The Central Limit Theorem for Markov Chains in Markovian Environments

In this paper, a central limit theorem for function of countable Markov chains in Markovian environments is investigated. Moreover, some sufficient conditions on the jointly Markov chains and sample function of the jointly Markov chains are given. At last, R_\theta-chains are systematically studied, some sufficient contions for the central limit theorem to hold for function of Markov chains in Markovian environments are obtained.