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1、AnidiosyncraticintroductiontostochasticprocessesClassnotesforMath216Notes-Fall2010JonathanC.MattinglySeptember22,20101FiniteStateMarkovChainsAdiscretetimestochasticprocess(Xn)n0isacollectionofrandomvariablesindexedbythenon-negativeintegersZ+=fn2Z:n0g.ThesetinwhichtheXntakevaluesiscalledthest
2、atespaceofthestochasticprocess.De nition.Astochasticprocess(Xn)n0isaMarkovchainifP(Xn+1=jjXn=in;;X0=i0)=P(Xn+1=jjXn=in)forallj;in;;i02I.De nition.AMarkovchainistimehomogeneousifforallk2Z+andi;j2IP(Xk+1=ijXk=j)=P(X1=ijX=j)UnlesswesayotherwisewewillalwaysassumethatallMarkovchainsaretimeho
3、mogeneous.Insuchcaseswewillwritepn(i;j)=P(Xn=jjX0=i)BytheMarkovpropertyonehasP(Xn=xn;Xn
