Glivenko–Cantelli theorem

In the theory of probability, the Glivenko–Cantelli theorem (sometimes referred to as the Fundamental Theorem of Statistics), named after Valery Ivanovich Glivenko and Francesco Paolo Cantelli, describes the asymptotic behaviour of the empirical distribution function as the number of independent and identically distributed observations grows.

[1] Specifically, the empirical distribution function converges uniformly to the true distribution function almost surely.

The uniform convergence of more general empirical measures becomes an important property of the Glivenko–Cantelli classes of functions or sets.

[2] The Glivenko–Cantelli classes arise in Vapnik–Chervonenkis theory, with applications to machine learning.

Applications can be found in econometrics making use of M-estimators.

are independent and identically distributed random variables in

with common cumulative distribution function

The empirical distribution function for

is the indicator function of the set

is a sequence of random variables which converge to

almost surely by the strong law of large numbers.

Glivenko and Cantelli strengthened this result by proving uniform convergence of

Theorem This theorem originates with Valery Glivenko[4] and Francesco Cantelli,[5] in 1933.

For simplicity, consider a case of continuous random variable

by strong law of large numbers, we can guarantee that for any positive

Combined with the above result, this further implies that

, which is the definition of almost sure convergence.

One can generalize the empirical distribution function by replacing the set

to obtain an empirical measure indexed by sets

is the indicator function of each set

Further generalization is the map induced by

on measurable real-valued functions f, which is given by Then it becomes an important property of these classes whether the strong law of large numbers holds uniformly on

with a sigma algebra of Borel subsets A and a probability measure

For a class of subsets, and a class of functions define random variables where

is the corresponding map, and Definitions Glivenko–Cantelli classes of functions (as well as their uniform and universal forms) are defined similarly, replacing all instances of

The weak and strong versions of the various Glivenko-Cantelli properties often coincide under certain regularity conditions.

The following definition commonly appears in such regularity conditions:

Theorems The following two theorems give sufficient conditions for the weak and strong versions of the Glivenko-Cantelli property to be equivalent.

Theorem (Dudley, Giné, and Zinn, 1991)[7] The following theorem is central to statistical learning of binary classification tasks.

Theorem (Vapnik and Chervonenkis, 1968)[8] There exist a variety of consistency conditions for the equivalence of uniform Glivenko-Cantelli and Vapnik-Chervonenkis classes.