First-passage times over moving boundaries for asymptotically stable walks

Denis Denisov, Alexander Sakhanenko, Vitali Wachtel

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    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.
    Original languageEnglish
    Pages (from-to)755-778
    JournalTheory of Probability and Its Applications
    Issue number4
    Publication statusPublished - 2018


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