Haldane's asymptotics for supercritical branching processes in an iid random environment

TL;DR

This paper extends Haldane's asymptotics to supercritical branching processes in iid random environments, showing that survival probabilities decay similarly to classical cases when away from criticality.

Contribution

It demonstrates that classical Haldane's asymptotics apply to branching processes in random environments under certain conditions, linking to perpetuities with vanishing interest rates.

Findings

Haldane's asymptotics hold in iid random environments for supercritical processes.

Survival probability decay matches classical Galton-Watson results away from criticality.

Connection established between branching processes and perpetuities with diminishing interest rates.

Abstract

Branching processes in a random environment are natural generalisations 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 an iid 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/subcritical regime. A central building block is a connection to and a limit theorem for perpetuities with asymptotically vanishing interest rates.