Some Strong Limit Theorems For Markov Chain Fields On A Tree

In recent years, the tree model has attracted a great deal of interest 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 stronglimit theorems is one of the central 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 stronglimit theorems for Markov chain fields on a kind of non-homogenous tree aregiven. This paper includes 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, the generalized random selection system was introducedto the Markov chains on a kind of non-homogenous tree. Some strong limit the-orems of generalized random selection system for Markov chains on a kind ofnon-homogenous tree are given.In the forth chapter, through constructing n-element product density functionsand non-negative martingales, we give some strong limit theorems on the gener-alized gambling system for Markov chains of continuous state space on a kind ofnon-homogeneous tree.In the fifth chapter, by applying the notion of likelihood ratio and the Laplacetransformation to the study of r ordered Markov chains of continuous state,somestrong deviation theorems based on Laplace transform for r ordered Markovchains of continuous state on a kind of non-homogeneous tree are given.In the last chapter, we sum up what we have done in this paper.