On the Maximal Displacement of a Critical Branching Random Walk

TL;DR

This paper analyzes the asymptotic behavior of the maximum displacement in a critical branching random walk with specific moment conditions, providing precise tail estimates and a conditional limit theorem.

Contribution

It establishes the tail distribution asymptotics for the maximum position in a critical branching random walk with finite moments, extending understanding of its extremal behavior.

Findings

The probability that the maximum exceeds x behaves like 6η²/(σ²x²) as x→∞.

A conditional limit theorem describes the distribution of the rightmost particle given survival.

The results depend on finite variance and moment conditions of offspring and jump distributions.

Abstract

We consider a branching random walk initiated by a single particle at location 0 in which particles alternately reproduce according to the law of a Galton-Watson process and disperse according to the law of a driftless random walk on the integers. When the offspring distribution has mean 1 the branching process is critical, and therefore dies out with probability 1. We prove that if the particle jump distribution has mean zero, positive finite variance $\eta^{2}$, and finite $4+\varepsilon$ moment, and if the offspring distribution has positive variance $\sigma^{2}$ and finite third moment then the distribution of the rightmost position $M$ reached by a particle of the branching random walk satisfies $P\{M \geq x\}\sim 6\eta^{2}/ (\sigma^{2}x^{2})$ as $x \rightarrow \infty$. We also prove a conditional limit theorem for the distribution of the rightmost particle location at time $n$ given that the process survives for $n$ generations.