Berry–Esseen’s bound and Cramér’s large deviation expansion for a supercritical branching process in a random environment

Abstract

Abstract. Let (Z n ) be a supercritical branching process in a random environment ξ = (ξ n ). We establish a Berry-Esseen bound and a Cramér's type large deviation expansion for log Z n under the annealed law P. We also improve some earlier results about the harmonic moments of the limit variable W = lim n→∞ W n , where W n = Z n /E ξ Z n is the normalized population size. Introduction and main resultsA branching process in a random environment (BPRE) is a natural and important generalisation of the Galton-Watson process, where the reproduction law varies according to a random environment indexed by time. It was introduced for the first time in Smith and Wilkinson [24] to modelize the growth of a population submitted to an environment. For background concepts and basic results concerning a BPRE we refer to Athreya and Karlin [4,3]. In the critical and subcritical regime the process goes out and the research interest is concentrated mostly on the survival probability and conditional limit theorems for the branching process, see e.g. [21]. In this article, we complete on these results by giving the Berry-Esseen bound and asymptotics of large deviations of Cramér's type for a supercritical BPRE.A BPRE can be described as follows. The random environment is represented by a sequence ξ = (ξ 0 , ξ 1 , ...) of independent and identically distributed random variables (i.i.d. r.v.'s); each realization of ξ n corresponds to a probability law {p i (ξ n ) : i ∈ N} on N = {0, 1, 2, . . . }, whose probability generating function is

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Berry-Esseen's bound for a supercritical branching process with immigration in a random environment

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