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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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

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

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

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Zentralblatt MATH identifier

Branching random walk consistent maximal displacement perturbed random walk small deviations


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.

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