Some Strong Limit Theorems For Markov Chain Fieldswith Countable States By A Kind Of Non-Homogeneous Tree

In recent years tree model has caused extensive interest in the fields of Physics, Probability theory, Information theory and Financial mathematics, tree indexed random process is one of the research direction for studying in the probability theory. And the limit theorems is one of the central questions for studying in the International Ptobability theory.In this paper, convergence almost everywhere was studied by constructing a non-negative supermartingale and martingale difference sequences, and using the Doob martingale and martingale difference sequences convergence theorem.In the second chapter,first we gave the definitions of the tree model and a special kind of non-homogeneous tree,then the definition of its non-homogeneous Markov chains was gaven.In the third chapter, some strong limit theorems for non-homogeneous Markov chains with countable states by a kind of special non-homogeneous tree were obtained. Furthermore,we gave convergence almost everywhere of Shannon-McMillan theorem.In the fourth chapter, the random selection system was introduced to the special kind of non-homogeneous trees, and some strong limit theorems of the random selection for non-homogeneous Markov chains system were obtained. In the last chapter,we summed up what we have done in this paper.