Earth mover/Wasserstein distance between a pdf and an empirical distribution

Question

This question is inspired by this much older question:

Convergence of an empirical distribution w.r.t. the Hellinger distance

Let $P$ be a continuous probability distribution on a compact subset of $\mathbb{R}^3$ and let $X_1,X_2,…,X_n$ be drawn i.i.d. according to $P$. Define the empirical distribution: $$\hat{P_n}(x) = \frac{1}{n} \sum_{i=1}^{n} 1_{X_i = x}$$

Let $d(P,Q)$ denote the earth-mover distance between two distributions $P$ and $Q$. Is there a nice expression for the expected distance between $\hat{P_n}$ and $P$?

Answer 1

See, e.g., section 2.1 of http://arxiv.org/pdf/1209.1077.pdf; also http://dx.doi.org/10.1007/BF02213456, which is cited in Villani's book as having particularly precise estimates.