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THE GENERALIZED ENTROPY ERGODIC THEOREM FOR NONHOMOGENEOUS MARKOV CHAINS INDEXED BY A HOMOGENEOUS TREE

Part of:Combinatorial probabilityGraph theory

Published online by Cambridge University Press:  30 January 2019

Huilin Huang
Huilin Huang*
Affiliation:
Department of Mathematics, Wenzhou University, Zhejiang, 325035, PR China E-mail: huilin_huang@sjtu.org
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Abstract

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In this paper, we extend the strong laws of large numbers and entropy ergodic theorem for partial sums for tree-indexed nonhomogeneous Markov chains fields to delayed versions of nonhomogeneous Markov chains fields indexed by a homogeneous tree. At first we study a generalized strong limit theorem for nonhomogeneous Markov chains indexed by a homogeneous tree. Then we prove the generalized strong laws of large numbers and the generalized asymptotic equipartition property for delayed sums of finite nonhomogeneous Markov chains indexed by a homogeneous tree. As corollaries, we can get the similar results of some current literatures. In this paper, the problem settings may not allow to use Doob's martingale convergence theorem, and we overcome this difficulty by using Borel–Cantelli Lemma so that our proof technique also has some new elements compared with the reference Yang and Ye (2007).

Keywords

generalized entropy ergodic theoremnonhomogeneous Markov chainstrong law of large numberstree

MSC classification

Primary:60C05: Combinatorial probability
Secondary:05C80: Random graphs

Information

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 34 , Issue 2 , April 2020, pp. 221 - 234
Copyright
Copyright © Cambridge University Press 2019

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