Strong Limit Properties For Discrete-Time And Continous-State Nonhomogeneous Markov Chains

Probability theory is a widly applicable discipline . It is the framework foundations of many applying subjects , such as Information theory , Mathematics Risk theory and Insurance theory for Actuaries etc . The strong limit theorems is one of the central question for studying probability . Markov process is an important stochastic processes. It has profound theoretical fundament , such as topology , theory of functions , functional analysis , modern algebra and geometry , and it has extensive applied area , such as physics , chemistry , biology , astronomy , computer ,. communication , management of economy . The research about homogeneous Markov chains has formed integrated theoretic system , the research about nonhomogeneous Markov chains has been researching ,but these study are limited to discrete-time and discrete-state nonhomogeneous Markov chains . This article is going to study the strong limit properties for discrete-time and continuous-state nonhomogeneous Markov chains .In the first chapter , we introduce the research and progresses about Markov chains . In the second chapter , we introduce the basic theory which needs to use in the subsequent chapters . The definition of discrete-time and continous-state Markov chains and the norm of the functionals of two variables is given . In the third chapter , we study the strong limit properties for discrete-time and continuous-state nonhomogeneous Markov chains . By applying the method of studying the strong law of large numbers for functionals of discrete-state Markov chains in recent years , adopting the definition of mathematical expectation in continuous-state and some special inequalities , etc . The convergence of discrete-time and continuous-state nonhomogeneous Markov chains are studied . The strong law of large numbers for functionals of two variables of discrete-time and continuous-state nonhomogeneous Markov chains are obtained , then get Shannon-McMillan theorem .The strong deviation theorems for the discrete-time and continuous-state nonhomogeneous Markov chains are given in the fourth chapter.All the conclusions drawn in this paper are on the basis of the existence of transition probability density.