Optimal matching and empirical measures

Source abstract

Using results from optimal matching, we find the exact order of convergence for β ( P n , P ) \beta ({P_n},P) and ρ ( P n , P ) \rho ({P_n},P) , where β \beta denotes the dual bounded Lipschitz metric, ρ \rho the Prokhorov metric and P n {P_n} the n n th empirical measure associated to P P , the uniform measure on the unit square. The results solve a long-open problem in empirical measures.