First-passage times over moving boundaries for asymptotically stable walks

TL;DR

This paper analyzes the first-passage times over moving boundaries for oscillating asymptotically stable random walks, establishing their tail behavior and convergence to stable meanders under certain conditions.

Contribution

It determines the tail behavior of first-passage times for a broad class of stable-like walks and boundary sequences, extending existing results to more general settings.

Findings

Derived the tail asymptotics of $T_g$ for all oscillating asymptotically stable walks.

Proved convergence of the conditioned walk to the stable meander.

Extended classical results to non-constant boundaries and stable limit laws.

Abstract

Let $\{S_n, n\geq1\}$ be a random walk wih independent and identically distributed increments and let $\{g_n,n\geq1\}$ 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 $\{c_n,n\geq1\}$ 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.