An Entropy Rate Theorem for a Hidden Inhomogeneous Markov Chain

Yao Qi-feng1, Dong Yun2, Wang Zhong-Zhi1, *
1 School of Mathematics and Physics, Anhui University of Technology, Ma'anshan, 243002, P.R. China
2 School of Mathematics, Maanshan Teachers' College, Ma'anshan, 243041, P.R. China

© 2017 Qi-feng et al.

open-access license: This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 International Public License (CC-BY 4.0), a copy of which is available at: This license permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

This research is supported in part by the NNSF of China (Grant No.11571142,11601191), the NNSF of Anhui Province (Grant No.1408085MA04,1608085QA03) and the RP of Anhui Provincial Department of Education (KJ2017A851) and the RIP for College Graduates of Anhui University of Technology (2015128). * Address correspondence to this author at the School of Mathematics and Physics, Anhui University of Technology, Ma'anshan, 243002, P.R. China: Tel: +13855507696; E-mails:;



The main object of our study is to extend some entropy rate theorems to a Hidden Inhomogeneous Markov Chain (HIMC) and establish an entropy rate theorem under some mild conditions.


A hidden inhomogeneous Markov chain contains two different stochastic processes; one is an inhomogeneous Markov chain whose states are hidden and the other is a stochastic process whose states are observable.

Materials and Methods:

The proof of theorem requires some ergodic properties of an inhomogeneous Markov chain, and the flexible application of the properties of norm and the bounded conditions of series are also indispensable.


This paper presents an entropy rate theorem for an HIMC under some mild conditions and two corollaries for a hidden Markov chain and an inhomogeneous Markov chain.


Under some mild conditions, the entropy rates of an inhomogeneous Markov chains, a hidden Markov chain and an HIMC are similar and easy to calculate.

Keywords: Hidden inhomogeneous Markov chains, Entropy rate, Stochastic process, Cesaro average, Ergodicily, Norm.