Central limit theorem and almost sure results for bivariate empirical W 1 distances

Philippe Berthet (1) , Jérôme Dedecker (2) , Florence Merlevède (3)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5
3 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées

In this paper we study the behavior of the Wasserstein distance of order 1 (also called Kantorovich distance) between the two marginal empirical measures of a stationary sequence of bi-variate random variables. We give sufficient conditions for the central limit theorem, the compact law of the iterated logarithm and the Maricinkiewicz-Zygmund strong law.

Mots clés: Kantorovich distance; empirical measure; central limit theorem; law of the iterated logarithm; stationary sequences; Banach spaces; Mathematics Subject Classification (2010): 60F05; 60F15; 60G10; 60B12

Domaines: Probabilités [math.PR]

Philippe Berthet, Jérôme Dedecker, Florence Merlevède. Central limit theorem and almost sure results for bivariate empirical W 1 distances. 2020. ⟨hal-02881842v1⟩