关于可列齐次马氏链转移概率的强大数定律

本文建立了关于可列齐次马氏链转移概率的一组强大数定律.关于有穷链及具有可列个结果的独立重复实验的相对频率的通常强大数定律是本文所得到的结果的一个简单的推论.在证明中,本文提出了研究强极限定律的一种新的纯分析方法,它与概率论中的通常方法截然不同.

THE STRONG LAW OF LARGE NUMBERS FOR TRANSITION PROBABILITIES OF DENUMERABLE HOMOGENEOUS MARKOV CHAINS

In this paper the following results have been proved:Theorem 1. Let x_n(n = 0, 1, 2,…) be a homogeneous Markov chain, S(k, n) be the number of occurrences of the state k and A (k, l, n) be the number of the state l occurring directly after the state k in the first n trials, and assume that P(D_k)>0, where D_k={ω:x_i=k for infinite i}, then holds almost everywhere in D_k, i.e.,Theorem 2. Let x_n(n = 0, 1, 2,…) be a homogeneous Markov chain, C be a irreducible class of the recurrent states, k ∈ C, S(k, n) be the number of occurrences of the state k and A(k, l, n) be the number of the state l occurring directly after the state k in the first n trials, if there exists j ∈C such that q_i = P(x_o = j) > 0, then holds almost everywhere in δ_j = {ω:x_o = j}, i.e.,In proof, the author has put forward a new purely analytical method for researches on strong limit theorems, a method of the function theory which is completely different from the usual methods in probability theory.