Mean convergence rates for Gaussian-smoothed Wasserstein distances and classical Wasserstein distances

Abstract

We establish upper bounds for the expected $p$-th power of the Gaussian-smoothed $p$-Wasserstein distance between a probability measure $\mu$ and the corresponding empirical measure $\mu_N$, whenever $\mu$ has finite $q$-th moment for some $q>p$. This generalizes recent results that were valid only for $q>2p+2d$. We provide two distinct proofs of such a result. We also investigate the optimality of these bounds by establishing a lower bound of order $N^{-1/2-\varepsilon}$ for a probability measure possessing finite moments of all orders. Finally, we exploit a third upper bound for the Gaussian-smoothed $p$-Wasserstein distance to derive new convergence rates for the classical $p$-Wasserstein distance in the critical regime where $\mu$ has finite $p$-th moment but infinite moments of order $q > p$, covering for instance the case of Zygmund classes $L^p(\log L)^\alpha$.