任意N值随机变量序列关于m阶非齐次马氏链的一类小偏差定理

A Class of Small Deviation Theorems for the Sequences of N-valued Random Variables with Respect to mth-order Nonhomogeneous Markov Chains

设{X_n,n≥0}为定义在概率空间(Ω,F,P)上在{1,2,…,N}中取值的随机变量序列.设Q为F上的另一概率测度,并且{X_n,n≥0}在Q下为m阶非齐次马氏链.设h(P|Q)为P关于Q相对于{X_n}的样本散度率距离.该文首先研究{X_n,n≥0}关于m阶非齐次马氏链的m+1元函数平均值的一类小偏差定理.作为推论,得到了{X_n,n≥0}关于m阶非齐次马氏链状态出现频率和熵密度的一类小偏差定理.最后,得到了m阶非齐次马氏链的若干强大数定律和Shannon-McMillan定理.

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 ruth-order nonhomogeneous Markov chain. Let h（P｜Q） be the sample divergence rate of P with respect to Q related to {Xn}. In this paper, the author first establishes a class of small deviation theorems for the averages of the functions of m＋ 1 variables of {Xn, n ≥ 0} with respect to ruth-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 {Xn, n ≥ 0} with respect to rnth-order nonhomogeneous Markov chains. Finally, the author gets several strong laws of large numbers and a Shannon-McMillan theorem for ruth-order nonhomogeneous Markov chains.