Limit laws for transient random walks in random environment on $\z$

Nathanaël Enriquez (1, 2) , Christophe Sabot (3) , Olivier Zindy (2, 4)

1 MODAL'X - Modélisation aléatoire de Paris X
2 LPMA - Laboratoire de Probabilités et Modèles Aléatoires
3 ICJ - Institut Camille Jordan
4 WIAS - Weierstraß-Institut für Angewandte Analysis und Stochastik = Weierstrass Institute for Applied Analysis and Stochastics [Berlin]

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.

Mots clés: Random walk in random environment; stable laws; fluctuation theory of random walks; Beta distributions

Domaines: Probabilités [math.PR]

Nathanaël Enriquez, Christophe Sabot, Olivier Zindy. Limit laws for transient random walks in random environment on $\z$. 2007. ⟨hal-00137770v2⟩