## Brazilian Journal of Probability and Statistics

### Maxima of branching random walks with piecewise constant variance

Frédéric Ouimet

#### Abstract

This article extends the results of Fang and Zeitouni [Electron. J. Probab. 17 (2012a) 18] 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 $O_{\mathbb{P}}(1)$ 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 [Electron. J. Probab. 20 (2015b) 40] when the law of the increments is not necessarily Gaussian. However, the proof we present here generalizes the approach of Fang and Zeitouni [Electron. J. Probab. 17 (2012a) 18] 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.

#### Article information

Source
Braz. J. Probab. Stat., Volume 32, Number 4 (2018), 679-706.

Dates
Accepted: February 2017
First available in Project Euclid: 17 August 2018

https://projecteuclid.org/euclid.bjps/1534492897

Digital Object Identifier
doi:10.1214/17-BJPS358

Mathematical Reviews number (MathSciNet)
MR3845025

Zentralblatt MATH identifier
06979596

#### Citation

Ouimet, Frédéric. Maxima of branching random walks with piecewise constant variance. Braz. J. Probab. Stat. 32 (2018), no. 4, 679--706. doi:10.1214/17-BJPS358. https://projecteuclid.org/euclid.bjps/1534492897

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