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

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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