Large deviations of extremes in branching random walk with regularly varying displacements

TL;DR

This paper investigates the large deviations of extreme positions in a branching random walk with heavy-tailed displacements, revealing clustering behavior and analyzing the maximum position in the nth generation.

Contribution

It introduces a detailed analysis of large deviations for extremes in branching random walks with regularly varying displacements, highlighting clustering phenomena.

Findings

Extreme positions form clusters in the limit

Large deviations of the maximum are characterized

Displacements have jointly regularly varying tails

Abstract

In this article, we consider a branching random walk on the real-line where displacements coming from the same parent have jointly regularly varying tails. The genealogical structure is assumed to be a supercritical Galton-Watson tree, satisfying Kesten-Stigum condition. We study the large deviations of the extremal process, formed by the appropriately normalized positions in the $n$-th generation and show that the large extreme-positions form clusters in the limit. As a consequence of this, we also study the large deviations of the maximum among positions at the $n$-th generation.