Glivenko–Cantelli theorem

In the theory of probability, the Glivenko–Cantelli theorem, named after Valery Ivanovich Glivenko and Francesco Paolo Cantelli, determines the asymptotic behaviour of the empirical distribution function as the number of independent and identically distributed observations grows. The uniform convergence of more general empirical measures becomes an important property of the Glivenko–Cantelli classes of functions or sets. 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.

Assume that ${\displaystyle X_{1},X_{2},\dots }$ are independent and identically-distributed random variables in ${\displaystyle \mathbb {R} }$ with common cumulative distribution function ${\displaystyle F(x)}$. The empirical distribution function for ${\displaystyle X_{1},\dots ,X_{n}}$ is defined by

...
Wikipedia