An extension of Shannon-McMillan theorem and some limit properties for nonhomogeneous Markov chains

Let {Xn, n >= 0} be a Markov chains with the state space S = {1, 2, ..., m}, and the probability distribution P(x0) [Pi]nk=1Pk(xkxk-1), where Pk(ji) is the transition probability P(Xk = jXk-1 = i). Let gk(i, j) be the functions defined on S x S, and let Fn([omega]) = (1/n)[Sigma]nk=1gk(Xk-1, Xk). In this paper the limit properties of Fn([omega]) and the relative entropy density fn([omega]) = -(1/n)[logP(X0) + [Sigma]nk=1logPk(XkXk-1] are studied, and some theorems on a.e. convergence for {Xn, n >= 0} are obtained, and the Shannon-McMillan theorem is extended to the case of nonhomogeneous Markov chains.