Limit laws for transient random walks in random environment on $\mathbb{Z}$

Enriquez, Nathanaël; Sabot, Christophe; Zindy, Olivier

doi:10.5802/aif.2497

Limit laws for transient random walks in random environment on

ℤ

[Lois limites pour les marches aléatoires en milieu aléatoire transientes sur

ℤ

]

Enriquez, Nathanaël ¹ ; Sabot, Christophe ² ; Zindy, Olivier ³

¹ Université Paris 10 Laboratoire Modal’X 200, avenue de la République 92000 Nanterre (France) Laboratoire de Probabilités et Modèles Aléatoires CNRS– UMR 7599 Université Paris 6 - Paris 7 Boîte Courrier 188 4, place Jussieu 75252 Paris Cedex 05 (France)
² Université de Lyon Université Lyon 1 INSA de Lyon – École Centrale de Lyon CNRS – UMR 5208 Institut Camille Jordan 43, boulevard du 11 novembre 1918 69622 Villeurbanne Cedex (France)
³ Université Paris 6 Laboratoire de Probabilités et Modèles Aléatoires CNRS – UMR 7599 Boîte Courrier 188 4, place Jussieu 75252 Paris Cedex 05 (France)

Annales de l'Institut Fourier, Tome 59 (2009) no. 6, pp. 2469-2508.

Résumé
Abstract

Nous considérons les marches aléatoires en milieu aléatoire sur $ℤ$ transientes et de vitesse nulle. D’après un résultat classique de Kesten, Kozlov et Spitzer, le temps d’atteinte du niveau $n$ converge en loi, après renormalisation, vers une variable aléatoire stable positive, mais ces auteurs n’obtiennent pas la description de son paramètre. Une preuve différente est présentée, qui permet d’obtenir une caractérisation complète de cette loi stable. Le cas d’environnements de Dirichlet s’avère être particulièrement explicite.

We consider transient random walks in random environment on $ℤ$ with zero asymptotic speed. A classical result of Kesten, Kozlov and Spitzer says that the hitting time of the level $n$ converges in law, after a proper normalization, towards a positive stable law, but they do not obtain a description of its parameter. A different proof of this result is presented, that leads to a complete characterization of this stable law. The case of Dirichlet environment turns out to be remarkably explicit.

MR Zbl | 3 citations dans Numdam

DOI : 10.5802/aif.2497

Classification : 60K37, 60F05, 60F17, 60E07, 60E10
Keywords: Random walks in random environment, stable laws, fluctuations theory for random walks, Beta distributions
Mot clés : marches aléatoires en milieu aléatoire, lois stables, théorie des fluctuations pour une marche aléatoire, lois Beta

Affiliations des auteurs :

Enriquez, Nathanaël ¹ ; Sabot, Christophe ² ; Zindy, Olivier ³

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     journal = {Annales de l'Institut Fourier},
     pages = {2469--2508},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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Enriquez, Nathanaël; Sabot, Christophe; Zindy, Olivier. Limit laws for transient random walks in random environment on $\mathbb{Z}$. Annales de l'Institut Fourier, Tome 59 (2009) no. 6, pp. 2469-2508. doi : 10.5802/aif.2497. http://www.numdam.org/articles/10.5802/aif.2497/

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