Strong Limit Theorems For Circular Markov Chains

Probability theory is a branch of mathematics dealing with the statistical law of chance phenomena.Its theorems and methods have been widely used in finance, insurance,economy and administration,industry and agriculture,medical science, disaster forecast,even in the social science.Many borderline courses have emerged by using the theorems and methods of probability theory,such as Information theory, Policymaking theory,Biology statistics,Financial mathematics and Actuaries etc. 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 homogenous Markov chains,many results have been obtained,which are mature enough to form a complete theoretical system.For the limit theory for nonhomogenous 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 circular often appears in practical use,the research on the strong limit theorems for circular Markov chains has great theoretical and practical significance.The purpose of this paper is to study the strong law of large numbers for circular Markov chains and the asymptotic equipartition property for circular Markov information sources,which is a more common case of nonhomogenous Markov chains in real life.In this paper,we first introduce the definition of three-order circular Markov chains.Then by applying the limit property for functions of nonhomogenous Markov chains,the strong limit theorem on the frequencies of occurrence of states for three-order circular Markov chains is established,and the strong law of large numbers on the frequencies of occurrence of states for three-order circular Markov chains and the asymptotic equipartition property for three-order circular Markov information sources are obtained.According to this,we state the strong law of large numbers on the frequencies of occurrence of states for circular Markov chains.At last,we give the asymptotic equipartition property for circular Markov information sources.These offer a theoretical basis for the applications of circular Markov chains in real life.