Birth and death process with one-side bounded jumps in random environment

Abstract

Let $\omega =({\omega }_i{)}_{i\in \mathbb{Z}}=({\mu }_i^L,...,{\mu }_i^1,{\lambda }_i{)}_{i\in \mathbb{Z}}$, which serves as the environment, be a sequence of i.i.d. random nonnegative vectors, with $L\ge 1$ a positive integer. We study birth and death process $N_t$ which, given the environment $\omega ,$ waits at a state $n$ an exponentially distributed time with parameter ${\lambda }_n+{\sum }_{l=1}^L{\mu }_n^l$ and then jumps to $n-i$ with probability ${\mu }_n^i/({\lambda }_n+{\sum }_{l=1}^L{\mu }_n^l),$ $i=1,...,L$ or to $n+1$ with probability ${\lambda }_n/({\lambda }_n+{\sum }_{l=1}^L{\mu }_n^l).$ A sufficient condition for the existence, a criterion for recurrence, and a law of large numbers of the process $N_t$ are presented. We show that the first passage time $T_1\stackrel{\mathcal{D}}{=}{\xi }_{0,1}+{\sum }_{i\le -1}{\sum }_{k=1}^{U_{i,1}}{\xi }_{i,k}+{\sum }_{i\le -1}{\sum }_{k=1}^{U_{i,1}+...+U_{i,L}}{\tilde{\xi }}_{i+1,k},$ where $(U_{i,1},...,U_{i,L}{)}_{i\le 0}$ is an $L$-type branching process in random environment and, given $\omega ,$ ${\xi }_{i,k},{\tilde{\xi }}_{i,k},i\le 0,k\ge 1$ are mutually independent random variables such that $P_{\omega }({\xi }_{i,k}\ge t)=e^{-({\lambda }_i+{\sum }_{l=1}^L{\mu }_i^l)t},t\ge 0.$ This fact enables us to give an explicit velocity of the law of large numbers.

9 pages

Cited by in corpus (2)

• Law of large numbers for random walk with unbounded jumps and BDP with bounded jumps in random environment
• Stationary distribution for birth and death process with one-side bounded jumps