First-passage times over moving boundaries for asymptotically stable walks

Let {S _n ,n≥ 1} be a random walk with independent and identically distributed increments, and let {g _n ,n≥ 1} be a sequence of real numbers. Let T _g denote the first time when S _n leaves (g _n , ∞). Assume that the random walk is oscillating and asymptotically stable, that is, there exists a sequence {c _n ,n≥ 1} such that S _n /c _n converges to a stable law. In this paper we determine the tail behavior of T _g for all oscillating asymptotically stable walks and all boundary sequences satisfying g _n = o(c _n ). Furthermore, we prove that the rescaled random walk conditioned to stay above the boundary up to time n converges, as n →∞, towards the stable meander.