Abstract

In this paper, we investigate some strong limit properties for Markov chains indexed by a class of infinite trees, which are the generalizations of the spherically symmetric tree. We first give the definition of the generalized spherically symmetric tree model, then obtain the strong law of large numbers for Markov chain indexed by such tree. Finally, we establish the Shannon-McMillan theorem for Markov chains indexed by generalized spherically symmetric tree by applying the Stolz theorem. The obtained properties are generalizations of some known results in the existing literature.

中文翻译：

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Shannon-McMillan 定理和广义球对称树索引马尔可夫链的强大数定律

摘要

在本文中，我们研究了由一类无限树索引的马尔可夫链的一些强极限性质，这些无限树是球对称树的推广。我们首先给出广义球对称树模型的定义，然后得到以该树为索引的马尔可夫链的强大数定律。最后，我们通过应用 Stolz 定理建立了广义球对称树索引马尔可夫链的 Shannon-McMillan 定理。获得的特性是现有文献中一些已知结果的概括。