非齐次马氏链泛函变换的若干极限定理

Some limit theorems on the transforms for functional of nonhomogenous Markov chain

设 { Xn,n≥ 0 }是一列非齐次马氏链 ,{ f (.,.) ,n≥ 1}是一列二元可测函数 ,{ Vn,n≥ 1}是一列可预报随机序列 .引入非齐次马氏链二元泛函停时变换的概念 ,即Γn =∑nk=1Vkfk(Xk-1,Xk) .利用鞅方法讨论了变换的强极限定理 ,得到 limn a-1n ∑nk=1Vk{ fk(Xk-1,Xk) - E[fk(Xk-1,Xk) | Xk-1]} =0 .作为特殊情形 ,将随机选择的概念拓展到非齐次马氏链中 。

Let {X n,n≥0} be a sequence of nonhomogenous Markov chains,{f(·,·),n≥1} a sequence of measurable functions and {V n,n≥1} a series of predicable random variables.The transformation for nonhomogenous Markov chains is defined as Γ n=∑nk=1V kf k(X k-1,X k).In this paper we consider the limit theorems on the transformation for Markov chains and get the result limna -1 n∑nk=1V k{f k(X k-1,X k)-E[f k(X k-1,X k)|X k-1]}=0 by means of martingle method.As a special case,some strong law of large numbers on the random selection for Markov chains is obtained.