Simultaneous concentration of order statistics

Understand

Let $\mu$ be a probability measure on $\mathbb{R}$ with cumulative distribution function $F$, $(x_{i})_{1}^{n}$ a large i.i.d.

• sample from $\mu$, and $F_{n}$ the associated empirical distribution function.
• The Glivenko-Cantelli theorem states that with probability 1, $F_{n}$ converges uniformly to $F$.
• In so doing it describes the macroscopic structure of $\{x_{i}\}_{1}^{n}$, however it is insensitive to the position of individual points.