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 AjtaiKomlosTusnady theorem https://mathscinet.ams.org/mathscinetgetitem?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 $\mathbb {R}^d$)?
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$\begingroup$ Not sure if that is your question, but of course under sufficient moments ($q$th moments for $q > 2p$) you have the same behavior as for the uniform distribution, which is shown in the paper by Fournier and Guillin "On the rate of convergence in Wasserstein distance of the empirical measure." $\endgroup$– SteveCommented Oct 4, 2022 at 7:31
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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.

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