Necessary and sufficient conditions for the convergence of the consistent maximal displacement of the branching random walk

TL;DR

This paper establishes the exact conditions under which the consistent maximal displacement in a supercritical branching random walk grows at a specific rate, refining previous asymptotic growth results.

Contribution

It provides a necessary and sufficient condition for the known asymptotic growth rate of the consistent maximal displacement in branching random walks.

Findings

Identifies precise conditions for the growth rate to hold

Refines understanding of displacement behavior in branching processes

Extends previous asymptotic results

Abstract

Consider a supercritical branching random walk on the real line. The consistent maximal displacement is the smallest of the distances between the trajectories followed by individuals at the $n$th generation and the boundary of the process. Fang and Zeitouni, and Faraud, Hu and Shi proved that under some integrability conditions, the consistent maximal displacement grows almost surely at rate $\lambda^* n^{1/3}$ for some explicit constant $\lambda^*$. We obtain here a necessary and sufficient condition for this asymptotic behaviour to hold.