ProbabilityTheoryStatisticsGiven i.i.d. data with distribution function . The Glivenko-Cantelli theorem states that the empirical distribution functionconverges uniformly in probability to the true distribution function: with rate of convergence this follows by approximating the function class by the finite class defined so that the elements of the class form a 1-D ε-net or brakcket of :
covers : with
It is always possible to partition into at most disjoint intervals such that 1) and 2) are satisfied. Then for any where the RHS holds uniformly over (does not depend of 𝟙 and can be controlled using a Maximal Inequality, which yields the bound.