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

Let {Xn, n⩾0} be a sequence of random variables on the probability space (Ω,F,P) taking values in the alphabet S={1,2,…,N}, and Q be another probability measure on F, under which {Xn, n⩾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⩾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⩾0} are obtained. In the proof an analytic technique in the study of a.e. convergence together with the martingale convergence theorem is applied.