Abstract
A countable Markov chain in a Markovian environment is considered. A Poisson limit theorem for the chain recurring to small cylindrical sets is mainly achieved. In order to prove this theorem, the entropy function h is introduced and the Shannon-McMillan-Breiman theorem for the Markov chain in a Markovian environment is shown. It's well-known that a Markov process in a Markovian environment is generally not a standard Markov chain, so an example of Poisson approximation for a process which is not a Markov process is given. On the other hand, when the environmental process degenerates to a constant sequence, a Poisson limit theorem for countable Markov chains, which is the generalization of Pitskel's result for finite Markov chains is obtained.
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Communicated by LIU Zeng-rong
Foundation item: the National Natural Science Foundation of China (19971072)
Biography: FANG Da-fan (1958-)
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Da-fan, F., Han-xing, W. & Mao-ning, T. Poisson limit theorem for countable Markov chains in Markovian environments. Appl Math Mech 24, 298–306 (2003). https://doi.org/10.1007/BF02438267
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DOI: https://doi.org/10.1007/BF02438267