Some Strong Limit Theorems Of Even-odd Markov Chains On Rooted Cayley Trees

Thirty years ago, random fields came into being. It抯 a subject of intersection of Probability and Statistical Physics. Random fields, together with other branches of probabilistic Physics, stands for an important aspects of a trend, which is the inter- permeating of Math, and Phys.. Professor Liu Wen and hi~ associates extend some strong limit theorems and Shannon-MeMillan theorem for classical Markov chains to Markov chains on Bethe trees and Cayley trees recently. In this paper, we continue the research. The paper is composed of three chapters. In chapter 1, we introduce the elementary knowledge on trees and random fields. We also testify, using a simple and explicit method, the existence and uniquess of even-odd Markov chains on rooted Cayley trees(for simplicity, call them even-odd Markov chains). The main results of chapter 2 are about strong limit theorems on frequencies of occurrences of states and ordered couples of states of even-odd Markov random fields. It consists of three sections. In the first section, we have a class of limit theorems on frequencies of occurrences of states and ordered couples of states, which have related expressed type. In the second section, we present a class of limit theorems on frequencies of occurrences of states, which are expressed by transition probabilities. In the third section, we give a class of small deviation theorems on frequencies of occurrences of ordered couples of states. In chapter 3, we introduce the notion of random comparison coefficient of random fields on rooted Caylcy trees(for simplicity, call them random fields)relative to Markov chains on it and give a class of strong deviation theorems on frequencies of occurrences of ordered couples of states of random fields. What is worth mentioning is that the limit theorems above are extensions of corresponding strong laws of large numbers of Markov random fields.