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

@article{AIF_2009__59_6_2469_0,
     author = {Enriquez, Nathana\"el and Sabot, Christophe and Zindy, Olivier},
     title = {Limit laws for transient random walks in random environment on $\mathbb{Z}$},
     journal = {Annales de l'Institut Fourier},
     pages = {2469--2508},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {59},
     number = {6},
     year = {2009},
     doi = {10.5802/aif.2497},
     zbl = {1200.60093},
     mrnumber = {2640927},
     language = {en},
     url = {https://numdam.org/articles/10.5802/aif.2497/}
}

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
SP  - 2469
EP  - 2508
VL  - 59
IS  - 6
PB  - Association des Annales de l’institut Fourier
UR  - https://numdam.org/articles/10.5802/aif.2497/
DO  - 10.5802/aif.2497
LA  - en
ID  - AIF_2009__59_6_2469_0
ER  -

%0 Journal Article
%A Enriquez, Nathanaël
%A Sabot, Christophe
%A Zindy, Olivier
%T Limit laws for transient random walks in random environment on $\mathbb{Z}$
%J Annales de l'Institut Fourier
%D 2009
%P 2469-2508
%V 59
%N 6
%I Association des Annales de l’institut Fourier
%U https://numdam.org/articles/10.5802/aif.2497/
%R 10.5802/aif.2497
%G en
%F AIF_2009__59_6_2469_0

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. https://numdam.org/articles/10.5802/aif.2497/

Bibliographie
Cité par

[1] Alili, S. Asymptotic behaviour for random walks in random environments, J. Appl. Probab., Volume 36 (1999), pp. 334-349 | DOI | MR | Zbl

[2] Alili, S. Persistent random walks in stationary environment, J. Stat. Phys., Volume 94 (1999), pp. 469-494 | DOI | MR | Zbl

[3] Ben Arous, G.; Černý, J. Dynamics of trap models, Ecole d’Été de Physique des Houches, Session LXXXIII “Mathematical Statistical Physics” (Lecture Notes in Math.), Elsevier, 2006, pp. 331-394

[4] Chamayou, J.-F.; Letac, G. Explicit stationary distributions for compositions of random functions and products of random matrices, J. Theoret. Probab., Volume 4 (1991), pp. 3-36 | DOI | MR | Zbl

[5] Dembo, A.; Zeitouni, O. Large deviations techniques and applications, Applications of Mathematics (New York), 38, Springer-Verlag, New York, 1998 | MR | Zbl

[6] Enriquez, N.; Sabot, C.; Zindy, O. A probabilistic representation of constants in Kesten’s renewal theorem, Probab. Theory Related Fields, Volume 144 (2009), pp. 581-613 | DOI | MR | Zbl

[7] Feller, W. An introduction to probability theory and its applications. Vol. II., Second edition, John Wiley & Sons Inc., New York, 1971 | MR | Zbl

[8] Goldie, C. M. Implicit renewal theory and tails of solutions of random equations, Ann. Appl. Probab., Volume 1 (1991), pp. 126-166 | DOI | MR | Zbl

[9] Goldsheid, I. Ya. Simple transient random walks in one-dimensional random environment: the central limit theorem, Probab. Theory Related Fields, Volume 139 (2007), pp. 41-64 | DOI | MR | Zbl

[10] Golosov, A. O. Limit distributions for random walks in a random environment, Soviet Math. Dokl., Volume 28 (1986), pp. 18-22

[11] Hu, Y.; Shi, Z.; Yor, M. Rates of convergence of diffusions with drifted Brownian potentials, Trans. Amer. Math. Soc., Volume 351 (1999), pp. 3915-3934 | DOI | MR | Zbl

[12] Iglehart, D. L. Extreme values in the $G I / G / 1$ queue, Ann. Math. Statist., Volume 43 (1972), pp. 627-635 | DOI | MR | Zbl

[13] Kawazu, K.; Tanaka, H. A diffusion process in a Brownian environment with drift, J. Math. Soc. Japan, Volume 49 (1997), pp. 189-211 | DOI | MR | Zbl

[14] Kesten, H. Random difference equations and renewal theory for products of random matrices, Acta Math., Volume 131 (1973), pp. 207-248 | DOI | MR | Zbl

[15] Kesten, H. The limit distribution of Sinai’s random walk in random environment, Physica A, Volume 138 (1986), pp. 299-309 | DOI | MR | Zbl

[16] Kesten, H.; Kozlov, M. V.; Spitzer, F. A limit law for random walk in a random environment, Compositio Math., Volume 30 (1975), pp. 145-168 | Numdam | MR | Zbl

[17] Mayer-Wolf, E.; Roitershtein, A.; Zeitouni, O. Limit theorems for one-dimensional transient random walks in Markov environments, Ann. Inst. H. Poincaré Probab. Statist., Volume 40 (2004), pp. 635-659 | DOI | Numdam | MR | Zbl

[18] Peterson, J.; Zeitouni, O. Quenched limits for transient, zero speed one-dimensional random walk in random environment, Ann. Probab., Volume 37 (2009), pp. 143-188 | DOI | MR

[19] Siegmund, D. Note on a stochastic recursion, State of the art in probability and statistics (Leiden, 1999) (IMS Lecture Notes Monogr. Ser., 36), Inst. Math. Statist., Beachwood, OH, 2001, pp. 547-554 | MR

[20] Sinai, Ya. G. The limiting behavior of a one-dimensional random walk in a random medium, Th. Probab. Appl., Volume 27 (1982), pp. 256-268 | DOI | MR | Zbl

[21] Singh, A. Rates of convergence of a transient diffusion in a spectrally negative Lévy potential, Ann. Probab., Volume 36 (2008), pp. 279-318 | DOI | MR | Zbl

[22] Solomon, F. Random walks in a random environment, Ann. Probab., Volume 3 (1975), pp. 1-31 | DOI | MR | Zbl

[23] Zeitouni, O. Random walks in random environment, Lectures on probability theory and statistics (Lecture Notes in Math.), Volume 1837, Springer, Berlin, 2004, pp. 189-312 | MR | Zbl

Cité par Sources :