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

Let $\omega=(\omega_i)_{i\in\mathbb Z}=(\mu^{L}_i,...,\mu^{1}_i,\lambda_i)_{i\in \mathbb Z}$, which serves as the environment, be a sequence of i.i.d. random nonnegative vectors, with $L\ge1$ 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 $\lambda_n+\sum_{l=1}^L\mu^{l}_n$ and then jumps to $n-i$ with probability ${\mu^i_n}/(\lambda_n+\sum_{l=1}^L\mu^{l}_n),$ $i=1,...,L$ or to $n+1$ with probability ${\lambda_n}/(\lambda_n+\sum_{l=1}^L\mu^{l}_n).$ 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 $T_1\overset{\mathscr D}{=}\xi_{0,1}+\sum_{i\le -1}\sum_{k=1}^{U_{i,1}}\xi_{i,k}+\sum_{i\le -1}\sum_{k= 1}^{U_{i,1}+...+U_{i,L}}\tilde{\xi}_{i+1,k},$ where $(U_{i,1},...,U_{i,L})_{i\le0}$ 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 $P_\omega(\xi_{i,k}\ge t)=e^{-(\lambda_i+\sum_{l=1}^L\mu^{l}_i)t},\ t\ge 0.$ This fact enables us to give an explicit velocity of the law of large numbers.