Abstract:
Let $\{S_n,\, n\geq1\}$ be a random walk with independent and identically
distributed
increments, and let $\{g_n,\,n\geq1\}$ be a sequence of real numbers.
Let $T_g$ denote the first time when $S_n$ leaves $(g_n,\infty)$.
Assume that the random walk is oscillating and asymptotically stable, that is,
there exists a sequence $\{c_n,\,n\geq1\}$ 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 $n\to\infty$, towards the
stable meander.
Keywords:random walk, stable distribution, first-passage time,
overshoot, moving boundary.