Recurrence of Multidimensional Persistent Random Walks. Fourier and Series Criteria

Abstract : The recurrence features of persistent random walks built from variable length Markov chains are investigated. We observe that these stochastic processes can be seen as Lévy walks for which the persistence times depend on some internal Markov chain: they admit Markov random walk skeletons. A recurrence versus transience dichotomy is highlighted. We first give a sufficient Fourier criterion for the recurrence, close to the usual Chung-Fuchs one, assuming in addition the positive recurrence of the driving chain and a series criterion is derived. The key tool is the Nagaev-Guivarc'h method. Finally, we focus on particular two-dimensional persistent random walks, including directionally reinforced random walks, for which necessary and sufficient Fourier and series criteria are obtained. Inspired by \cite{Rainer2007}, we produce a genuine counterexample to the conjecture of \cite{Mauldin1996}. As for the one-dimensional situation studied in \cite{PRWI}, it is easier for a persistent random walk than its skeleton to be recurrent but here the difference is extremely thin. These results are based on a surprisingly novel -- to our knowledge -- upper bound for the Lévy concentration function associated with symmetric distributions.
Type de document :
Pré-publication, Document de travail
Liste complète des métadonnées

Littérature citée [41 références]  Voir  Masquer  Télécharger
Contributeur : Yoann Offret <>
Soumis le : jeudi 7 décembre 2017 - 16:16:11
Dernière modification le : vendredi 8 juin 2018 - 14:50:07


Fichiers produits par l'(les) auteur(s)


Domaine public


  • HAL Id : hal-01658494, version 1
  • ARXIV : 1712.02999



Peggy Cénac, Basile De Loynes, Yoann Offret, Arnaud Rousselle. Recurrence of Multidimensional Persistent Random Walks. Fourier and Series Criteria. 2017. 〈hal-01658494〉



Consultations de la notice


Téléchargements de fichiers