Law Of Large Number For Even-Odd Markov Chain Fields And A Three Times Circulation Markov Chain Indexed By A Tree

Random fields on trees are applications on trees of theory of random process-a new mathematical model, which developed from coding and encoding problem in information theory. Assuming there is a sequence of {X_n}, whether the appearing frequency of state and state couple obey the strong law of large numbers is the key of a good coding and encoding method, so this domain is always being a researching emphases for many scholars.The tree model has recently drawn increasing interest from specialists in physics, probability and information theory. In recent years, Professor Liu Wen and Professor Yang Weiguo and their associates have done much work in studying Markov chains on trees and obtained fruitful results.The purpose of this paper is to study the strong limit theorem for even-odd Markov chain fields and a three times circulation Markov chain indexed by a tree. In this paper, firstly, we introduce the basic theory which needs to use in the subsequent chapters, for example, the definition and property of martingale, Markov chain and Markov chain indexed by a tree, and so on. Secondly, we prove a known result by using the method of constructing martingale. Then, we give strong limit theorems for even-odd Markov chain fields on a Cayley tree. This theorem could extend to common- homogeneous tree. On the basic of this, we give strong laws of large numbers on the frequencies of states and state couple. In the end, we give strong limit theorems and strong laws of large numbers on the frequencies of states and state couples for a three times circulation Markov chain indexed by a tree.