Branching random walk with infinite progeny mean: a tale of two tails

Ray, Souvik ; Hazra, Rajat Subhra ; Roy, Parthanil ; Soulier, Philippe (2020) Branching random walk with infinite progeny mean: a tale of two tails arxiv 1909.08948 .

Full text not available from this repository.

Official URL:


We study the extremes of branching random walks under the assumption that the underlying Galton-Watson tree has infinite progeny mean. It is assumed that the displacements are either regularly varying or they have lighter tails. In the regularly varying case, it is shown that the point process sequence of normalized extremes converges to a Poisson random measure. In the lighter-tailed case, however, the behaviour is much more subtle, and the scaling of the position of the rightmost particle in the nth generation depends on the family of stepsize distribution, not just its parameter(s). In all of these cases, we discuss the convergence in probability of the scaled maxima sequence. Our results and methodology are applied to study the almost sure convergence in the context of cloud speed for branching random walks with infinite progeny mean. The exact cloud speed constants are calculated for regularly varying displacements and also for stepsize distributions having a nice exponential decay.

Item Type:Article
Source:Copyright of this article belongs to author(s).
Keywords:Branching Random Walk; Galton-Watson Tree With Innite Progeny Mean; Cloud Speed; Point Processes; Extremes.
ID Code:121158
Deposited On:12 Jul 2021 08:12
Last Modified:12 Jul 2021 08:14

Repository Staff Only: item control page