Some Strong Deviation Theorems For Markov Chain Fields On A Tree

During recent decades,the tree model has gained great concerns among scien-tists from various research fields such as physics,probability theory,informationtheory etc..Moreover,stochastic process indexed by a tree has become a hot topicin the field of the probability theory in recent years. On the other hand, the re-search of the deviation theorem has held an important position in the developmentprocess of probability theory, and the strong deviation theorem is one of the cen-tral issues of the international probability theory.In this paper, through constructing non-negative martingales and applies Doob'smartingale convergence theorem to the research of a.e. convergence, some strongdeviation theorems for Markov chain fields on a tree are given. This paper in-cludes six chapters:The first chapter is introduction, introducing the researching purpose and mean-ings of this paper, and the work that existed.The second chapter is preparative knowledge. We give the concept of the treeand the definition of a kind of non-homogeneous tree.In the third chapter, we first give a class of strong deviation theoremswith respect to m-ordered non-homogeneous Markov chains on a kind of non-homogenous tree.As corollaries,we give some strong deviation theorems for theentropy density and the frequency of occurrence of the states with respect to m-ordered non-homogeneous Markov chains on a kind of non-homogenous tree.In the forth chapter, a class of random approximation theorems for randomsums on a kind of non-homogenous tree are obtained by using the tools of theconditional generating functions and the tailed-probability generating function.In the fifth chapter, we give the strong deviation theorem for functional of con-tinuous state Markov chains on a kind of non-homogeneous tree.In the last chapter, we sum up what we have done in this paper.