Poisson Limit Theorem for Countable Markov Chains in Markovian Environments
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摘要: 研究了马氏环境中的可数马氏链,主要证明了过程于小柱集上的回返次数是渐近地服从Poisson分布。为此,引入熵函数h,首先给出了马氏环境中马氏链的Shannon-Mc Millan-Breiman定理,还给出了一个非马氏过程Posson逼近的例子。当环境过程退化为一常数序列时,便得到可数马氏链的Poisson极限定理。这是有限马氏链Pitskel相应结果的拓广。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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Key words:
- Poisson distributions /
- Markov chains /
- random environments
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