Branching random walk with infinite progeny mean: a tale of two tails

TL;DR

This paper investigates the extreme values of branching random walks with infinite progeny mean, analyzing tail behaviors and convergence of scaled maxima under different tail assumptions of displacements.

Contribution

It provides a detailed analysis of the asymptotics and limit distributions of the maxima in branching random walks with infinite progeny mean, considering various tail behaviors.

Findings

Convergence of normalized extremes to a Poisson random measure in the regularly varying case

Identification of exact scaling for maxima under different tail behaviors

Existence of a non-trivial limit when the tail index exceeds 1

Abstract

We study the extremes of branching random walks under the assumption that the underlying Galton-Watson tree has infinite progeny mean. It is assumed that the displacements are either regularly varying or they have lighter tails. In the regularly varying case, it is shown that the point process sequence of normalized extremes converges to a Poisson random measure. We study the asymptotics of the scaled position of the rightmost particle in the $n$-th generation when the tail of the displacement behaves like $\exp(-K(x))$, where either $K$ is a regularly varying function of index $r> 0$, or $K$ has an exponential growth. We identify the exact scaling of the maxima in all cases and show the existence of a non-trivial limit when $r> 1$.