A Class Of Strong Limit Theorems For Markov Chain Fields On A Non-homogeneous 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 non-homogeneous tree are given. Thispaper includes five 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, firstly, the definition of m-ordered asymptotic circulan-t Markov chains on a non-homogenous tree is given. Then some strong lim-it theorems of m-ordered asymptotic circulant Markov chain fields on a non-homogenous tree are given. As corollaries, we give some strong laws of largenumbers for the frequency of occurrence of the states of m-ordered asymptoticcirculant Markov chains on a non-homogenous tree.In the forth chapter, we give some limit properties of arbitrary informationsource on a non-homogeneous tree.In the last chapter, we sum up what we have done in this paper.