The Shannon–McMillan theorem for Markov chains in Markovian environments indexed by homogeneous trees

Huilin Huang, Weiguo Yang and Zhiyan Shi

Communications in Statistics - Theory and Methods, 2018, vol. 47, issue 21, 5286-5297

Abstract: This article presents the definition of Markov chain indexed by homogeneous trees in Markovian environment. Then we mainly study the strong limit theorems for a Markov chain indexed by homogeneous trees in Markovian environment. We also establish the strong law of large numbers and the Shannon–McMillan theorems for finite Markov chains indexed by a homogeneous tree in a Markovian environment with finite state space. We only prove the results on a Bethe tree and then just state the analogous results on a rooted Cayley tree. There are abundant achievements in research of Markov chains in determined environments, but few results about this topic of Markov chains indexed by trees in random environment. The results of this manuscript are very meaningful to give a good start to establish the strong limit properties for Markov chains indexed by trees in different random environments.

Date: 2018
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DOI: 10.1080/03610926.2017.1388405

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