Rate of growth of a transient cookie random walk

Abstract

We consider a one-dimensional transient cookie random walk. It is known from a previous paper (BS2008) that a cookie random walk $(X_n)$ has positive or zero speed according to some positive parameter $\alpha >1$ or $\leq 1$. In this article, we give the exact rate of growth of $X_n$ in the zero speed regime, namely: for $0<\alpha<1$, $X_n/n^{(α+1)/2}$ converges in law to a Mittag-Leffler distribution whereas for $\alpha=1$, $X_n(\log n)/n$ converges in probability to some positive constant.