马氏环境中可数马氏链的Poisson极限率

基金项目: 国家自然科研基金资助项目(19971072)

Poisson Limit Theorem for Countable Markov Chains in Markovian Environments

1.
Department of Mathematics, Yueyang Normal University, Hunan 414000, P. R. China;
2.
Department of Mathematics, Shanghai University, Shanghai 2000436, P. R. China

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。
- Poisson分布 /
- 马氏链 /
- 随机环境
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.
- Poisson distributions /
- Markov chains /
- random environments

[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.
[2]	Wang Y H.A Compound poisson Convergence Theorem[J].Ann Probab,1991,19:452-455.
[3]	Pitskel B.Poisson limit law for markov chains[J].Ergod Th Dynam Sys,1991,11,501-513.
[4]	Nawrotzki K.Discrete open systems or markov chains is a random environment[J].J Inform Process Cybernet,1981,17:569-599.
[5]	Nawrotzki K.Discrete open systems or Markov chains in random environment[J].J Inform Process Cybernet,1982,18:83-98.
[6]	Cogburn R.Markov chains in random environments:the case of Markovian environments[J].Ann Probab,1980,8:908-916.
[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.
[8]	Cornfeld I,Fomin S,Sinai Ya G.Ergodic Theory[M].New York:Springer-Verlag,1982.