Weak convergence of empirical Wasserstein type distances

Philippe Berthet (1) , Jean Claude Fort (2)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5

We estimate the Wasserstein type distance between two continuous distributions F and G on R such that the set {F = G} is a finite union of intervals, possibly empty or R. The positive cost function ρ is not necessarily symmetric and the sample may come from any joint distribution H on R 2 having marginals F and G with light enough tails with respect to ρ. The rates of weak convergence and the limiting distributions are derived in a wide class of situations including the classical distances W1 and W2. The key new assumption in the case F = G involves the behavior of ρ near 0, which we assume to be regularly varying with index ranging from 1 to 2. Rates are then also regularly varying with powers ranging from 1/2 to 1 also affecting the limiting distribution, in addition to H.

Mots clés: Central limit theorems; Generalized Wasserstein distances; Empirical processes; Strong approximation; Dependent samples; Non-parametric statistics; Goodness of fit tests

Domaines: Probabilités [math.PR]; Statistiques [math.ST]

Philippe Berthet, Jean Claude Fort. Weak convergence of empirical Wasserstein type distances. 2018. ⟨hal-01838700v1⟩