A note on multitype branching process with bounded immigration in random environment

In this paper, we study the total number of progeny, W, before regenerating of multitype branching process with immigration in random environment. We show that the tail probability of |W| is of order t .. as t . ., with . some constant. As an application, we prove a stable law for (L-1) random walk in random environment, generalizing the stable law for the nearest random walk in random environment (see .Kesten, Kozlov, Spitzer: A limit law for random walk in a random environment. Compositio Math., 30, 145.168 (1975).).