Rate of growth of a transient cookie random walk

Anne-Laure Basdevant (1) , Arvind Singh (1)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

We consider a one-dimensional transient cookie random walk. It is known from a previous paper that a cookie random walk $(X_n)$ has positive or zero speed according to some positive parameter $\alpha >1$ or $\le 1$. In this article, we give the exact rate of growth of $(X_n)$ in the zero speed regime, namely: for $0<\alpha <1$, $X_n/n^{\frac{\alpha+1}{2}}$ converges in law to a Mittag-Leffler distribution whereas for $\alpha=1$, $X_n(\log n)/n$ converges in probability to some positive constant.

Mots clés: rates of transience; cookie or multi-excited random walk; branching process with migration

Domaines: Probabilités [math.PR]

Anne-Laure Basdevant, Arvind Singh. Rate of growth of a transient cookie random walk. 2007. ⟨hal-00135952⟩