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November 2018 Maxima of branching random walks with piecewise constant variance
Frédéric Ouimet
Braz. J. Probab. Stat. 32(4): 679-706 (November 2018). DOI: 10.1214/17-BJPS358


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.


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Frédéric Ouimet. "Maxima of branching random walks with piecewise constant variance." Braz. J. Probab. Stat. 32 (4) 679 - 706, November 2018.


Received: 1 August 2016; Accepted: 1 February 2017; Published: November 2018
First available in Project Euclid: 17 August 2018

zbMATH: 06979596
MathSciNet: MR3845025
Digital Object Identifier: 10.1214/17-BJPS358

Rights: Copyright © 2018 Brazilian Statistical Association


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Vol.32 • No. 4 • November 2018
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