Stationary Distribution And Limit Distribution For Markov Processes

This thesis is devoted to stationary distribution and limit distribution for Markov processes (chains).In recent several decades, the theory and application of Markov chains on general state spaces have developed quickly, one reason is that the theory of small sets and splitting technique were introduced into the theory of the stability for general state space Markov chains which enables many stability results of Markov chains on countable state space being enlarged to general state space. And another reason is wide applications of nonlinear time series analysis and Monte-Carlo method, which stimulate the advancement of theoretical study. Especially, the study of φ-irreducible and the Feller chains have grown more perfect. But there have been few studies for non-irreducibility, non-Feller and non-homogeneous Markov chains and yet we frequently use the theory of such Markov chains in applications. In Chapter 2 and 3, we study the stationarity of non-irreducibility and non-Feller chains. Both Chapter 6 and 7 take the limit distribution of non-homogeneous Markov chains as background of research. Chapter 4 investigates the stationarity of Markov processes on Polish space by coupling methods, which has gained much attention during the past few years. Chapter 5 discusses a.s. central limit theorem for non-stationary Markov chains.The organization of detailed information in the paper is as follows:In Chapter 1, we introduce the main results in this paper briefly.Chapter 2 gives a necessary and sufficient condition for existence of stationary distribution of so called "generalized irreducible" Markov chains. We know that the irreducibility and the Feller property are the usual conditions in dealing with stability. But this kind of Markov chains has neither irreducibility nor the Feller property. And we use generalized petite sets instead of petite sets as the verification condition, which is more convenient for applications as in many cases compact set is generalized petite but not petite.In the proofs of the main theorems we also obtain that, for non-irreducible Markov chains, the relationship between petite sets and uniformly transient sets. Furthermore, we also present a necessary and sufficient condition for the existence of stationary distribution for the chains which have Harris decomposition.In Chapter 3, we discuss the stationarity of nonlinear AR processes with Markov switching, which has been used widely in economy field and finance field. Three aspects of the problem are considered:In section 3.2, the focus is on AR model with additive noise, and the correspondent Markov chain is non-irreducible and non-Feller. But being different from Chapter 2, here the discussion is based on the "uniform countable additively condition", using the technique of skeleton chain and the approximation of Lp functions by compact supported continuous functions, we prove the existence of stationary distribution and higher moment of the model.In section 3.3, we first set up the results of §3.2 to nonlinear AR model with conditional heteroscedasticity and then establish the CLT and LIL of the Markov chain of this AR model by adding a condition that the error variable has a positive density. Moreover, we give a method for judgment that a compact set is petite under uniform countable additively condition to overcome the difficulties that the Markov chain of the model is not a Feller chain.In Section3.4, by Kaplan’s condition for a general state space Markov chain and Lya-punov’s methods we give some sufficient conditions which lead that the stationary distribution doesn’t exist.Chapter 4 proves the existence and uniqueness of stationary distribution for Markov chains on Polish space by coupling method and the duality representation of KRW metric. Further-more, the estimate of convergence rate are presented. Then, by applying the results to diffusion processes, some new criteria are obtained. By Lyapunov’s methods, in last section, we discuss the existence and uniqueness of stationary distribution for diffusion processes with Markov switching. And then, the results are used to the Hopfield type stochastic neural networks.Chapter 5 deals with a.s. central limit theorem for non-stationary Markov chains. In order to overcome the non-stationary property, we first give a a.s. central limit theorem for stationary Markov chains by mixing condition. Then by the technique of "shift operator" and "harmonic function" we show that the result is still true when the initial distribution is not a stationary distribution.In Chapter 6, we first consider the connections between convergence in distribution of non-homogeneous Markov chains and composition convergence of a sequence of probability measures on some semigroups.Then three aspects of the work of composition convergence are considered.First, we discuss the relationship between composition convergence and strong composi-tion convergence of a sequence of probability measures on a countable discrete H-semigroup and then a conjecture in[106] is verified.Subsequently, in the case of being same distribution, strong Kloss convergence criterion is extended in algebraic and topological structure. Finally we discuss the structure of some accumulation point sets of a probability measures sequence on a locally compact H semigroup with compact kernel and the result of Maksimov is extended in algebraic structure.Chapter 7 investigates the limit distribution and convergence rate of Huber-Dutter(HD) estimator which is a notable robust estimators, for linear model with non-homogeneous Markov chain noise. Under some regular conditions, it is shown that HD estimators are asymptotically normal with convergence rate n-1/2 which is satisfactory compared with the case of i.i.d. noise.Throughout this chapter, we mainly use martingale theory and methods.