An Exact Asymptotics for the Moment of Crossing a Curved Boundary by an Asymptotically Stable Random Walk

Abstract

Suppose that {Sn, n 0} is an asymptotically stable random walk. Let g be a positive function and Tg be the first time when Sn leaves [−g(n), ∞). In this paper we study asymptotic behavior of Tg. We provide integral tests for function g that guarantee P(Tg > n) ∼ V (g)P(T 0 > n), where T 0 is the first strict descending ladder epoch of {Sn}.

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