The Open Mathematics, Statistics and Probability Journal
2017, 8 : 19-26Published online 2017 September 30. DOI: 10.2174/1876527001708010019
Publisher ID: TOSPJ-8-19
RESEARCH ARTICLE
An Entropy Rate Theorem for a Hidden Inhomogeneous Markov Chain
* 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: ; wzz30@ahut.edu.cn
ABSTRACT
Objective:
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.
Introduction:
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.
Results:
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.
Conclusion:
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.