Let $X_1, X_2, \ldots$ be iid random variables with common distribution $\gamma$, the standard Gaussian distribution on $\mathbb {R}$, and let $\mu_n = \frac 1n \sum_{i=1}^n \delta_{X_i}$, $n \geq 1$, be the empirical measures based on the sample $X_1, X_2, \ldots$. Is there anything known on the rate of convergence of $E (W_p (\mu_n, \gamma))$ as $n \to \infty$, where $W_p$ is the Wassertein distance with exponent $p \geq 1$? Same question if the $X_i$'s are standard Gaussian in $\mathbb{R}^k$.
1 Answer
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A complete literature can be found in the article of Fournier and Guillin https://link.springer.com/article/10.1007/s0044001405837

$\begingroup$ I was interested in the exact rates in this Gaussian example. Does this reference address this issue? $\endgroup$ Commented Jul 16, 2021 at 12:23

$\begingroup$ The rate in the onedimensional case is different from the one holding in higher dimensions. For $\mathbb{R}$, I suggest to have a look at the book by Bobkov and Ledoux: Onedimensional empirical measures, order statistics, and Kantorovich transport distances. $\endgroup$ Commented Jul 16, 2021 at 14:55

$\begingroup$ Thank you, but there is nothing in higher dimension in the BobkovLedoux Memoir? $\endgroup$ Commented Jul 17, 2021 at 9:03

$\begingroup$ Unfortunately, no. The reason is simple, the situation in one dimension is relatively different from higher dimensional case. In one dimension a Bahadur representation type of theorem can be used whereas no such thing exists (yet) for $d \geq 2$. Bobkov and Ledoux decided to treat only the 1D case in their book. $\endgroup$ Commented Jul 19, 2021 at 13:16

1$\begingroup$ There is recent progress on the exact rates for multidimensional Gaussians: arxiv.org/abs/1911.07579 $\endgroup$– DanCommented Jul 24, 2021 at 12:00