Branching random walk with infinite progeny mean: a tale of two tails - MODAL'X - UMR 9023 - Modélisation aléatoire de Paris Nanterre
Article Dans Une Revue Stochastic Processes and their Applications Année : 2023

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

Résumé

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 $n^{th}$ 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.
Fichier principal
Vignette du fichier
1-s2.0-S0304414923000467-main.pdf (1.75 Mo) Télécharger le fichier
Origine Fichiers éditeurs autorisés sur une archive ouverte
licence

Dates et versions

hal-03145352 , version 1 (09-09-2024)

Licence

Identifiants

Citer

Souvik Ray, Rajat Subhra Hazra, Parthanil Roy, Philippe Soulier. Branching random walk with infinite progeny mean: a tale of two tails. Stochastic Processes and their Applications, 2023, 160, pp.120-160. ⟨10.1016/j.spa.2023.03.001⟩. ⟨hal-03145352⟩
49 Consultations
3 Téléchargements

Altmetric

Partager

More