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.

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