Strong Laws Of Large Numbers For Functionals Of Countable Nonhomogeneous Markov Chains

The strong limit theorems for random variables is one of the central question for studying probability and it also is the foundation of other branches of probability. Probability has been widely used in many related realms. 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. In this paper, we will study a strong law of large numbers for univariate functions and bivariate functions of countable nonhomogeneous Markov chains. The results promoted some known conclusions.The article include five chapters. In the first chapter, we introduce the relative background on this paper, the main methods used in this paper and what will be studied in this paper, then give some simple expressions of the work which have been done . In the second chapter, we introduce the basic theory which needs to use in the subsequent chapters., for example, the definition and functions of conditional expect and Markov Chains. From the third chapter to the forth chapter is the main part of the paper. In the third chapter, The definition of the functional of countable nonhomogeneous Markov chains is first introduced. Then by giving the known conclusion as the lemma, using truncation method and applying the nature of the conditional expection and Jensen inequality, we obtain a class of strong laws of large numbers for univariate functions of countable nonhomogeneous Markov chains. In the fourth chapter,we will study the Shannon-McMillan-Breiman theorem for countable nonhomogeneous Markov chains by applying the convergence in Cesaro sense and the strong laws of large numbers of functions of two variables for countable nonhomogeneous Markov chains. Finally, We will give the simple sum-up about this paper in the last chapter.