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