树指标马氏链的等价定义

The Equivalent Definition of T-indexed Markov Chains

国内外关于树指标随机过程的研究已经取得了一定的成果.Benjamini和Peres首先给出了树指标马氏链的定义.Berger和叶中行研究了齐次树图上平稳随机场熵率的存在性.杨卫国与刘文研究了树上马氏场的强大数定律与渐近均分性.杨卫国又研究了一般树指标马氏链的强大数定律.为了以后更有效的研究树指标随机过程的一系列相关问题,本文在分析研究前人成果的基础上,给出了树指标马氏链的等价定义,并用数学归纳法证明了其等价性.

There have been some works on tree-indexed stochastic processes at home and abroad. Benjamini and Peres have given the notion of the T-indexed Markov chains. Berger and Ye have studied the existence of entropy rate for shift invariant random fields on a homogeneous tree. Yang and Liu have studied the strong laws of large numbers and Asymptotic Equipartition Property （AEP） for Markov chains field on trees. Yang has studied the strong laws of large numbers for T-indexed Markov chains. In order to study a series of related problems about tree-indexed stochastic processes efficiently, this paper presents the equivalent definition of T-indexed Markov chains based on the analysis to the predecessors, and proves the equivalence of it by using mathematics inductive method.