一类被有限非齐次马氏链控制的条件独立的随机变量序列的强极限定理

A Strong Limit Theorem for Conditionally Independent Random Variables Sequence Controlled by a Finite Nonhomogeneous Markov Chain

设Ｅ＝｛１，２，…，Ｎ｝，｛（Ｋｋ，Ｙｋ），ｋ≥１｝是在Ｅ×Ｅ中取值的随机向量序列，其中｛Ｙｋ，ｋ≥１｝是非齐次马氏链，对于任意的ｎ≥２，（Ｘ１，…，Ｘｎ）在给定（Ｙ１，…，Ｙｎ）的情况下条件独立，且Ｘｉ的条件分布仅依赖于Ｙｉ的值．设ｉ∈Ｅ，Ｓｎ（ｉ），Ｑｎ（ｉ）分别表示序列Ｘｉ，…，Ｘｎ与Ｙｎ，…，Ｙｎ中的ｉ的个数．本文用分析方法研究关于Ｓｎ（ｉ）与Ｑｎ（ｉ）的强极限定理．

Let E= {l, 2, '', N} and { (Xk, Yk ), k ≥ 1 ) be a sequence of random vectors taking values in Ex E, where { Yk, k ≥ 1 } is anonhomogeneous Markov Chain. For any n ≥ 2, X1, …, Xu are conditionally independent under the condition that (Y1,…, Yn ) is given and the conditional distribution of Xi dependent only on the value of Yi. For i∈ E, let Sn(i) and Qn(i) denote respectively the numbers of i in the sequences X1,…, Xn and Y1, …, Yn. In this paper, by using an analytic method, a strong limit theorem concerning Sn(i)and Qn(i) is studied.