; TeX output 2003.09.30:1708썠:-iӍ{-{+'"V 3 cmbx10RANDOM2DtYNAMICALSYSTEMS lmrON2ORDEREDTOPOLOGICALSP\AtCES8"K`y 3 cmr10HansfG.KellererUniv!ersityfofMunic!h²+Junef30,2002wݍ4SEK`y cmr10Let,D( b> cmmi10X 0ercmmi7nq~;UPn!", cmsy100)bGearandomdynamicalsystemanditsstatespacebe AqendoweddwithareasonabletopGology*.InsteadofcompletingthestructureAqascommonbysomelinearity*,)thisstudystresses{motivqatedinparticu-Aqlarixbyeconomicapplications{orderaspGects.1IftheunderlyingrandomAqtransformationsjaresuppGosedtobeorder-preserving,thisresultsinafairlyAqcomplete theory*.'Firstofall,theclassicalnotionsofandfamiliarcriteriaAqforJrecurrenceandtransiencecanbGeextendedfromdiscreteMarkovJchainAqtheory*.2wThebmostimpGortantfactisprovidedbyexistenceanduniquenessofAqaGloGcally niteinvqariantGmeasureforrecurrentsystems.m0ItallowstoderiveAqergoGdictheoremsaswellastointroGduceanattractorinanaturalway*.DTheAqclassi cationsiscompletedbydistinguishingpGositiveandnullrecurrenceAqcorrespGonding,respectively*,tothecaseofa niteorin niteinvqariantmea-Aqsure;equivqalently*,ythisEramountsto niteorin nitemeanpassagetimes.AqF*orpGositiverecurrentsystems,moreover,strengthenedversionsofweakAqconvergenceUUaswellasgeneralizedlawsoflargenumbGersareavqailable.l,-sInttroYduction. RThe6randomdynamicalsystemsstudiedinthispapMerev!olvesin=6one-sideddiscretetimewithindepMenden!tandstationaryincrements.NThereforestheTformalismneededformoregeneralmoMdels(seethemonographsb!yKifer[26]sor9Arnold[2])isdispMensable,andtheprocess(# b> 3 cmmi10Xz2cmmi8nP;bnX$!", 3 cmsy100)9canbein!troducedasans\7iteratedffunctionsystem"wjXzn= HznP(XznK cmsy8 |{Ycmr81)for#n2N1;swhere(HznP;1n2N)isasequenceofi.7i.d.-transformationsofthestatespaceE,sindepMenden!t`oftheinitialE{vdDaluedvariableXz0.oIfnospMeci cstructureinEhastosbMeltak!enintoaccount,x1thisisjustanotherwaytointroMduceahomogeneousMarkovsc!hainfonEG(seevonWeeizsfacker[39]).-sIf,ho!wever,statespaceandmappingsaresuppMosedtoha!veappropriatelinearit!ys(orzsmoMothness)properties,them!ultiplicativezergodictheoremyieldsadditionalin-ssigh!t,LprovidedtherelevdDan!tintegrabilityconditionsaresatis ed.LAnothercommonsmoMdelf{studiedinparticularb!yBarnsleyandElton(seee.g.2[6,f5,14])f{supposessEto?-bMeacompletemetricspaceandthemappingsHzn }tosatisfyana!verage?-con-stractivit!ye.NButwinbMothcasesresearchconcernsprimarilyexistenceresp.Nuniquenesssof?aanequilibriumandconsequencesonthelong-termbMeha!viourofthesystem.InsbMothfcases,too,therequiredmomen!tconditionsarebynomeansnecessarye.-sTeocexemplifytheproblemsleftopMen,*c!hoosecE\=LRz+ ,andrestrictHzn toanesmaps,fi.e.considerthe\7autoregressiv!emoMdel"kXzn= UznPXzn1+nVznPfor(_n2Ns ffU?a': cmti102000MWMathematicsSubje}'ctClassi cations.X6Primary60J05,60H25;"3secondary54H20,54F05.q1*썠:-iӍ{-swith*asequenceofi.7i.d.)R2A+x{vdDalued*variables(UznP;1Vzn).)If(Szn;bnNA0)denotes sthe8lrandomw!alkwithincrements logUzn{(11),then{withoutanymomentsconditionsonUzn #andunderw!eakbMoundednessconditionsonVzn{thefollo!wingstric!hotomyfisestablishedinthepreprin!t[24]:s(a) LiffSzn! 1,then(XznP;bn0)is\7pMositiv!erecurrent7",s(b) iff(SznP;bn 0)oscillates,then(XznP;bn0)is\7n!ullrecurrent7",s(c) iffSzn! +1,then(XznP;bn0)is\7transien!t".-sInDtheexistingliterature(seeinparticularthesurv!eysbyVeervdDaat[38]andby sEm!brechts/Goldie[15]andGoldie/Maller[20])thereisnearlynodistinctionbMet!weenscases\(b)and(c),Zthoughacommontreatmen!tofcases(a)and(b)isinfactmuchsmoreY^adequate. Butitisnottheane(andtopMological)structurethatallo!wssanaturalextensionofclassicalnotionsandcen!tralcriteriafromdiscreteMarkovsc!haintheorytoanuncountablestatespace.iWhileunderatopMologicalstructuresalonedthereisavdDariet!yofde nitionsfortransienceand(nullorpMositive)recurrences(seeRe.g.thevdDariousnotionsinTw!eedie[37]andMeyn/Tweedie[32]),|theresultssobtainedzinthepreprin!t[25]provethatorderandtopMologycombinedprovideanidealsframew!orkforafairlycompletetheoryofrandomdynamicalsystemsasconsideredsinfthispapMer.-sTeoco!ver,however,autoregressivemoMdelsofhigherdimensionorstoc!hasticrecur-ssionsQofhigherorder,bassumingEStobMetotallyordered,asisthecasein[24]or[25],sistoMorestrictiv!e.ÖInthesequel,therefore,thesepreprintsaredisregarded,andthesstatespaceissuppMosedtobean!y(partially)orderedtopologicalspace.oInessence,sthereisonlyonerestriction: aE_:issuppMosedtobeboundedfrombelo!w,yaconditionsthatu8islargelyinaccordancewiththeoryasw!ellaswithapplications.JRIndeed,tosgiv!e+aonlytwotypicalexamples:ZmostworkonproMductsofi.7i.d.matrices+aisactuallysrestrictedtononnegativ!ematrices(seee.g.\Hfognas/Mukherjea[22,7Chapter4]),swhileeconomicproMcesses,0modelingdams,insurancerisk,queues,storage,tracsetc.,(see Re.g.Asm!ussen[3,*VPartC]) *ingeneralhave0asnaturallowerbMound.,Thesesandotherexamples,}moreo!ver,justifytherestrictiontomappingsHzn thatrespMectsthefstructureofE,i.e.areorder-preservingandcon!tinuous.-sTheDpreciseassumptionsonthestatespaceEarecollectedatthebMeginningofsSection0.-TheyaremetnotonlyintheclassicalcaseE7=`RdA+x,2butasw!ell,forsinstance,!b!y(roMoted)treemodels.Thesigni canceoftheorderisre ectedb!ythessingularroleofordercon!vexsetsandfunctionsofbMoundedvdDariation,.'leadingtostheRbasicclassesN%n 3 eufm10NV(E)andV|(E),respMectiv!elye.,Theextensionofthesenotionsfromstotalڻtopartialorderisstraigh!tforward,apartڻfromthefactthatmonotonesetsorsfunctionsKneednotbMemeasurableinthegeneralcase. Butthearisingproblems,sas;somelesscommonnotionsandfactsconcerningorderedtopMologicalspaces,aRarespMostponedd:tothe nalsection.YRandomisin!troMducedbyspMecifyinganelementsinBthespace)m#R 3 cmss10N[E]ofdistributionsonthespaceH[E]oforder-preservingcon!tinuousstransformationsofE.Thisde nesa(coupled)familyofhomogeneousMark!ovchainss(XxAn$;1n 0),fdepMendingontheinitialstatex 2E.-sIfKthein!terestisnotlimitedtostationarydistributionsandmoregenerallyrecur-srence|propMertiesarestudied,٫theexistenceofnotnecessarily nitein!vdDariant|measuressfortheunderlyingtransitionk!ernelisjustasinteresting.ٮTeoassureuniqueness,[bMe-sing]essen!tialforergoMdictheorems,Krestrictiontosuitably\7irreducible"]systemsissinevitable.Asfitturnsout,cKtheadequatenotion{in!troMducedatthebeginningofq2썠:-iӍ{-sSectionJ1{emplo!ystheorderinsteadofthetopMologyofE.QIthastobeemphasized sherethatthis,asallsubsequen!tnotions,isin!vdDariantunderconjugation,i.e.undersorder-preservingohomeomorphisms(notdistinguishing,forinstance,statespacesRz+sand[0;11[).Sinceingeneralnonaturalmetricisa!vdDailableinE,toanswerthebasicsquestions>ofasymptoticequivdDalenceoft!wos>proMcesses(XxAn$;1n`0)s>and(XyAn;n`0)stranslations*Bf:E2!Rm!ustbMeused. irAgain,9topologyhastobereplacedb!ysorder:whiletthedi erencesjf-(XxAn$) ]f(XyAn)jtneednotcon!vergettozeroforbMoundedscon!tinuous֦functionsf-,theyev!enaresummable,iffbisofbMoundedvdDariation.nThesfundamen!talfinequality(1.4)anditscorollariesarecrucialinthesequel.-sAk rsttconsequenceisthezero-onela!w(2.1),8applyingtoallsetsinthealgebrasNV(E)Aandconcerningtheprobabilit!yofanin nitenumbMerofvisitsbytheproMcesss(XznP;bnm0).ZESincethisvdDalueisindepMenden!toftheinitiallaw,i.e.toRadonmeasures.Theonlyresultsrelatedto(3.4),skno!wnsofar,+(areduetoBabillotetal.{[4],whostudyorder-preservinganesys-stemspofaspMecialstructureandundersuitablemomen!tconditions(seetheremarkssfollo!wingp(9.3)).:Inaddition,ytheyderivealimittheoremforoMccupationtimesassconsideredfinthenextsection.-sWithoutKpMostulatingthein!vdDariantKmeasuretobe nite,2theergodictheoremssinmSection4ha!vemtoconcernratios,߻regardingbMoundedfunctionswithcompactssuppMort.WhileOEthev!ersionformeansin(4.2)isrestrictedtofunctionsinK(~(E),`thespMoin!twisezversionin(4.3)canbMeestablishedforfunctionsofboundedvdDariationandstheiruniformlimits,%de ningaclassR(E)thatingeneralism!uchlargerthanK(~(E).sAsxaconsequencerecurren!tsetscansimplybMedescribedasthoseha!vingpositiv!esin!vdDariant&.measure. Moregenerallye,?(4.4)c!haracterizesrecurrenceresp.transienceofsagiv!ensystemthroughhittingprobabilitiesaswellasthroughthepMotentialkernels{Vincompleteanalogytothew!ell-knownVcriteriafordiscreteMark!ovVchains.ThessectionCconcludesb!yderivingastrongversionofirreducibilityandapMeriodicityCinsthefrecurren!tcase.-sFerom$NtheabMo!ve$Ndescriptionofrecurren!tsetsitiseasilydeducedthat,>Swithprob-sabilit!y1,4thesetoflimitpMointsofarecurrentsystemequalsthesuppMortMDofthesin!vdDariantLmeasure,indepMenden!tlyoftheinitiallaw..By(5.2)itisjusti edtocallsMVthe\7attractor"ofthesystem,whetheris niteornot.SinceMVneednotq3%썠:-iӍ{-sbMecompact,ltostateitsself-similarit!yingeneralinvolvesapassagetotheclosure. sNev!erthelessitallowstocharacterizetheattractorasthesmallestnonemptyclosedssetthatismappMedin!toitselfbythemappingsHzn Fwithprobability1.Amoreex-splicitc!haracterizationoftheattractorisestablishedattheendofSection5:>MscanqbMeiden!ti edwiththesetofconstantsintheclosedsemigroupgeneratedbythesmappingsthatde nethesystem,i.e.;thatarecon!tainedinthesuppMortNofr.Ifsthestatespaceisonlypartiallyordered,actually\7loMcalboundedness7"ofE\EhastosbMefsupposedhere(seethe nalsection).-sAsmen!tionedabMove,theexistingliteratureconcernsalmostexclusivelypMositivesrecurren!t8systems,Ni.e.Ythecaseofa niteinvdDariantmeasure,NastreatedinSection6.sThere=aren!umerousstudiesconsideringE+=Rz+ Z withitstotalorderandrandomstransformationspreservingthisorderas,forinstance,Alpuim/A!thayde[1],Goldies[19],3bHelland/Nilsen[21],Lundetal.[30],Yeaha!v[40].Thereis,moreo!ver,3banumbMersof$opapMersdealingalsowithonlypartiallyorderedstatespacesasBougerol/Picards[10],hDiaconis/FereedmanY![12,Section3],Glasserman/Yeao[18],Jarner/Tw!eedie[23],sMairesse[31], Rac!hev/SamoroMdnitsky[34],Rac!hev/TeoMdorovic[35],andinparticularsBrandtetal.[11,Section1.3](where,ho!wever,ametriccompatiblewiththeordersispMostulated).Thefullbene tfromorderandmonotonicit!yturnsoutundertimesrev!ersal,i.e.WreplacingleftbyrightcompMositionoftheunderlyingrandomtrans-sformations.$Thehresulting(non-Mark!ovian)h\7dualproMcess"(Yn0An.;bnN0)hincreasesspMoin!twise,Iand)?itslimitY.satis esazero-onela!w,distinguishingpMositiv!eandnullsrecurrence,Xwhere5inthepMositiv!erecurrentcase(6.7)revealsthestationarydistri-sbutionvasthela!wofYn.NoCombinedwiththeasymptoticbMehaviouroftheoriginalsproMcessE8(X0AnP;1n 0)thisyieldsthecriterion(6.8)andtheuni edc!haracterizationofspMositiv!efrecurrence,nullrecurrence,andtransienceattheendofthesection.-sAnotherTcriterionforpMositiv!eresp. nullrecurrencecanbMeestablishedbycon-ssideringfthehittingtimesTB ofsetsB' thataredeterminedb!ytheorder.WhilesforMFincreasingin!tervdDals[x;b1]itisshownalreadyin(1.2)thatE(T:u[x;j]),vregardlesssof/theinitialla!w,isalways nite,(7.1)exhibitsadi erentbMehaviourfordecreasingsin!tervdDals[0;1x].Now,.withT<yDB denotingthehittingtimeofBfortheproMcessstartingsat*yd,2[foran!yx2Mthe*meanpassagetimeE(Tx[0;x]G)is niteifandonlyiftheָssystemPispMositiv!erecurrent.+ThemaintoMolintheproofis(asligh!textensionof)thesrecurrenceT!theoremofKac.Section7concludesb!ythetopMologicalanalogue(7.3)sofstheclassicalresultonmeanpassagetimesfordiscreteMark!ovschains(hereagainsloMcalfboundednessen!ters).-sInQSection8theratioergoMdictheoremsareconsiderablystrengthenedforpositiv!esrecurren!t?systems.gInthiscasethelawszn ^=L(Yn0An.)ofthedualproMcess,)hencesalso thela!wsL(X0AnP)`@=zn,fare easilyseentocon!verge weaklytothestationarysdistribution#.By(1.6)thisextendstoarbitraryinitialla!wsandfromfunctionsinsC(E)tothoseintheclassR(E).#Theresultingcon!vergenceforalldecreasingsetsssuggestsɛthein!troMductionofametricdfordistributionsonthespaceEk-(lackingsan!yrMmetricstructure)suchthatmetricconvergenceisstrictlystrongerthanweakscon!vergence. HEmployingtanideainDubins/Fereedman[13](apparen!tlythe rststreatmen!t؏ofiteratedfunctionsystems,thoughrestrictedtothecaseE=^@[0;11]),itsis4thenpMossibletov!erifyevend(znP;1)!04in(8.3){withgeometricconvergenceswhenev!erthestatespaceisbMounded.ThesectionconcludesbyprovingthemixingspropMert!yBforthestationaryversion(XznP;bn 0)Bandderivingfromitin(8.5)afairlysgeneralfla!woflargenumbMers,againindependen!toftheinitiallaw.q49썠:-iӍ{--sBasicassumptionthroughouttheprecedingsectionsis{apartfromtheexistence sofqminRE{qtheirreducibilit!yofthesystem.>Underatotalorderingthismeansinsfactlnorealrestriction: simplyreducethestatespacetothoseelemen!tsthatcanbMesreac!hedfrom0inthesenseof(1.1).)ButthisproMcedureneednotworkingeneral,sas#isdemonstratedinSection9.SThearisingproblems,B*ho!wever,can#bMesettledassabMo!ve,wheneverthesystemisstrictlyorder-preserving.JThisisafamiliarh!ypothesissinEanemoMdels,Yforinstance,whereitconcernstheoMccurrenceofzeroen!triesinthesassoMciatedrandommatrix.WFeorstrictlyorder-preservingsystemsthereductionw!orkssas inthetotallyorderedcaseandleadstoastatespacethatisinadditionloMcallysbMoundedf(asrequiredforonehalfof(5.7)and(7.3)).-sThe/`concludingSection10collectsthenecessarymaterialonorderedtopMologicalsspaces.Thefundamen!talfactneededhereistheexistenceofanopMenbaseconsistingsofordercon!vexsets.,`ThisisasimpleconsequenceofNac!hbin'sextensiontheorem,si.e.LofTietze'sextensiontheoremcarriedo!vertoorderedtopMologicalspaces.ResultssnottobMefoundinthestandardreference[33]concerntheAlexandro!vcompacti -scation1oforderedtopMologicalspaceswithalo!wer1boundin(10.5),citsconsequencessforKtheappro!ximationbynonnegativedecreasingfunctionsfromK(~(E)in(10.6)ands(10.7),wandN nallytheexplicitrepresen!tationofsetsinthealgebraNV(E)resp.Թofsfunctions inthev!ectorlatticeV|(E)seenfromtheviewpMointofuniversalmeasura-sbilit!yfin(10.8).-s0.7Preliminaries.ThroughoutZSections1{9thestatespaceEJisanorderedstopMologicalspace(thede nitionbeingrecalledinSection10),whereregardingthestopMologyfitissucien!ttosupposethats(E1) LE|%': 3 cmti10islopcalFlycompactandsecondcountable,swhilefregardingtheorderitisnecessarytosuppMosethats(E2) LE|hasalowerbpound,denotedby0.sIfEl== E[ f1gistheAlexandro!vcompacti cationofE(1bMeinganisolatedpoin!tsif>|Eiscompact),dtheorderisextendedfromEtoE inaccordancewith(E2)b!ysletting5W1bMeanupperboundofEa.Thismak!esE againanorderedtopologicalsspacefifandonlyif(see(10.5))s(E3) LthedepcreasinghulFlofacompactsubsetofE|iscompact,sen!tailingUcompactnessinparticularforeachintervdDal[0;1x].4Inthesequelconditionss(E1)f{(E3)willbMetak!enforgranted,callingEGadmissibleinthiscase.-sThetopMologicalnotationsareasusual:NG(E)/NF(E)/NK(E)/NB(E)denote,srespMectiv!elye,70thecclassofopen/closed/compact/BorelsubsetsofE,70whileC(E)c/sK(~(E)/BU@(E)stand,.respMectiv!elye,fortheclassofboundedcon!tinuous/compactlyssuppMortedfcon!tinuous/Borelfunctionsf8c: E9!R.-sTheR0follo!wingnotationsrefertotheorder:IfAisasubsetofE,cthenAq% cmsy6#andA"sdenote,frespMectiv!elye,itsdecreasingandincreasinghull,i.e.u-Az#_= C0Cu cmex10C[LZx2A$[0;1x]and/FAz"= C0C[LZx2A[x;b1]:sAOiscalled(orpder)convexifA$3xy2AOimplies[x;1yd]$AOor,zequivdDalen!tlye,if sA=A#9\A"UT.w@Arro!ws.arealsousedtodenote,Pforinstance,b!yNG#UT(E)theclassofsdecreasingfsetsinNG(E)orb!yBU@"(E)theclassofincreasingfunctionsinBU@(E).q5N썠:-iӍ{--sTeoݩextendthenotionofbpoundedvariationݩfromtotaltopartialorder,zletV|(E) sbMeftheclassoffunctionsf8c2 BU@(E)satisfying`suptCnJ̟C0XƟ۹k62t : cmbx9N/jf-(xȮk6+1)nf(xȮk#)j :xz1ʫxz2:1::wlCo,<1㍑sandȹNV(E)theclassofsetsB{)2NB(E)with1B <2V|(E).DUniv!ersalnullsetsdis-sregarded,[V|(E)7Listhelinearspacegeneratedb!ythebMoundedmonotonefunctions,swhileNV(E)isthealgebrageneratedb!ytheconvexsets(see(10.8)).gTheclosuresof&V|(E)withrespMecttotheuniformnorm,yieldingtheclassofrpegular&functionss(ha!vingD nitelimitsfromleftandrighteverywhereonEa)underatotalorderingsandfincludingK(~(E)inthegeneralcase(see(10.6)),isdenotedb!yR(E).-sM(E)2denotestheclassofallloMcally nitemeasuresonNB(E),Iwhic!hdueto(E1)sareRadonmeasuresandcanbMeiden!ti edwiththepositiv!elinearfunctionalsonsK(~(E).dXIf}fdenotesthein!tegralofafunctionf-,thenM(E)isendo!wed}withsthevdDague(w!eak*)topMologye,xgeneratedbythemappings7!f;bf2K(~(E);(thescorrespMondingcon!vergenceisdenotedby!|vr.OnthesubspaceMz1(E)ofprobabil-sit!y2?measuresthisinducestheweak(narrow)topMologye,U5generatedbythemappingss 7!f;bf8c2C(E);fthecorrespMondingcon!vergencefisdenotedb!y!wD.-sThe=spaceC[E]b:=C(E;1E)=ofcon!tinuous=mappingsfromEPtoE,rendo!wed=withsthe#compact-opMentopologye,BisaP!olishspacedueto(E1),andthiscarrieso!ver#tostheclosedsubspaceH[E]oforder-preservingcon!tinuousmappingsh :E9!E,bMeingsacen!tralobjectofthispapMer.{(UndercompositionH[E]isasubsemigroupofC[E],swhere{compMositioniscon!tinuous{bytheloMcalcompactnessofE andthusmeasurablesb!yfthesecondcountabilityofC[E].-sThe_-mainobjectofthispapMeristhespaceN[E]ofdistributionsonH[E].ThessemigroupstructureofH[E]inducesacon!volutioninN[E],1makingthisspaceassemigroupfitself.OccuringpMo!wersfaredenotedb!yrnZ,hence,&CZ5DHl[Erظ]L8f-(h(x))1rznZ(dh) =&CZ ٟHl[Erظ]$U:::2a&CZ;“Hl[Erظ]RgVf-(hz1.n:::;nhzn{(x))r(dhz1):::l(dhznP)sfor.x 2E;bf72C(E).andn 2N,%while.r0 Kistheunitmeasure"Ȯh ?withhbMeingstheiden!tymap.1SinceH[E]issecondcountable,sthesuppMortNiswell-de nedfors2 N[E].-sNo!wgthestoMchasticmoMdelcanbeformallyin!troduced.3Letbegiv!enanadmissiblesspacefEGand,onsomeprobabilit!yspace( ;1NA;P),s{ a;sequenceofindepMenden!trandomvdDariablesHzn Vw:' !H[E];withidenticalsdistributionf2 N[E],s{ afrandomvdDariableXz0ʫ: !EGthatfisindepMenden!tof(HznP;1n 2N).sThisfde nesanorpder-preservingrandomdynamicalsystemfb!yݍjXzn= HznP(Xzn1)for#n2N1:sThereforethedistributionof(XznP;bn/0)iscompletelydeterminedb!y;^andthe sinitial"la!wz0 D3=/L(Xz0). R=IfinparticularXz0=/x,thiswillbMeexpressedb!ythesnotationf(XxAn$;bn 0),i.e.ݍpXzx:jn /= Hzn<n:1::;nHz15(x)for#x2EandEn01:sTh!usfforgeneralz0fjconditionalprobabilitiesaregivenbyy}PzxH((XznP;bn 0)2B)=P((Xzx:jn$;n0)2B)swithfananalogousequationforconditionalexpMectations.q6`̠썠:-iӍ{--sAsusual,theinitialla!wislargelyofsecondaryimpMortance,andtheprimary scompMonen!tisthedistributionr.Therefore,;wheneverpMossible,;itwillbebrie ysreferredtothe(dynamicpal7)system(E;1r).QAllnotionstobMede nedinSectionss1{9willdepMendonr,butthisdependencewillbesuppressedinrelatednotationss(asfforthesuppMortNCzabo!ve),fbecauseXissupposedtobe xed.-sClearlye,(XznP;bn|0)isahomogeneousMark!ovchain.ItstransitionkernelPstransformsf(nonnegativ!e)functionsf8c2 BU@(E)intoPVf"givenby.a|6PVf-(x) =&CZ ٟHl[Erظ]$Uf(h(x))1r(dh)for#x2EsandfameasureonEGin!toP+givenbyjSPV(B) =&CZ ٟZEar(h(x)2B)1(dx)for#BK2NB(E);swhic!hfinthed nitecaseequalsdQPV(B) =&CZ ٟHl[Erظ]"(h(x)2B)1r(dh)for#BK2NB(E):-sPishbaFeellerk!ernel,twhichhbinadditiontransformsincreasingfunctionsin!tofunc- stionsfofthesamet!ypMe.By(E2)thisimpliesinparticular:-sQ- 3 cmcsc10Q(0.1)'Lemma dThesepquence("z05PVn-;bn 0)isstopchasticalFlyincreasing,i.e.1ꍑqE(f-(Xz0:jn1)) E(f(Xz0:jnP))for#c0f8c2BU@z"(E)1:-sQProof.nThisfisimmediatefromtheequations1E(f-(Xz0:jn1)) =&CZ ٟZEaPVzn1 "f(0)1PV(0;dx); C&/E(f-(Xz0:jnP)) =&CZ ٟZEaPVzn1 "f(x)1PV(0;dx):FNH 3 msam10He©-sFinallye,ithastobMemen!tionedthatthepassagefromthedistribution tothe sk!ernelfP+ingeneralisnotinjective.-s1.Irreducible}systems.ZLet(E;1r)bMeadynamicalsystemasin!troducedinsthe)uprecedingsection.g Then,J9toclassifyitasrecurren!tortransient,J9requiressomescomm!unicationMAstructuretopreventthestatespacefromsplittingintodi erentsclasses.ItAturnsouttobMesucien!tthat,hstartingfromtheminimalstate0,eac!hsstatefxwithpMositiv!eprobabilitycanbMereachedorexceededinthefollowingsense:-sQ(1.1)nDefinition oThe%system(E;1r)ݨiscalled\7(upwarpds)irreducible7",Difforsan!yfx 2EGthereissomen2Nsuc!hthat1ꍑnnPVzn-(0;1[x;])=P(Xz0:jn x)=rznZ(h(0)x)>01:-sIt aisimmediatefrom(0.1)thatthesystem(E;1rnZ)isirreducibleforalln2N, swhenev!erfthisholdsforonen 2N.-sThe;question,`whetherirreducibilit!ycanbMeaccomplishedbysuitablyreducingsthestatespace,sispMostponedtoSection9.ϖAsanexampleconsiderthe\7Can!torq7t썠:-iӍ{-ssystem7",assigningbmassr(fhzidg)J=K"}1"}fe@PA2~tothet!wobmappingshz1 N:x7!x=3and shz2ʫ: x7!x=3n+2=37:thefadequatestatespaceinthepresen!tsettingisE9= [0;11[.-sInZspiteofitsw!eakappMearance,ntheconditionin(1.1)hasstrongrecurrencesimplicationsfforincreasingin!tervdDals:-sQ(1.2)zProposition(IfSthesystem(E;1r) isirrpeducible,qthenSforarbitrparyinitialslawandalFlx 2Eb܍s(a)P(lim1sup!En!1#XfXzn xg)=11;噚s(b) E(T:u[x;j]) <11;swherpeTB X:= inf0ifn 2N:Xzn2Bg(1)forBK2NB(E).-sQProof. xtBy\assumption#@:=rnZ(h(0)x)>0\forsomen,whic!hyieldsby smonotonicit!yfandindepMendenceDE[P(T:u[x;j]> kX?n).tPz0(T:u[x;j]> kX?n)E.tC0CYßqG0i(yzn;bn2N)2EzN F:-sFeorfan!yx 2EGandflA2NsatisfyingP(X0'͍lx)>0thereforenHdfXz0:jl xgB\C 2C0\ן۹m2N4R=C0C[AA۹n2NXffXz0:jm kyznPgC ּ6=;1;&shencefform =lݏinfparticularfXz0:jl xgB\C 2C0[ן۹n2N0"fXz0:jl yznPgC ּ6=;1:0isThisfimpliesx 2A#forfA:=fyzn:n2Ng.>H-sTheffollo!winginequalityisthecrucialtoMolinthesequel:q8 썠:-iӍ{--sQ(1.4)vTheoremLpet^thesystem(E;1r) beirreducibleandsuppose0 f8c2BU@"(E). sThenforalFlx 2E`C0CXpL<_n0QE(f-(Xzx:jn$)nf(Xz0:jnP))NE(Tz0[x;j])bsup % -n0%E(f(Xz0:jnP))1; C荑swherpeT<VyDB X:= inf0ifn 2N:XyAn2Bg(1)foryo:2E|andBK2NB(E).-sQProof.hIffóisreplacedb!yf{^NkX?;bkb2 N,@thenthecorrespMondingdi erencesonsthe.left-handsideincreaseforkFg!(1tothedi erenceinquestion,Qi.e.w`f\ma!ybMesassumedntobMebounded.5Inaddition,6lb!y(1.2b)thestoppingtimeT:= T0[x;j]}maybMeָsassumedtobMe niteev!erywhere.Then,=фE(1Anm&cf-(HT.:+m1?n:1::;nHT.:+1P(Xz0:jT)))M>фE(1Anm&cf-(HT.:+m1?n:1::;nHT.:+1P(x)))M>=фP(AznmɌ)1E(f-(Xzx:jm));swhere@theinequalit!yisaconsequenceoffnbMeingincreasing,gwhilethe nalequal- sit!yeusesthefactthat(HT.:+1};1:::l;1HT.:+mS)isindepMendentofTanddistributedass(Hz1;1:::l;1Hzm).But,ff"isnonnegativ!eandthus*;C0CX:5<_0mn\=f-(Xz0:jm)C0CX"ҟZ0m;jT.:+mnORMf(Xz0:jT.:+mS)=C0CX"ҟ<_0mn7H1Anm&cf(Xz0:jT.:+mS)1:$sThisfleadsb!yintegrationandinsertingfrom()to^lC0CXn<_0mnנE(f-(Xzx:jm)nf(Xz0:jm))RKX^lC0CXn<_0mnנE(f-(Xzx:jm))BC0CX<_0mn6P(AznmɌ)1E(f(Xzx:jm))KX=^lC0CXn<_0mnנ(1nP(AznmɌ))1(E(f-(Xzx:jm)f(Xz0:jm))+E(f(Xz0:jm)))1;swhic!hfby(0.1)impliesC0CX0<_0mnV P(T nnm)1E(f-(Xzx:jm)f(Xz0:jm))sfC0CX0<_0mnV P(T> nnm)1E(f-(Xz0:jm))BsfE(TV)1E(f-(Xz0:jnP)):sSincefP(T nnm)"1fforn!1,theassertionfollo!ws.qXH -sApartfromarisingmeasurabilit!yproblemstheprecedingresultcanbMeeasilysextendedftoalargerclassoffunctions:-sQ(1.5)Proposition5LpetYthesystem(E;1r) beirreducibleandsupposef8c2 V|(E).sThenforalFlx 2Es(a)ϟC0CXaɟ<_n0fE(jf-(Xzx:jn$)nf(Xz0:jnP)j) <11;sandforarbitrparyinitiallaws(b)C0CX <_n0$jf-(XznP)nf(Xz0:jnP)j <1a.s.g:q9 y썠:-iӍ{--sQProof.UAccording]to(10.8)therearebMoundedincreasinganduniv!ersallymea- ssurablenfunctionsfziy0suc!hthatfBp=fz12rfz2.SincethevdDariablesXxAn 9: !ENaresuniv!ersallymeasurableaswell,providedtheunderlyingprobabilityspace( ;1NA;P)sisZ7assumedtobMecomplete,+theproofof(1.4)w!orksforfzid,+too.PInviewof(1.2b)sthisfpro!vesassertion(a),whichbyFeubiniimpliesassertion(b).VώH-sFeort someapplications,~wheresummabilit!yofthedi erencesmaybMereplacedbyscon!vergenceftozero,theclassofadmissiblefunctionscanbMeenlargedoncemore:-sQ(1.6)R&PropositionhLpet/thesystem(E;1r) beirreducibleandsupposef8c2 R(E).sThenforarbitrparyinitiallaw獒f-(XznP)nf(Xz0:jnP)!0a.s.g:-sQProof.nThisfisimmediatefrom(1.5b)andthede nitionofR(E).1H -sIt%hastobMeemphasizedthat,ev!enunderatotalordering,thelastresultma!yfailsforbZfunctionsf8c2 C(E)..Teoconstructanondegenerate(i.e.recurren!t)counterexam-spleMconsiderthefollo!wingsystem:PonE9= Rz+ letthesuppMortN$aof9consistofthesfractionalflinearmappings獑}zhz1ʫ: x7!xn+1 and%Fhz2: x7!x1=(xn+1)1:sStartingfwithnz0ʫ:= 0c!hoMoseasequence(nȮk#;1kb0)suc!hthatYmlim1sup Y_k6!1P(AȮk#) =1 with)JAȮk.9:=C0C[L<_ni?kAacmr61ڹEandtransience.ϧIngeneral,#theprobabilit!ythattheproMcesss(XxAn$;bn!0)visitsasetB&2!NB(E)in nitelyoftenneitherobMeysazero-onela!wnorsisuindepMenden!tofthestartingpoin!txdo2Eq{uascanbeseen,forinstance,b!ythesCan!torsystemevenforappropriatecountablesets.Butanapplicationofthe nalsresultfofSection1yields:-sQ(2.1)Theorem aLpetcthesystem(E;1r)beirreducibleandsupposeBru2NV(E).sThenrPClim1sup! zZn!1+$/fXzn2 BgC 2=N0or611;bswherpethevalueisindependentoftheinitiallaw.%10 썠:-iӍ{--sQProof.nChoMosingff8c= 1B X2V|(E)in(1.6)pro!vesklim1sup!q%n!1:fXzn2 Bg=lim1sup!n!1&bfXz0:jn2Bg=:Az0 a.s.s:+sApplying6thisresulttothedistributionofX0g1asinitialla!wandtheshiftedsequence s(HznP;1n >1)fyields~XAz0ʫ= lim1sup!n!1&bfHzn<n:1::;nHz25(0) 2Bga.s.:sRepMeatingtheargumen!tshowsA0!tobMecontainedinthecompletedtaild eldof s(HznP;1n 2N),fandtheassertionfollo!ws.H-sThefprecedingresultjusti es:-sQ(2.2)/Definition 1Letthesystem(E;1r){bMeirreducible.vThenB2NV(E)isscalledf\7rpecurrent",fif^d:bPClim1sup! zZn!1+$/fXzx:jn /2 BgC 2=1forfone(orall)\Rx2Et;7sandf\7trpansient"otherwise.-sTheffollo!wingconsequenceofrecurrencewillbMeneeded:-sQ(2.3) WLemma WrLpetthesystem(E;1r) beirreducibleandBK2 NV(E)berecurrent.sThenforany nitesubsetAofE|therpeexistssomen 2NsuchthatPSP(Xzx:jn /2 BforalFl'ʰx2A)>01:-sQProof.nAccordingfto(1.6)xr1BN>(Xzx:jn$)n1B(Xz0:jnP) !0a.s.%forfallI2 x2A1:sSincefAis niteandB3 isrecurren!t,thisimpliesNlim1sup!TI#n!1qfXzx:jn /2 Bforfall& x2Ag=lim1sup!n!1&bfXz0:jn2Bg= a.s.:7sThereforeftheprobabilitiesinquestionev!ensumupto1.nbH -sItisimmediatefrom(1.2)thateac!hincreasingsetB86=\;isrecurrent.$Feorde-screasingfsetsthefollo!wingcriterionisavdDailable:-sQ(2.4)Proposition LpetDthesystem(E;1r)~beirreducible.qThenforarbitrarysinitiallawasetBK2 NB#UT(E)isrpecurrentifandonlyif lC0CP؟<_n0*P(Xzn2B)=1.-sQProof. TDueWto(2.1),%theconditioniscertainlynecessarye.ToWpro!veitssu-sciencye,ĉletOBbMetransien!tandassumewithoutrestrictionXz0 H.=*0.ThereforethesvdDariableZ):=`C0CP۹n2N&1BN>(X0AnP)satis es#`:=P(Zl7))<1forsuitablel2`N.xhUsingsthefstoppingtime~$Tc.:=inffn 2N:؟C0CX"ҟ<_1mn7H1BN>(Xz0:jm)=l7)gswithfrespMectto(HznP;1n 2N),fitfollo!wsasintheproofof(1.4)that%11 썠:-iӍ|R*P(Z( k+pl7))h#=zP(T< 11;؟C0CX"ҟ۹n2N,G1B !o(HT.:+nn:::;nHT.:+1P(Xz0:jT))(k+1)1l7))M󍍍h#zP(T< 11;؟C0CX"ҟ۹n2N,G1B !o(HT.:+nn:::;nHT.:+1P(0))(k+1)1l7))ljh#=zP(Z( l7))P(Z(k+n1)1l7)):dsTherefore P(Z( k+pl7))#zk#forfall7tkb2Nsandfth!usindeedE(Zȁ) <1. H-sTheftopMologicalstructureen!tersinthecentralclassi cation:-sQ(2.5)Definition uAnuirreduciblesystem(E;1r)c(ortheproMcess(XznP;bn0)sor{thek!ernelPV)iscalled\7rpecurrent",Nif{NV(E)con!tainsacompactrecurrentsetKȁ,sandf\7trpansient"otherwise.-sInSection4,$usingthein!vdDariantmeasure,itwillbMesho!wnthatthesetKÚrequiredsinfthisde nitionma!yinfactbMepostulatedtobeanin!tervdDal[0;1yd].-sTheassumption(E3),connectingcompactnessandorderinSection0,isessen!tialsforfthefollo!wingcriterion:-sQ(2.6)WTheoremLpet&thesystem(E;1r) beirreducible.ThenforarbitraryinitialslawthefolFlowingcponditionsareequivalent:ds(1),(E;1r)istrpansient:G;噚s(2)Xzn! 1a.s.g;s(3)tܟC0CX?֟<_n0DP(Xzn2 Kȁ)<1foralFl3> K(2NK(E)1:-sQProof.The equivdDalence(1),(2)follo!wsfromtheexistenceofasequenceof ssets\KȮl^2NK(E)suc!hthateachK_2NK(E)isincludedinsomeKȮl!.ٿTheimplications(1))(3)isaconsequenceof(2.4),bMecausethecompactsetKȁ,dueto(E3),ma!ysbMefassumedtobedecreasing.The nalimplication(3))(1)isstraigh!tforward.9H-sRestrictingcondition(3)todecreasingsetsKȁ,itfollo!wsby(0.1)thatthesystems(E;1rnZ)fisrecurren!tforalln 2N,fwheneverthisholdsforonen 2N.-sFeorXanapplicationconsideran\7exc!hangeproMcess"ashasbMeenstudiedin[21]sandfinitsbMestaccessibleformisgiv!enbytherecursiond]BXzn= (Xzn1n1)_UznPfor(_n2Nswithasequenceofi.7i.d.dvdDariablesUzn 0.Here,Xzn1&istheutilit!yofsomeequip- smen!tinuseattimen01,losingoneunitduringpMeriodn,andUzn theutilit!yofasnewequipmen!tavdDailableattimen.QThecorrespMondingmappingsh :x7!(x1)_u,srestrictedftothestatespacedWE9= fx0:P(Uznx)>0g1;sbMelongtoH[E]forallu)12E.Withۣ2N[E]bMeingtheassociateddistribution,Nthe ssystemf(E;1r) isob!viouslyirreducible.TheexplicitrepresentationY(Xz0:jn= (Uz1.n(n1))_:1::;_(Uzn11)_UznPfor(_n2N%12 q썠:-iӍ{-simpliesfb!yindepMendenceغ|P(Xz0:jn yd)=C0CY<_0m0.sFeorfE9= Rz+ bMothcasesarepossible:"s(1) nifPUznP;bn&2N;ha!vethecommondensityfz1(x)&=(xz2+1)2 \|,thenFV(yd)&= yZ ȉfe ty+n1>simplies@aZ C0CXj<_n0~P(Xz0:jn yd)=C0CXO<_n02y(ȉfe ty+nnF}=1forfall2Qeyo:>01;^&si.e.thefproMcessisrecurren!t;@as(2) ifvfthedensit!yisreplacedbyfz2(x) =2x(x+1)3 \|,thenvfFV(yd) =( y33ȉfe ty+n1 )z2+S,andvfitsfollo!wsfsimilarlythattheproMcessistransient.-sIt}shouldbMenotedthat{inspiteofthecon!trastingasymptoticbeha!viour{the svdDariablesUzn g,bMeha!vesimilarlyinbMothcasesasfarasitconcernstheexistenceofsmomen!ts,@duetofz1(x)ofz2(x)2fz1(x)forxo1(foracon!tinuationseeSectionss3,f4,and6).-sIn2general,٥asindiscreteMark!ov2chaintheorye,٥itmaydemandsomee orttosdecide(whetherasystem(E;1r)&isrecurren!tortransient.e]FeoranexampleconsidersE=[0;11[<withxassigningmassKo91o9fe@PA2vtothemappingsde nedb!yhz1(x)=(xҬ+1)=2sandfhz2(x) =x2fj(solvdDableb!yconjugation).-s3.H_IntvLariantmeasures.U4AsindiscreteMark!ovchaintheorye,acentralquestionsconcernstheexistenceanduniquenessofin!vdDariantmeasuresintherecurren!tcase.sIn)accordancewiththetopMologicalassumptions,6thisquestionwill{andm!ust{sbMe75treatedwithintheclassM(E)oflocally nitemeasures.ITheeasytaskhereissexistence,^,where9kgeneralresultsb!yFeoguel[17]apply.Sincetheargumen!tscanbMessimpli ed,fduetomonotonicit!ye,theproMofofthefollowingassertionisoutlined:-sQ(3.1)'Proposition dLpetthesystem(E;1r) beirreducibleandsupposeغt 0 go:2K(~z#}(E) with)hC0CX9b<_n0M.E(gd(Xz0:jnP))=11: 6sThenthemepasures%zn2 M(E)de nepdby;Pz%znP(f-) :=C0CXO<_0m0.sThenfor0 f8c2BU@#(E)withsuppf KkandalFlx2E3鍍YC0CXj3<_0my4(1K;(Xzx:jm)n1K(Xz0:jm))1C. C0X<_0m-(E;1r)EandexhaustE߿b!yanincreasingsequenceofsetsKȮl 2NK#UT(E),wheres(KȮl!)>0Fand09(KȮl!)>0FandinadditionrecurrenceofKȮlma!ybMeassumed.~Thensanfapplicationof(3.3)yieldsconstan!ts Ȯl.suchthatVy^z09(1Ki?l f-) = Ȯl(1Ki?lf-)for#0f8c2BU@z#(E)1:%15=썠:-iӍ{-sInsertingfگ=1Km ;1m6=l7);pro!ves Ȯl)tobMeaconstant indepMendentofl>andthus, swithfKz0fjasde nedin(10.6a),+z09(1Ki?l f-) = n(1Ki?lf-)for#f8c2Kz05:sThiswequationextendstof(2lK(~(E),{bMecausetheuniformappro!ximationoff3bysasequence(fȮk#;bky2:N)fromKz0 Uaccordingto(10.6a)canbMecarriedoutwithallsfunctions6bvdDanishingoutsidea xedcompactset(otherwisem!ultiplybysomegȮlX*froms(10.7)).Thefassertionfollo!wsforlA! 1.*H-sAstusualtheactuallyone-dimensionalfamilyofin!vdDarianttmeasuresbs2M(E)swillfbMebrie ycalledtheinvariantmepasurefinthesequel.-sIt}hastobMeemphasizedthat,ev!enunderatotalordering,theuniquenessstate-smen!tpHin(3.4)concernsloMcally nitemeasuresonlye.;TopHexhibitacounterexample,sc!hoMosefE9= [0;11[andletXassignmassKٙ1ٙfe@PA2 6tothepiecewiseanemapsde nedby6khz1(x) ==1=ڟȉfey t3 (2xn+1) and%Fhz2(x) ==1=ڟȉfey t2 x_(2x1)1:EsThen(E;1r)Jisclearlyirreducible,andtheuniformdistributiononE;Giseas-sily~c!heckedtobMea niteinvdDariantmeasure. $WithasinitiallawtheseriessC0CP(S<_n0<P(Xzn 3 yd)&sdiv!ergesforallyL>0,uandth!us(E;1r)isrecurren!tby(2.6).sOnftheotherhand,itisnothardtoc!heckfthatz0:= C0CXOZx2D(x1"zxHwith.DX:= E\BQsde nesanotherin!vdDariantmeasure,which,however,isd niteonlye.WInciden!tally,stheP0existenceofbMotha niteandanin nite,anbutd nite,in!vdDariantP0measureinthissexample#pro!vestheDoMeblin-HarristheoryofrecurrentMarkovchainsonanabstractsstatefspacetobMeinadequateinthepresen!tsetting.-sFinallye,itmhastobMemen!tionedthat,againev!enunderatotalordering,thereissno;con!verseof(3.4),`)i.e.therearetransientsystemswithauniquenontrivialandsloMcally5 nitein!vdDariant5measure.pFeoranexampleconsiderthetransien!tcase(2)ofstheCexc!hangeproMcessfromSection2.tItiseasilyseenthatameasure2M(E)Cissin!vdDariantfifandonlyifthefunction(yd) :=([0;1y])fsatis es ]K(yd) =(y+n1)1FV(y)=(y+n1)1( y33ȉfe ty+1 )z25for)Dyo: 01:sTherefore]z0(yd) =y2de nes]asolution,,whileforan!yothersolutionbyiterationh(yd) =(y+nn)1( -y33ȉfe ty+n)z2for'yo: 0andFn2N1:7sFeorf0 yo:m2Nthispro!videsbymonotonicitythebMounds9e@e(yd) =(y+nn)1( -y33ȉfe ty+n)z2ʫ (10B+n)( -y33ȉfe ty+n)z2ʫ= (1)ydz2( ndȉfe ty+nh)z25;"h@D(yd) =(y+nn)1( -y33ȉfe ty+n)z2ʫ (m+n)1( -y33ȉfe ty+n)z2ʫ= (1)1ydz2(33m+n33ȉfe t)y+nW)z25;swhic!hfforn !1fleadto(yd) =(1)1y2andfth!ustotheasserteduniqueness.%16썠:-iӍ{--s4.7ErgoYdictheorems.kThej rsttheoreminthissectionrequiressomeprepa- sration,fconcerningagainasymptoticindepMendenceoftheinitialla!w:-sQ(4.1),Proposition mLpetZthesystem(E;1r)berecurrentwithinvariantmeasuresandsupppose0 f8c2BU@#(E)withf> 0. vThenforarbitrparyinitiallawIniBC0CXz;<<_0m0,then(3.3)implies&eC0CX_<_0mSincethisconver-sgenceisdominatedb!ytheconstant1,itcontinuestoholdwhenintegratedbythesinitialfla!w.SH-sNo!wfameanergoMdictheoremcanbeestablished:-sQ(4.2)Theorem wLpetn}thesystem(E;1r)berecurrentwithinvariantmeasuresandsuppposef;bgo:2 K(~(E). vThenforarbitraryinitiallawIn]C0CXn<_0m 0ma!ybMeassumedsinfthesequel.s(2)Sincetheuniformappro!ximationoffbyasequence(fȮk#;bkb2 N)fromKz0Zaccord-singsto(10.6a)canbMecarriedoutfrombelo!wandabo!veswithallfunctionsvdDanishingsoutside՘a xedcompactset(otherwisem!ultiplybysomegȮl`from(10.7)),efTmaybMesrestrictedftoKz0.s(3)Thereforeitissucien!ttoconsiderthecase0 fNr2K(~#}(E),whereinadditionsf8c> 0fma!ybMeassumed(otherwisereplacef"byf+ngd).s(4)Th!us!fOandgbMothsatisfytheassumptionsof(4.1),@whichjusti es nallythesrestrictionftotheinitialla!w"z0.-s2.No!w<_0m0.ThenproMceedasfollo!ws:s(1) l$Feorf+2oV|(E)c!hoMosearepresentationf+=ofz1 $ fz2 vaccordingto(10.8)andsa;constan!t Ifz1(0)}_fz2(0),so;thatfۺ=f-0g1=}f-0g2 withuniv!ersallymeasurablesdecreasingfunctionsf-0g1 c:=)_ J$ifz20andf-0g2 c:= J$ifz10. {IfinadditionssuppDf-0|i Kȁ,(ascanbMeac!hievedDbymultipyingwith1K;,(thenitfollowsasinsthefproMofof(1.5a)thattheargumen!tsfor(3.3)workaswellforthefunctionsf-0|i.s(2) QFeor(f82|R(E)appro!ximateuniformlybyasequence(fȮk#;bk;2|N)fromV|(E),swhic!hfcanbMecarriedoutfrombelo!wandabo!vefwithsuppfȮk nKJK/Jforallk2Ns(otherwisefm!ultiplyagainby1K;).㖴H-sAsa rstapplicationoftheseergoMdictheoremsconsidertherecurren!tcase(1)sofftheexc!hangeproMcessfromSection2.HereP(X0An yd)=83y=ڟȉfe ty+nn"implies!vhQ]E(f-(Xz0:jnP)) =W1=ڟȉfe tn&CZQf1H8=0$}=f(yd)1C gyHȉfe tn3+n1Cmں2(dydfor#n2Nusandfth!us|<4ZuE(f-(Xz0:jnP))1C. E(gd(Xz0:jn))!&CZQfٺ1H l0f(yd)1dy7ğC. 'h&ZQf'i1H0 g(y)dydfor#f;bgo:2 K(~(Rz+x):qsSincethiscon!vergencecarrieso!vertothecorrespMondingquotien!tsin(4.2),theinvdDari-san!twmeasureissimplytheLebMesguemeasurerestrictedtoRz+x.By(4.3)thisimpliessthat,4regardless>oftheinitialla!w,theproMcess(XznP;bnd0)>is\7equidistributed">onsRz+x,fi.e.~R]C0CXc,W<_0m-1Iq2(Xzm)!(Iz1)=(Iz2)a.s.9sforfsubin!tervdDalsIȮkofRz+ ofpMositiveand nitelength.-sBy\meansof(4.3)recurrenceofasetBp2NV(E)canbMeseentobesimplysequivdDalen!t-to(B) >0.uMore-generallye,28extending(2.1)and(2.4)andreplacingsetssb!yffunctions,thefollowingdichotomyholds:-sQ(4.4)rlTheorem Lpet,thesystem(E;1r)berecurrentwithinvariantmeasure.sThenfor0 f8c2V|(E)andanyx 2Es(a) f8c> 0impliesuX3C0CXhx<_n0|}Nf-(Xzx:jn$) =1a.s.&6andHC0CXY;ޟ<_n0m@E(f(Xzx:jn$))=11;%18(_썠:-iӍ{-s(b) f8c= 0impliesX3C0CXhx<_n0|}Nf-(Xzx:jn$) <1a.s.&6andHC0CXY;ޟ<_n0m@E(f(Xzx:jn$))<11:-sQProof.(a)ڢChoMoseK*62aNK#UT(E)withCR X~ZK@fd>0andapply(4.3)tothefunc- stionsf1K;f"and1K (inV|(E).Thisyieldsthe rst(andsecond)assertion.-s(b)fUsethein!vdDarianceoftoobtaines&CZnFCZEx˟CC0Xڟ<_n0 E(f-(Xzx:jn$))C(dx)=C0CX"ҟ<_n0)'PVzn-f8c= 01:sThereforethein!tegrandvdDanishesalmosteverywhereandthusby(1.5a)is nitesforfallx 2E.Thisfpro!vesthesecond(and rst)assertion.nDH-sTeogether,@(4.4)&zand(2.6)implythatthet!wo&zfamiliarcriteriaforrecurrenceresp.stransience{fromdiscreteMark!ov{chaintheorycarryovertothepresentsettinginstheffollo!wingform:s(1) LIff(E;1r) isrecurren!t,thenforx2supptheassertion>PzxH(Xzn2 Gin nitelyfoftenJ])=11;shence EzxH(jfn 0:Xzn2Ggj)=11;Usholds,fwhenev!erGisanopMenneighbMorhoodfofx.s(2) LIffXistransien!t,thenforarbitraryx 2EGthefassertion.EzxH(jfn 0:Xzn2Kȁgj)<11;shence ^PzxH(Xzn2 Kȁin nitelyfoftenK)=01;Usholds,fwhenev!erKnisacompactsubsetofE.-sClearlye,Bunderatotalorderingthecompactrecurren!tsetin(2.5)canbMerequiredstobMeanin!tervdDal.Nowitispossibletoextendthissimpli cationtothegeneralcase:-sQ(4.5)Theorem _kLpetc"thesystem(E;1r)pberecurrent.rThenthereexistsyI2pEssuchthatK(= [0;1yd]isrpecurrent.-sQProof. ԳSincei`(1.3)pro!videsy.2OE withinvdDariantmeasure([0;1yd])O>0,thesassertionfisanimmediateconsequenceof(4.4a).UH-sExplicitlye,thisdpro!vesanirreduciblesystem(E;1r) tobMerecurrentifandonlyifzC0CX:<_n0?}P(Xz0:jn yd)=1forfsome>yo:2Et:Ռ-sAs%anotherconsequenceof(4.4),arecurren!tsystemcanbMeshowntobMeirre-sducible^KandapMeriodic^Kinav!erystrongsense.Theessentialstepconcernsdecreasingsin!tervdDals:-sQ(4.6)Lemma 8Lpet$$thesystem(E;1r)Bberecurrentwithinvariantmeasure.sThenanyyo:2 E|with([0;1yd])>0satis esP(Xzy:jn yd)>0foralmostalFlV8n01:%19;썠:-iӍ{--sQProof.nThefMark!ovpropMertyandmonotonicityimplythatDX:= fn0:P(Xzy:jnyd)>0gsis3anadditiv!esemigroup.Thereforeitissucienttoproved%=13forthegreatest scommondivisordofDM.oNo!wtheassumptiond >1yieldsP(X<ynd+1 yd)=0forallsn 0fandth!usby(1.5a)lC0CXf<_n02P(Xz0:jnd+1 yd)<11:.sBut2thisleadstoacon!tradiction,ebMecausetheprobabilitiesP(X0An K[yd)decreasebys(0.1)fandsumuptoin nit!yby(4.4a). NH-sItPhastobMeaddedthatunderatotalordering([0;1yd])>0PinfactimpliessP(XyAn yd)>0iforalln0.gIndeed:9theiassumptionr(h(y)>y)=1iyieldssP(XyAn >4yd)=1Xforalln42N,jwhileX([0;1y])>0XimpliesXyAn 4ycin nitelyoftenswithfprobabilit!y1.-sNo!wf(4.6)canbMeconsiderablyextended:-sQ(4.7),Proposition mLpetZthesystem(E;1r)berecurrentwithinvariantmeasures. vThenforcponvexsetsBK2 NB(E)with(B)>0andarbitrparyinitiallawP(Xzn2 B)>0foralmostalFlV8n01:-sQProof.SFix rstyg2Ewith([0;1yd])>0,Hasisjusti edb!y(1.3).Thenapplys(4.4a)fand(4.1)togetkb2 Nsuc!hthats(1)MP(XȮk.9 yd)>01:sNext,fuse(4.6)toobtainlA2 Nsuc!hthats(2)P(Xzy:jn yd)>0for#nl Z:sFinallye,fapply(4.4a)tof8c= 1B and(2.3)toA=f0;1ydgto ndm2Nsuc!hthats(3)MP(Xzz:jm k2 Bforfall& zyd)>01:sCom!biningf(1){(3)itfollowseasilythat8P(Xzx:jn /2 B)>0for#nk++nl+m1:FNH-s5.The2attractor.Theffollo!wingnotationwillbMeusedinthesequel:-sQ(5.1))Definition Letjthesystem(E;1r)bMerecurren!twithinvdDariantmeasures.ThenftheclosedsetM70g).||ThentheinclusionML(!d)isob!viouslysequivdDalen!tfto5s(1)iC0CXyV<_n0Z1Gi?k (XznP(!d)) =1 whenev!er@pUGȮk~\nM0æforGȮk\nM0sandM\n[x;1yd] 6=;.-sQProof. {TheOFin!tervdDal[x;b1]isrecurrentby(1.2a),y~hencesatis es([x;1])$>0sb!y?J(4.4b).Now(1.3)providesyn 2 wEwith([x;1yd])>0,ewhic!himpliesthesecondsassertionfdueto(E~nnM1) =0.H-sThe/`self-similarit!yoftheattractorisadirectconsequenceofthetopMologicalsassumptions.!7Since/MCneednotbMecompact,-!ho!wever,it/involvesapassagetothesclosure:-sQ(5.4),Proposition mLpetZthesystem(E;1r)berecurrentwithinvariantmeasures. vThenitsattrpactorMsatis es$u#Mfx 2E9:h(x)xforfall&Qeh2Ng1:-sTheterm\7attractor"doMesnotmeanthat,regardlessoftheinitialla!w,theev!ent s\7Xzn2 M 8ev!entually"ٙhasprobabilit!y1{ascanbMeseen,forinstance,b!ytheCantorssystemG(E;1r),?where~hzi8 [E3nM1]aEnM forGi=1;b2.yInthepresen!tsettingtheresisfasubstitute:-sQ(5.6) Proposition vIf7thesystem(E;1r)sisrpecurrent7withattrpactorM1, thensforarbitrparyinitiallawЬ2P(Xzn2 M1z" foralmostalFlOn0)=11:-sQProof.:Teosettle rstthenecessarymeasurabilit!y,notethatMZisdcompact sandth!usM1"= C0CS 3/۹lK2N!hKȁ"RwlwithKȮl,o NK(E),HhenceKȁ"Rwl(|2NF(E)asfollo!wsbysequentialscompactnessfofE.ThereforeM1" -Yisinfactoft!ypMeFz.Since(5.5)implieshh[M1z"] (h[M1])z"_Mz"forfall7Xh2NpE;sthe:proMcess(XznP;bnF^0):sta!ysinM1" Q-almostsurelywheneverenteringthisset.IZButsb!yf(1.2a)thisentranceoMccurswithprobability1./H-sFeora nalc!haracterizationofMletN 2-(QMfeYN0Y)denotethe(closed)semigroupsgeneratedfb!yN.Thenthefollowingcriterionholds:-sQ(5.7),Proposition mLpetZthesystem(E;1r)berecurrentwithinvariantmeasuresanddenotebyjDQthecpanonicalinjectionofEEmintoH[E].4JThentheattractorMssatis ess(a) jv(x) 2QMfeYN0ne) x2M1,s(b) jv(x) 2QMfeYN0ne( x2M1, lprpovidedE|islopcalFlybounded(see(10.2)).-sQProof.(a)By(5.5)theinclusionh[M1] M}holdsforh 2N,henceforh 2QMfeYN0c.sTh!usftheassertionfollowsfromh[M1] =fxgfforh =jv(x).-s(b)fByde nitionofthetopMologyinH[E]ithastobesho!wnthatЬ/Gfh 2Nz g:h[Kȁ]Gg6=;%22e썠:-iӍ{-sfornGXZ2NG(E)con!tainingx,hencesatisfying(G)>0,andarbitraryK 2NK(E). sSinceEOisloMcallybounded,K>canbeco!veredbya nitenumbMerofboundedopenssetsandth!usKLA# Rforsome nitesubsetAofE.Since,cmoreover,0canbMesincluded9inAandGcanbMesupposedtobecon!vex9by(10.4),Nitissucienttoproveͤw^fh 2Nz g:h(yd)2Gforfall&Qeyo:2Ag6=;1:sButftranslatedin!toprobabilitiesthisisimpliedbymP(Xzy:jn2 Gforfall&Qeyo:2A)>0forfsome>n2N1;swhic!hffollowsindeedfrom(4.4a)and(2.3).dH -sSinceatotallyorderedspaceisloMcallybounded,/inthiscaseb!y(5.7)simplysM0,thisisimmediatefrom(2.6). GH-sClearlye,wcompactnesskLofthestatespacealw!ayskLimpliespMositiv!erecurrence;~moresgenerally:-sQ(6.3)FProposition pA\n irrpeduciblesystem(E;1r)ћispositiverecurrentundersepachofthefolFlowingconditions:s(a) r(h[E]isrpelativelycompacte) >0,s(b) E|cpontainsamaximalelementyd.-sQProof. (a)QChoMoseasequenceofsetsKȮlK2wNK(E)suc!hthateachK@2wNK(E) sisfincludedinsomeKȮl!. Thentheassumptionconcernstheunionofthesetssfh2H[E]:h[E]KȮl!g,whic!hduetoE[zandKȮl۰bMeingoftypMeKz ^andGȮ,re-sspMectiv!elye,v6aretagainoftypMeGȮ. 7Thissettlesthequestionofmeasurabilityand%23wߠ썠:-iӍ{-sensuresf b:= r(h[E]Kzm)>0forsomem,whic!hinviewoftheestimate|B P(Xzn2 Kzm)P(HznP(x)2Kzm forfall.V)x2E)= forfall2 n2Nspro!vesftherecurrenceof(E;1r) b!y(2.6).InvdDarianceof,moreover,impliesy{o(Kzm) =&CZ ٟZEar(h(x)2Kzm)1(dx) n(E)ysandfth!us(E) <1fasasserted. -s(b)Themaximalit!yofxandthemonotonicityofh?]2H[E]proveh(0)?]xandsh[E]Ϫ=fxgtobMeequivdDalen!tstatements.@~SincernZ(h(0)Ϫx)>0forsomenϪ2N,sb!yf(a)thesystem(E;1rnZ){andthusalso(E;1r) {ispMositiverecurrent.1JH-sIt~isaconsequenceof(6.3a)and(6.2)thattheexamplesfollo!wing(1.6)and(3.4)sbMothareev!enpositiv!erecurrentcounterexamples;vontheotherhandthelatteronespro!vesfneitherconditionin(6.3)tobMenecessaryforpositiv!erecurrence.-sTeooobtainanecessaryandsucien!tconditionideasfromqueuingtheorywillbMestak!enfupthatcanbMetracedbackto[28,29]andleadtothefollowingnotion:-sQ(6.4)zyDefinition oLetl(HznP;1n[2N)bMethegeneratingsequenceofthesystems(E;1r).ThenftherandomvdDariables|Yznx:jn }:= Hz1.n:1::;nHzn{(x)for#n0sde nefthe\7dualprpocess"fbMelongingtox 2E. -sClearlye,"theWdistributionsofXxAn |andYnxAn agreefor xedn,but(YnxAn;bn0)sneed4notbMeaMark!ov4chain.'HMoreover,/whilethejointdistributionoftheproMcesss(XxAn$;bn 0);depMendsonlyonthek!ernelPmandnotontheunderlyingdistributionr,sthisffailsingeneralfortheproMcess(YnxAn;bn 0).-sConsideredinthecompacti cationEdtofEpalldualproMcessescon!vergetoacom-smonflimit:-sQ(6.5)Proposition wIfbthesystem(E;1r) isirrpeducible,Jtherebisarpandomvari-sableYy: !E Lsuchthat|s(a)sYzn0:jn "Y nppointwise:1;噚s(b)Yznx:jn }! Y na.s.)9%foralFlMw1x2Et:-sQProof.Tt(a)[SincetheorderedtopMologicalspaceEyissequen!tiallycompactand sthe]sequence(Yn0An.(!d);bn0)]isincreasing,,itcon!verges]inviewof(10.4)toalimitsYn(!d)cforeac!h!.Since,moreo!ver,E Piscmetrizable,themappingY!:F2 !E Pissmeasurable.-s(b)>_Whenev!erYn(!d) =1,S.monotonicity>_yieldsYnxAn(!d) !1>_forallx,S.toMo.0Other-swiseb+c!hoMoseacountablebaseofE,owhichmaybMeassumedtoconsistofconvexsetssGȮk#;bkb2 N;b!y(10.4). Since(1.5a){andthus(1.5b){carriesoverfrom(XxAn$;bn 0)stof(YnxAn;bn 0),duetoL(XxAn$)=L(YnxAn),thisimplies|t}1Gi?k (Yznx:jn)n1Gi?k(Yzn0:jn.) !0a.s.%forfallI2 kb2N1:sThereforefYnxAn(!d) !Yn(!)fforalmostall! satisfyingYn(!) 6=1.Qw"H%24썠:-iӍ{--sThefvdDariableYUobMeysazero-onela!wcharacterizingthetypMeofrecurrence:-sQ(6.6)Proposition MLpetZthesystem(E;1r)beirreducible.XsThen,v~withYɃbeing sde nepdby(6.5),s(a) P(Yy= 1)=0 if(E;1r)isppositiverecurrent,s(b) P(Yy= 1)=1 otherwise.-sQProof.C}(a)+kLetbMethestationarydistributionandc!hoosefunctionsgȮl!;blA2 N; saccordingto(10.7).QIden!tifyinggȮlwithitscontinuousextensiontoE .yieldsbys(6.5b)fandthein!vdDarianceofE(gȮl!(Yn))=xlimyn!1u&CZHZE>E(gȮl!(Yznx:jn))1(dx)8󍍍=xlimyn!1u&CZHZE>E(gȮl!(Xzx:jn$))1(dx)X=y1gȮl!forfall5s-lA2 N:sThefassertionfollo!wsbythepassagelA! 1.-s(b1) {If(E;1r) isn!ullrecurrentandistheinvdDariantmeasure,+Citfollowssimilarlye,sno!wfapplyingFeatou,Vs~E(gȮl!(Yn))(E)mJlim1infdn!1x&CZL/ZEE(gȮl!(Yznx:jn))1(dx)m=Jlim1infdn!1x&CZL/ZEE(gȮl!(Xzx:jn$))1(dx)Xm=J1gȮl,o< 1forfall2QelA2N:sIn}viewof(E) =1}thereforeE(gȮl!(Yn)) =0}foralllA2 N,andtheassertionfollo!wssagainfforlA! 1.-s(b2)fIf nally(E;1r) istransien!t,then(6.5a)and(2.6)imply3P(Yy2 Kȁ)=,limn!18P(Yzn0:jn 92K)=,limn!18P(Xz0:jn2K)=0for#K(2 NKz#UT(E)1;6fshencefforallK(2 NK(E).`H-sFeor1anapplicationofthiscriterionconsideroncemoretheexc!hangeproMcesssfromD*Section2,Wdenotingagainb!yFɀtherelevdDantdistributionfunction.Theexplicitsrepresen!tationYy=^sup  n2N(Uzn<n(n1))zssho!wsfthatpP(Yy yd)=C0CY<_n0%FV(y+nn)>0LsifGandonlyifFV(yd)>0GandtheseriesC0CP)<_n0$(1{F(y?+n))Gcon!verges.Thereforesthe`Fsystem(E;1r) ispMositiv!erecurrentifandonlyifthevdDariablesUznP;bn 2N;`Fhaveas nitefexpMectation(forextensionssee[22]).-sTheffollo!wingresultisrelatedtoa\7contractionprinciple7"in[27]:-sQ(6.7)Theorem4If]thesystem(E;1r) isppositiverecurrent,itsstationarydistri-sbutionisgivenbyL(Yn),withYYbpeingde nedby(6.5).%25썠:-iӍ{--sQProof. Applicationof(6.5)and(6.6)tothesequence(HznP;1n>1)yieldsa svdDariablefYn0suc!hthatэoيHz2.n:1::;nHzn{(0)"Ynz0=(and4nP(Ynz0G= 1)=01:sTherefore,{b!yDthecontinuityofthemappingsHz1(!d),{thevdDariablesY3andHz1(Yn0=()sagree'almostsurelye. SinceYn0OisindepMenden!tofHz1|+anddistributedasYn,thiscom-smonfla!wisindeedastationarydistribution. H-sFinallye,fthecriterion(2.6)forrecurrenceresp.transiencecanbMecompleted:-sQ(6.8)UTheorem NLpetthesystem(E;1r)T>berecurrent.TThenforarbitraryinitialslawthefolFlowingcponditionsareequivalent:s(1).(E;1r)isnulFlrpecurrentQ|;噚s(2)P(Xzn2 Kȁ)!0foralFl3> K(2NK(E)1:-sQProof. C1. G$Inestablishingthat(1)implies(2)clearlyKF2~NK#UT(E)ma!ybMesassumed.Byf(6.5)thisyieldsamP(Xzn2 Kȁ)P(Xz0:jn2Kȁ)=P(Yzn0:jn 92K)!P(Yy2K)swithfP(Yy2 Kȁ)=0b!y(6.6).-s2.Con!verselye,fletcondition(2)bMesatis ed.Thentheestimate[P(Xzn2 Kȁ)P(Xz0ʫx)1P(Xzx:jn /2K)for#K(2NKz#UT(E)1;swithfP(Xz0ʫ x)>0forsomex2EGb!y(1.3),yieldskP(Yznx:jn }2 Kȁ)=P(Xzx:jn /2K)!0for#K(2NKz#UT(E)1;shencefforallK(2 NK(E).Againb!y(6.5)and(6.6),thisveri es(1).H&H-sNo!wftheresultsof(2.6)and(6.8)canbMesummarizedasfollows:%H-s(E;1r) pMositiv!efrecurrent f, P(X0An! 1)=0 and%FP(Yn0An 9!1)=0,-s(E;1r) n!ullfrecurrent , P(X0An! 1)=0 and%FP(Yn0An 9!1)=1,-s(E;1r) transien!t7a, P(X0An! 1)=1 and%FP(Yn0An 9!1)=1.%H-s7. Mean6passagetimes. 6xAs(2statedin(1.2),theexpMectedtimetoen!tersanincreasingin!tervdDalis niteinanycase. ?Incontrast,hdecreasingintervdDalscansserv!etodistinguishpMositivefromnullrecurrence.Thisreliesonthewell-knownsrecurrence9theoremofKac,^whic!hismostlystatedunderunnecessaryrestrictions.sInfthepresen!tsettingthefollowingdichotomycanbMeestablished:-sQ(7.1)rlTheorem Lpet,thesystem(E;1r)berecurrentwithinvariantmeasure.sThen,withthenotationsTB 9(andTVx Basde nepdinSection1,s(a) E(TVx[0;x]G) <1,if(E;1r) isppositiverecurrentand([0;1x]) >0,ָs(b) E(TVx[0;x]G) =1,if(E;1r) isnulFlrpecurrent(andx2E|arbitrpary).%26썠:-iӍ{--sQProof.nh(a) Consider rstthecasethatxisamaximalelemen!tofE.ݿThenby s(1.2b) .E(TzVx[0;x]G) E(TzVxfxg )=E(TzVx[x;j])<11:rsOtherwiseTc!hoMosey<2+[x;b1]nfxgTandm2Nsuc!hthatP(X0Am yd)>0. jIfs(XznP;bn0)o-isthestationaryproMcessbelongingto(E;1r),^thenitfollo!wsfroms([0;1x]) >0fb!ytheMarkovpropMertyandmonotonicitythatGP(Xz0ʫ x;1Xzm kyd)>0:sIffmisc!hosenminimalwithrespMecttothisinequalitye,thenkQP(Xz0ʫ x;1XȮl,ox;Xzm kyd)=0for#00ffortheev!entEBA :=fXz0ʫx;1XȮlc=,o2[0;x]for0 mg:sWithftheincreasingfunctionngd(z{I) :=E(TzVz[0;x]G)for#z2EstherecurrencetheoremofKacinitsfamiliarv!ersionandtheMarkovpropMertyimplyu|P(T:u[0;x]< 1)-= &CZRfXq0*xg T:u[0;x]qxdP8󍍍- &CZRZA6T:u[0;x]qxdP-= &CZRZA6(mn+gd(Xzm))1dP- P(A)1(mn+gd(x)):GsThereforefgd(x) <1,ashadtobMesho!wn.-s(b)"Theextendedv!ersionoftherecurrencetheoremofKacneededhereconcernsstheVin nitemeasure%-:=| P7 P :1::onC0CN~2<_n0&NB(E),whic!hVisshiftinvdDariantsb!ytheinvdDarianceof.NowthestandardproMofinthecaseofprobabilitymeasuressisfeasilyc!heckedftow!orkaswellfor%,resultinginuLc&CZUZB`aE(T<VyDBN>)1(dyd) =&CZ ٟZEaP(T<VyDB X<1)(dyd)forfall2QeBK2NB(E):sInapplyingthisequationtoBJ=i[0;1x]itmeansnorestrictiontoassumeagains([0;1x]) >0,bMecause٪otherwiseanapplicationof(4.4b)tof8c= 1:u[0;x]wandtheMark!ovspropMert!y^yieldP(TVx[0;x]==1)>0.EUnder^thisassumption,nowinviewof(4.4a),eݍsP(T<Vy '͍[0;x]< 1)=1fforallyo:2 EGandth!usbymonotonicity'tE(TzVx[0;x]G)1([0;x])GD&CZv[0;x]E(T<Vy '͍[0;x]G)1(dyd)G=D&CZvZELP(T<Vy '͍[0;x]< 1)1(dyd)G=D(E)Gswithf([0;1x]) <1=(E).H%27G썠:-iӍ{--sItJistrivialthatassertion(b)extendstotransien!tsystems,bMecauseinthiscase sagainfP(TVx[0;x]= 1)>0.ָ-sThep nalresultofthissectionrequiresauniformv!ersionof(7.1a),:nnotrestrictedstofdecreasingin!tervdDals:-sQ(7.2)Proposition  Lpetɞthesystem(E;1r) bepositiverecurrentwithstationarysdistribution_.LpetmoreoverBK2 NB(E)beconvexwith(B) >0_andyo:2 EKbe xed.sThenthestoppingtime/`T:= inf]8fn 2N:Xzx:jn /2Bfor0xydgshasa niteexppectation.-sQProof.SinceBKisrecurren!tby(4.4a),/anapplicationof(2.3)withA3=f0;1ydg syieldsfm 2Nsuc!hthat# :=P(Xzx:jm k2Bfor0xyd)>01:sSincethesystem(E;1r0)withr0MZ=̯rm meetstheassumptionsasw!ellandthecor-srespMonding0ma!ybMeassumedinthessequel.ThenftherecursionLSz0ʫ:= 0and/FSȮk6+1 :=inf]8fn>SȮk.9:Hzn<n:1::;nHSi?k+1(yd)ygsde nes1asequenceofstoppingtimeswithrespMectto(HznP;1n2N),/whic!h1by(4.4a)sma!yfbMeassumedtobe nite.By(7.1a)itfollo!wsasintheproofof(1.4)thats(1)xʇE(SȮk~nSȮk61) =E(T<Vy '͍[0;yI{]Rm)<1for#kb2N1:sMoreo!ver,ftheev!ents!XAȮk.9:= fHSi?k+1(x)2Bfor0xydgsb!yftheconventionm =1fsatisfys(2)6P(AȮk#) =#for#kb01:sByWIconstructionthevdDariables1Aq0 ;1:::l;11Ai?k1;bSȮk6+1вSȮkzareindepMenden!tfor xedkX?.sFinallye,ftheestimatemTNC0CXZHqGk60-C0CY<qG0ikYg(1n1A8:i V)1(SȮk6+1nSȮk#)B+1썑sholds,XbMecauseEfor xed!therigh!t-handsideequalsSȮk#(!d).+1,ifEkisthe rstindexswithМ!2QAȮk#,)andisin nite,ifthereisnosuc!hindex.\~Bycancellingforeachk(thesfactor#withi =kbthe#bMoundforT%yisincreased,dandthesummandsarecomposedofsindepMenden!tffactors.By(1)and(2)thisyields=E(TV) #z1 /E(T G2NG(E)with!7x2G1;s(b) if(E;1r) isnulFlrpecurrentandx2E|arbitrpary,thenn>E(TzVx:jG) =1forsome?G2NG(E)with!7x2G1;sprpovidedE|islopcalFlybounded(see(10.2)).-sQProof.(a)SinceGb!y(10.4)maybMeassumedtobecon!vex,1sthisisaspecial scasefof(7.2).-s(b)LBytheloMcalboundednessthereexistGz0 2NG(E)andy/2Esuc!hthatsx 2Gz0ʫ[0;1yd],fwhereyo:6= xma!ybMeassumed,becauseotherwiseb!y(7.1b)gE(TzVx:jGq0 J?) E(TzVx[0:x]G)=11:sBut7thenGz0%\:![yd;b1]=;7ma!ybMeassumedaswell,#jbMecauseotherwiseGz0 ;canbe sdecreasedXatoGz0n[yd;b1].No!wletm 2NXasatisfy# :=P(XxAm ky)>0.IfXamisc!hosensminimal,fthengVP(Xzx:jl /= x;1Xzx:jm kyd)P(Xzx:jmlMy)=0for#00fortheev!entT!A :=fXzx:jl f= /2Gfor0monA,%theMark!ovpropMertyandmonotonicitye, sagainfb!y(7.1b),yield6E(TzVx:jG) P(A)1(mn+E(T<Vy '͍[0;yI{]Rm)) =1:FNH-sAsinthecon!textof(5.7)itisanopMenproblem,%>whetherlocalboundednessises- ssen!tialk=forassertion(b)(see,whowever,(9.5)).%Moreover,assertionk=(b)extendsagainstoFtransien!tsystems,~bMecauseinthiscaseP(TVx G Z_=:1)>0FwheneverGisrelativelyscompact.-s8.F\urther|limittheorems.Feromy theresultsofSection6itiseasilyderiv!edsthat^thedistributionsofXznP;bn0;^con!vergeinthepMositiverecurrentcaseweaklystoQuthestationarydistribution(andotherwisevdDaguelyto0). Actuallye,|9theclassofsfunctions,fforwhic!hconvergenceholds,isconsiderablylarger:-sQ(8.1)Proposition  Lpetɞthesystem(E;1r) bepositiverecurrentwithstationarysdistribution. vThen,forarbitrparyinitiallawz0andwithzn:= z05PVn-,gznPf8c! f-foralFl4kf2R(E)1:%29Ǡ썠:-iӍ{--sQProof.cSincef4isbMounded,applicationof(1.6)withinitialvdDariableXz0kp=lxz0 sresp.Xz0ʫ= xfyieldsVRxznPfnf8c= &CZ ٟZE^&CZ!1ğZE+LE(f-(Xzxq0:jn O)f(Xzx:jn$))1z0(dxz0)(dx) !0:FNH-sThisfresultimpliesinparticularthat\P(Xzn= x)!(fxg)forfall2Qex2Et:sUnderatotalorderingthisfactiseasilyseentoimplyuniformcon!vergenceofthesdistribution|*functionsFznP(yd) :=P(Xzny)|*tothelimitFV(y) :=([0;1y]).Teo|*extendsthisfresulttothegeneralsetting,anappropriatemetriconMz1(E)isneeded:-sQ(8.2)'Proposition dThede nitionqed(z1;1z2) :=supZ9fjz1(B)nz2(B)j :BK2NBz#UT(E)gsyieldsametriconMz1(E)withthefolFlowingprpoperties:s(a)Sjd(z1;1z2)=sup-jfjz1fnz2f-j :f8c2BU@z#(E)with!70f1g1;噚s(b)td(znP;1) !0 implies5lzn!wD;s(c)Dthemapping 7!PVisacpontractionR:-sQProof.n(a)fThisfollo!wsfromzidf8c= CR[0;1]zi(f-(x) >yd)1dy. -s(b)fBy(a)uznPf8c! f-for$If2BU@z#(E)with#J0f11;swhic!himpliesconvergenceforfo2Kz0 andthusforfo2K(~(E)accordingto(10.6a).sBesidesfzn!wDthispro!vesfdtobMeindeedametric.-s(c)fBy(a)thisfollo!wsfromPVf8c= CRHl[Erظ]-F(fnh)1r(dh).gH-sAnqapplicationof(10.3)toE^yieldsd("zxH;1"zy ) =1qforx 6=yd,osho!wingqmetriccon-sv!ergenceTingeneraltobMemuchstrongerthanweakconvergence.Usingthenotionofsa'\7splittingpMoin!t"(seezpinthefollo!wingproMof),gasintroMducedbyDubins/Fereedmans[13]mforE9= [0;11]andextendedb!yBhattacharya/Majumdar[8,x9]tohigherdimen-ssion,fthew!eakconvergencefrom(8.1)canbMestrengthenedtometricconvergence:-sQ(8.3)uTheorem If/thesystem(E;1r)isppositiverecurrentwithstationarydis-stribution,thend(z05PVzn-;1) !0foralFl3> z0ʫ2Mz1(E):-sQProof. ?c1.6Applying܄(1.3),* c!hoMosezWwith([0;1z{I])>0܄and,applying(5.3),sc!hoMoseft> 0suchthatqs([0;1z{I])n^([z;b1]) >=H:sThen,forYgiv!en">0Yandwithzn L:=z05PVn-,thereexistsa nitesubsetAofEssatisfyingU?(E~nnAz#UT) =H"and/FznP(EnnAz#UT)=H" forfall0Qen2N1;%30썠:-iӍ{-sasfollo!wsfrom(1.3)and(8.1),appliedtof8c= 1^bA# 38.ܸMoreover,againby(8.1),there sexistsfkb2 Nsuc!hthat]P(Xzx:jk / z{Iforfall&̮x2Az#UT)jand0]P(Xz0:jkz{I)=H;swherefthe rstinequalit!yusesthefactthatby(1.5a)y"PCfXz0:jn z{IgnnC ;C0\*x2A#.~DfXzx:jn /z{IgCϠ؟C0CX"ҟZx2A)M(P(Xz0:jnz)nP(Xzx:jn /z{I))!01:=asSinceQthevdDaluesd(znP;1) =d(zn1PM;PV)Qb!y(8.2c)formadecreasingsequence,bitisssucien!t9toproved(Ȯk6n K;1) !0.#Therefore,passing9fromGtork,thenotationcansbMefsimpli edb!yassumingkb= 1inthesequel.-s2.FeorfarbitraryBK2 NB#UT(E)thereareno!wtwopMossibilities:s(1) LInfthecasez2 B,i.e.[0;1z{I]B,considerKHzO:= fh2H[E]:h[Az#UT][0;1z{I]g;ssatisfyingfA#_ h1 \|[B]2NB#UT(E)forh2Hz .ThereforeNjznP(B)n(B)j"&CZzBHl[Erظ]jzn1(hz1 \|[B])n(hz1[B])j1r(dh)"(1nj)1d(zn1;)B+=H"forfall2Qen 2N1;sbMecause|b!ythe rstpartr(H[E]7nHz6u) 17and|onHzUtheintegrandisbMoundedsb!yf=H".s(2) LInfthecasezh=2 eB,i.e.[z{I;b1] E~nnB,freplaceHzL)b!yHzC:= fh2H[E]:h(0)z{Ig1;ssatisfyingBhh1 \|[B]R=;forh2HӾ. Thenitfollo!wssimilarlye,hinfactsomewhatssimpler,fthatcRjznP(B)n(B)j (1nj)1d(zn1;)forfall2Qen 2N1:sCom!biningfbMothcasesandputting b:= 1n}thisfyieldsFd(znP;1)  nd(zn1;)+=H"sandfth!usbyrecursion|7d(znP;1) < znp!o:= d03=O.ːThereforeothemixingpropMert!yofwithrespecttotheproductmea-ssure4qC0CN$۹n2Z'y3carriesa:o!vertod0withrespMecttoitsimageby=O.Sincethisobviously-sisfthedistributionof(X0AnP;bn 0),ftheassertionfollo!ws.|yH-sTeobMecomplete,,:ithastobemen!tionedthatingeneralthetaild eldofs(XznP;bn0),~`ev!enunderstationaritye,~`neednotbMetrivial. JAcounterexampleisspro!videdbytheCantorsystem,whereXzn1AcanbMereconstructedfromXzn elwithsprobabilit!y1, [andthusthetaild eldof(XznP;bn 0)coincideswiththefull eldsgeneratedfb!ytheproMcessuptosetsofprobability0.-sNo!wfafairlygenerallawoflargenumbMerscanbederiv!ed:-sQ(8.5)Theorem ALpet8Hthesystem(E;1r)fbepositiverecurrentwithstationarysdistribution. vThenforarbitrparyinitiallawF1.ȉfe tnS&C0CX <_0m0.Thentheinitialla!w"zxandthefunctionf8c=1F aressuitedfforacoun!terexample.-s9. Strictly:order-preservingsystems. ThroughoutjSections1{8thessystemT(E;1r)-?hasalw!aysTbMeensupposedtobeirreducible.Teoseethecrucialrolesofthisassumptionconsidertheexamplefollo!wing(3.4).*Ifinthiscasethestatesspaceʪisenlargedfrom[0;11[to[0;1](extendingthemappingshzi/con!tinuously),ӻthesde nitionT="z1 yieldsanotherin!vdDariantmeasure.{ThisobservdDationsuggestsasrestrictionfoftheoriginalstatespace:-sQ(9.1).Definition Feoranarbitrarysystem(E;1r) the\7rpeduced?statesppace"issgiv!enfbythesubspaceҴq0forfsome2n2Ng1:-sIfEpistotallyordered,ҴW+Cb-L0=andconsiderhz02SH[E]withhz0(xz0)/=2Ҵr+Cb-L 01:sTherefore,fwiththenotationvH(yd) :=fh2H[E]:h(xz0)ydgfor#yo:2Et;seac!h*hz0 y2H[E]withhz0(xz0)d=2Ҵ>Cb-LE is*con!tainedintheunionofallsetsH(yd)withsr(H(yd))~=0. SinceOthesetsH(yd)decreaseforincreasingy,IthisunioncanbMesreplacedb!yacountableone,dueto(E1),andth!usisarnullsetitself.HThereforeshz0(xz0) 2ҴCb-LEforP%ralmostallhz0ʫ2 H[E]and,aereplacingxz0)b!yasequence(xȮk#;bkb2N)swithf[0;1xȮk#] "ҴCb-LE ,thissettles().-sAs%thisconsiderationsho!ws,totreatonlyirreduciblesystemsmeansnorealsrestrictionunderatotalordering.This,ho!wever,doMesnotholdforageneralstatesspace,Lasthefollo!wingexamplesshow.mChoMoseEYk=R2A+ candletZ]besupportedsb!y$constantmappings(resultinginindepMendentvdDariablesXznP;bn80),takingtheirsvdDaluesi0inthetotallydisorderedsubsetDX:= fx2E9:xz1+xz2ʫ=1gi0onlye.vIfassignsspMositiv!eUHmasstoeachconstantinadensesubsetofDM,thereducedstatespaceҴ7Cb-LEsiseasilyseentobMenolongerlocallycompact.wIfontheotherhand2hasnopoin!tsmassesfatall,ob!viouslyzĸh[Ҵ}Cb-LE]n\ҴCb-LE0t= ; fforfralmostallb\h2H[E]1:%33"썠:-iӍ{--sThisdobservdDationisthemotivationtoin!vestigatedsystems(E;1r)FKthatarenot snecessarilyirreduciblebutstrictlyorder-preserving.lxBeforein!troMducingthisnotionsthefnecessarymeasurabilit!yhastobMesettled:-sQ(9.2)~Proposition ThesubsppaceJ[E]consistingofalFlmappingsh2H[E]ssuchthatH{h(xz1) 2HȮlisequivdDalen!ttotheexistenceofsetsGzi2NG(E)satisfyingsh[Kȁl|iI] GziandGz1]Gz2ʫS.Sincethesetsfh2H[E]:h[Kȁl|iI]GzidgareopMen,thesassertionfisestablished,H-sNo!wfthecentralassumptionforthissectioncanbMemadeprecise:-sQ(9.3)Definition hThensystem(E;1r)Xiscalled\7strictly\orpder-preserving",ifstheffollo!wingtwoconditionsaresatis ed:֍s(a)ør(J[E]) =11;噚s(b),rznZ(h(0) >0)>0forfsome>n2N1:-sClearlye,fcondition(a)isofrelevdDanceonlyinconjunctionwithcondition(b). -sTeoBdiscussbrie yanimpMortan!tspecialcase,considergeneralizedautoregressiv!esmoMdelsWonE9= RdA+x,g[where issupportedb!yanemapsh :x7!AxR+b.ÙIfWAandbsare%compMosedofthe(nonnegativ!e)vdDariablesaȮik -andbzid,?Orespectiv!elye,then%conditionss(a)fand(b)aresatis edassoMonas֍s(a09){P(azi1 +n:1::;+naȮid )> 0)=1for#1id1;s(b09)P(bz15:1::,bȮdO> 0)>01:sThisexampleshouldbMecomparedwiththemodelin[4].dWhilethestatespace sthere;isenlargedtoE;=Rdߨ,a(onlymappingsh:x7!ax҆+b;withstrictlypMositiv!esscalar /factorsaareadmitted,(impMosinginadditionstrongmomen!tconditionsonthesvdDariablesfaandb.-sThefnotionsin(9.1)and(9.3)arerelatedb!ythefollowingfacts:-sQ(9.4)/Lemma sIf1thesystem(E;1r)isstrictlyorpder-preserving,then1anyx2ҴCb-LEssatis es֍s(a)P(Xz0:jn> x)>0forsome?n2N1;%34#&썠:-iӍ{-噚.s(b),x h2N \nJ[E]1:-sQProof.SinceJ[E]isstableundercompMosition,r(J[E])=1implies srk(J[E])=1forallk'2N.LjInthesequeldenotethesuppMortofrk tb!yNk _andsitsfelemen!tsbyhȮk#.-s(a)fBytheassumptionsonxand(E;1r) therearelA2Nandm2Nsuc!hthat.wrzl:(hȮl!(0) x)>0 and%Frzm6(hzm(0)>0)>01:sWithfn =l+nmand Jzav:= f(hȮl!;1hzm):hȮlnhzm (0)>xg䍑sthisfimplies>/ rznZ(hznP(0) >x)=rzlC& nrzm g(Jza)rzl nrzm g(hȮl!(0)x;bhzm(0)>0)>01:-s(b)|By(a)thereareh0Am v2qNmTwithh0Am(0)>0andh0An 2Nn ^\dJ[E]withsyo::= h0AnP(0)>x.Thenfh0An<nh0Am (0)>y andth!usMJȮb-:= f(hznP;1hzm):hzn<nhzm (0)>ydgsde nesanopMensubsetofH[E]٭H[E]in!tersectingthesuppMortofrn4o ٭rm6.SThereforerzn+m{(hzn+mɌ(0) >yd)=rznɮ nrzm g(JȮb")>01;shencefinparticularyo:2ҴCb-L E .-s(c)ChoMoseh0g1H=handh0An asin(b).Thenh0g1ah0An{(0)>h(x),janditfollo!wsassabMo!ve,fconsideringno!w Jzc9 := f(hz1;1hznP):hz1.nhzn{(0)>h(x)g;sthatfh(x) 2ҴCb-LE .?߬H-sThefcrucialpropMertiesofthereducedstatespacefollo!wreadily:-sQ(9.5)'Proposition dIfthesystem(E;1r) isstrictlyorpder-preserving,thens(a)Ҵ Cb-LEYislopcalFlycompact;噚s(b)"xh[Ҵ}Cb-LE] ҴCb-LEforralmostalFlkh2H[E]1;s(c)ҴhCb-LEislopcalFlyboundedI/:-sQProof.}](a)Ҵ) Cb-LE isadecreasingsubsetofE,Ihenceb!y(9.4b)opMeninELandthus sagainfloMcallycompact.-s(b)?nSinceҴ]Cb-LE5issecondcoun!tableandby(9.4b)coveredbythefamilyofopMensetssfx2ҴVCb-LEh:xandEz0 Mlbeaclosedsubspace. sThenanyfunctionfz0ʫ2 C"(Ez0)cpanbeextendedtoafunctionf8c2 C"(E).-sQProof.nSeef[33,Corollary3.4andTheorem3.6].ʧH-sThegfollo!wingfactisanimmediateconsequenceof(10.3)andthereforesuppliedswithfitssimpleproMof:-sQ(10.4)PropositionvLpetEKIbe(asubspaceof$d)acompactOTS.Thentheclasssofcponvexopensetsisabaseofthetopology.-sQProof. Assume"EntobMecompactandapply(10.3)toEz0 D=fxz1;1xz2g"withsxz1 `6=xz2 ZtoseethatC"(E)separatesthepMoin!tsofE.ThereforeC"(E)inducesasHausdor |topMologyinE9coarserthantheunderlyingcompacttopologye,7andth!ussbMothtopologiesagree.Sincesetsfxj2E E:a K(2NK(E)1:-sQProof. x1.)Let rstE bMeanOTS@andconsiderasetKn2zNK(E)NK(Ea). sThenJdenotingb!yR KtheordergraphofE 5andbypz1 theprojectionofE=gEson!tofits rstfactorleadstoKȁz#(|= pz1[1(EzЂnKȁ)\Rz]2NK(Eza)1;swheref1B= 2 bKȁ#;andth!usindeedKȁ#(|2 NK(E). -s2. Letcon!verselycondition()bMesatis edandR~bede nedasabo!ve. Then,stofpro!ve(EЂnEa)nR|tobMeopen,c!hoosean!y(xz1;1xz2)inthisset.s(1) LIfffxz1;1xz2g E,thenthereareGzio2NG(E)NG(Ea)fsuc!hthatPZtxzio2 Gzidand3 Gz1.nGz2 (E~E)nR(EzЂEza)nRz:s(2)֪IfUfxz1;1xz2g 6E,thenxz1ʫ=1andxz2ʫ2E.SinceEisloMcallycompact,thereisa sneigh!bMorhood5Gz22%NG(E)NG(Ea)5ofxz2v9withclosureK2%NK(E). KBycondition(),stherefore,Kȁ#2NK(E) asw!ell,andth!usGz1\:=E ngKȁ#2NG(Ea) isaneighbMorhoodsoffxz1.Thenitiseasilyc!heckedfthat(Gz1.nGz2)\RS= ;.tSH-sNo!wfanappropriateversionofStone'stheoremcanbMeestablished:-sQ(10.6)IProposition FLpetEbealocalFlycompactOTSsatisfyingcondition()sof(10.5). vThens(a)FbKz0ʫ:= ffz1.nfz2:0fzio2K(~z#}(E)gsisadensesubsppaceofK(~(E)withrpespecttotheuniformnorm,s(b)PK(~(E) R(E)1:%37&T썠:-iӍ{--sQProof. JSinceE \accordingto(10.5)isacompactOTS,itfollo!wsasinthe sproMoffof(10.4)thatxCz:j0n2:= ffz-:j1\nfz-:j2g:0fz-:ji2Cz#(Eza)gͤsseparatesthepMoin!tsofEa.;AllfurtherconditionsinStone'stheoremareobviouslyssatis ed,andth!usCg0 F,isdenseinC(Ea).Letnowf2o K(~(E)bMegivenandf- abMeitsstrivialfextensiontoEa.Thentherearenonnegativ!efunctionsf-'͍ik 2 C#(E)with(kfz-:jk\nf-zk !0for+fz-:jkg:=fz-:j1k ҂fz-:j2k 6;swhereb!yaddingsuitableconstantsf-'͍ikl(1)M=0canbMeachieved.WTherestrictionof s(f-'͍ik wXn1=kX?)+ tofEGyieldsnonnegativ!efunctionsfȮik 2 C(E)suchthat.XkfȮk~nf-k !0for+fȮk.9:=fȮ1k ҂fȮ2k 6:sSincethesuppMortoffȮik isasubsetofff-'͍ik h1=kX?gandthissetiscompactinviewsofqf-'͍ikl(1) =0,|[thefunctionsfȮik zDareinfactcon!tainedinK(~(E).XThisproves(a)and,sdueftoKz0ʫ V|(E),also(b).!H-sThefnextresultrequiressomecoun!tability:-sQ(10.7)A2Proposition gLpetEObealocalFlycompactandsecondcountableOTSssatisfyingcpondition()of(10.5). vThentherearefunctionsgȮl,o2 K(~#}(E)suchthats(1)0 gz1ʫgz2:1::۱!11;噚s(2)`eachK(2 K(~(E)isincl7)udedinfgȮl,o=1gf-orMsomel Z:-sQProof.BythetopMologicalpropertiesofE%/thereisasequenceofsetsKȮl,o2 NK#UT(E)ssuc!h=thateachK2/ NK(E)isincludedinsomeKȮl!.cSinceE accordingto(10.5)isascompactOTS,(10.3)appliesandyieldsfunctionsf-'͍l %27C#(Ea)withf-'͍ljKȁ#RwlU=1andsf-'͍l(1) =1.IfffȮl.denotestherestrictionoff-'͍l &toE,thefunctionsOgȮl:=(fe-+1 _n:1::;_nfe-+%lJ4)B^1for#lA2 Nsmeetfallrequiremen!ts.=H-sThe nalresultconcernstheclassesV|(E)andNV(E)in!troMducedinSection0.sClearlye,&V|(E)wincludesthelinearspaceofalldi erencesfz1fz27withbMoundedfunc-stionsfziGv2BU@"(E),_andth!usNV(E)includesthealgebraofall niteunionsofsetssBz1.nnBz2fjwithfBzio2 NB"UT(E).Con!versely:-sQ(10.8)A2Proposition gLpetEObealocalFlycompactandsecondcountableOTSswithlowerbpound0.IfU_(E)andNU(E)denotetheclassofuniversalFlymepasurablesfunctionsonandsubsetsofE,rpespectively,thens(a) epachf8c2 V|(E)isadi erencefz1.nfz2withboundedfunctions0 fzio2U_"k(E),s(b) epachBK2 NV(E)isa niteunionofsetsBz1.nnBz2withBzio2NU"UT(E).-sQProof. (a)9Assumewithoutrestrictionf-(0)n=0. *WAs9underatotalorder-sing0Nf^ canthenbMerepresen!tedasf;=f-+ f- zwithboundedincreasingfunctionssf- b 0;bo:2f;1+g;fde nedb!y᲍Ff-zһ(x) :=sup-jCn!~xC0X1r۹k62NHcu(f-(xȮk6+1)nf(xȮk#))jz R: xz1ʫxz2:1::۱x1Co Q:%38'f썠:-iӍ{-sTeofv!erifyf- b2 U_(E),considertheBorelmeasurablefunctionPgdz (xz1;1xz2;:::l;x) :=1:ufxq1*xq2:::\xg>A1C0CXN+۹k62Ne&.(f-(xȮk6+1)nf(xȮk#))jzsonytheSuslinspaceEN "JE. ~Thenthesetofx*2E( satisfyingyaninequalit!y sf-һ(x)~> equalstheprojectionofthesetof(xz1;1xz2;:::l;x)~2EN 3iEgsatisfy-singgd (xz1;1xz2;:::l;x)*J> Qon!tothesecondfactorandisthereforeaSuslin,>!hencesuniv!ersallyfmeasurable,setforall b 0.-s(b) sInthespMecialcasef"=f1B [thefunctionsf- .arein!teger-vdDalued.Thisyieldssthefrepresen!tationfBK= C0C[L۹k62N&O(ff-z+ T kX?gnnff-z kX?g)1;IYswhereftherigh!t-handsideisinfacta niteunion.,)H-sItuhastobMemen!tionedthatnonullsetsinterveneunderatotalordering,?bMecausesinfthiscaseincreasingfunctionsareBorelmeasurable.;REFERENCES l[1]2sAlpuim,M.,Athayde,E.(1990).x 3 cmmi10"K`y 3 cmr10"V cmbx10': cmti10!", cmsy10 b> cmmi10K`y cmr10t : cmbx9K cmsy82cmmi8 |{Ycmr8 O!cmsy7 0ercmmi7ٓRcmr7q% cmsy6;cmmi6Aacmr6E