Critical branching processes in random environment and Cauchy domain of attraction
Résumé
We are interested in the survival probability of a population modeled by a critical branching process in an i.i.d. random environment and in the growth rate of the population given its survival up to a large time n. We assume that the random walk associated with the branching process is oscillating and satisfies a Doney-Spitzer condition P(S-n > 0) -> rho, n -> infinity, which is a standard condition in fluctuation theory of random walks. Unlike the previously studied case rho is an element of(0, 1), we investigate the case where the offspring distribution is in the domain of attraction of an asymmetric stable law with parameter 1, which implies that rho = 0 or 1. We find the asymptotic behaviour of the survival probability of the population and prove a Yaglom-type conditional limit theorem for the population size in these two cases.