Branching random walk and log-slowly varying tails

We investigate a branching random walk with independent and identically distributed heavy-tailed displacements. The offspring distribution is supercritical and satisfies the Kesten-Stigum condition. Our focus is on the case where the displacement law does not belong to the max-domain of attraction of an extreme value distribution. We demonstrate that when the tails of the displacements are such that the absolute value of their logarithm is a slowly varying function, the extremes of the process can still be effectively analyzed. Specifically, after applying a non-linear transformation, the extremes of the branching random walk converge to a clustered Cox process.