The Classification Of State, Law Of Large Numbers And Theory Of Ergodicity For Markov Chains In Random Environments

In recent years, the Markov chain in random environments is developing rapidly and has been a very important research area in the stochastic processes. It is necessary requirement that leads the theoretical research about stochastic processes to develop in depth and width, and it also has very important theoretical meaning and broad applied prospects. This paper studies the general theory of Markov chains in random environments. It consists of four chapters. The first chapter is the introduction. In this chapter, we primarily introduce some basic concepts and developments of general theory of Markov chains in random environments. Emphatically we introduce achievements related to this paper and give a brief account of the main results in it. The second chapter is about the classification of states and some properties on Markov chains in random environments. In this chapter, we mainly definite all kinds of state, and completely partition the state space X of the original chains Xρ. By studying the invariable function, the closed sets and the mutual property, we draw many conclusions on the relationships in all kinds of state and their relative properties. The third chapter is about the strong law of large numbers and the law of large numbers for Markov chains in Markovian environments. In this chapter, we primarily study the strong law of large numbers and law of large numbers of some special functions for Markov chains in unidirectional infinitely Markovian environments, and give some sufficient conditions on the jointly Markov chains and the sample function of the jointly Markov chains. The fourth chapter is about the ergodic properties of Markov chains in random environments. In this chapter, we introduce the concepts of weak ergodicity, uniformly weak ergodicity, strong ergodicity, uniformly strong ergodicity of Markov chains in double infinitely random environments when the "starting time "is arbitrarily given point. We have discussed all kinds of relationship among these ergodicities, and prove that θρ? chains is weak ergodicity if P (θ) is limited to inevitable exit sets.