TY - JOUR

T1 - First-passage times over moving boundaries for asymptotically stable walks

AU - Denisov, Denis

AU - Sakhanenko, Alexander

AU - Wachtel, Vitali

PY - 2018

Y1 - 2018

N2 - 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.

AB - 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.

U2 - 10.4213/tvp5181

DO - 10.4213/tvp5181

M3 - Article

VL - 63

SP - 755

EP - 778

JO - Theory of Probability and Its Applications

JF - Theory of Probability and Its Applications

SN - 0040-585X

IS - 4

ER -