Some Limit Theorems Of Nonhomogeneous Markov Chains And Tree-Indexed Markov Chains

Markov chain is a mathematic model describing practical problems.It is also a special stochastic process.Asymptotically circular Markov chains are a kind of nonhomogeneous Markov chains which are common in life.Relative entropy is a very basic and important concept in information theory.It is an asymmetric measure of the difference between two probability distributions.Some scholars have studied the existence conditions of the sample relative entropy rate of a class of nonhomogeneous Markov chains.In this paper,we further study the strong limit theorem for relative entropy density rates between two asymptotically circular Markov chains on the basis of the predecessors.With the development of the information theory,the tree model has increasing interest from all circles of society.Tree-indexed Markov chain is a new mathematical theory system with the mixture of tree and Markov chain.It is a kind of important tree-indexed stochastic processes,which has been widely used in biology,computer science and financial engineering.Therefore,the research on limit theory of tree-indexed Markov chains has not only great theoretical significance,but also has high application value.This doctoral dissertation focuses on the strong law of large numbers and the entropy theorem for nonhomogeneous Markov chains indexed by a homogeneous tree and second-order nonhomogeneous Markov chains indexed by a two-rooted tree.Hidden Markov model is the generalization of the definition of classical Markov chains.It has become a powerful tool to solve problems in areas such as data mining,pattern recognition and biological genetic information.Although some progress has been made in the theoretical research of hidden Markov model,the existing theoretical knowledge is not enough to solve all the problems because the practical problems are much more complex than the mathematical models.We encounter more cases that Markov chains are nonhomogeneous in the actual application,such as dynamic image processing,risk assessment and stock prediction.In these cases,we need to establish nonhomogeneous hidden Markov models.Therefore,the fourth chapter of this paper has great practical significance in studying the strong law of large numbers for nonhomogeneous hidden Markov models taking values in general state space.There are eight chapters in this doctoral dissertation.In chapter 1,we introduce the background,the meaning,main results and innovations of this study.In chapter 2,we give an introduction of the basic knowledge and review the theoretical results obtained on asymptotically circular Markov chains,relative entropy density rates,hidden Markov models and tree-indexed Markov chains.In chapter 3,we study the existence conditions for relative entropy density rate of asymptotically circular Markov chains that take values in the finite state.In chapter 4,we first give the definition of nonhomogeneous hidden Markov models that take values in general state space,and then prove some properties and equivalent properties of those models.Finally,we establish the strong law of large numbers for nonhomogeneous hidden Markov models,where the observable chain take values on~d.As corollaries,we obtain the strong law of large numbers for countable nonhomogeneous hidden Markov models.In chapter 5,we establish the strong law of large numbers and Shannon-McMillan theorem for a class of nonhomogeneous Markov chains indexed by a homogeneous tree by applying the strong limit theorem of nonhomogeneous Markov chains indexed by a homogeneous tree,which extend the corresponding results of asymptotic even-odd Markov chains indexed by a homogeneous tree.In chapter 6,firstly,we introduce the definition of second-order nonhomogeneous Markov chains indexed by a two-rooted tree.Then by applying the limit property for ternary functions of nonhomogeneous Markov chains,the strong law of large numbers on the frequencies of occurrence of states and ordered couples of states and Shannon-McMillan theorem for this Markov chains are established.In chapter 7,we study some equivalent properties and a property for the bifurcating Markov chains indexed by a binary tree taking values in general state space.In chapter 8,we summarize the doctoral dissertation and make a prospect to the future.