Some Strong Laws Of Markov Chain Fields Indexed By A Non-homogeneous Tree

During recent years, the tree model has attracted extensive interest in the fieldsof physics, probability theory and information theory etc.. Markov chain fieldindexed by a tree is a special stochastic field. Actually it is a stochastic walkindexed by a tree. In recent years, tree indexed stochastic process has become oneof the research directions for studying in the probability theory. On the other hand,the research of the strong law of large numbers has held an important position inthe development process of probability theory, and the strong law of large numbersis 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 non-homogeneous tree are given. Thispaper 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 special kind of non-homogeneous tree.In the third chapter, we give some strong laws of large numbers of non-homogeneous hidden Markov model on a special kind of non-homogenous tree.In the forth chapter, we give a class of strong laws of large numbers aboutcountable non-homogeneous markov chain transferring between sets on a specialkind of non-homogenous tree.In the fifth chapter, we give a strong deviation theorem of arbitrary informationsource on a special kind of non-homogenous tree by introducing the concept ofrelative entropy density deviation. In the last chapter, we sum up what we havedone in this paper.