Extremes of Multi-type Branching Random Walks: Heaviest Tail Wins

TL;DR

This paper analyzes a multi-type branching random walk with heavy-tailed displacements, deriving the limiting distribution of particle positions and the asymptotic behavior of the rightmost particle, revealing the dominance of the heaviest tail.

Contribution

It establishes the weak limit of the point process of particle positions in a multi-type branching random walk with heavy tails, identifying the limit as a scale-decorated Poisson process.

Findings

Limiting point process is a scale-decorated Poisson point process.

Asymptotic distribution of the rightmost particle is derived.

Heavy-tailed displacements dominate the extremal behavior.

Abstract

We consider a branching random walk on a multi($Q$)-type, supercritical Galton-Watson tree which satisfies Kesten-Stigum condition. We assume that the displacements associated with the particles of type $Q$ have regularly varying tails of index $\alpha$, while the other types of particles have lighter tails than that of particles of type $Q$. In this article, we derive the weak limit of the sequence of point processes associated with the positions of the particles in the $n^{th}$ generation. We verify that the limiting point process is a randomly scaled scale-decorated Poisson point process (SScDPPP) using the tools developed in \cite{bhattacharya:hazra:roy:2016}. As a consequence, we shall obtain the asymptotic distribution of the position of the rightmost particle in the $n^{th}$ generation.