On weighted branching processes in random environment

Abstract

In this paper we investigate the nonnegative martingale W n=Z n/μ n( U ), n⩾0 and its a.s. limit W, when ( Z n ) n⩾0 is a weighted branching process in random environment with stationary ergodic environmental sequence U=( U n ) n⩾0 and μ n ( U) denotes the conditional expectation of Z n given U for n⩾0. We find necessary and sufficient conditions for W to be nondegenerate, generalizing earlier results in the literature on ordinary branching processes in random environment and also weighted branching processes. In the important special case of i.i.d. random environment, a Z log Z -condition turns out to be crucial. Deterministic and nonvarying environments are treated as special cases. Our arguments adapt the probabilistic proof of Biggins’ theorem for branching random walks given by Lyons (1997) to our situation.