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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 $T_g$. Weprovide integral tests for function $g$ that guarantee $P(T_g>n)\simV(g)P(T_0>n)$ where $T_0$ is the first strict descending ladder epoch of$\{S_n\}$

Exact asymptotics for the instant of crossing a curve boundary by an asymptotically stable random walk