Convergence of the empirical measure in expected Wasserstein distance: non asymptotic explicit bounds in $\mathbb{R}^d$

We provide some non asymptotic bounds, with explicit constants, that measure the rate of convergence, in expected Wasserstein distance, of the empirical measure associated to an i.i.d. $N$-sample of a given probability distribution on $\mathbb{R}^d$. We consider the cases where $\mathbb{R}^d$ is endowed with the maximum and Euclidean norms.