The asymptotics of the probability of nonextinction of a multidimensional branching process in a random environment

We study a multi-type Galton–Watson process in a random environment generated by a sequence of independent identically distributed random variables. For this process we show that under some conditions on the generating functions of offspring distributions the asymptotics of the probability of non-extinction at time $n$ has the order $n^{-1/2}$ as $n\to\infty$.
This research was supported by the Russian Foundation for Basic Research, grants 96–01–00338, 96–15–96092, and by INTAS–RFBR, grant 95–0099.