Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

The infinite valley for a recurrent random walk in random environment

Nina Gantert, Yuval Peres, and Zhan Shi

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Abstract

We consider a one-dimensional recurrent random walk in random environment (RWRE). We show that the – suitably centered – empirical distributions of the RWRE converge weakly to a certain limit law which describes the stationary distribution of a random walk in an infinite valley. The construction of the infinite valley goes back to Golosov, see Comm. Math. Phys. 92 (1984) 491–506. As a consequence, we show weak convergence for both the maximal local time and the self-intersection local time of the RWRE and also determine the exact constant in the almost sure upper limit of the maximal local time.

Résumé

Nous prouvons que les mesures empiriques d’une marche aléatoire unidimensionnelle en environnement aléatoire convergent étroitement vers la loi stationnaire d’une marche aléatoire dans une vallée infinie. La construction de cette vallée infinie revient à Golosov, voir Comm. Math. Phys. 92 (1984) 491–506. En applications, nous obtenons la convergence étroite du maximum des temps locaux et du temps local d’intersections de la marche aléatoire en environnement aléatoire; de plus, nous identifions la constante représentant la “limsup” presque sûre du maximum des temps locaux.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 46, Number 2 (2010), 525-536.

Dates
First available in Project Euclid: 11 May 2010

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1273584133

Digital Object Identifier
doi:10.1214/09-AIHP205

Mathematical Reviews number (MathSciNet)
MR2667708

Zentralblatt MATH identifier
1201.60096

Subjects
Primary: 60K37: Processes in random environments 60J50: Boundary theory 60J55: Local time and additive functionals 60F10: Large deviations

Keywords
Random walk in random environment Empirical distribution Local time Self-intersection local time

Citation

Gantert, Nina; Peres, Yuval; Shi, Zhan. The infinite valley for a recurrent random walk in random environment. Ann. Inst. H. Poincaré Probab. Statist. 46 (2010), no. 2, 525--536. doi:10.1214/09-AIHP205. https://projecteuclid.org/euclid.aihp/1273584133


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