Some Problems Of Markov Chains In Random Environments

In general, while doing the theoretical research for Markov chains in random environments, we generalize the existing results in the determine environments to the random environments. But many new concepts and new methods are created during the research, which is the essence of the theory of Markov chains in random environments.In this paper, the limit properties of the relative frequency of Markov chains in single infinite Markov environment are studied by the classic analytical methods-interval subdivision method. But there is difficulties in dealing with the corresponding problems of the double infinite Markov chains. In view of this, some limit properties of Markov source in bi-infinite random environments are given by constructing a nonnegative martingale which converges almost surely. One of the specific models of the Markov chains in random environments, that is branching process in random environments with immigration, is studied in this paper. We generalize the existing results in the Galton-Waston branching process with immigration to the branching process in random environments with immigration.In this paper, we obtain the following conclusions:1、The limit properties of relative frequency of markov chains in single infinite Markov environments are studied, and the upper (lower) bound of the upper (lower) limit is obtained.2、A strong limit theorem of the average of ternary functions for Markov chains in bi-infinite random environments is giving by constructing a nonnegative martingale which converges almost surely. As corollaries, some limit properties of relative entropy density of these chains are obtained, and the environment sequences are not required to be a Markov chains. The properties of Markov chain transition probability is studied, and the geometric average of the strong limit theorem represented by inequalities is obtained. As a corollary, the arithmetic average of the strong limit theorem represented by inequalities is obtained.3、The branching process in random environments with immigration is study. Under certain conditions, its total number of then generation individuals’conditional probability generating function converges to an appropriate stable distribution. Let Wn=Zn/Πf=0n-1mj, then there exists an integrable random variable W, makes Wn convergence to W, which converges almost surely and also according to the mean square.