The extremal point process for branching random walk with stretched exponential displacements

TL;DR

This paper studies the behavior of particles in a branching random walk with stretched exponential displacements, providing a new limit theorem for the position of the rightmost particle and describing the extremal process.

Contribution

It introduces a novel analysis of the extremal process for branching random walks with stretched exponential displacements, including a new limit theorem for the maximum particle position.

Findings

Point process convergence near the rightmost particle

New limit theorem for the maximum particle position

Precise large deviations outside the one big jump regime

Abstract

We investigate a branching random walk where the displacements are independent from the branching mechanism and have a stretched exponential distribution. We describe the positions of the particles in the vicinity of the rightmost particle in terms of point process convergence. As a consequence we give a~new limit theorem for the position of the rightmost particle. Our methods rely on providing precise large deviations for sums of i.i.d. random variables with stretched exponential distributions outside the so-called one big jump regime.