Convergence in law of the maximum of nonlattice branching random walk

Abstract

Let $η^*_n$ denote the maximum, at time $n$, of a nonlattice one-dimensional branching random walk $η_n$ possessing (enough) exponential moments. In a seminal paper, Aidekon demonstrated convergence of $η^*_n$ in law, after recentering, and gave a representation of the limit. We give here a shorter proof of this convergence by employing reasoning motivated by Bramson, Ding and Zeitouni. Instead of spine methods and a careful analysis of the renewal measure for killed random walks, our approach employs a modified version of the second moment method that may be of independent interest.

Contains a sketch of the lattice case. To appear in AIHP B