About the prohorov distance between the uniform distribution over the unit cube in R dand its empirical measure

Abstract

We compute the almost sure order of convergence of the Prokhorov distance between the uniform distribution P over [0, 1] d and the empirical measure associated with n independent observations with (common) distribution P. We show that this order of convergence is n -1/d up to a power of log(n). This result extends to the case where the observations are weakly dependent.