Some Results about Multidimensional Branching Processes with Random Environments

Abstract

A multidimensional branching process with random environments is considered. Two results are proven about this process. The first proves that all nonzero states of the process are transient. Since the process in question is not Markov, the proof of this result is more involved than in the classical case. Our second result deals with the extinction of the process when we are in the critical case. We prove as in the classical theory that extinction occurs $\operatorname{w.p.}$1.