Extreme positions of regularly varying branching random walk in random environment

Abstract

In this article, we consider a Branching Random Walk (BRW) on the real line where the underlying genealogical structure is given through a supercritical branching process in i.i.d. environment and satisfies Kesten-Stigum condition. The displacements coming from the same parent are assumed to have jointly regularly varying tails. Conditioned on the survival of the underlying genealogical tree, we prove that the appropriately normalized (depends on the expected size of the n-th generation given the environment) maximum among positions at the n-th generation converges weakly to a scale-mixture of Frechét random variable. Furthermore, we derive the weak limit of the extremal processes composed of appropriately scaled positions at the n-th generation and show that the limit point process is a member of the randomly scaled scale-decorated Poisson point processes (SScDPPP). Hence, an analog of the predictions by Brunet and Derrida [19] holds.