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) Πnk=1Pk(xk|xk−1), where Pk(j|i) is the transition probability P(Xk = j|Xk−1 = i). Let gk(i, j) be the functions defined on S × S, and let Fn(ω) = (1n)Σnk=1gk(Xk−1, Xk). In this paper the limit properties of Fn(ω) and the relative entropy density fn(ω) = −(1n)[logP(X0) + Σnk=1logPk(Xk|Xk−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.