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1、Chapter8TransienceandrecurrenceWehavedevelopedsubstantialstructuralresultsforψ-irreducibleMarkovchainsinPartIofthisbook.PartIIisdevotedtostabilityresultsofever-increasingstrengthforsuchchains.InChapter1,wediscussedinaheuristicmannertwopossibleapproachestothest
2、abilityofMarkovchains.Thefirstofthesediscussedbasicideasofstabilityandinstability,formulatedintermsofrecurrenceandtransienceforψ-irreducibleMarkovchains.Theaimofthischapteristoformalizethoseideas.InmanywaysitiseasiertotellwhenaMarkovchainisunstablethanwhenitiss
3、table:itfailstoreturntoitsstartingpoint,iteventuallyleavesany“bounded”setwithprobabilityone,itreturnsonlyafinitenumberoftimestoagivensetof“reasonablesize”.Stablechainsarethenconceivedofasthosewhichdonotvanishfromtheirstartingpointsinatleastsomeoftheseways.There
4、aremanywaysinwhichstabilitymayoccur,rangingfromweak“expectedreturntoorigin”properties,toconvergenceofallsamplepathstoasinglepoint,asinglobalasymptoticstabilityfordeterministicprocesses.Inthischapterweconcen-trateonratherweakformsofstability,orconverselyonstron
5、gformsofinstabil-ity.OurfocushereisonthebehavioroftheoccupationtimerandomvariableηA:=∞n=1I{Φn∈A}whichcountsthenumberofvisitstoasetA.IntermsofηAwestudythestabilityofachainthroughthetransienceandrecurrenceofitssets.UniformtransienceandrecurrenceThesetAiscalledu
6、niformlytransientifforthereexistsM<∞suchthatEx[ηA]≤Mforallx∈A.ThesetAiscalledrecurrentifEx[ηA]=∞forallx∈A.Thehighlightofthisapproachisasolidarity,ordichotomy,theoremofsurprisingstrength.171172TransienceandrecurrenceTheorem8.0.1.SupposethatΦisψ-irreducible.Then
7、either(i)everysetinB+(X)isrecurrent,inwhichcasewecallΦrecurrent,or(ii)thereisacountablecoverofXwithuniformlytransientsets,inwhichcasewecallΦtransient,andeverypetitesetisuniformlytransient.ProofThisresultisprovedthroughasplittingapproachinSection8.2.3.Wealsogiv
8、eadifferentproof,notusingsplitting,inTheorem8.3.4,wherethecoverwithuniformlytransientsetsismademoreexplicit,leadingtoTheorem8.3.5whereallpetitesetsareshowntobeuniformlytransientifth
