Acta mathematica scientia,Series B

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THE ERGODICITY FOR BI-IMMIGRATION BIRTH AND DEATH PROCESSES IN RANDOM ENVIRONMENT

Hu Dihe; Zhang Shulin   

  1. School of Mathematics and Statistics, Yunnan University, Kunming 650091, China
    School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China
  • Received:2005-06-21 Revised:2005-12-05 Online:2008-01-20 Published:2008-01-20
  • Contact: Hu Dihe

Abstract:

The concepts of bi-immigration birth and death
density matrix in random environment and bi-immigration birth and
death process in random environment are introduced. For any
bi-immigration birth and death matrix in random environment
$Q(\theta)$ with birth rate $\lambda<$ death rate $\mu$, the
following results are proved, $(1)$ there is an unique $q$-process
in random environment, $\bar P(\theta^*(0); t)=(\bar
p(\theta^*(0); t, i, j), i, j\geq0)$, which is ergodic, that is,
$\lim\limits_{t\rightarrow \infty}\bar p(\theta^*(0); t, i,
j)=\bar \pi(\theta^*(0); j)\geq 0$ does not depend on $i\geq 0$
and $\sum\limits_{j\geq 0}\bar \pi (\theta^*(0); j)=1$, $(2)$
there is a bi-immigration birth and death process in random
environment $(X^*=\{X_t, t\geq 0\}, \xi^*=\{\xi_t, t\in(-\infty,
\infty)\})$ with random transition matrix $\bar P(\theta^*(0); t)$
such that $X^*$ is a strictly stationary process.

Key words: Density matrix in random environment, random transition matrix, Markov process in random environment, bi-immigration birth and death density matrix in random environment, bi-immigration birth and death process in random environment

CLC Number: 

  • 60J27
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