One-dimensional empirical measures, order statistics, and Kantorovich transport distances

This work is devoted to the study of rates of convergence of the empirical measures μ n = 1 n ∑ k = 1 n δ X k μ _n = 1/n ∑ ₖ₌₁^n δ X_k , n ≥ 1 n ≥ 1 , over a sample ( X k ) k ≥ 1 {(X_k)}k ≥ 1 of independent identically distributed real-valued random variables towards the common distribution μ μ in Kantorovich transport distances W p W_p . The focus is on finite range bounds on the expected Kantorovich distances E ( W p ( μ n , μ ) ) E(W_p(μ _n,μ )) or [ E ( W p p ( μ n , μ ) ) ] 1