One-dimensional second-Wasserstein convergence criterion
= One-dimensional second-Wasserstein convergence criterion
For probability measures $\mu_n,\mu$ on the real line with finite second moments, $W_2(\mu_n,\mu)\to0$ exactly when $\mu_n$ converges <weak convergence of probability measures>[weakly] to $\mu$ and their second moments converge. The common-quantile coupling is optimal and satisfies
$$
W_2(\mu_n,\mu)^2
=\int_0^1|F_{\mu_n}^{-1}(u)-F_\mu^{-1}(u)|^2\,du.
$$