关于树指标马氏链强极限定理的若干研究

Some Researches of Strong Limit Theorems for Markov Chains Indexed by Trees

树指标随机过程是近年来概率论的研究方向之一,已引起了概率论、物理学、计算机等学科的广泛关注.树指标马氏链是一类重要的树指标随机过程.近年来,作者与其合作者利用研究概率论强极限定理的新方法,在树指标马氏链的强极限定理方面得出一系列研究成果,其中包括树指标马氏链的若干强大数定律和Shannon-McMillan定理.本文系统地介绍了这些工作,特别地,详细介绍了作者在树指标马氏链的强大数定律和Shannon-McMillan方面的工作,并结合一个定理的证明,介绍所用的方法.

The theory of stochastic processes indexed by tree is one of the research branches in probability theory recently. The tree model has drawn increasing interest from specialists in physics, probability, and information theory. Markov chains indexed by tree are one important class of stochastic processes indexed by tree. In recent years, we with coauthors, by using a new approach of studying the strong limit theorems in probability theory, have obtained a class of strong limit theorems for Markov chains indexed by tree, including some strong law of large numbers and Shannon-McMillan theorems for these Markov chains. In this paper, we review those results systematically, especially, we review in detail our results of strong law of large numbers and Shannon-McMillan theorems indexed by tree. By proving a theorem, we present our approach.