Large Excursions of Reflected L\'evy Processes: Asymptotic Shapes

TL;DR

This paper explores the geometric properties of reflected Lévy process excursions, revealing different asymptotic behaviors depending on the process's drift and tail conditions, with implications for stable and Pareto distributions.

Contribution

It extends existing results by analyzing the asymptotic shapes of excursions under various conditions, including stable attraction and heavy-tailed scenarios, with new convergence results.

Findings

Long excursions in oscillating processes can have small height despite long lifetime.

In negative-drift cases, long excursions are asymptotically equivalent in height and lifetime.

Excursions conditioned on large height or lifetime converge to a Pareto-distributed jump process.

Abstract

This paper primarily investigates the geometric properties of excursions of L\'evy processes reflected at the past infimum with long lifetime or large height. For an oscillating process in the domain of attraction of a stable law, our results state that excursions with a long lifetime need not have a large height. After a suitable scaling, they behave like stable excursions with lifetime or height greater than one. These extend the related results in Doney and Rivero [Prob. Theory Relat. Fields, 157(1) (2013) 1-45]. In contrast, for the negative-drift case we prove that under a heavy-tailed condition, long lifetime and large height are asymptotically equivalent. Conditioned on either event, excursions converge under spatial scaling to a single-jump process with Pareto-distributed jump size and size-biased jump time. Moreover, after a suitable time rescaling, the effect of the negative drift becomes apparent.