Branching random walk and log-slowly varying tails

Abstract

We study a branching random walk with independent and identically distributed, heavy tailed displacements. The offspring law is supercritical and satisfies the Kesten-Stigum condition. We treat the case when the law of the displacements does not lie in the max-domain of attraction of an extreme value distribution. Hence, the classical extreme value theory, which is often deployed in this kind of models, breaks down. We show that if the tails of the displacements are such that the absolute value of the logarithm of the tail is a slowly varying function, one can still effectively analyse the extremes of the process. More precisely, after a non-linear transformation the extremes of the branching random walk process converge to a cluster Cox process.