R-POSITIVITY, QUASI-STATIONARY DISTRIBUTIONS AND R

December6,1996R-POSITIVITY,QUASI-STATIONARYDISTRIBUTIONSANDRATIOLIMITTHEOREMSFORACLASSOFPROBABILISTICAUTOMATAP.A.Ferrari,H.KestenandS.MartnezUniversidadedeS~aoPaulo,CornellUniversityandUniversidaddeChileAbstract.Weprovethatcertain(discretetime)probabilisticautomatawhichcanbeabsorbedina\nullstatehaveanormalizedquasi-stationarydistribution(whenrestrictedtothestatesotherthanthenullstate).Wealsoshowthattheconditionaldistributionofthesesystems,giventhattheyarenotabsorbedbeforetimen,con-vergestoanhonestprobabilitydistribution;thislimitdistributionisconcentratedonthecongurationswithonlynitelymany\activeoroccupiedsites.AsimpleexampletowhichourresultsapplyisthediscretetimeversionofthesubcriticalcontactprocessonZdororientedpercolationonZd(foranyd1)asseenfromthe\leftmostparticle.Forthisandsomerelatedmodelsweproveinadditionacentrallimittheoremforn12timesthepositionoftheleftmostparticle(conditionedonsurvivaltilltimen).ThebasictoolistoprovethatoursystemsareR-positive-recurrent.1.Introductionandprincipalresults.LetfXngn0beaMarkovchainonacountablestatespaceS,withanabsorbingstates0.Weshalldealexclusivelywiththediscretetimecaseinthispaper,butwebelievethatalltheresultsandproofshaveanaloguesinthecontinuoustimecase.WeshallwriteS0forSnfs0g.Asusual,wedenotetheprobabilitymeasuregoverningXwhenconditionedtostartatX0=xbyPx;thenforanyprobabilitymeasureonSP=Xx2S(x)Px=PisthemeasurewhichgovernsXwhentheinitialdistributionis.ExandEdenoteexpectationwithrespecttoPxandP,respectively.Thetransitionproba-bilitiesareP(x;y)=PxfX1=ygandPn(x;y)=PxfXn=yg:(1:1)1:1ItisconvenienttointroducetherestrictionbPofPtoS0S0.Becauses0isabsorbingwethenalsohave(bP)n(x;y)=PxfXn=yg=Pn(x;y)forx;y2S0:1991MathematicsSubjectClassication.Primary60J10,60F05Secondary60K35.Keywordsandphrases.AbsorbingMarkovchain.Quasi-stationarydistribution.Ratiolimittheorem.Yaglomlimit.R-positivity.Centrallimittheorem..TypesetbyAMS-TEX12P.A.FERRARI,H.KESTENANDS.MARTINEZThroughoutweassumethatbPisirreducible,i.e.,forallx;y2S0thereexistsann=n(x;y)withPn(x;y)0:(1:2)1:3TheabsorptiontimeisT=T(s0)=inffn0:Xn=s0g;(1:3)1:4andweassumethatabsorptioniscertain.Thismeansthatforsomex2S0(andhenceallx)PxfT1g=1:(1:4)1:5Thefactthats0isabsorbingmeansofcoursethatXn=s0forallnTandPn(s0;x)=(s0;x);n0:(1:5)1:6Anormalizedquasi-stationarydistributionforXisaprobabilitymeasureonS0whichsatisestheinvarianceconditionbPn=r(n);(1:6)1:7where,necessarily,r(n)=Xy2S0bPn(y)=PfTng:(1:7)1:8Notethatthesedistributionsareconditionallyinvariant,inthesensethatPfXn=yjTng=(y)forally2S0:Anormalizedquasi-stationarydistribution0iscalledminimalifE0T=inffET:anormalizedquasi-stationarydistributiong:(1:8)1:9Theinterestinnormalizedquasi-stationarydistributionsarisesfromthefactthatifforsomeinitialdistributiononS0,PfXn=ygPfTng!(y);y2S0;(1:9)1:10forsomeprobabilitydistributiononS0,thenisnecessarilyanormalizedquasi-stationarydistribution.(SeeSenetaandVere-Jones(1966),Theorem4.1).Ifthelimitin(1.9)existsweshallcallitaYaglomlimit,becauseYaglom(1947)provedtheexistenceofthislimitforsubcriticalbranchingprocesses(whenisconcen-tratedononepointx).Suchalimitisalsocalledaconditionallimitdistributionintheliterature.Thereisanextensiveliteraturediscussingtheexistenceofnormal-izedquasi-stationarydistributionsandtheYaglomlimit;seeFerrarietal.(1995)forsomereferences.AnadditionalrecentreferenceisRobertsandJacka(1994).Recently,Ferrarietal.(1995)gaveanecessaryandsucientconditionfortheQUASI{STATIONARYDISTRIBUTIONSANDRATIOLIMIT3existenceofanormalizedquasi-stationarydistributionforchainsXwhichsatisfyinadditionto(1.2)and(1.4)theconditionlimx!1PxfTtg=0forallxedt1:(1:10)1:12Thisofcoursemeansthatforallt1and0,PxfTtgforallbutnitelymanyx.(ActuallyFerrarietal.(1995)dealswiththecontinuoustimecase,buttheirresultscarryovertodiscretetime;seeKesten(1995),TheoremAforastatementofthediscretetimeresult.)Unfortunately,(1.10)israrelyfullledforchainsXwhichdescribeinteractingparticleswhichalsohaveaspatialpositionorwhichcanbeofinnitelymanytypes.Theabsorbingstateisthestateinwhichnoparticleispresent.Typicallyinsuchmodelsabsorptioninunittimefromanyoftheinnitelymanystatesinwhichonlyoneparticleispresentisboundedawayfrom0.Innitelymanystates,becausethesingleparticlecanhaveinnitelymanypositionsortypes;theabovephenomenonoccursiftheprobabilityforaparticletodieinonetimeunitisuniformlyboundedawayfrom0.Asimplespecialexample(whichwasinfacttheprincipalmotivationforthispaper)isthesubcriticalcontactprocessororientedpercolationonZdasseenfromthe\leftmostparticle(withstatespaceSacertaincollectionofnitesubsetsofZdcontainingf0gplustheemptyset).Ouraimhereistoprovetheexistenceofanormalizedquasi-stationarydistributionandaYaglomlimitforaclassofinteractingparticlesystemsandprobabilisticautomatawhichincludesorientedpercolationonZd,d1.ThiswillbedonebyprovingthosechainsR-positive-recurrent.(AmoredetaileddescriptionofsomeoftheseexamplesandtheroleoftheleftmostparticleisgivenbeforethestatementofTheorem2below;fulldetailsareinSect.4.)Pakes(1995)investigatedsomeother