Asymptotically optimal Wasserstein couplings for the small-time stable domain of attraction
Abstract
We develop two novel couplings between general pure-jump L\'evy processes in 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.
Keywords
Cite
@article{arxiv.2411.03609,
title = {Asymptotically optimal Wasserstein couplings for the small-time stable domain of attraction},
author = {Jorge González Cázares and David Kramer-Bang and Aleksandar Mijatović},
journal= {arXiv preprint arXiv:2411.03609},
year = {2025}
}
Comments
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 L\'evy processes); to appear in AIHP; for a short YouTube video describing the results, see https://youtu.be/76eJD6a8Kko?si=5OkdWw4AiNp0P1po