  Suppose that $\{S_n,\ n\geq0\}$ is an asymptotically stable random walk. Let
$g$ be a positive function and $T_g$ be the first time when $S_n$ leaves
$[-g(n),\infty)$. In this paper we study asymptotic behaviour of $T_g$. We
provide integral tests for function $g$ that guarantee $P(T_g>n)\sim
V(g)P(T_0>n)$ where $T_0$ is the first strict descending ladder epoch of
$\{S_n\}$
