Brazilian Journal of Probability and Statistics

Maxima of branching random walks with piecewise constant variance

Frédéric Ouimet

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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
Received: August 2016
Accepted: February 2017
First available in Project Euclid: 17 August 2018

Permanent link to this document
https://projecteuclid.org/euclid.bjps/1534492897

Digital Object Identifier
doi:10.1214/17-BJPS358

Mathematical Reviews number (MathSciNet)
MR3845025

Zentralblatt MATH identifier
06979596

Keywords
Extreme value theory branching random walks time inhomogeneous environments

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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