Asymptotic Stationary Property Of Markov Operators

Markov chains have significant applications in some areas, such as probability, queuing theory, Monte Carlo method, stochastic differential equation, iterated function system and statistical physics. Ergodic theory of Markov chains is the theory of investigating the asymptotic properties of Markov chains. The main researching subjects of Ergodic theory are the existence of invariant probabilities and asymptotic stationary properties.In general, Markov chains can be considered from two points of view. They can be investigated by purely probabilistic and purely analytic methods.In our paper we use the second approach. In analytic approach, an important tool is the Markov operator which is defined on the space of all Borel measures or functional space by transition probability of Markov chains. We can get the asymptotic properties of Markov chains by investigating the asymptotic properties of Markov operators. Asymptotic stationary property is the best asymptotic property for Markov chains. The pivot of this paper is investigating asymptotic stationary properties of two different Markov operator which is defined on polish space, i.e. a separable, complete metric space. There are total four chapters in this paper.The existence of invariant probabilities is one of the most significant researching subjects of the asymptotic theory of Markov chains, and it is the precondition of the asymptotic stationary properties for Markov chains. In the first chapter, we summarize some sufficient conditions for the existence of an invariant measure for Markov operators defined on different spaces which can be found in the literatures. There are three spaces:compact spaces, locally compact and separable spaces, and polish space. In the case of a compact space, we introduce the proof of existence as follows:first, we construct a positive, invariant functional defined on the space of all continuous and bounded functions, and then using the Risez representation theorem. We can have an invariant measure. In the case of locally compact and separable spaces, first, we emphasize the idea of A.Lasota and Yorke [2] which is using the concept of nonexpansiveness and lower bound technique. Secondly, we introduce an necessary and sufficient condition for the existence of an invariant measure for Markov-Feller operator by J.B.Lasserre[10]and a sufficient condition which is easy validated by one-step transition probability of Markov chains[11]. When X is a polish space, the ideas above break down, since a positive functional may not correspond to a measure. Scholars of polish school, represented by A.Lasota, prove the existence theory for Markov operator by the concept of tightness. J.Myjak and T.Szarek[12,13,14,15] construct a series of suitable concentration properties to prove the existence of invariant probabilities for Markov operator.We are interested in asymptotic stationary properties for Markov operator defined on polish space. Every nonexpansive and semiconcentrating Markov operator admits an invariant measure and every nonexpansive and concentrating Markov operator is asymptotically stable (see T.Szarek[18]). As a concentrating Markov operator is semiconcentrating, we suspect the condition of asymptotic stationary property for nonexpansive and semiconcentrating Markov operator. In the second chapter, we get an important property of nonexpansive and semiconcentrating Markov operator (lemma2.2). using this property and the lemma from A.Lasota and Yorke [2](lemma2.1),we can prove that Every nonexpansive and semiconcentrating Markov operator as asymptotically stable.The third chapter focuses on E-chains defined on polish space, i.e. A Markov chain possesses an equicontinuous Markov transition function. T.Szarek[13]get the condition of existence invariant measure for E-chain defined on polish space, i.e. there exists z∈X such that for every open set (?)containing z,limsup(1/n∑(?)P'(x,(?))>0 for some x∈X. Starting from this condition, we get two different class of asymptotically stationary. First, enhanced the condition of the existence, i.e. for every x∈X,lim sup(1/n∑P'(x,(?))> 0, we can prove that E-chain admits a unique invariant probability measureμ, supported on Z, moreoverμPnâ†'wμ,as nâ†'∞for every probability measure such that suppμ(?) Z.(Z=∪supp Pn(z,·)).This condition is a generalization of T.Szarek[13]. Second, considering the set Q={x∈X:(1/n∑(?) P' tight},we get the class of asymptotically stationary about set Q and some properties of the set Q. For example, Q contains support of the invariant probability measure, Q is G(?) set, and Borel set.The last chapter makes researches on the long-run behavior of a Markov chain which the state space is compact space. A usual way to study the long-run behavior of a Markov matrix P is to investigate the limiting behavior of the n-step matrix. However, the limiting matrix of may not exist. Mei-Hsui Chi[39]gave a different approach and investigate which is the limit of the spaces of the n-step possibility distributions. For the finite Markov chains, Mei-Hsui Chi[39]proves that the limiting space of a Markov nxn matrix P contains exactly all steady-state distribution and periodic-state distribution. In this chapter, if the state space is compact space, using the concept of locally convex and the Krein-milman theorem, we get the same results.