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 $\mathbb{R}^k$.


A complete literature can be found in the article of Fournier and Guillin https://link.springer.com/article/10.1007/s00440-014-0583-7

  • $\begingroup$ I was interested in the exact rates in this Gaussian example. Does this reference address this issue? $\endgroup$ Jul 16 '21 at 12:23
  • $\begingroup$ The rate in the one-dimensional 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: One-dimensional empirical measures, order statistics, and Kantorovich transport distances. $\endgroup$ Jul 16 '21 at 14:55
  • $\begingroup$ Thank you, but there is nothing in higher dimension in the Bobkov-Ledoux Memoir? $\endgroup$ Jul 17 '21 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. $\endgroup$ Jul 19 '21 at 13:16
  • $\begingroup$ There is recent progress on the exact rates for multidimensional Gaussians: arxiv.org/abs/1911.07579 $\endgroup$
    – Dan
    Jul 24 '21 at 12:00

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