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Age-dependent branching processes in random environments

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Science China Mathematics 2008年第10期51卷 1807-1830页

作者： LI YingQiu LIU QuanSheng College of Mathematics and Computing Science Changsha University of Science and TechnologyChangsha 410076China LMAM Universit′e de Bretagne SudCampus de TohannicBP57356017 VannesFrance

We consider an age-dependent branching process in random environments. The environments are represented by a stationary and ergodic sequence ξ = (ξ 0, ξ 1,…) of random variables. Given an environment ξ, the process is a non-homogenous Galton-Watson process, whose particles in n-th generation have a life length distribution G(ξ n ) on ?+, and reproduce independently new particles according to a probability law p(ξ n ) on ?. Let Z(t) be the number of particles alive at time t. We first find a characterization of the conditional probability generating function of Z(t) (given the environment ξ) via a functional equation, and obtain a criterion for almost certain extinction of the process by comparing it with an embedded Galton-Watson process. We then get expressions of the conditional mean E ξ Z(t) and the global mean EZ(t), and show their exponential growth rates by studying a renewal equation in random environments.

关键词： age-dependent branching processes random environments probability generating function integral equation extinction probability exponential growth rates of expectation and conditional expectation random walks and renewal equation in random environments renewal theorem 60J80 60K37 60K05

The Moments of First Entrance Time Distributions of Markov Chains in random environments

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Chinese Quarterly Journal of Mathematics 2011年第1期26卷 51-55页

作者： LI Shou-wei ZHU Dong-jin School of Economics and Management Southeast UniversityNanjing 211189China Department of Mathematics Anhui Normal UniversityWuhu 241003China

After defining generating functions,this paper discusses their properties,and then provides a sufFcient and necessary condition for a finite property of the moments of first entrance time distributions of Markov chains in random environments by generating ***,the paper obtains relevant conclusions of the moments of first entrance time distributions.

关键词： random environments taboo set generating functions

POISSON LIMIT THEOREM FOR COUNTABLE MARKOV CHAINS IN MARKOVIAN environments

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Applied Mathematics and Mechanics(English Edition) 2003年第3期24卷 298-306页

作者：方大凡王汉兴唐矛宁 Department of Mathematics Yueyang Normal University Department of Mathematics Shanghai UniversityShanghai 200436P.R.China Department of Mathematics Shanghai University

A countable Markov chain in a Markovian environment is considered.A Poisson limit theorem for the chain recurring to small cylindrical sets is mainly *** 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 *** 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.

关键词： Poisson distributions Markov chains random environments