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

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 nth 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. (c) 2023 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). MSC: primary 60J80; 05C81; secondary 60G70

Automated summary

We study the extremes of branching random walks where the underlying Galton-Watson tree has infinite progeny mean. The displacements are either regularly varying or have lighter tails. In the regularly varying case, we find that the sequence of normalized extremes converges to a Poisson random measure. We also analyze the scaled position of the rightmost particle in the nth generation when the tail of the displacement behaves like exp(-K(x)), and identify the exact scaling of the maxima in all cases.