Some Strong Limit Theorems Of Random Process Indexed By A Tree

Markov chain is a mathematic model describing practical problems. It is also a special random process. Markov chain theory in scientific research, develop production, improve technology, social services, and other areas, has become a powerful mathematical tool widely used in physics, chemistry, computers, communications, stochastic service and many other spheres, and has made a very fruitful results. With the development of the information theory, the tree model has drawn increasing interest from special in physics, probability and information theory. Random fields on trees is the random process be extended to the case of the tree index, and Markov chain fields indexed by trees is a special random fields on trees. In recent years, Professor Liu Wen and Professor Yang Weiguo and their associates do much work in studying Markov chains on trees and obtain fruitful results.The main purpose of this thesis is to study a limit property of the geometric average of random transition probability for a nonhomogenous Markov chains indexed by a tree and a strong deviation theorem for nonsymmetric Markov chain fields on a tree. This thesis includes five chapters. In chapter 1 and chapter 2, we give an introduction of the basis notions main results and approaches used in the paper. Chapter 3 and chapter 4 are the main results. Chapter 5 is the end. In chapter 3, we mainly research a limit property of the geometric average of random transition probability for a nonhomogenous Markov chains indexed by a tree. By introduced the notion of log-likelyhood ration of stochastic sequences, we obtain a limit property of the geometric average of random transition probability for a nonhomogenous Markov chains indexed by a tree. This result is an extension of the limit property of the geometric average of random transition probability for a nonhomogenous Markov chains. In chapter 4, we mainly research a strong deviation theorem for nonsymmetric Markov chain fields on a tree. By using the definition of nonsymmetric Markov chain fields on a tree and constructing a non-negative martingale, we obtain a strong deviation theorem for nonsymmetric Markov chain fields on a Cayley tree.