Abstract
ABSTRACT In this paper, we introduce the definition of tree-indexed Markov chains in random environment with countable state space, and then study the realization of Markov chain indexed by a tree in random environment. Meanwhile, we establish a class of small deviation theorems for functional of random fields on double Cayley tree in random environment. Finally, We prove the asymptotic equipartition property (AEP) for tree indexed Markov chains in Markovian environment with finite state space.
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