Let(ξn)n^∞=obeaMarkovchainwiththestatespaceχ={1,2...,b},(gn(x,y))n^∞=lbefunctionsdefinedonχ×χandFmn,bn(W)=bn/1mn+bn∑k=mn+1gk(ξk-1,ξk).InthispaperthelimitpropertiesofFmn,bn(w)andthegeneralizedrelativeen-tropydensityfmn,bn(w)=-(1/bn)logp(ξmn,mn+bn)arediscussed,andsometheoremsona.s.convergencefor(ξn)n^∞=0andthegeneralizedShannon-McMillan(AEP)theoremonfinitenonhomogeneousMarkovchainsareobtained.
