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Applied Mathematics and Mechanics

, Volume 24, Issue 3, pp 298–306 | Cite as

Poisson limit theorem for countable Markov chains in Markovian environments

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

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 

Chinese Library Classification

O211.62 

2000 MR Subject Classification

60J05 60J10 

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References

  1. [1]
    Seyast'yanov B A. Poisson limit law for a scheme of sums of independent random variables[J].Theory of Probability and Its Applications, 1972,17(4):695–699.CrossRefGoogle Scholar
  2. [2]
    Wang Y H. A Compound Poisson convergence theorem[J].Ann Probab, 1991,19:452–455.zbMATHMathSciNetGoogle Scholar
  3. [3]
    Pitskel B. Poisson limit law for Markov chains[J].Ergod Th Dynam Sys, 1991,11,501–513.zbMATHMathSciNetCrossRefGoogle Scholar
  4. [4]
    Nawrotzki K. Discrete open systems or Markov chains in a random environment[J].J Inform Process Cybernet, 1981,17:569–599.zbMATHMathSciNetGoogle Scholar
  5. [5]
    Nawrotzki K. Discrete open systems or Markov chains in random environment[J].J Inform Process Cybernet, 1982,18:83–98.zbMATHMathSciNetGoogle Scholar
  6. [6]
    Cogburn R. Markov chains in random environments: the case of Markovian environments[J].Ann Probab, 1980,8:908–916.zbMATHMathSciNetGoogle Scholar
  7. [7]
    Blum J R, Hanson D L, Koopmans L H. On the strong law of large numbers for a class of stochastic processes[J].Z Wahrsch Verw Gebrete, 1963,2:1–11.zbMATHMathSciNetCrossRefGoogle Scholar
  8. [8]
    Cornfeld I P, Fomin S, Sinai Ya G.Ergodic Theory[M]. New York: Springer-Verlag, 1982.Google Scholar

Copyright information

© 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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