Extremal Process of Last Progeny Modified Branching Random Walks

TL;DR

This paper studies the extremal process of a modified branching random walk where the last generation's particle positions are adjusted by an independent random variable, revealing how tail properties of this variable influence the process's asymptotic behavior.

Contribution

It provides a detailed description of the asymptotic behavior of the extremal process in last progeny modified branching random walks based on the tail properties of the modification variable.

Findings

Characterization of extremal process depending on tail behavior of Y

Asymptotic results for different tail regimes

Insights into the impact of modifications on branching random walk extremes

Abstract

We consider a last progeny modified branching random walk, in which the position of each particle at the last generation $n$ is modified by an i.i.d. copy of a random variable $Y$. Depending on the asymptotic properties of the tail of $Y$, we describe the asymptotic behaviour of the extremal process of this model as $n \to \infty$.