First-passage times over moving boundaries for asymptotically stable walks

Let $\{S_n,n\ge 1\}$ be a random walk wih independent and identically distributed increments and let $\{g_n,n\ge 1\}$ be a sequence of real numbers. Let $T_g$ denote the first time when $S_n$ leaves $(g_n,\mathrm{\infty })$. Assume that the random walk is oscillating and asymptotically stable, that is, there exists a sequence $\{c_n,n\ge 1\}$ such that $S_n\mathrm{/}{c}_n$ 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 $n\to \mathrm{\infty }$, towards the stable meander.