Weak convergence of the extremes of branching L\'evy processes with regularly varying tails

TL;DR

This paper investigates the weak convergence of the maximum values of supercritical branching Lévy processes with heavy-tailed jumps, revealing distinct behavior from classical branching Brownian motion and establishing a limit theorem for their order statistics.

Contribution

It introduces the first weak convergence results for extremes of branching Lévy processes with regularly varying tails, highlighting differences from Brownian motion cases.

Findings

Weak convergence of the renormalized process $\\mathbb{X}_t$ is established.

A limit theorem for the order statistics of the process is derived.

The behavior differs significantly from branching Brownian motion with light tails.

Abstract

In this paper, we study the weak convergence of the extremes of supercritical branching L\'evy processes $\{\mathbb{X}_t, t \ge0\}$ whose spatial motions are L\'evy processes with regularly varying tails. The result is drastically different from the case of branching Brownian motions. We prove that, when properly renormalized, $\mathbb{X}_t$ converges weakly. As a consequence, we obtain a limit theorem for the order statistics of $\mathbb{X}_t$.