the Wasserstein metric
/ VAH-ser-shtine /
The Levy-Prokhorov metric metrizes weak convergence but is blind to how far mass actually moves and to the size of tails. Many applications, statistics, optimal transport, the convergence of empirical measures, machine learning, want a distance that does measure the cost of transporting mass and that controls moments. The Wasserstein metrics, built from optimal transport, are exactly that. They have become one of the most heavily used tools in modern probability and its applications.
Fix p >= 1 and a metric space (S, d). The p-Wasserstein distance between probability measures mu, nu (with finite p-th moments) is W_p(mu, nu) = ( inf_pi integral d(x, y)^p dpi(x, y) )^{1/p}, where the infimum runs over all couplings pi of mu and nu, i.e. all joint laws with marginals mu and nu. It is the minimal average transport cost, measuring distance by d^p, to morph mu into nu, hence 'earth mover's distance'. For p = 1 the Kantorovich-Rubinstein duality gives the clean alternative form W_1(mu, nu) = sup { integral f dmu - integral f dnu : f is 1-Lipschitz }. On R, W_p has the explicit quantile formula W_p(mu, nu)^p = integral_0^1 |F_mu^{-1}(u) - F_nu^{-1}(u)|^p du, transport on the line just matches quantiles.
Crucially W_p metrizes a STRONGER convergence than weak convergence: W_p(mu_n, mu) -> 0 iff mu_n => mu AND the p-th moments converge, integral d(x_0, x)^p dmu_n -> integral d(x_0, x)^p dmu. So Wasserstein convergence is weak convergence plus convergence of moments up to order p; equivalently, weak convergence plus uniform integrability of the p-th powers. This is precisely why it is the right tool when you care about expectations, not just distributions, and why a CLT or empirical-measure bound stated in W_2 or W_1 is genuinely more informative than one stated in Levy-Prokhorov.
On R let mu = delta_0 and nu_n = (1 - 1/n) delta_0 + (1/n) delta_n. Then nu_n => mu weakly (and Levy-Prokhorov to 0), but W_1(nu_n, mu) = (1/n) * n = 1 for all n, it never converges, because the escaping mass travels distance n. The Wasserstein metric SEES the runaway mass that weak convergence ignores; this is exactly the uniform-integrability failure.
Wasserstein detects escaping mass that weak convergence misses; W_1 = 1 even as nu_n => delta_0.
W_p requires finite p-th moments to be finite at all, and W_p convergence is strictly stronger than weak convergence (it adds p-th moment convergence). Do not use it to metrize weak convergence on the whole of P(S); use Levy-Prokhorov or bounded-Lipschitz for that.