Limit Theorems for Birth and Death Process with One-Side Bounded Jumps in Random Environment

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.