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As we all know , the Markov chain is one of the most important stochastic processes , and its application throughout the field of industry, agriculture, economics , insurance , biology , medicine, engineering and social sciences . A key issue affecting the Markov chain is the transition matrix , also known as the Markov chain environment . Just as the natural environment is random , classic Markov chain environment randomized , Markov chain in random environment , referred to as MCRE MCRE can be divided into according to MCRE ( ie the environment depends on the time ) , in accordance with an empty MCRE ( environment dependent on space) and in accordance with the time and space MCRE ( the environment both time-dependent and space - dependent ) , this article only study according to MCRE . Randomized environment , a lot of results in the classic can do a simple promotion , but some need to do a lot of work . This paper is to review this previous work , the text is divided into four parts : The first part introduces MCRE history and Research ; second part of the study of Markov processes , Martingale and stationary process three types of classical stochastic process of mutual relations, including inclusion and exclusion relations between them , the introduction of the concept of \chain structure constructed Markov chains in random environments and Skew Product Markov Chains to discuss the nature of the associated probability characteristic function ; discussed in section IV GW tree dimension, and a random model of the environment branched chain , discussed the formula of mathematical expectation and variance of the branching process in random environment .
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