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Branching processes in a random environment are a natural generalisation ofGalton-Watson processes. In this paper we analyse the asymptotic decay of thesurvival probability for a sequence of slightly supercritical branchingprocesses in a random environment, where the offspring expectation convergesfrom above to $1$. We prove that Haldane's asymptotics, known from classicalGalton-Watson processes, turns up again in the random environment case,provided that one stays away from the critical or subcritical regimes. Acentral building block is a connection to and a limit theorem for perpetuitieswith asymptotically vanishing interest rates.

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