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1、FIRSTPASSAGEANDRECURRENCEDISTRIBUTIONSBYT.E.HARRIS1.Introduction.WeconsiderMarkovchainswithdenumerablestates,designatedby0,1,2,•••,andwithtransitionprobabilitiesindependentoftime.Lettingxo,xu■■■bethestatesafter0,1,•••steps,wedefine(1.1)PM(i,j)=P(xn=jx0=i),n-0,1,.••,whereP(^4
2、ß)standsforthecondit
3、ionalprobabilityofA,givenB.Weas-sumethatforeachiandjthereisanintegern=n(i,j)suchthat(1.2)P(¿,j)>0forn=n(i,j).LetNabethefirst-passagetimefromitoj;Ntjisthesmallestpositiveintegernsuchthatxn=j,ifx0=i.Ifthereisnonsuchthatxn=j,theniV,-,-=oo.Ifj=i,wespeakoftherecurrencetimeforthestatei.Weshallusuall
4、ymaketheassumption(1.3)E(Nit)<<*>.If(1.2)holds,then(1.3)(whichistrueforall*ifitistrueforanyi)impliestheexistenceofasetofstationaryprobabilitiesir¡>0satisfying12»,-lim—EP(r)(*Í),n-»oonr=-0(1.4)»/-£*rP°>(r,j),r=0oo3=0SeeFeller[4,Chap.15]fortherelevanttheory.Letdabedefinedastheprobabilitythatthesta
5、te,initiallysupposedtobei,takesonthevaluejatleastoncebeforereturningtoi.Thequantitiesdaturnouttobeveryuseful.In§2wederivesomeidentitiestobeusedinthesequel.In§3weconsiderthedistributionoftherecurrencetimeNkkundertheassumptions(1.2)and(1.3),for"rare"states—i.e.,statesforwhichtt*issmall.Since(assumi
6、ngthatthereareinfinitelymanystates)nomatterhowthestatesarenum-bered,wemusthaveit*—»0ask—>&>,wecanspeakofthedistributionofNkkforlargek.ItisshownthatReceivedbytheeditorsFebruary20,1952.471Licenseorcopyrightrestrictionsmayapplytoredistribution;seehttp://www.ams.org/journal-terms-of-use472T.E.HARRIS[
7、November(1.5)P(TrkekoNkk>u)=eko(e-»+€*(«)),u>0,whereek(u)-^0ask—»■»foreachfixedm>0.In§4wegiveexplicitexpressionsfortheir¡,da,andformeanrecurrenceandfirst-passagetimes,inthecasewheretheMarkovchainisarandomwalk;thatis,P¡¿+i=pi,P$-i=l—pi-ThemethoddependsontherepresentationofarandomwalkasaBrownianpar
8、ticlemovingamongsuitablyselectedpoints.§5givesamorepreciseformofLemma3forrandomwalksandamethodforgettingmomentsoffirst-passagetimesinrandomwalks.§6givesarathercuriouscorrespondencebetweenrandomwalksandtrees.Theauthorwa
