Glivenko-Cantelli lemma

Let $X_{1},\ldots,X_{n}$ be iid as $X$ with (unknown) distribution function $F$ . Let $\omega$ be the outcome and $F_{n}(x,\omega)$ be the empirical distribution function based on observations $X_{1}(\omega),\ldots,X_{n}(\omega)$ . Then, as $n\rightarrow\infty$ ,

\sup_{-\infty<x<\infty}|F_{n}(x,\omega)-F(x)|\rightarrow 0\mbox{ a.s.},

where a.s. means almost surely.

Title	Glivenko-Cantelli lemma
Canonical name	GlivenkoCantelliLemma
Date of creation	2013-03-22 14:33:24
Last modified on	2013-03-22 14:33:24
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	4
Author	CWoo (3771)
Entry type	Theorem
Classification	msc 62G20