A fixed-point equation approach for the superdiffusive elephant random walk - Archive ouverte HAL Accéder directement au contenu
Pré-Publication, Document De Travail Année : 2023

A fixed-point equation approach for the superdiffusive elephant random walk

Résumé

We study the elephant random walk in arbitrary dimension $d\geq 1$. Our main focus is the limiting random variable appearing in the superdiffusive regime. Building on a link between the elephant random walk and P\'olya-type urn models, we prove a fixed-point equation (or system in dimension two and larger) for the limiting variable. Based on this, we deduce several properties of the limit distribution, such as the existence of a density with support on $\mathbb R^d$ for $d\in\{1,2,3\}$, and we bring evidence for a similar result for $d\geq 4$. We also investigate the moment-generating function of the limit and give, in dimension $1$, a non-linear recurrence relation for the moments.
Fichier principal
Vignette du fichier
2308.14630.pdf (1.09 Mo) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)

Dates et versions

hal-04190858 , version 1 (30-08-2023)
hal-04190858 , version 2 (18-04-2024)

Identifiants

Citer

Hélène Guérin, Lucile Laulin, Kilian Raschel. A fixed-point equation approach for the superdiffusive elephant random walk. 2024. ⟨hal-04190858v2⟩
36 Consultations
80 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More