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.
Article PDF
Similar content being viewed by others
References
Bakhvalov, N.S.: On approximate calculation of multiple integrals (in Russian) Vestnik Mosk. Ser. Mat. Mekh. Astron. Fiz. Khim, 4, 3–18 (1959)
Bretagnolle, J., Massart, P.: Classes de fonctions d'entropie critique. C.R. Acad. Sci., Paris, Ser. 1 302, 363–366 (1986)
Bretagnolle, J., Massart, P.: Hungarian constructions from the non-asymptotic view point. Ann. Probab. (in press)
Collomb, G.: Uniform complete convergence of the kernel predictor. Z. Wahrscheinlichkeitstheor. Verw. Geb. 66, 441–460 (1984)
Doukhan, P., Portal, F.: Principe d'invariance faible pour la fonction de répartition empirique, dans un cadre multidimensionnel et mélangeant. Probab. Math. Statist. 8, 117–132 (1987)
Dudley, R.M.: Distances of probability measures and random variables. Ann. Math. Stat. 38, 1563–1572 (1968)
Dudley, R.M.: The speed of mean Glivenko-Cantelli convergence. Ann. Math. Stat. 40, 40–50 (1969)
Dudley, R.M.: Metric entropy of some classes of sets with differentiable boundaries. J. Approximation Theory, 10, 227–236 (1974)
Dudley, R.M.: Central limit theorems for empirical measures. Ann. Probab. 6, 899–929 (1978): correction 7, 909–911 (1979)
Federer, H.: Geometric measure theory. Berlin Heidelberg: Springer 1969
Gäenssler, P.: A note on a result of Dudley on the speed of Glivenko-Cantelli convergence. Ann. Math. Stat. 41, 1339–1343 (1970)
Hocking, J.G., Young, G.S.: Topology. Reading, Mass: Addison-Wesley (1961)
Hoeffding, W.: Probability inequalities for sums of bounded random variables. J. Am. Stat. Assoc. 58, 13–30 (1963)
Komlós, J., Major, P., Tusnády, G.: An approximation of partial sums of independent RV's and the sample D.F.I. Z. Wahrscheinlichkeitstheor. Verw. Geb. 32, 111–131 (1975)
Okamoto, M.: Some inequalities relating to the partial sum of binomial probabilities. Ann. Inst. Stat. Math. 10, 29–35 (1958)
Skorohod, A.V.: On a representation of random variables. Theor. Probab. Appl. 21, 628–632 (1976)
Zuker, M.: Speeds of convergence of random probability measures. Ph. D. dissertation, M.I.T. 1974
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Massart, P. About the prohorov distance between the uniform distribution over the unit cube in R dand its empirical measure. Probab. Th. Rel. Fields 79, 431–450 (1988). https://doi.org/10.1007/BF00342234
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00342234