关键词: 小偏差定理, m 阶非齐次马氏链, Shannon-McMillan定理

Abstract:

Let {Xn, n ≥ 0} be a sequence of random variables on the probability space (Ω, F, P) taking values in alphabet S={1, 2, ^… , N}. Let Q be another probability measure on F, under which {Xn, n ≥ 0} is an mth-order onhomogeneous Markov chain. Let h(P|Q) be the sample divergence rate of P with respect to Q related to {X_n}. In this paper, the author first establishes a class of small deviation theorems for the averages of the functions of m+1 variables of {X_n}, n ≥ 0} with respect to $m$th-order nonhomogeneous Markov chains. As corollaries, the author obtains the small deviation theorems for the frequency of occurrence of the states and the entropy density of {X_n}, n ≥ 0} with respect to mth-order nonhomogeneous Markov chains. Finally, the author gets several strong laws of large numbers and a Shannon-McMillan theorem for mth-order nonhomogeneous Markov chains.

Key words: Small deviation theorems, Markov chains, Shannon-McMillan theorem