Wasserstein and total variation distance between marginals of L\'evy processes

TL;DR

This paper derives bounds for Wasserstein and total variation distances between marginals of Lévy processes, including Gaussian approximations, and explores connections to other metrics, with applications in statistical lower bounds.

Contribution

It introduces new upper bounds for distances between Lévy process marginals, including for infinite activity jumps, and relates these to other probability metrics.

Findings

Upper bounds for Wasserstein distances derived

Upper bounds for total variation distances established

Connections to Zolotarev and Toscani-Fourier distances shown

Abstract

We present upper bounds for the Wasserstein distance of order $p$ between the marginals of L\'evy processes, including Gaussian approximations for jumps of infinite activity. Using the convolution structure, we further derive upper bounds for the total variation distance between the marginals of L\'evy processes. Connections to other metrics like Zolotarev and Toscani-Fourier distances are established. The theory is illustrated by concrete examples and an application to statistical lower bounds.