markov chains and stochastic stability ergodicity

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1、Chapter13ErgodicityInPartIIwedevelopedtheideasofstabilitylargelyintermsofrecurrencestructures.Ourconcernwaswiththewayinwhichthechainreturnedtothe“center”ofthespace,howsurewecouldbethatthiswouldhappen,andwhetheritmighthappeninafinitemeantime.PartIIIisdevotedt

2、otheperhapsevenmoreimportant,andcertainlydeeper,con-ceptsofthechain“settlingdown”,orconverging,toastableorstationaryregime.InourheuristicintroductiontothevariouspossibleideasofstabilityinSection1.3,suchconvergencewaspresentedasafundamentalidea,relatedinthed

3、ynamicalsys-temsanddeterministiccontextstoasymptoticstability.Wenotedbriefly,in(10.4)inChapter10,thattheexistenceofafiniteinvariantmeasurewasanecessaryconditionforsuchastationaryregimetoexistasalimit.InChapter12weexploredinmuchgreaterdetailthewayinwhichconver

4、genceofPntoalimit,ontopologicalspaces,leadstotheexistenceofinvariantmeasures.Inthischapterwebeginasystematicapproachtothisquestionfromtheotherside.Giventheexistenceofπ,whendothen-steptransitionprobabilitiesconvergeinasuitablewaytoπ?Wewillprovethatforpositiv

5、erecurrentψ-irreduciblechains,suchlimitingbehaviortakesplacewithnotopologicalassumptions,andmoreoverthelimitsareachievedinamuchstrongerwaythanunderthetightnessassumptionsinthetopologicalcontext.TheAperiodicErgodicTheorem,whichunifiesthevariousdefinitionsofpos

6、itivity,summarizesthisasymptotictheory.Itisundoubtedlytheoutstandingachievementinthegeneraltheoryofψ-irreducibleMarkovchains,eventhoughweshallprovesomeconsiderablystrongervariationsinthenexttwochapters.Theorem13.0.1(AperiodicErgodicTheorem).SupposethatΦisan

7、aperiodicHarrisrecurrentchain,withinvariantmeasureπ.Thefollowingareequivalent:(i)ThechainispositiveHarris:thatis,theuniqueinvariantmeasureπisfinite.(ii)Thereexistssomeν-smallsetC∈B+(X)andsomeP∞(C)>0suchthatasn→∞,forallx∈CPn(x,C)→P∞(C).(13.1)313314Ergodicity(

8、iii)ThereexistssomeregularsetinB+(X):equivalently,thereisapetitesetC∈B(X)suchthatsupEx[τC]<∞.(13.2)x∈C(iv)ThereexistssomepetitesetC,someb<∞andanon-negativefunctionVfiniteatsomeonex0∈X,satisfying∆V(x):=P