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

We establish upper bounds for the expected Gaussian-smoothed $p$-Wasserstein distance between a probability measure $\mu$ and the corresponding empirical measure $\mu_N$, whenever $\mu$ has finite $q$-th moments for any $q>p$. This generalizes recent results that were valid only for $q>2p+2d$. We provide two distinct proofs of such a result. We also use a third upper bound for the Gaussian-smoothed $p$-Wasserstein distance to derive an upper bound for the classical $p$-Wasserstein distance. Although the latter upper bound is not optimal when $\mu$ has finite $q$-th moment with $q>p$, this bound does not require imposing such a moment condition on $\mu$, as it is usually done in the literature.