First-passage times over moving boundaries for asymptotically stable walks

Let{ Sn,n ≥ 1 } - random walk with independent equally distributed increments and let { gn,n ≥ 1 } - a sequence of real numbers. Denote byTg first moment when Sn coming out of ( gn, ∞ ). Suppose that the random walk is oscillating and asymptotically stable, i.e. there is a sequence{ cn,n ≥ 1 } such that Sn/ cnconverges to a sustainable law. In this article, we will define tail behavior.Tg for all oscillating, asymptotically stable walks and all boundary sequences satisfying gn= o ( cn). Further, we will prove that a scaled random walk, under the condition that the boundary does not intersect beforenconverges with n → ∞to sustainable meander.