Haldane's asymptotics for Supercritical Branching Processes in an iid Random Environment

Abstract

Branching processes in a random environment are a natural generalisation of Galton-Watson processes. In this paper we analyse the asymptotic decay of the survival probability for a sequence of slightly supercritical branching processes in a random environment, where the offspring expectation converges from above to 1. We prove that Haldane's asymptotics, known from classical Galton-Watson processes, turns up again in the random environment case, provided that one stays away from the critical or subcritical regimes. A central building block is a connection to and a limit theorem for perpetuities with asymptotically vanishing interest rates.