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

Summary 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.