Asymptotically optimal Wasserstein couplings for the small-time stable domain of attraction

TL;DR

This paper introduces two new couplings for pure-jump Lévy processes to analyze their convergence rates in Wasserstein distance, providing matching upper and lower bounds in the small-time regime.

Contribution

It presents novel couplings for Lévy processes and derives sharp convergence rate bounds in Wasserstein distance for processes attracted to stable laws.

Findings

Convergence rate is polynomial in the normal attraction domain.

Convergence rate is slower than a slowly varying function in the non-normal domain.

Upper and lower bounds on convergence rates typically match.

Abstract

We develop two novel couplings between general pure-jump L\'evy processes in $\R^d$ and apply them to obtain upper bounds on the rate of convergence in an appropriate Wasserstein distance on the path space for a wide class of L\'evy processes attracted to a multidimensional stable process in the small-time regime. We also establish general lower bounds based on certain universal properties of slowly varying functions and the relationship between the Wasserstein and Toscani--Fourier distances of the marginals. Our upper and lower bounds typically have matching rates. In particular, the rate of convergence is polynomial for the domain of normal attraction and slower than a slowly varying function for the domain of non-normal attraction.