| As branches of stochastic process, Markov chains in random environments is one of the most active areas in probability today,and have been well developed. Especially, some of the concrete research models such as the branching process and the branching random walks have been deeply studied. Based on the research of Coburn, Orey studied the problems of Markov chains in double bi-infinite environments and presented a series of open problems, and aroused many probability scholar's widespread interest. Domestic, the scientific research groups led by professor Dai Yonglong, and Hu Dihe respectively have done many valuable work in Markov chains in random environments such as classification of state, the existence of invariant measure so on. However, the above work need to further studied and developed for the completeness and richness of the general theory of the area.To solve the open problem presented by Orey, Li Yingqiu first introduced the concepts ofπ?irreducible, recurrence and transience, gave the criterion of the recurrence of the chains in bi-infinite environments and discussed the existence of invariant measure ofπ? irreducible chains. R.L.Tweedie had discussed stochastic stability of Markov chains. Liu Wen discussed the strong limit theorem by the analysis method and used this method in a series of classic Makov chains result, obtained the new proof method.This paper consists of seven parts. In chapter 1 the history and the development of the Markov chains in random environments are introduced. In chapter 2, 3, 4 the irreducibility, invariant measures and ergodicity of Markov chains in random environments are given. In chapter 5, 6 the strong limit theorems for transition probabilities and functional in Markov chains in Markov environments are discussed. In chapter 7 the construction of the Markov chains in random environment is solved by using the stochastic transition matrix. In addition, the concepts of communicating state and strong communicating state are introduced.
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