# Convergence of empirical measures in Wasserstein distance

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 $$\mathbb{R}^k$$.

• 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. Jul 16 '21 at 14:55
• 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. Jul 19 '21 at 13:16