Research On Generalized Entropy Ergodic Theorems And Related Issues For Tree-indexed Markov Chains And Nonhomogeneous Markov Chains

Probability is a mathematical subject which studies the random phenomena and their regularity.Its main purpose is to reveal the regularity contained in various random phenomena.Limit theory is an important branch of probability,and it is also the important foundation for other probability projects and mathematical statistics.The famous former Soviet Union Mathematician Kolmogorov once said in ?Independent Random Variables and Limit Theory?,“the value of probability theory could only be revealed by limit theory,and it is impossible to understand the real meaning of the true probability concepts without limit theorems.”Tree-indexed Markov process is the application of the stochastic process theory on trees,which arised from code problems of information theory.The researching of treeindexed Markov chains is an important project of probability in recent years,and the achievements has caught the general attentions in probability,physics and computer science.Tree-indexed Markov chains is a kind of Markov process defined on a tree.Since tree is a kind of uncontrollable graph,the method to research tree-indexed Markov chains is different from that for classical Markov process.In recent years,scholars research the limit theorems for tree-indexed Markov chains by constructing a likelihood ratio with a parameter or martingale,and then they could prove the existence of limit almost everywhere by the convergence of likelihood ratio almost everywhere or Doob's martingale convergence theorem.Scholars have proved limit theorems for a class of classical tree-indexed Markov chains.This dissertation generalizes some theorems for tree-indexed Markov chains,which studies the convergence theorems for tree-indexed Markov chains defined between any two levels,which includes a class of strong limit theorem and strong law of large numbers,generalized Shannon-McMillan theorem and generalized small deviation theorem for Markov chains.The main contents of this dissertation are as follows.In chapter 1,we introduce the outline of this dissertation,give the researching background of tree-indexed Markov chains,which includes the research subjects and the achievements for tree-indexed Markov chains,the concepts about ShannonMcMillan theorem and its importance in information theory,gives the definitions,notations in probability and information theory,and lists Shannon-McMillan theorems,existence theorems for sample relative entropy rate and small deviation theorems which have been obtained.In chapter 2,we study the generalized entropy ergodic theorem for tree-indexed homogeneous Markov chains.Firstly,we introduce the lemmas used in the following sections.Then,we prove the strong law of large numbers for the delayed sum of states of tree-indexed finite homogeneous Markov chains and the generalized entropy ergodic theorem for tree-indexed homogeneous Markov chains.In chapter 3,we study the generalized entropy ergodic theorem for tree-indexed nonhomogeneous Markov chains.There are some lemmas in the first section.Then we obtaine the strong law of large numbers for the delayed sum of states of tree-indexed finite nonhomogeneous Markov chains and prove the generalized entropy ergodic theorem for the tree-indexed nonhomogeneous Markov chains.In chapter 4,we study the generalized entropy ergodic theorem for the Markov chains indexed by an infinite tree with uniformly bounded degree.We give the lemmas and some notations different from the chapters before in the first section since there is no any relationship between the numbers of vertex of neighbouring levels.Then,we prove the main theorems,the strong law of large numbers for the delayed sum of states and the generalized entropy ergodic theorem.As corollaries,we get the entropy ergodic theorem for the Markov chains indexed by an infinite tree with uniformly bounded degree and the main theorems in chapter 2.In chapter 5,we study the generalized entropy ergodic theorem for the m th-order nonhomogeneous Markov chains indexed by an m rooted Cayley tree.We give the lemmas and some of their corollaries in the first section.And we prove the strong law of large numbers and the generalized entropy ergodic theorem in the second section.In chapter 6,we obtain the existence theorem of the generalized relative entropy rate for the nonhomogeneous Markov chains.We get the equivalent form of the generalized sample relative entropy from the equivalent definition of nonhomogeneous Markov chains and some lemmas firstly.And then,we prove the existence theorem of the generalized sample relative entropy rate for nonhomogeneous Markov chains.In chapter 7,we prove the existence theorem of the generalized relative entropy rate for the second-order nonhomogeneous Markov chains.Firstly,we introduce the equivalent form of generalized sample relative entropy for the second-order nonhomogeneous Markov chains and some lemmas.Then,we prove the existence theorem for the generalized relative entropy rate for the second-order nonhomogeneous Markov chains.In chapter 8,we prove the generalized small deviation theorem for nonhomogenous Markov chains.We introduce some lemmas firstly,and then,prove a class of generalized small deviation theorems for nonhomogeneous Markov chains and some corollaries.