First-passage times over moving boundaries for asymptotically stable walks

Let {S_n, n≥1} be a random walk wih 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 behaviour 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.