On weighted branching processes in random environment

In this paper we investigate the nonnegative martingale Wn=Zn/μn(U), n⩾0 and its a.s. limit W, when (Zn)n⩾0 is a weighted branching process in random environment with stationary ergodic environmental sequence U=(Un)n⩾0 and μn(U) denotes the conditional expectation of Zn given U for n⩾0. d 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. important special case of i.i.d. random environment, a Z log Z-condition turns out to be crucial. inistic and nonvarying environments are treated as special cases. guments adapt the probabilistic proof of Biggins’ theorem for branching random walks given by Lyons (1997) to our situation.