Let $X_1, X_2, X_3, \ldots \in \text{Random}(\Omega \to \mathbb{R})$ each be a D3161: Random real number such that
(i) | $X_1, X_2, X_3 \ldots$ is an D3358: I.I.D. random collection |
(ii) | $F : \mathbb{R} \to [0, 1]$ is a D205: Probability distribution function for $X_1$ |
Then
\begin{equation}
\lim_{N \to \infty} \sup_{x \in \mathbb{R}} \left| \frac{1}{N} \sum_{n = 1}^N I_{\{ X_n \leq x \}} -F(x) \right|
\overset{a.s.}{=} 0
\end{equation}