Branching Random Walks, Stable Point Processes and Regular Variation

TL;DR

This paper establishes conditions under which branching random walks with regularly varying displacements converge to stable point processes, extending classical results and confirming recent predictions in the field.

Contribution

It provides a sufficient condition for point processes to belong to the superposition domain of attraction of stable processes and explicitly characterizes their weak limits in branching random walks.

Findings

Derived explicit weak limit representations for branching random walks.

Extended Durrett's 1983 results to broader settings.

Confirmed predictions of Brunet and Derrida (2011) for this model.

Abstract

Using the language of regular variation, we give a sufficient condition for a point process to be in the superposition domain of attraction of a strictly stable point process. This sufficient condition is then used to obtain an explicit representation of the weak limit of a sequence of point processes induced by a branching random walk with jointly regularly varying displacements. As a consequence, we extend the main result of Durrett (1983) and verify that two related predictions of Brunet and Derrida (2011) remain valid for this model.