Nous considérons les marches aléatoires en milieu aléatoire sur transientes et de vitesse nulle. D’après un résultat classique de Kesten, Kozlov et Spitzer, le temps d’atteinte du niveau 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 with zero asymptotic speed. A classical result of Kesten, Kozlov and Spitzer says that the hitting time of the level 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.
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
@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 = {http://www.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 - http://www.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 http://www.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. http://www.numdam.org/articles/10.5802/aif.2497/
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