An extension of Shannon-McMillan theorem and some limit properties for nonhomogeneous Markov chains
Let \((X_n, n\geq 0)\) be a Markov chain with state space \(S=(1,2,\dots,m)\) and joint distribution \(P(X_0=x_0, X_1=x_1,\dots,X_n=x_n)= P(x_0,x_1,\dots,x_n)>0\), \(x_i\in S\), \(i=1,2,\dots,n\). Then the relative entropy density \(f_n(\omega)\) is given by \(nf_n(\omega)=-\log P(X_0,X_1,\dots,X_n)\). Let \(g_k(i,j)\) be a real-valued function defined on \(S\times S\) and let \(nF_n(\omega)= \sum^n_{k=1}g_k(i,j)\). Some almost sure limit theorems have been proved for \(F_n(\omega)\), and Shannon-McMillan theorem on the entropy has been extended to a class of nonhomogeneous Markov chains.
scientific article; zbMATH DE number 894791
scientific article