The Markov approximation of the sequences of N-valued random variables and a class of small deviation theorems

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 divergence rate of P with respect to Q related to {Xn}. In this paper the Markov approximation of {Xn, n[greater-or-equal, slanted]0} under P is discussed by using the notion of h(P Q), and a class of small deviation theorems for the averages of the bivariate functions of {Xn, n[greater-or-equal, slanted]0} are obtained. In the proof an analytic technique in the study of a.e. convergence together with the martingale convergence theorem is applied.