Maxima of branching random walks with piecewise constant variance

TL;DR

This paper analyzes the maximum of branching random walks with piecewise constant variance, extending previous Gaussian results and providing a more robust proof approach that highlights how variance profiles affect asymptotic behavior.

Contribution

It introduces a new, simplified proof method for BRWs with piecewise constant variance, extending Gaussian case results and clarifying the influence of variance profiles.

Findings

Asymptotics of the maximum are derived with an OP(1) error.

Variance profile significantly influences the leading order and correction terms.

The proof approach is more robust and easier to understand than previous methods.

Abstract

This article extends the results of Fang & Zeitouni (2012a) on branching random walks (BRWs) with Gaussian increments in time inhomogeneous environments. We treat the case where the variance of the increments changes a finite number of times at different scales in [0,1] under a slight restriction. We find the asymptotics of the maximum up to an OP(1) error and show how the profile of the variance influences the leading order and the logarithmic correction term. A more general result was independently obtained by Mallein (2015b) when the law of the increments is not necessarily Gaussian. However, the proof we present here generalizes the approach of Fang & Zeitouni (2012a) instead of using the spinal decomposition of the BRW. As such, the proof is easier to understand and more robust in the presence of an approximate branching structure.