Several Strong Laws For Markov Chain Fields Indexed By Trees

In recent years,the tree model has attracted a great deal of interest among scientists from various research fields such as physics,probability theory and information theory etc..Moreover,stochastic process indexed by a tree has become a hot topic in the field of probability theory in recent years.On the other hand,the strong law of large numbers is one of the central issues of the international probability theory.In this paper,through constructing non-negative martingales and applies Doob's martingale convergence theorem to the research of a.e.convergence,a class of strong laws for Markov chain fields on a non-homogeneous tree are given.This paper includes six chapters:The first chapter is introduction,introducing the researching purpose and meanings of this paper,and the work that existed.The second chapter is preparative knowledge.We introduce the concept of the tree and give the definition of a special kind of non-homogeneous tree.In the third chapter,we give some limit properties of k-ordered non-homogeneous Markov chain on a special kind of non-homogeneous tree.In the forth chapter,we give some strong deviation theorems of non-homogeneous Markov chain on a special kind of non-homogenous tree.In the fifth chapter,we give some strong deviation theorems for functional of m-ordered continuous state Markov chains on a special kind of non-homogenous tree.In the last chapter,we sum up what we have done in this paper.