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We provide a Central Limit Theorem for the Monge-Kantorovich distance betweentwo empirical distributions with size $n$ and $m$, $W_p(P_n,Q_m)$ for $p>1$ forobservations on the real line, using a minimal amount of assumptions. Weprovide an estimate of the asymptotic variance which enables to build a twosample test to assess the similarity between two distributions. This test isthen used to provide a new criterion to assess the notion of fairness of aclassification algorithm.

A Central Limit Theorem for $L_p$ transportation cost with applications to Fairness Assessment in Machine Learning