Recurrence rates and hitting-time distributions for random walks on the line

Abstract

We consider random walks on the line given by a sequence of independent identically distributed jumps belonging to the strict domain of attraction of a stable distribution, and first determine the almost sure exponential divergence rate, as $\varepsilon\to0$, of the return time to $(-\varepsilon,\varepsilon)$. We then refine this result by establishing a limit theorem for the hitting-time distributions of $(x-\varepsilon,x+\varepsilon)$ with arbitrary $x\in\mathbb{R} $.