We study a birth and death process $\{N_t\}_{t\ge0}$ in i.i.d. random environment, for which at each discontinuity, one particle might be born or at most $L$ particles might be dead. Along with investigating the existence and the recurrence criterion, we also study the law of large numbers of $\{N_t\}$. We show
that the first passage time can be written as a functional of an $L$-type branching process in random environment and a sequence of independent and exponentially distributed random variables. Consequently, an explicit velocity of the law of large numbers can be given.
The project was supported by the National Nature Science Foundation of China (Grant No. 11501008) and the Nature Science Foundation of Anhui Province (Grant No. 1508085QA12).
WANG Huaming. Limit Theorems for Birth and Death Process with One-Side Bounded Jumps in Random Environment[J]. CHINESE JOURNAL OF APPLIED PROBABILITY AND STATIST, 2019, 35(1): 51-62.