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1、Chapter12InvarianceandtightnessInoneofourheuristicdescriptionsofstability,inSection1.3,weoutlinedapictureofachainsettlingdowntoastableregimeindependentofitsinitialstartingpoint:wewillshowinPartIIIthatpositiveHarrischainsdopreciselythis,andoneroleofπistodescribe
2、thefinalstochasticregimeofthechain,aswehaveseen.Itisequallypossibletoapproachtheproblemfromtheotherend:ifwehavealimitingmeasureforPn,thenitmaywellgenerateastationarymeasureforthechain.Wesawthisdescribedbrieflyin(10.4):andourmaingoalnowistoconsiderchainsontopologi
3、calspaceswhichdonotnecessarilyenjoythepropertyofψ-irreducibility,andtoshowhowwecanconstructinvariantmeasuresforsuchchainsthroughsuchlimitingarguments,ratherthanthroughregenerativeandsplittingtechniques.Wewilldeveloptheconsequencesofthefollowingslightlyextendedf
4、ormofbound-ednessinprobability,introducedinChapter6.TightnessandboundednessinprobabilityonaverageAsequenceofprobabilities{µk:k∈Z+}iscalledtightifforeachε>0,thereexistsacompactsubsetC⊂Xsuchthatliminfµk(C)≥1−ε.(12.1)k→∞ThechainΦwillbecalledboundedinprobabilityona
5、verageifforeachinitialconditionx∈Xthesequence{Pk(x,·):k∈Z+}istight,wherewedefinek1iPk(x,·):=P(x,·).(12.2)ki=1Wehavethefollowinghighlightsoftheconsequencesofthesedefinitions.28812.1.Chainsboundedinprobability289Theorem12.0.1.(i)IfΦisaweakFellerchainwhichisbounded
6、inprobabilityonaverage,thenthereexistsatleastoneinvariantprobabilitymeasure.(ii)IfΦisane-chainwhichisboundedinprobabilityonaverage,thenthereexistsaweakFellertransitionfunctionΠsuchthatforeachxthemeasureΠ(x,·)isinvariant,andPn(x,f)→Π(x,f),asn→∞,forallboundedcont
7、inuousfunctionsf,andallinitialconditionsx∈X.ProofWeprove(i)inTheorem12.1.2,togetherwithanumberofconsequentsforweakFellerchains.Theproofof(ii)essentiallyoccupiesSection12.4,andisconcludedinTheorem12.4.1.WewillseethatforFellerchains,andevenmorepowerfullyfore-cha
8、ins,thisap-proachbasedupontightnessandweakconvergenceofprobabilitymeasuresprovidesaquitedifferentmethodforconstructinganinvariantprobabilitymeasure.Thisisexemplifiedbythelinea
