The Strong Law Of Large Numbers For MTH Countable Nonhomogeneous Markov Chains

Markov chain is a mathematic model describing practical problems,it has got rich progress in many areas such as economics,biology,stochastic service system,computer science. The research about limit theorems and ergodic properties has been researching in recent years. The definition of mth-order nonhomogeneous Markov chain is an extension of the definition of Markov chain. Mth-order Markov information source is an important information source .So the research about the theory of mth-order Markov chain is very important.The article is going to study the ergodic properties and strong law of large numbers about mth-order countable nonhomogeous 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 subsequent chapters. In the third chapter,we introduce the definition of mth-order countable nonhomogeneous Markov chains on the basis of the second chapter. In the fourth chapter,we study the strong law of large numbers for mth-order countable nonhomogeneous Markov chains .Firstly,we get a strong law of large numbers of the functions of m+1th variables of mth-order countable nonhomogeneous Markov chains by means of martingable method and the ergodic coefficient. Then constraining number sequence {a_n,n≥m}, we obtain a strong law of large numbers for functionals of countable nonhomogeneous Markov chains based on obtained results. It is an extension of Liu Guoxin's result. In the fifth chapter,firstly we quote the concept of absolute average strong ergodic which Yang Weiguo has introduced,then we introduce the concept of absolute average strong ergodic in the form of transition probability and give its sufficient condition.Finally,we discuss its application in Markov function system and information theory.