Rates of convergence of empirical measures in Wasserstein distance

Question

Let $X_1, X_2, \ldots$ be iid random variables in $\mathbb {R}^d$ with common distribution $\mu$, and $\mu_N = \frac 1N \sum_{k=1}^N \delta_{X_k}$, $N \geq 1$, the associated empirical measures. If $\mu$ is uniform on $[0,1]^d$, the optimal (in $d$ and $p$) rates of convergence of $E(W_p^p(\mu_N, \mu))$ as $N \to \infty$, where $W_p$, $p \geq 1$, is the Wasserstein distance, are known from the Ajtai-Komlos-Tusnady theorem https://mathscinet.ams.org/mathscinet-getitem?mr=779885 What kind of rates, if possible sharp, are known under some strong (for example exponential) moment of the law $\mu$ of the $X_k$'s, specifically when $d \geq 2$? For example, what about $\mu$ the standard normal (in $\mathbb {R}^d$)?

Answer 1

There are some result for Gaussian samples on a series of paper by Michel Ledoux and coauthors: On Optimal matching of Gaussian samples, On optimal matching of Gaussian samples II and on Optimal matching of Gaussian samples III. They also know sharp rates, of course there is dependency on both the power of the cost and the dimension of the problem.