## 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

#### 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
Accepted: January 2018
First available in Project Euclid: 4 March 2019

https://projecteuclid.org/euclid.bjps/1551690038

Digital Object Identifier
doi:10.1214/18-BJPS391

Mathematical Reviews number (MathSciNet)
MR3919027

Zentralblatt MATH identifier
07057451

#### 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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