A central limit theorem for Lp transportation cost on the real line with application to fairness assessment in machine learning

We provide a central limit theorem for the Monge-Kantorovich distance between two empirical distributions with sizes n and m, W-p(P-n, Q(m)), p >= 1, for observations on the real line. In the case p > 1 our assumptions are sharp in terms of moments and smoothness. We prove results dealing with the choice of centring constants. We provide a consistent estimate of the asymptotic variance, which enables to build two sample tests and confidence intervals to certify the similarity between two distributions. These are then used to assess a new criterion of data set fairness in classification.