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1、Chapter14f-Ergodicityandf-regularityInChapter13weconsideredergodicchainsforwhichthelimitlimEx[f(Φk)]=fdπ(14.1)k→∞existsforeveryinitialconditionandeveryboundedfunctionfonX.Anassumptionthatfisboundedisoftenunsatisfactoryinapplications.Forexam-ple,fmaydenoteacostfunctioninanoptimalc
2、ontrolproblem,inwhichcasef(Φn)willtypicallybeacoercivefunctionofΦnonX;inqueueingapplications,thefunctionf(x)mightdenotebufferlevelsinaqueuecorrespondingtotheparticularstatex∈Xwhichis,again,typicallyanunboundedfunctiononX;instoragemodels,fmaydenotepenaltiesforhighvaluesofthestoragel
3、evel,whichcorrespondtooverflowpenaltiesinreality.Thepurposeofthischapteristorelaxtheboundednessconditionbydevelopingmoregeneralformulationsofregularityandergodicity.Ouraimistoobtainconvergenceresultsoftheform(14.1)forthemeanvalueoff(Φk),wheref:X→[1,∞)isanarbitraryfixedfunction.AsinC
4、hapter13,itwillbeshownthatthesimplestapproachtoergodictheoremsofthiskindistoconsidersimultaneouslyallfunctionswhicharedominatedbyf:thatis,toconsiderconvergenceinthef-norm,definedasνf=sup
5、ν(g)
6、g:
7、g
8、≤fwhereνisanysignedmeasure.Thegoalsdescribedaboveareachievedinthefollowingf-NormErgod
9、icTheoremforaperiodicchains.Theorem14.0.1(f-NormErgodicTheorem).SupposethatthechainΦisψ-irreducibleandaperiodic,andletf≥1beafunctiononX.Thenthefollowingcondi-tionsareequivalent:(i)Thechainispositiverecurrentwithinvariantprobabilitymeasureπandπ(f):=π(dx)f(x)<∞.336f-Ergodicityandf-
10、regularity337(ii)ThereexistssomepetitesetC∈B(X)suchthatτC−1supEx[f(Φn)]<∞.(14.2)x∈Cn=0(iii)ThereexistssomepetitesetCandsomeextended-valuednon-negativefunctionVsatisfyingV(x0)<∞forsomex0∈X,and∆V(x)≤−f(x)+bIC(x),x∈X.(14.3)AnyofthesethreeconditionsimplythatthesetSV={x:V(x)<∞}isabsor
11、bingandfull,whereVisanysolutionto(14.3)satisfyingtheconditionsof(iii),andanysublevelsetofVsatisfies(14.2);andforanyx∈SV,Pn(x,·)−π→0(14.4)fasn→∞.Moreover,ifπ(V)<∞,thenthereexistsafiniteconstantBfsuchthatforallx∈SV,∞Pn(x,·)−π≤B(V(x)+1).(14.5)ffn=0ProofTheequivalenceof(i)and(ii)follow
12、sfromTheorem14.1.1andTheo-rem14.2
