A Limit Theorem Of Optimal Code Rate For Non-Homogeneous Markov Information Source And The Strong Law Of Large Numbers For Non-Homogeneous Markov Chain On Cayley Trees

Markov chain is a mathematic model describing practical problems. It has got rich progress in many areas such as economics, biology, stochastic service systems, computer science and so on. Markov information source is an important information source, so it is important meaning to further study the theory of Markov information. At the same time, with the development of the information theory, the tree model has drawn increasing interest from specialist in physics, probability and information theory. Random fields on trees are applications on trees of theory of stochastic process—a new math model, which developed from coding and encoding problem in information theory. In recent years, Professor Liu Wen and Professor Yang Weiguo and their associates do much work in studying Markov chains on trees and obtain fruitful results.The purpose of this paper is to study the limit of optimal code rate for non-homogeneous Markov information source and the strong law of large numbers for non-homogenous Markov chains on Cayley trees. Firstly, we introduce the basic theory which needs to use in the subsequent chapters, for example, the definition and functions of martingle, of the difference of martingle, of conditional expect and Markov Chains and so on. Secondly, under the condition of transition matrices of non-homogeneous Markov information source converging mean to an irreducible transition matrix, we obtain the limit of optimal code rate for non-homogeneous Markov information source by using the asymptotic equipartition property for non-homogeneous Markov information source, and generalize a recent result. In the end, by constructing a non-negative martingale, we apply the Doob's Martingale convergence theorem and some special inequalities to study and obtain the strong law of large numbers of non-homogenous Markov chains on Cayley trees. This result is an extension of the strong law of large numbers for a non-homogenous Markov chains.