Ergodic Theory Of Discrete-time Markov Chains

Ergodic Theory of Markov chains is the theory of investigating the asymptotic properties of Markov chains. Markov chains have significant application in some areas, such as probability, stochastic differential equation, queueing theory, statistical physics, Monte Carlo numerical method and iterated function system. The main researching subjects of Ergodic theory are the existence of invariant probabilities, ergodicity and asymptotic stationary properties.This paper deals with some important problems in Ergodic Theory of Discrete-time Markov chains. It is mainly through Markov operator which is defined by one-step transition probability of Markov chains to investigate the ergodic properties of Markov chains. There are total four chapters in this paper.The first chapter makes researches on the Ergodic Decomposition of Markov-Feller operators on locally compact separate metric space. Ergodic Decomposition Theorem states that ergodic measures are 'basic elements' of invariant probabilities space and every invariant measure can be expressed by the integral of elementary ergodic measures. From this point, the research work of Ergodic Decomposition can make contribution to the characterization of the existence of invariant measures and asympototic properties. Yosida[49] provides Ergodic Decomposition Theorem of Markov operators with invariant probabilities on compact spaces. Radu Zaharopol[11] investigates the same question on locally compact separate metric spaces and generalizes the conclusion of Yosida[49]. Moreover, the supports of ergodic measures are given. In the first chapter, we provide the Ergodic Docomposition theorem for Markov operators withσ-finite invariant measure in locally compact separate metric space. Moreover, we have proved that there are finite elementary ergodic measures when Markov operators are strictly constricted. The main method is to find the equivalent invariant probability of theσ-finite invariant measure by Birkhoff average ergodic theorem and then obtain elementary ergodic measures by Yosida Ergodic Decomposition theorem.The second chapter offers the relationship between irreducibility and ergodicity of Markov chains in uncountable state space. In countable state space, irreducibility is defined by the connection of all the states while it is characterized by the reachable of Markov chains to open sets in uncountable state spaces. Ergodicity means that invariant sets are negligible sets or total sets. Irreducibility is weaker than ergodicity. D.Revuz[6] proves that ergodicity is equavalent to irreducibility and aperiod in finite state spaces. In this chapter, we give the equivalent condition of irreducibility and ergodicity of conservated Markov operators. That condition speaks that the support of invariant measure is the whole space.The third chapter focuses on the unique ergodic property of Markov operators in locally compact separate metric space. The unique ergodic property says that there is only one stationary measure of Markov operators. It is a property which is weaker than asympototic stationary property, but stronger than the existence of invariant measures. So the research work of unique ergodic property is significant in Ergodic Theory. P.Walters [12] investigates continuous transformations in compact metric space and gives three conditions equavalent with unique ergodic property. Radu Zaharopol[11] proves that a sufficient and necessary condition which guarantees Markov-Feller operator is unique ergodic is that there is only one control generic point. In this chapter, the average ergodic theorem and uniform ergodic theorem that are equivalent with unique ergodicity are given and proved. This part is a generalization of the research work on the unique ergodic theorem of continuous transformations in compact space by P.Waltes[13].The last chapter makes researches on the existence of invariant measures of Markov-Feller operators in Polish space. At first, Lasota-Yorke[10] gave sufficient conditions of existence of invariant measures and asympototic stationary properties of Markov operators in locally compact separate metric space. The conditions are characterized by the iterated transition probabilities. S.Meyn, R.Tweedie[8] proved that Foster-Lyapunov condition guarantees the existence of invariant measures in locally compact spaces. This condition only includes the one-step transition probability and then it is easy to use. T.Szarek[33,34,35,36,27,9] investigated the two questions in Polish space. Various conditions with types of iterated transition probabilities are given to prove the existence of invariant measures and asympototic stationary properties. In this chapter, we continue the research work in Polish space and give one sufficient condition, which states that there is an invariant probability if the Markov-Feller operator is equ-continuous in one point. It is convenient to use. Moreover, the proof gives a method to find the invariant probability. This condition is a generalization of T.Szarek[27].