关于齐次树指标马氏链的广义熵遍历定理  

The Generalized Entropy Ergodic Theorem for Homogeneous Markov Chains Indexed by a Homogeneous Tree

作　　者：杨洁[1] 杨卫国[1] YANG Jie,YANG Weiguo(Faculty of Science,Jiangsu University,Zhenjiang 21201)

机构地区：[1]江苏大学理学院,镇江212013

出　　处：《工程数学学报》2018年第3期295-307,共13页Chinese Journal of Engineering Mathematics

基　　金：国家自然科学基金(11571142)~~

摘　　要：本文主要研究有限状态齐次树指标Markov链的强大数定律和广义熵遍历定理.熵遍历定理研究的是信息论中信源的渐近均分割性,树指标Markov链是近年来概率论的研究方向之一.首先,参照非齐次Markov链广义熵密度概念,本文给出了树指标Markov链的广义熵密度的定义.然后,通过构造一组期望值为1的随机变量,利用Markov不等式和Borel-Cantelli引理,证明得到了定义在树指标Markov链上一类随机变量的延迟平均的强极限定理.最后,利用上述定理的推论,我们证明得到了Cayley树上有限状态Markov链状态出现次数的延迟平均的强大数定律和广义熵遍历定理.本文的结果是对一些已有结果的推广.In this paper, we study the generalized entropy ergodic theorem for Markov chains indexed by a homogeneous tree. The entropy ergodic theorem studies the asymptotic equipartition property of information source in the information theory, and the theory of stochastic processes indexed by tree has become one of the research branches in probability theory recently. We first introduce the definition of the generalized entropy density. Then we prove the strong limit theorem of certain random variables by constructing a single parameter class of random variables with means 1 and using the Markov inequality and the Borel-Cantelli lemma. Finally, from the corollaries of the above theorem, we obtain the strong law of large numbers for the delayed average of the number of occurrences of some state and the generalized entropy ergodic theorem for finite Markov chains indexed by a Cayley tree, which generalize some known results.

关 键 词：CAYLEY树 MARKOV链 强大数定律 广义熵遍历定理 

分 类 号：O211.4[理学—概率论与数理统计]