Coupling,Probability Distance And Markov Processes

In this paper,we discuss the coupling and probabilistic distance of the Markov processes(chains)and the recurrence of the continuous-time Markov processes.The Markov processes is a stochastic process that has been well studied and is thriving.With the development of modern science and technology,more and more attention has been paid to the Markov processes in many applications,especially the random service system which can be seen everywhere in life,from which many Markov processes can be abstracted.For example,quasi-birth and death processes,random walk Markov chains and so on,have attracted many scholars' attention.The first chapter is the introduction,which first introduces the research back-ground and current research situation at home and abroad,and then outlines the or-ganization structure and main results of this paper.Chapter 2 gives some definitions and related knowledge needed in this paper,mainly Markov chains,including the definitions and main properties of countable state space Markov chains and general state space Markov chains.Coupling is introduced.Several common concepts of coupling are introduced.Several examples of coupling are given.Several optimal coupling of ? and its existence theorem are introduced.The probabilistic distance,the minimum probabilistic distance and the integral probabilistic distance are introduced.This paper is mainly divided into two parts.The first part includes the third,fourth and fifth chapters.The coupling method is used to study the Markov processes.The second part,namely the sixth chapter,studies the recurrence of continuous-time Markov processes in general state space.In chapter 3,we study the convergence of non-homogeneous Markov chain-s in general state space.The study of the convergence of time-homogeneous Markov chains is mature,but it is not enough in the application of Markov chains Monte Carlo(MCMC)method.There is no stationary distribution in the non-homogeneous Markov chains.We need to find its limit distribution.Therefore,it is necessary to discuss the convergence of the non-homogeneous Markov chains.Dobrushin-Isaacson-Madsen theorem is a well-known result for the convergence of non-homogeneous Markov chains in countable state space,but in simulated annealing theory,the con-vergence of non-homogeneous Markov chains in countable state space is insufficient in practical application.we first introduce the classical theorem Dobrushin-Isaacson-Madsen,and s-tudy the convergence of non-homogeneous Markov chains in finite state space by Dobrushin-Isaacson-Madsen theorem.Then the convergence of non-homogeneous Markov chains in general state space is studied by using probabilistic distance and coupling method,and a sufficient condition for the convergence of non-homogeneous Markov chains is obtained.It is more widely used than Dobrushin-Isaacson-Madsen theorem.Dobrushin-Isaacson-Madsen theorem can be regarded as a special case and an inference of the theorem.Then,the convergence of non-homogeneous Markov chains in general state space is studied by using the f-norm representation and cou-pling method.A sufficient condition for the convergence of non-homogeneous Markov chains similar to this theorem is obtained.Dobrushin-Isaacson-Madsen theorem can al-so be regarded as an inference of this theorem.The fourth chapter comes from the theorem of 14.0.1of Meyn S.P,Tweedie R L'Monographs Markov Chains and Stochastic Stability:let P be ergodic Markov chains,? be the only stable distribution of P and measurable function f?1,and?(f)<?,Then there is a full absorption set of X0((?)X)so that P is limited to X0 and is traversed by f-by analogy.the condition ?(f)<? is strengthened to?(fp)<?(there is p>1),and the conclusion of Markov chains f-geometric ergod-icity is obtained by coupling method.In chapter 5,we study the ergodicity of continuous-time Markov processes.First,by using the coupling method,we obtain the following sufficient conditions for the f-ergodicity determination of continuous-time Markov processes in general state space.Then,by using the coupling method,the conclusion of continuous-time general state space Markov processes f-geometric ergodicity is obtained.In chapter 6,the recurrence of continuous-time Markov processes in general state space is studied.Firstly,some methods for determining uniform extraordinary recurrent sets are discussed.Secondly,some methods for determining the recurrence of ? irre-ducible continuous-time Markov processes are discussed.Finally,some conclusions related to fine sets for determining the recurrence of Markov processes are obtained.