ProbabilityTheoryStatistics Given i.i.d. data with distribution function . The Glivenko-Cantelli theorem states that the empirical distribution function 𝟙 converges uniformly in probability to the true distribution function: with rate of convergence this follows by approximating the function class 𝟙 by the finite class 𝟙 defined so that the elements of the class form a 1-D ε-net or brakcket of :

  1. 𝟙𝟙
  2. covers : with 𝟙𝟙𝟙

It is always possible to partition into at most disjoint intervals such that 1) and 2) are satisfied. Then for any 𝟙𝟙𝟙 𝟙𝟙𝟙𝟙𝟙𝟙𝟙𝟙where the RHS holds uniformly over (does not depend of 𝟙 and can be controlled using a Maximal Inequality, which yields the bound.

The tighter bound can also be shown.