Limit laws for transient random walks in random environment on ℤ

Nathanaël Enriquez[1]; Christophe Sabot[2]; Olivier Zindy[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 (2009)

  • Volume: 59, Issue: 6, page 2469-2508
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Enriquez, Nathanaël, Sabot, Christophe, and Zindy, Olivier. "Limit laws for transient random walks in random environment on $\mathbb{Z}$." Annales de l’institut Fourier 59.6 (2009): 2469-2508. <http://eudml.org/doc/10461>.

@article{Enriquez2009,
abstract = {We consider transient random walks in random environment on $\mathbb\{Z\}$ 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.},
affiliation = {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); 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); Université Paris 6 Laboratoire de Probabilités et Modèles Aléatoires CNRS – UMR 7599 Boîte Courrier 188 4, place Jussieu 75252 Paris Cedex 05 (France)},
author = {Enriquez, Nathanaël, Sabot, Christophe, Zindy, Olivier},
journal = {Annales de l’institut Fourier},
keywords = {Random walks in random environment; stable laws; fluctuations theory for random walks; Beta distributions; random walks in random enviroment; fluctuations theory for random walk; the Dirichlet environment},
language = {eng},
number = {6},
pages = {2469-2508},
publisher = {Association des Annales de l’institut Fourier},
title = {Limit laws for transient random walks in random environment on $\mathbb\{Z\}$},
url = {http://eudml.org/doc/10461},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Enriquez, Nathanaël
AU - Sabot, Christophe
AU - Zindy, Olivier
TI - Limit laws for transient random walks in random environment on $\mathbb{Z}$
JO - Annales de l’institut Fourier
PY - 2009
PB - Association des Annales de l’institut Fourier
VL - 59
IS - 6
SP - 2469
EP - 2508
AB - We consider transient random walks in random environment on $\mathbb{Z}$ 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.
LA - eng
KW - Random walks in random environment; stable laws; fluctuations theory for random walks; Beta distributions; random walks in random enviroment; fluctuations theory for random walk; the Dirichlet environment
UR - http://eudml.org/doc/10461
ER -

References

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