Some Limit Theorems For Countable Non-homogeneous Markov Chains

Stochastic process is a quantitative desc ription of the dynamic relationships of a series of random events. It has a clos e relationship with other branches of mathematics, and has been widely used in many fields. As a kind of i mportant stochastic process, Markov process has a profound theoretical foundation, and it also has a broad application space. For exam ple, it can be app lied not only to related disciplines such as topology, geometry, modern algebra, functional analysis and function theory, but also to m any other fields such as computer, communication and economic management. Markov process pl ays an unusual role in num erical calculation, information theory, automatic control of biological sciences, which makes people pay more and more attention to Ma rkov process theory and the application research of it.Probability limit theory is one of the main branches of probability theory; it is also the important foundation of other branches of probability theory and mathematical statistics. In modern times, the study of strong limit theorem of random variable sequence has been one of the central issues of the probability lim it theory. This paper is to discuss the limit nature of countable nonhomogeneous Markov chains part and Moving A verage, thus introduci ng the concept of relative entropy. By analysing the limit properties of likelihood ratio, the existing strong lim it theorem are extended. This paper also gives a category of the strong lim it theorem expressed in inequality, which can be treated as an extension of the strong deviation theorem.This book is com posed of six chapters. Chapter One is the introduction part, which introduces the backgr ound of the topic. Chapter Two briefly introduces the basic conception and relevant resear ches on t heories about countable nonhomogeneous markov chains lim it. Chapter Three gives som e strong lim it theorems of markov chains. In Chapter F our, a category of the strong law of lar ge Numbers about countable nonhomogeneous markov chains is further discussed. The fifth chapter conducts a further study on c ountable state of triple loop asym ptotic equipartition type of markov chains based on the research done in Chapter Three and Chapter Four. The last chapter is the conclusion part which also expresses the present author’s prospect towards the research.