Applied Mathematics and Mechanics

, Volume 24, Issue 3, pp 298–306

Poisson limit theorem for countable Markov chains in Markovian environments

• Fang Da-fan
• Wang Han-xing
• Tang Mao-ning
Article

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.

Key words

Poisson distributions Markov chains random environments

O211.62

60J05 60J10

References

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© Editorial Committee of Applied Mathematics and Mechanics 2003

Authors and Affiliations

• Fang Da-fan
• 1
• 2
• Wang Han-xing
• 2
• Tang Mao-ning
• 2
1. 1.Department of MathematicsYueyang Normal UniversityYueyang, HumanPR China
2. 2.Department of MathematicsShanghai UniversityShanghaiPR China

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