stability structures drift and regularity

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1、Chapter11DriftandregularityUsingthefinitenessoftheinvariantmeasuretoclassifytwodifferentlevelsofstabil-ityisintuitivelyappealing.Itissimple,anditalsoinvolvesafundamentalstabilityrequirementofmanyclassesofmodels.Indeed,intimeseriesanalysisforexample,astandardstartingpoint,rath

2、erthananendpoint,istherequirementthatthemodelbestationary,anditfollowsfrom(10.4)thatforastationaryversionofamodeltoexistweareineffectrequiringthatthestructureofthemodelbepositiverecurrent.Inthischapterweconsidertwootherdescriptionsofpositiverecurrencewhichweshowtobeequivalen

3、ttothatinvolvingfinitenessofπ.Thefirstisintermsofregularsets.RegularityAsetC∈B(X)iscalledregularwhenΦisψ-irreducible,ifsupE[τ]<∞,B∈B+(X).(11.1)xBx∈CThechainΦiscalledregularifthereisacountablecoverofXbyregularsets.WeknowfromTheorem10.2.1thatwhenthereisafiniteinvariantmeasureand

4、anatomα∈B+(X)thenE[τ]<∞.AregularsetC∈B+(X)asdefinedby(11.1)hasααthepropertynotonlythatthereturntimestoCitself,butindeedthemeanhittingtimesonanysetinB+(X)areboundedfromstartingpointsinC.Wewillseethatthereisasecond,equivalent,approachintermsofconditionsontheone-step“meandrift”

5、
∆V(x)=P(x,dy)V(y)−V(x)=Ex[V(Φ1)−V(Φ0)].(11.2)XWehavealreadyshowninChapter8andChapter9thatforψ-irreduciblechains,drifttowardsapetitesetimpliesthatthechainisrecurrentorHarrisrecurrent,anddrift256Driftandregularity257awayfromsuchasetimpliesthatthechainistransient.Thehighpoint

6、sinthischapterarethefollowingmuchmorewiderangingequivalences.Theorem11.0.1.SupposethatΦisaHarrisrecurrentchain,withinvariantmeasureπ.Thenthefollowingthreeconditionsareequivalent:(i)Themeasureπhasfinitetotalmass;(ii)ThereexistssomepetitesetC∈B(X)andMC<∞suchthatsupEx[τC]≤MC;(1

7、1.3)x∈C(iii)ThereexistssomepetitesetCandsomeextended-real-valued,non-negativetestfunctionV,whichisfiniteforatleastonestateinX,satisfying∆V(x)≤−1+bIC(x),x∈X.(11.4)When(iii)holdsthenVisfiniteonanabsorbingfullsetSandthechainrestrictedtoSisregular;andanysublevelsetofVsatisfies(11.

8、3).ProofThat(ii)isequivalentto(i)isshownbycombiningTheorem10.4.10withTheorem11.1.4