About the prohorov distance between the uniform distribution over the unit cube in Rdand its empirical measure
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- Massart, P. Probab. Th. Rel. Fields (1988) 79: 431. doi:10.1007/BF00342234
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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.