Brazilian Journal of Probability and Statistics

Necessary and sufficient conditions for the convergence of the consistent maximal displacement of the branching random walk

Bastien Mallein

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Abstract

Consider a supercritical branching random walk on the real line. The consistent maximal displacement is the smallest of the distances between the trajectories followed by individuals at the $n$th generation and the boundary of the process. Fang and Zeitouni, and Faraud, Hu and Shi proved that under some integrability conditions, the consistent maximal displacement grows almost surely at rate $\lambda^{*}n^{1/3}$ for some explicit constant $\lambda^{*}$. We obtain here a necessary and sufficient condition for this asymptotic behaviour to hold.

Article information

Source
Braz. J. Probab. Stat., Volume 33, Number 2 (2019), 356-373.

Dates
Received: June 2017
Accepted: January 2018
First available in Project Euclid: 4 March 2019

Permanent link to this document
https://projecteuclid.org/euclid.bjps/1551690038

Digital Object Identifier
doi:10.1214/18-BJPS391

Mathematical Reviews number (MathSciNet)
MR3919027

Zentralblatt MATH identifier
07057451

Keywords
Branching random walk consistent maximal displacement perturbed random walk small deviations

Citation

Mallein, Bastien. Necessary and sufficient conditions for the convergence of the consistent maximal displacement of the branching random walk. Braz. J. Probab. Stat. 33 (2019), no. 2, 356--373. doi:10.1214/18-BJPS391. https://projecteuclid.org/euclid.bjps/1551690038


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