Strong Limit Theorems For Asymptotic Circular Mth-Order Markov Chains

Markov chain is a mathematic model describing practical problems, it got rich progress in many areas such as economics,biology, stochastic service system, computer science and stochastic fractal.The research about limit theory and ergodic properties of nonhomogeneous Markov chains has been researching in recent years.The definition of mth-order Markov chains is the generalization of Markov chains. Mth-order Markov information source is an important information source in information theory. Many of practical information sources, such as language and image information, are often an mth-order Markov information source, and always nonhomogeneous.Since the case that the mth-order transition matrices of mth-order nonhomogeneous Markov chains are asymptotic circular often appears in practical use, the research on the strong limit theorem for asymptotic circular mth-order Markov chains has great theoretical and practical significances.The purpose of this paper is to study the strong law of large numbers and the asymptotic equipartition property for asymptotic circular mth-order Markov chains,which is a more common case of mth-order nonhomogeneous Markov chains in real life.This article includes five chapters.In the first chapter, we introduce the research and progress about Markov chains.In the second chapter, we introduce the basic theory which needs to use in subsequent chapters.In the third chapter, we study the strong limit theorem for asymptotic circular second-order Markov chains.In the fourth chapter, we introduce the definition of asymptotic circular mth-order Markov chains on the basis of the third chapter. Then by applying the limit property for the functions of m+1 variables of mth-order nonhomogeneous Markov chains,the strong limit theorem on the frequencies of occurrence of states for asymptotic circular mth-order Markov chains is established.According to this, the strong law of large numbers on the frequencies of occurrence of states for asymptotic circular mth-order Markov chains is obtained. Finally, we give the asymptotic equipartition property for asymptotic circular mth-order Markov chains.As corollaries,the strong law of large numbers on the frequencies of occurrence of states and the asymptotic equipartition property for mth-order nonhomogeneous Markov chains are obtained.We give the simple sum-up in the last chapter.