First-passage times over moving boundaries for asymptotically stable walks

2018/01/12 by Denisov, Denis, Sakhanenko, Alexander, Wachtel, Vitali
#60G50 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1801.04136

Abstract

Let \S_n, n≥1\ be a random walk wih independent and identically distributed increments and let \g_n,n≥1\ be a sequence of real numbers. Let T_g denote the first time when S_n leaves (g_n,∞). Assume that the random walk is oscillating and asymptotically stable, that is, there exists a sequence \c_n,n≥1\ such that S_n/c_n converges to a stable law. In this paper we determine the tail behaviour of T_g for all oscillating asymptotically stable walks and all boundary sequences satisfying g_n=o(c_n). Furthermore, we prove that the rescaled random walk conditioned to stay above the boundary up to time n converges, as n→∞, towards the stable meander.

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