Second Wasserstein distance ($W_2(\mu ,\nu )$)

For probability measures with finite second moments on a metric space,

For probability measures ${\mu }_n,\mu$ on the real line with finite second moments, $W_2({\mu }_n,\mu )\to 0$ exactly when ${\mu }_n$ converges weakly to $\mu$ and their second moments converge. The common-quantile coupling is optimal and satisfies