Some Limit Theorems Of Uniformly Bounded Trees In Random Environment

The strong limit theorem is one of the important issues in the study of probability theory and has attracted the attention of many researchers.Many researchers gave some the strong limit theorems for Markov chains and the strong limit theorems for Markov chains indexed by an infinite tree.So far,the research result of tree-indexed Markov chains in random environment is far less than those of Markov chains and tree-indexed Markov chains.In the paper,we will study the Markov chains indexed by an infinite tree in random environments,mainly discussing the strong law of large numbers and Shannon-Mc Millan theorem for Markov chains indexed by an infinite tree with uniformly bounded degree in Markovian environments and a class of small deviation theorems for functionals of random fields on a tree with uniformly bounded degree in Markovian environment.The main contents of this article are arranged as follows:In chapter 1,we give the background development status of subject.And the main research content and the arrangement of subsequent chapters are introduced.In chapter 2,some basic knowledge required for this article are introduced.Then,we give the definitions and properties of the tree-indexed Markov chains and the tree-indexed Markov chains in random environment.Finally,some known strong limit theorems for the tree-indexed Markov chains and the tree-indexed Markov chains in random environment are given.In chapter 3,the strong law of large numbers and the Shannon-Mc Millan theorem for Markov chains indexed by an infinite tree with uniformly bounded degree in Markovian environment are studied.Firstly,some the strong limit theorems of Markov chains indexed by an infinite tree with uniformly bounded degree in Markovian environment are introduced.Then,we establish the strong law of large numbers for Markov chains indexed by an infinite tree with uniformly bounded degree in Markovian environment.Finally,Shannon-Mc Millan theorem for Markov chains indexed by an infinite tree with uniformly bounded degree in Markovian environment is given.In chapter 4,a class of strong deviation theorems for random fields on uniformly bounded trees in a Markovian environment is established.Firstly,consider using a class of asymptotic loglikelihood ratio as a measure of the approximation of the Markov chains indexed by an infinite tree with uniformly bounded degree and random fields on a tree with uniformly bounded degree in Markovian environment.Then,a class of strong deviation theorems for random fields on uniformly bounded trees in a Markovian environment is established by the method of constructing martingales and the Doob convergence theorem of martingales.In chapter 5,the author summarizes the whole text,and indicates reflects on the areas for improvement.