In this paper, we mainly study a class of small deviation theorems for Markov chains indexed by an infinite tree with uniformly bounded degree in Markovian environment. Firstly, we give the definition of Markov chains indexed by a tree with uniformly bounded degree in random environment. Then, we introduce the some lemmas which are the basis of the results. Finally, a class of small deviation theorems for functionals of random fields on a tree with uniformly bounded degree in Markovian environment is established.

中文翻译：

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随机环境中具有一致有界度的树上随机场函数的一类小偏差定理

在本文中，我们主要研究了马尔科夫链在马尔可夫环境中的一类小偏差定理，该链由具有一致有界度的无限树索引。首先，我们给出了随机环境下由具有一致有界度的树索引的马尔可夫链的定义。然后，我们介绍了一些作为结果基础的引理。最后，建立了马尔可夫环境下一致有界度树上随机场泛函的一类小偏差定理。