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

Abstract

We develop two novel couplings between general pure-jump Lévy processes in ${\mathbb{R}}^d$ 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 Lévy 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.

45 pages, 2 figures; the new Section 2.5 in the revised version applies the results of the paper to the class of augmented stables processes (this new broad class, introduced in Section 2.5, contains many widely studied classes of Lévy processes); to appear in AIHP; for a short YouTube video describing the results, see https://youtu.be/76eJD6a8Kko?si=5OkdWw4AiNp0P1po