Strong Limit Theorems For Asymptotic Circular Markov Chains

Probability theory is a widely applicable discipline. It is the framework foundations of many applying subjects. The strong limit theorem is one of the central questions for studying probability. Markov process is an important stochastic process. It has profound theoretical fundament and extensive applied area. The limit theory for Markov chains is one of the basic areas on Markov processes'research. For the limit theory for homogeneous Markov chains, many results have been obtained , which are mature enough to form a complete theoretical system. For the limit theory for nonhomogeneous Markov chains, researchers have done much work in recent years, such as Yang and Liu's work on the limit theorems for nonhomogeneous Markov chains. Since the case that the transition matrices of nonhomogeneous Markov chains are asymptotic circular often appears in practical use, the research on the strong limit theorems for asymptotic circular Markov chains has great theoretical and practical significances. The purpose of this paper is to study the strong law of large numbers for asymptotic circular Markov chains and the asymptotic equipartition property for asymptotic circular Markov chains, which is a more common case of nonhomogeneous Markov chains in real life.In this paper, we first introduce the definition of asymptotic circular Markov chains. Then by applying the limit property for dual functions of nonhomogeneous Markov chains, the strong limit theorem on the frequencies of occurrence of states for asymptotic circular Markov chains is established, and the strong law of large numbers on the frequencies of occurrence of states for asymptotic circular Markov chains is obtained. Finally we give the asymptotic equipartition property for asymptotic circular Markov chains. These offer a theoretical basis for the applications of asymptotic circular Markov chains in real life.