# 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

where a.s. means almost surely.

Title Glivenko-Cantelli lemma GlivenkoCantelliLemma 2013-03-22 14:33:24 2013-03-22 14:33:24 CWoo (3771) CWoo (3771) 4 CWoo (3771) Theorem msc 62G20