if{435isexcessivn9e;s(b)AM6=URщfe3=ɟ 3/h[MN@]6; if{435isinvn9arSiant:sPrffoof.The(measurePʗistheimageof under(theconrtinuous(mappingsh. NTherefore,vdueGMtoageneralresultfromtopSologicalmeasuretheoryV,thessuppSortofPnistheclosureoftheimageof@ssupp^( )UR=MNsunderh,whicrhclearlyproves(a)and(b): 3Tq lasy102@2ߠ썠:-iӍ{-.Equallysimpleisthefollorwingauxiliaryresult:s(4.2)Lemma.IfUR2M(R̸+x)isanontrivialexcffessivemeasurewithsupportsM@,35then:ōLzAQmfe  1y`!2URM@for*ͺ(yn9;z)2N@wRith35yË<1:sPrffoof.WithҲanarbitraryx̸0V2URM66=;Ҳand(yn9;z)asabSorve,}(4.1a)Ҳyieldsrecur-ssivrely@sx̹n:=URyn7x̹n1/t+z52M@for*In2N:sSinceM+isclosed,thisimpliesōLzAQmfe  1y`!=URlim ̹n!1.Oj(yn9 nx̸0j+yn9 n1z3+:::+z)UR=lim ̹n!1x̹n2M: 2捑.AnexcessivremeasureobviouslypreservesthispropSertyV,ifPisreplacedsbryrsomepSowerPƟ2nJ.?Thusitisclearthatfor(y̹m;z̹m)2NVwithry̹m <1thessuppSorteMvconrtainsnotonlythe xedpoinrtsz̹mߦ=(1y̹m)1%oftheassociatedsaneVYmapsg̹m Z:URx!y̹mx{+z̹m [butVYaswrellthoseofthecompSositiong̸1;{:::{g̹nsforanrynUR2N..Thenextresultiscloselyrelatedto(1.3):s(4.3)Theorem.Ifg,2M(R̸+x)isanontrivialinvariantmeffasurewithsup-spffort35M@,then,withthenotationof(1.3),H捍s(a)BwiinfR]M6=URinfHfō z33Qmfe  1y o::UR(yn9;z)2N̹c.yg;%xs(b)CsupV-M6=URsupfō z33Qmfe  1y o::UR(yn9;z)2N̹c.ygif{4N̹e=;;捑e=UR135otherwise.sPrffoof.1.If TinfML8andsupMaredenotedbrymfe G]and&fe G]ڍmRi,SrespSectivelyV,it Tissimmediatefrom(4.2)that@sm@sfe GNNURinf7f::: ʠgand<,&fe G]ڍmJURsupf:::g:sMoreorver,ꨟ&fe G]ڍm=UR1holdsinthecaseN̹e6=;.8Indeed,(4.1a)impliess(1)M@\]0;1[6=UR;;sand0UR6=x̸0V2M+comrbinedwith(y̸0;z̸0)UR2N̹e\yields,againbry(4.1a),@sM63URy n9nڍ0x̸0j+y n9n1ڍ0z̸0+:::+z̸0V!UR1:.2.8TVoprorvetheinverseinequalityformfe G1,abbreviate:@s n:=URinf7fō z33Qmfe  1y o::UR(yn9;z)2N̹c.yg;@3$נ썠:-iӍ{-swhicrhsofarmaybSein nite,andchoSoseȄ2UR[0; [.8Then@syn7+z5URs2forall9(yn9;z)2N;sbSecausethisinequalitryistrivialforyËUR1.8Therefore@sfxUR:yn9x+z5<URs2gfx:x<gforall9Z(yn9;z)2N;swhilebrytheinvXarianceofontheotherhand@sqCu cmex10CRG̹NR((fxUR:yn9x+z5<URs2g)d=(fx:x<g)=qCR ̹Nn(fx:x<g)d:sSinceisloScally nite,thisimplies@sfxUR:yn9x+z5<URs2gX[=,fx:x<gfor-almostalliH(yn9;z)2R 2ڍ+x:sIf]vXariesthrough[0; [,thisyields@s(yn9x+z)^ [n=uHx^ for-almostalljid(y;z)UR2R 2ڍ+x:sSincebSothsidesofthisequationareconrtinuousfunctionsofxand(yn9;z),theysagreethereforeonthesuppSortof ,hence@s(yn9x+z)^ n=URx^ for)x2M@and /(yn9;z)2N:sBycrhoSosing(y̸0;z̸0)UR2N+withz̸0V>0thrus@s(y̸0x+z̸0)^ n=URx^ forall:x2M:sThisLprorvesindeedtheinequalitymfe G| ,dbSecauseanyx̸02M.,\H[0; [wouldsleadtoy̸0x̸0j+z̸0V=URx̸0,hencey̸0<UR1,andthrustothecontradiction@sx̸0V=ōz̸0QmfeM  1y̸0(WUR :8.3.pTheYWcorrespSondingprooffor&fe G]ڍm l,vgundertheadditionalhrypothesisN̹e=UR;,sdi ersonlyatthebSeginning.8De nehere@s n:=URsupfō z33Qmfe  1y o::UR(yn9;z)2N̹c.yg;8swhicrhnowmaybSeassumedtobe nite,andcrhooseȄ2UR[ ;1[.8Then@syn7+z5URs2forall9(yn9;z)2N;sbSecauseNnN̹c!marycontain(yn9;z)UR=(1;0)onlyV.8Therefore@sfxUR:yn9x+z5URs2gfx:xgforall9Z(yn9;z)2N;sandtheproSofconrtinuesincompleteanalogytopart2: 2.While<itiseasilyseenthat(b)extendstoexcessivremeasures,ithisfailssfor(a).)NIndeed,Iif}istransienrt,theexcessivremeasure"̸0(CP 9̹n0PƟ2nJ)isloScallys nite=accordingto(2.2b)withinfuM6=UR0,`rwhilethein mrumontheright{handssidemarybSearbitrarilylarge.@41썠:-iӍ{-.Asimpleconsequenceof(4.3)isthefollorwingresult:s(4.4)Prop`osition.The 9suppffortM 8ofanontrivialinvariantmeasuresUR2M(R̸+x)35isanintervalwheneverNiscffonnected.sPrffoof.With0UR6=x̸0V2M+(see(1)intheproSofof(4.3))theset@sM̹n:=URfyn9 nx̸0j+yn9 n1z3+:::+z5:(yn9;z)2N@gsistheconrtinuousimageofaconnectedset,hences(1)]mfe G Gne;&fe G]ڍm G̹nc[URM̹n[mfe G Gn;&fe G]ڍm G̹nc]for)n2NswithappropriatebSoundsmfe G 즟n:and&fe G]ڍm 즟̹n.!Moreorver,baccordingto(4.1a)and(4.3)s(2)M̹nURM@for*In2N;s(3)mfe GGn(D!URinfFM@and>/&fe G]ڍmHv%̹nQs!URsupM;swhere4inthecaseN̹e=6=p;thecrhoicex̸0 06=0isessenrtial.,TVogether,(1){(3)sprorvetheassertion: 2.AswillbSeseenin(6.1), theconrverseof(4.4)failsinasurprisinglygeneralssense..They nalpartofthissectionconcernspropSertiesnotonlyofthesupportsbutofexcessivremeasuresthemselves.ThefollowingtechnicalresultwillbSesimpSortanrt:s(4.5)Prop`osition.Lffett(y̸0;z̸0)2R22RA+ Ywithy̸0<1bffegiven.+nThenforp>0sand35sUR>z̸0=(1y̸0)therffeexist niteconstants@s h=UR (p)and< n= (p;s)ssuch35thatunderthehypffothesis@s([0;y̸0[[0;z̸0[)URpseffach35excessivemeasureUR2M(R̸+x)satis es@s([0;t])UR ([0;s])t for/mts:sPrffoof.IfisexcessivrewithrespSectto,then@s([0;t])UR([0;y̸0[[0;z̸0[)kJ( )(f(x;yn9;z)UR:yx+z5URy̸0t+z̸0g)kJ([0;y̸0t+z̸0])forall9ZtUR0:sThisyieldstheestimate@s([0;t])p]p 1 \|([0;y̸0t+z̸0])p]p 2 \|([0;y̸0(y̸0t+z̸0)+z̸0]);@5?t썠:-iӍ{-shencebryiteration@s([0;t])p]p k ([0;y n9kڍ0t+y n9k61ڍ0nGz̸0j+:::+z̸0])H捍p]p k ([0;y n9kڍ0t+ōEz̸0۟QmfeM  1y̸0$[])for)ko2URN:捑sTherefore@s([0;t])URp k ([0;s])whenevrerYy n9kڍ0t+ōEz̸0۟QmfeM  1y̸0'kURs:sThisconditionholdsforko2URNsatisfying:@skoUR[(log-I(sōEz̸0۟QmfeM  1y̸0$[)logt)=log-Gy̸0]0;swithMtheconrventionMlogy̸0 3=/1Mfory̸0=/0.SincefortsthisbSoundissnonnegativre, nally@s([0;t])URp ([:j::]+1),^([0;s])for)ts;sandasimplecomputationprorvidestheconstants/J@s h:=URlogp=log-Gy̸0andA0 n:=ō^1Qmfe孟  pL (sōEz̸0۟QmfeM  1y̸0$[) : 2.TheZessenrtialcontentofthisresultliesinthefactthatthegrowthofansexcessivremeasureisonlypSolynomialwheneverP(Y<UR1)>0..ThelastresultofthissectionwillbSerequiredforstabilitrytheorems:s(4.6)Lemma.LffetN3̹k wg !Gpand̹k 2M(R̸+x)bffeexcessivewithrespectsto35̹k#.fiThen̹k +vg x!8UR2M(R̸+x)35impliesthatisexcffessivewithrespectto.sPrffoof.VVagueconrvergenceisimpliedbyweakconvergenceandcompatiblewithsformingproSductmeasures,hence@s̹k: ̹k +vg x!8 :sBymonotoneapprorximationthisyields@sqCRIgn7d( )URliminf%̹k6!1>qCRFIgd(̹k: ̹k#)for)0gË2C5(R̸+"R 2ڍ+x):sTherefore,withP̹k :denotingthekrernelcorrespSondingto̹k#,@sPfc=w"AqCRfG(yn9x+z)(dx)(dy;dz)cw"Aliminf̹k6!1ϏqCRz8fG(yn9x+z)̹k#(dx)̹k(dy;dz)c=w"Aliminf̹k6!1Ϗ̹k#P̹kfcw"Aliminf̹k6!1Ϗ̹k#fc=w"AfGfor*P0URfQ2K,`(R̸+x);@6K썠:-iӍ{-swhicrhprovesPUR: 2.ItshouldbSemenrtionedthatthisresultdoesnotcarryorvertoinrvXariantsmeasures,ascanbSeshorwnbysomewhatinvolvedexamples.*s5.Existenceanduniquenessofinv@ariantmeasuressTVo,PderivreergoSdictheoremsintherecurrentcase,the\sojournprffocess";>belongingto(X̹nP)̹n0 and[0;t],si.e.@s tBX̹n:=URX̹T;cmmi6nfor4n0;swherffeET̸0 =<}T̸1<:::arffeEtherandomtimeswhen(X̹nP)̹n0Xisin[0;t]andthesnotation352tX2xRAn mis35useffdinthecaseX̸0V=URx..FVorREeasyreferenceasimpleconclusionfromprobabilisticpSotenrtialtheorysisstatedexplicitly:s(5.2)Lemma.Lffet+ berecurrentandUR2M(R̸+x)bffeexcessive.[If[2t'denotessthe35rffestrictionofto[0;t];33x33feR5<URt<1,35thens(a)2t35isinvariantwithrffespect35to2tP,s(b)35isinvariantwithrffespect35toP.sPrffoof.1.8IfI̹A ȌforAUR2B]m(R̸+x)denotesthekrernelp@sI̹A(x;)UR:=1̹A(x)"̹xHfor.Qx2R̸+x;sthecrucialpSoinrtistheinequalitys(1)(I̹ACPl̹n0'(PI RAacmr6+ºnAʫ) nP)URs(seee.g.8IX,(62.4)and(31.6),in[8])..2. \Multiplied_bryPI̹A =fromtherightandspSecializedtoAUR=[0;t],{(1)_yields@s( t tP)(B){j(P)(B){j(B){=j t6(B)for)BX2URB]m([0;t]):@7X|썠:-iӍ{-sThisprorves(a),bSecause2t|tis niteand2tPnisastoScrhastickernel..3.FVor^70fb2K,`(R̸+x)crhoSosenowt>xfeR%withsuppWfb[0;t]anddenotestherestrictionoff2to[0;t]bry2t|tfG.8Then(a)implies؍@sfX9=k tmt tfX9=k( t tP) tfXQk(P)fGfor*P0URfQ2K,`(R̸+x);soncemorebry(1),andthisyieldstheinequalityURPnneededfor(b): 2.Norw,theexistenceofaninvXariantmeasurecanbSeprovedintheusualswray<(seee.g.-?[14]),Pzundersomesimpli cationduetothemonotonicitryV.MoresgenerallyV,thefollorwingversionisrequiredinSection7:s(5.3)Prop`osition.If35isrffecurrent35andxfeR5<URt<1,35themeffasures؍@s%̹nP(B)UR:=CP㋟̸0m0;swhicrh#bispSossibleinviewoft>xfeR c2. With#b̹m :=L(X̹m)thisimpliesbrymono-stonicitry@sP(X̹m+lVURt)gKqCR̸0xs P(X xڍl URt)̹m(dx)gKqCR̸0xs P(X sڍlURt)̹m(dx)*=gK#P(X̹m ZURs)for)m0:؍sWiththenormingconstanrts@sr̹n:=URCP㋟̸0mA̸0mA̸0A̸0mx ՍI>s;tV :=URliminf%̹n!1AܰCPNj̸0mx I>s;tV :=URlimsup)W̹n!1CUfCPO㟟̸0m6standsforsort,monotonicitryinxyieldss(2)1߸[0;rUR0smallenoughtosatisfy@sȄ<URs^(ts)^(txfeR W)sandcrhoSosefQ2URK,`(R̸+x)suchthat@s1߸[0;rsfor35alCmostallxUR2[0;t].sPrffoof.1.!Since([0;t])is niteand,bry(4.3a)and(1.3a),strictlypSositivre,sthe$restriction2tofto[0;t]marybSeassumedtobenormalized,shencebrys(5.2a)rtobSeastationarydistributionfor2tw>P.)?AssumeinthesequelthatX̸0ݿB10 썠:-iӍ{-sisdistributedaccordingto(thetrivialextensionof8)2t.Then(2tX̹nP)̹n0isasstationaryproScess,hencetheclassicalergodictheoremensuresthat/Js(1)lim'Q̹n!1ōA1A-KQmfe  nIdCPU@̸0mqܱ=9qCR߸[0;t]r>6sucrhthats(1) n:=UR([0;t])>0and<,  0ʧ:= 09([0;t])>0;s(2)((CFnCܞ 0)̹t)UR=0;swhicrhdispSossibleaccordingto(5.4a).If2tJ0and2t20denotetherestrictionsofs 0and20ito[0;t],themeasures 212tband( 20uU)21 \z2tF20areconrtainedinM̸1(R̸+x)sbrym(1)andaccordingto(5.5a)agreeforsets[0;s]with(s;t)3T2Cܞ20,hencemonsB]m([0;t]),bry](2).7Therefore2ty)and2t20arelinearlydepSendenrt,which]fortUR!1sextendstoand209: 2.Twrocommentsonthisresultareinorder:s|UniquenessholdswithintheclassofloScally nitemeasuresonlyV,&asisseenݿB11 U썠:-iӍ{-salreadyo bryadeterministicexample:AIfY=h1=2=Zܞ,%theno 0isrecurrentswith]inrvXariantmeasure="̸1._Sincex!x=2a+1=2isabijectionofthessetAs:=Q\]1;1[,horwever,thede nition209(B)s:=jA>\Bjyieldsanothers(n9{ nite)inrvXariantmeasure.s|IndthetransienrtcasenontrivialinvXariantmeasures!2M(R̸+x)dmaybSesabsenrt,0as"followsfrom(4.3a)inthecaseYPx1,0orpresent,0ascanbSeshownsinthecaseY=UR n2]0;1[bryalimitingproScedure..ItgisanotherquestionhorwtogettheinvXariantmeasurefrom(5.6).ItissnotvhardtotranslatetheequationP=vYinrtoanintegralequationforthesfunctionF(t)UR:=([0;t]),butingeneralitisimpSossibletosolvreit.PassingtosLaplacetransformssimpli esatleasttheinrtegralequation:s(5.7)Prop`osition.Lffet35andbegivenaccordingto(5.6).fiThen:'s(a)@wk n9(u)UR:=qCR e ux=(dx)<1for)u>0;s(b)up35toascffalar, nisuniquelydeterminedbytheequation@s n9(u)UR=qCR (uy)e uzS(dy;dz)for)u>0:sPrffoof.While(a)isasimpleconsequenceof(4.5),(b)follorwsfrom@sqCRIe ux;(P)(dx)UR=qCR qRe u(xyI{+zV)((dx)(dyn9;dz)sand(5.6),bSecausetheLaplacetransformdetermines: 2.IfaY0;sbutisstillrarelysolvXable.*s6.Mainprop`ertiesoftheinv@ariantmeasuresThemeasurethat(5.6)assignstoarecurrenrtdistribution}actuallystandssforaone{dimensionalfamilyV. BNevrerthelessitwillbSebrie ycalled\theinvXari-sanrtL$measure"inthisandthefollowingsection.]SAspSointedoutattheendofstheprecedingsectionitsquanrtitativedeterminationisingeneraloutofreacrh.sThrusitisimpSortanttoobtainatleastaqualitativedescription..AX rstpropSertryofthesupportMBofenrteredalready:f(1.3)and(4.3)scomrbinetotheequationsEٍ@infQ$Mgy=zxzfeR>=inf7fō z33Qmfe  1y o::UR(yn9;z)2N̹c.yg;"x@ssupT$Mgy=z&feR]ڍx>=sup[Nfō z33Qmfe  1y o::UR(yn9;z)2N̹c.ygif!ڦN̹e=;;s=UP1otherwise.ݿB12 v썠:-iӍ{-sIt isanaturalquestiontoaskunderwhicrhconditions,jlessrestrictivethansin(4.4),9thewholeinrtervXal[xfeRR;&feR]ڍxP]isexhaustedbyM@.|Asurprisinglygeneralsanswrerisgivenbythefollowingresult(foraspSecialcasesee[2]):s(6.1)Theorem.LffetÞeberecurrentwithP(Y.=`0)=0and&feR]ڍxѮ=1.Thensthe35invariantmeffasure35hasthesuppffort@sM6=UR[xfeRR;1[:sPrffoof.1. Byc8thehrypSothesisP(Y=UT0)UR=0c8poinrts(0;z)UT2Ncannotc8beisolated,shence@sx@sfeRJv=URinfFfō z33Qmfe  1y o::UR(yn9;z)2N@with$0UR0;@s0URxx̹kxUR(1y̸0) k#(xx̸0)for)x̹kx=URg̸1j:::g̹ni?k (x̸0):sClearlyV,thisimpliesxUR2M+bry(4.1a)and(4.2)..2.ItOsucestoconstructn̹k6+1O&andg̹ni?kƸ+j1>;:::ʜ;g̹ni?k+1՗fromn̹krandg̸1;:::ʜ;g̹ni?ksundertheadditionalassumptionx̹k <uzx,6FbSecauseotherwisethede nitionsn̹k6+1U`=URn̹k :wrorks.8NowthehrypSothesis&feR]ڍx L=1enrters,providing@sg n90ڍi,=UR(y n90ڍid;z 0ڍi)2N @with*y n90ڍi,>0for)1imssucrhthat@s̹kx:=URy̸1:::y̹ni?k ?j(g n90ڍ1j:::g n90ڍm l(x̸0)x̸0)UR>xx̹k#:sIndeed,Dthis2(follorwsfromlimsup*4̹n!1BX2xq0RAn4=1,bSecausetherelationsY̹n wV>0sand(Y̹nP;Z̹n)UR2N+holdalmostsurelyV..3.8Letnorwl2URNbSede nedbys(1)y n9lK1ڍ0l}̹kx>URxx̹kURy n9lڍ0̹k#:sThentheconstructioncanbSeconrtinuedbyn̹k6+1U`=URn̹k:+l7+mand@sg̹i,:=URf`Cnō${g̸0;forOn̹kx<URin̹k:+lC;Yy gߍn90i(ni?kƸ+jlK);pforOyn̹k:+l<URin̹k6+1:f`CoݿB13썠:-iӍ{-sIndeed,sincex̸0isa xedpSoinrtofg̸0V=UR(y̸0;z̸0),byanity@sx̹k6+1x̹k{=)g̸1j:::g̹ni?k g n9lڍ0(g n90ڍ1:::g n90ڍm l(x̸0)){)g̸1j:::g̹ni?k g n9lڍ0(x̸0){=)y̸1:::y̹ni?k ?jy n9lڍ0j(g n90ڍ1:::g n90ڍm l(x̸0)x̸0){=)y n9lڍ0̹k#:sBy(1)thisimpliestherequiredinequalities@sx̹k6+1U`=URx̹k:+y n9lڍ0̹kxx;@sxx̹k6+1U`=UR(xx̹k#)y̸0y n9lK1ڍ0l}̹kx<(1y̸0)(xx̹k#): 2.ClearlyV,mtheRconditionP(YW=0)=0Risessenrtialforthisresult,asisseensfromI}thetrivialcaseY-=0,a3whereis(amrultipleof8)thedistributionofZܞ..TheargumenrtusedinthefollowingproSofcanbetracedbacrktoKarlin[22];sitappliestoloScally nitemeasuresaswrell:s(6.2)Theorem.Lffet_!berecurrentwithP(YD="0)=0.ThentheinvariantsmeffasureBoiseitherabsolutelycffontinuousorsingularwithrespecttoLebesguesmeffasure.sPrffoof.FVromtheequation@s(P)(B)UR=qCR ̹yI{>0,(ō33BEz33Qmfe1  y j)dfor)ʡBX2B]m(R̸+x)捑sitisclearthatimpliesPIVaswrell.If,' therefore,isdecompSosedsinrtoa=theabsolutelycontinuouspart̹candthesingularpart̹sn<,|itfollowsfrom@s̹c.yPLn+̹snUR0andz̸0UR0sucrhthat @s(ō33BEz̸033Qmfe"h%  }hx̸0$΋)UR=0and<,̹y (R̸+ nōBEz̸0۟Qmfe"h%  }hx̸0'y3)=0:\/sThecorrespSondingargumenrtfor̹zVisevensimpler: 2.It)isaneasyconsequenceof(1.4)thattheindepSendenceofYcandZissessenrtialin(b).+Itapplies,however,intheadditiveormultiplicativemoSdel.sMoreorver,AitshouldbSemenrtionedthat(a)and(b)arebynomeanssucientsforsingularitryof(inthiscontextseeSection10)..Thefollorwingresultiseasilyestablishedformeasurescontaininganatomswithmaximalmass(see[9]).SincethismaryfailforloScally nitemeasures,sthegeneralproSofgetsmoreinrvolved:s(6.4)Theorem.LffetMberecurrentwithP(Y=UR0)=0andxfeR *<&feR]ڍx .O/Thenthesinvariant35meffasureisnonatomic.sPrffoof.1.+Itfollorwsasin(6.2)thatiseithernonatomicorpurelyatomic.Itssuces,therefore,toshorwthatinthesecondcase~эs(1)Z1=URxURfeR L(1Yp);sthrus(inviewof(1.4)obtainingacontradictiontothehypSothesisxfeR<b&feR]ڍx M.`TVosthisendconsiderthecounrtableset@sRn:=URfx2R̸+ q:(fxg)>0g:ݿB15ɺ썠:-iӍ{-sBytheinrvXarianceofitsatis esI@s0UR=(R̸+"nRJ)=CP㋟̹x2R#](fxg)P(x;R̸+nRJ);shencethede nitionCACb@sPI(x;x 09)UR:=P(x;fx 09g)for)x;x 0#2RsyieldsaMarkrovchainonthestatespaceRwithinvXariantmeasure#A:Cbs@sG(x)UR:=(fxg)for)x2RJ:.2. FVory`xy`feRn<H?t<1y`de ne2t Cb *P%and2t# }Cbs *Minanalogyto2t ,P&and2tinSection5.sThen͟2t SrCbPisagainastoScrhastickernelwithstrictlypSositiveand niteinvXariantsmeasurei2t# Cbs k3|. $ByiclassicalMarkrovichaintheorythisimpliesthatallstatessx2RL\[0;t] are(pSositivre)recurrentwithrespSectto2t iCbPj,hencealsorecurrentswithrespSectto]CbP .8)FVortUR!1thisextendstoallstatesxUR2RJ.TherecurrenrtsMarkrov8kernelfCbPisinadditionirreducible."Indeed,Leachrestrictionof#MCbstoassingleclassyieldsagainaninrvXariantmeasureforPCbP:QanditstrivialextensiontosR̸+ qanUVinrvXariantmeasureforP.xThusbytheuniquenesspropSertyoftherescanexistonlyoneclass.Therefore,XagainbryclassicalMarkovchaintheoryV,sevreryn9{ nite(notnecessarilyloScally niteW!){.excessivemeasureforΝCbP6isinsfactamrultipleof#Cbs k..3.8NorwbythehypSothesisP(Y=UR0)=0thesetsI@sA(x;x 09)UR:=f(yn9;z):yn9x+z5=URx 09g;UPx2RJ;sare{almostdisjoinrtfor xedx209,hence@sCPM ̹x2Rc7#Cb`iHPi(x;x 09)UR=CP㋟̹x2R#](A(x;x 09))1forall9Zx 0#2RJ;si.e.rtheequidistributiononR4isexcessivreforCbP 2.Therefore,accordingtoparts2oftheproSof,s(2)#SCbs(x)UR=1forall9Zx2RsmarybSeassumedinthesequel..4.ZSinceisloScally nite,thesupportM6ofinviewof(2)mrustconsistsofisolatedpSoinrts,henceRmustcontainxfeR L=URminM@.8Nowby(1.2a)@syn9xn9feR 3+z5URxURfeRfor{almostalll@(yn9;z);swhicrhinviewofP(Y=UR0)=0implies@syn9x+z5>URxURfeRfor{almostalll@(y;z)whenevrerJxUR>xfeR :sBytheinrvXarianceof#Cbsthisyields@s1PS=cwkCPp̹x2R:CblPg(x;xfeRP)PS=cwkCPp̹x2Rv(f(yn9;z)UR:yx+z5=URxURfeR g)PS=cwk(f(yn9;z)UR:yxfeR 3+z5=xfeR g);ݿB16썠:-iӍ{-sand(1)isestablished: 2.The?remarkfollorwing(6.1)showsagainthattheconditionP(Y?=0)=0sisessenrtialfor(6.4)..The/< nalresultofthissectionisastrongstabilitrystatement,@avXalidundersanappropriatenormalization:s(6.5)Theorem.Lffet]fberecurrentwithinvariantmeasureandassumesNQ3I̹k LNwg m3!.3If#̹k m32M(R̸+x)isnontrivialandexcffessivewithrespectto̹k#,sthen@s(̹k#([0;t])) 1 \z̹k +vg x!8(([0;t])) 1wRhenevn9erStUR>xfeRand(ftg)=0:sPrffoof.By)(1.3a)thereexists(y̸0;z̸0)UR2N with)y̸0V<1satisfyingtheinequalitrysz̸0=(1y̸0)UR0smarybSeassumed(slightlyincreasingy̸0andz̸0ifnecessary).uWBywreakconver-sgencetherefore@spUR:=infF k62N(G̹k#(f(yn9;z):yË0smaryObSeassumednext.Then(4.5)(withtherolesofsandtinterchanged)sprorvidesconstants 7and suchthats(1)̹k#([0;s])UR Q̹k([0;t])s for.qstandall/?ko2N:sDueto̹kx6=UR0thisimplies̹k#([0;t])>0,hences(2)̹k#([0;t])UR=1 for ko2Nand<,([0;t])=1smarygbSeassumed nallyV.By(1), moreover,thegsequence(̹k#) k62N$jisuniformlysloScally nite,hence(asintheproofof(5.3a))eacrhsubsequence(20RAk#) k62N!con-stains?avXaguelyconrverging?subsubsequence(200RAk#) k62N . 7By(4.6)and(5.6)itsslimitisoftheforms2assigningmeasure0tothesetftg.8By(2)thisimplies@sȄ=URs2([0;t])=lim ̹k6!1-ʬ 00ڍk#([0;t])=1;sandtheassertionfollorws: 2.This(resulto ersanapprorximationmethoSdfortheinvXariantmeasuresassoSciatedwitharecurrenrtdistribution:8choSose@s̹kx:=ōM1Qmfe  k?A"߸(0;0)+(1ō/1۟Qmfe  k )for)ʡko2URNsand`observrethataninvXariantmeasure̹k for̹kcanbSedeterminedmoreorslessexplicitly(seeSection8).*ݿB17洠썠:-iӍ{-s7.Ratioergo`dictheoremssAJ rstinformationonthe uctuationofarecurrenrtsequence(X̹nP)̹n0Dbbymeanssof(itsinrvXariant(measureisgivrenbythefollowingmeanergoSdictheoremforsratios,holdingwithoutanryassumptionontheinitiallaw:s(7.1)Theorem.Ifis35rffecurrentwithinvariantmeasure,thenO@sCPM ̸0mxfeRDwith(ftg)=0andconsiderthemeasures%̹n pde nedsinW(5.3).@If(%̹ni?k ?j) k62NwisanryvXaguelyconvergingsubsequence,s(5.3b)and(5.6)simplyOs(1)%̹ni?k ?jfQ!URs2fGforall:f2K,`(R̸+x);swheretheconstanrt]satis es@ss2([0;t])UR=lim ̹k6!1-ʬ%̹ni?k ?j([0;t])=1;shenceisindepSendenrtofthesubsequence.Since(1)extendsfromK,`(R̸+x)tosK̹ (R̸+x)brymonotoneapproximation,therefore@s%̹nPf̹i,!URs2f̹idfor,mi=1;2sbry(5.3a).8FinallyV,theconstant]disappSearsbytakingquotients: 2.TheTproSofofapoinrtwiseTanalogueof(7.1)ismruchTmoreinrvolved.TheT rstsstepconsistsinprorvingacounterpartof(5.4):s(7.2)Lemma.Lffet35berecurrentwithinvariantmeasureandfortin@sD:=UR]xfeRR;1[sdenote35by2ttherffestrictionofto[0;t].fiThentheset@sDS 0h:=URft2D:P(CS UU̹n0fX xڍn =tg)=035fGoralCmostallvxUR2[0;t]g;swith35denotingLffebesgue35measure,satis ess(a)@wk(D6nDS 0)UR=0;s(b)2tPis35a\2tFelClerkernel"35whenevertUR2DS20,35i.e.@s tBPfQ2URC5tս B([0;t])for)f2C5tս B([0;t]):sPrffoof.1.8X2xRAn:j(!n9)tismeasurablein(t;x;!)forallnUR2N,hence@sAUR:=CS  n2N#Off(t;x;!n9)UR:0xtand,X xڍn:j(!)=tgsisameasurablesetwith@s(A̹x;! 7)UR=0forall9Z(x;!n9);ݿB18f썠:-iӍ{-sbSecausethesectionsA̹x;!"arecounrtable.8Therefore,byFVubini@s( P)(A̹t)UR=0for){almostall>4t2DS:sThisprorves(a),bSecausethelastequationisequivXalentto@sP(A̹t;x %)UR=0for){almostall>4x2[0;t]:.2.8TVovrerify(b), xtUR2DS20:andfQ2C5tս B([0;t]).Thentheset@sB̸1V:=URfx2[0;t]:P(CS UU̹n0fX xڍn =tg)>0gsisa{nrullset,andwiththenotation@sBX:=URfx2[0;t]:f2disconrtinuousat\̻xgsthisholdsaswrellfortheset@sB̸2V:=URfx2[0;t]:UP tP(x;B)>0g:sIndeed,ꨟ2t|tisbry(5.2a)invXariantwithrespSectto2t|tP,hence@sqCRK tLZP(x;B) t(dx)UR= tt for 0t for 04xUR2[0;t]:sIndeed,*thisgisaconsequenceof(5.5b),lettingsvXarythroughacounrtablesdensesubsetofC2ܞ0RAt.8Thruss(1)% xڍnNgË!UR t gn9forwg2URC5tս B([0;t])for){almostall>4x2[0;t]:.2. Theastationaritryof(2tX̹nP)̹n0-isimmediatefrom(5.2a).TVoprorveathesergoSdicitryitsucestoverify[ōB81AQmfe  nKԟCPXl ̸0m[thecentralresultofthispapSercanbeestablished,`holdingagainwith-soutanryassumptionontheinitiallaw:s(7.4)Theorem.If35isrffecurrent35withinvariantmeffasure35,then@sCPM ̸0mx̸0andsupp[If̹i,[0;t] for i=1;2;s(2)tsatis esconditionfV()inʤ(7:3):sThereforetheclassicalergoSdictheorem,comrbinedwithFVubini,yields/Js(3)ō133Qmfe  niCP*(̸0mxfeRsatisfyGA(ftg)=0.NThensbry(7.4)withprobability1r@sCPM ̸0m0)=1;swhicrhimpliestheinclusion@sL(!n9)URsuppka.s..2.RTVo!prorvetheconversedenotebyL̹t(!n9)theanalogueofL(!)forthessequence(2tX̹nP)̹n0.8Clearly@sL(!n9)UR=CS x fe玑s xfeR :sThishisobrvious,ifX̸0 mlisdistributedasin(7.3),bSecauseinthiscasewithsprobabilitry1@s tBX̹n2URsuppk\[0;t]forall9ZnUR0:sNorw~anapplicationof(2.6)toanyfunctionfQ2URK,`(R̸+x)withfG(x)=xon[0;t]sshorwsthatthedistributionofX̸0isactuallyirrelevXant: 2.TVogether,<>(2.2)Ňand(7.5)implythatthetrwoŇmaincrharacterizationsofsrecurrence/transienceFCfromclassicalMarkrovFCchaintheorycarryovertoanesrecursionsinthefollorwingform:s|Ifoisrecurrenrt,thenforxUR2suppkalways@sP xH(X̹n2URGin nitelyoftenQ)=1;shence@sE xH(jfnUR0:X̹n2Ggj)=1;sprorvidedGisanopSenneighbSorhoodofx.s|Ifoistransienrt,thenforxUR2R̸+  always@sE xH(jfnUR0:X̹n2Kܞgj)<1;shence@sP xH(X̹n2URKܞin nitelyoftenQ2)=0;ݿB22(Ġ썠:-iӍ{-sprorvidedKFisacompactsubsetofR̸+x..The{ nalresultofthissectionshorwsthatanerecursionsintherecurrentscasearenotonlyirreduciblebutalsoapSeriodicinastrongsense:s(7.6)Prop`osition.LffetLberecurrentwithinvariantmeasure.:Thenforseveryx̸0 2݆R̸+ #yandeffachopensubsetG̸0 ofR̸+ #ywith(G̸0)݆>0thereexistssn̸0V2URN35suchthat@sP xq0 s(X̹n2URG̸0)>0for35all:Ynn̸0:sPrffoof.0. cIfX̹n B!Cxa.s. ,thensuppWC=fxg(e.g. cbry(7.5)).Sincethesassertionistrivialinthiscase,xfeR L<UR&feR]ڍxwillbSeassumedhenceforth..1.8Ina rststepmUR0andsUR>xfeR LcanbSefoundsucrhthats(1)P(X xڍm Z2URG̸0)>0for)x)ڟfeR3 ~x0G\forsomemfollorwsfrom(7.5)andextendstoasneighrbSorhoodIofxfeR c,bSecausePƟ2misagainaFVellerkernelandthusPƟ2m 1̹Gq0Pisslorwersemicontinuous..2.8InthenextstepkoUR0canbSefoundsucrhthats(2)P(xfeR URXnxq0D>k c0:sIndeed, (X2xq0RAn e)̹n0hitsv]xfeRR;1[inviewofxfeRhk<D#&feR]ڍxandvstarysafterwardsin[xfeRR;1[saccording}to(1.2a),ihencevisits[xfeRR;s[in nitelyoftenbry(7.5){allthiswithsprobabilitry1..3.8Ina nalsteplUR0canbSefoundsucrhthats(3)P(xfeR URX xڍl 0 and*,P(xfeRX xڍlK+1S0for)x)ڟfeR3 ~xsandlUR0suchthat@sP(X sڍ1URt)>0and<,P(X tڍl؟0:sThenbry(1.2a)andmonotonicity@sP(xfeR URX xڍl 0:sWith̸1V:=URL(X2xRA1:j)moreorver@sP(xfeR URX xڍlK+1Sl cl c0for)x̸1V2[xfeRR;t]sand@s̸1([xfeRR;t])UR=P(xfeR X xڍ1 t)P(X sڍ1t)>0:ݿB233r썠:-iӍ{-.4..iByDcomrbiningk2astepsaccordingto(2),ыitimeslstepsandjxtimesl+j1ssteps:accordingto(3),andmstepsaccordingto(1),theMarkrov:propSertysyields@sP xq0 s(X̹n2URG̸0)>0for)n=k+il7+j(l+1)+m:sSincei;j%UR0arearbitraryV,therequiremenrtismetby@sn̸0V:=URk+(l71)l+m: 2.In_viewof(7.6)thefollorwingopSenproblemcanbeposed:@Nisitpossibletosstrengthen;(7.1)toastrongratiolimittheoremasvXalidforirreducibleandsapSeriodicrecurrenrtrandomwalk? sReferences""K`y 3 cmr102.0sBarnsleye,cM.,Elton,J.: A=new=classofMark!ov=proMcessesforimageencoding.0sAdv.Appl.Prob.'"V 3 cmbx1020,f14{32(1988)"8.0sDellac!herie, C.,Meyer,Pe.A.:Probabilit"DesetpMotentiel, ChapitresIX{XI.Paris:0sHermannf1983"9.0sDubins,+L.,Fereedman,D.:In!vdDariantvprobabilitiesforcertainMark!ovvproMcesses.0sAnn.Math.Stat.37,f837{848(1966) 14.0sFeoguel,S.:TheergoMdictheoryofpositiv!eoperatorsoncon!tinuousfunctions.0sAnn.Sc.Norm.SupMer.Pisaf27,19{51(1973) 22.0sKarlin,RS.:vRandom=w!alksarisinginlearningmoMdels.Pac.J.=Math.3,R725{7560s(1953)ݿB24A;ʔ Cu cmex107"Vff cmbx103Tq lasy100N cmbx12.@ cmti12-!", cmsy10,g cmmi12+XQ cmr12'"V 3 cmbx10"K`y 3 cmr10t : cmbx9K cmsy82cmmi8 |{Ycmr8;cmmi6Aacmr6H