Extreme values of critical and subcritical branching stable processes with positive jumps

TL;DR

This paper analyzes the maximum reach of particles in a critical or subcritical branching stable process with positive jumps, providing asymptotic results for the extremal behavior of such processes.

Contribution

It introduces asymptotic formulas for the maximum particle location in branching stable processes with positive jumps under critical or subcritical conditions.

Findings

Asymptotic behavior of maximum particle location derived

Results applicable to processes with stable Lévy jumps

Provides insights into extremal process dynamics

Abstract

We consider a branching stable process with positive jumps, i.e. a continuous-time branching process in which the particles evolve independently as stable L{\'e}vy processes with positive jumps. Assuming the branching mechanism is critical or subcritical, we compute the asymptotics of the maximum location ever reached by a particle of the process.