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)....

We prove a strong law of large numbers for functionals of nonhomogeneous Markov chains. The approach is analytic and different from the usual one.

Let {Xn, n[greater-or-equal, slanted]0} be a sequence of random variables on the probability space ([Omega],F,P) taking values in the alphabet S={1,2,...,N}, and Q be another probability measure on F, under which {Xn, n[greater-or-equal, slanted]0} is a Markov chain. Let h(P Q) be the sample...