On the Maximal Displacement of a Critical Branching Random Walk

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 $4+\epsilon$ 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 $P\{M\ge x\}\sim 6{\eta }^2/({\sigma }^2{x}^2)$ as $x\to \infty$. 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.

corrected error in proof of Theorem 1

References in corpus (2)

• Tightness for a family of recursion equations
• The branching Brownian motion seen from its tip

Cited by in corpus (2)

• Maximal Displacement of Critical Branching Symmetric Stable Processes
• Critical Branching Brownian Motion with Killing