Glivenko–Cantelli theorem

Called The Gliwenko - Cantelli theorem, and fundamental theorem of statistics or Fundamental Theorem of Statistics ( by Valeri Ivanovich Gliwenko and Francesco Cantelli, 1933), is a mathematical theorem, which states that the empirical distribution function of a one-dimensional sample with probability one uniformly to the actual distribution function converges.

Statement

Be independent, identically distributed random variables with the distribution function.

The random variable is then the corresponding empirical distribution function. The diamond symbol indicates the number of elements of the subsequent amount.

It is defined as the largest deviation of the empirical distribution of the underlying distribution of the random variable with respect to all forms

Then we have that the difference with probability 1, ie almost surely converges to zero:

Idea of ​​proof

One can prove the theorem of Glivenko - Cantelli, for example, with the law of large numbers.