Asymptotic shape of the concave majorant of a L\'evy process

TL;DR

This paper investigates the asymptotic behavior of the concave majorant of Lévy processes, revealing how its shape statistics fluctuate and depend on the tail properties of the Lévy measure as the observation window grows large.

Contribution

It provides the first distributional limit theorems for the shape statistics of the concave majorant of any Lévy process, utilizing a novel stick-breaking representation.

Findings

Fluctuations of length and shape statistics depend on Lévy measure tails.

Asymptotic dependence varies with tail behavior.

Established limit theorems for shape statistics as T approaches infinity.

Abstract

We establish distributional limit theorems for the shape statistics of a concave majorant (i.e. the fluctuations of its length, its supremum, the time it is attained and its value at $T$) of any L\'evy process on $[0,T]$ as $T\to\infty$. The scale of the fluctuations of the length and other statistics, as well as their asymptotic dependence, vary significantly with the tail behaviour of the L\'evy measure. The key tool in the proofs is the recent representation of the concave majorant for all L\'evy processes using a stick-breaking representation.