First-passage times over moving boundaries for asymptotically stable walks

Denis Denisov; Alexander Sakhanenko; Vitali Wachtel

paper

First-passage times over moving boundaries for asymptotically stable walks

arXiv:1801.04136

Abstract

Let ${S_{n}, n \geq 1}$ be a random walk wih independent and identically distributed increments and let ${g_{n}, n \geq 1}$ be a sequence of real numbers. Let $T_{g}$ denote the first time when $S_{n}$ leaves $(g_{n}, \infty)$ . Assume that the random walk is oscillating and asymptotically stable, that is, there exists a sequence ${c_{n}, n \geq 1}$ 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 $n \to \infty$ , towards the stable meander.

20 pages