On the Survival Probability and a Functional Limit Theorem for Branching Processes in Random Environment

#### E.E. Dyakonova, J. Geiger, V.A. Vatutin

2004, v.10, Issue 2, 289-306

ABSTRACT

Let $(Z_n)_{n\ge 0}$ be a branching process in i.i.d. random environment. We consider a generalization of the so-called critical case assuming that the distribution of the logarithmic conditional mean offspring number is attracted without centering to a stable law. We show that subject to moment assumptions the exact asymptotics of $P\{Z_n>0\}$ is proportional to $n^{-(1-\rho)} L_1(n)$, where $\rho\in (0,1)$ can be expressed in terms of the index and the skewness parameter of the attracting stable law and $L_1$ is some slowly varying function. Moreover, we prove a conditional functional limit law for the suitably rescaled generation size process.

Keywords: branching process,random environment,survival probability,functional limit theorem,conditioned random walk