Empirical process

Remarks

For each fixed $x$, $n F_n(x)$ is binomial with parameters $n$ and $F(x)$. So $E[G_n(x)] = 0$ and $\operatorname{Var}(G_n(x)) = F(x)(1 - F(x))$, and by the central limit theorem $G_n(x)$ converges in distribution to a normal law with these parameters. For the uniform law on $[0, 1]$, Donsker's theorem says that $G_n$ converges in distribution, as a random function, to a Brownian bridge. A non-example: if $X_i = X_1$ for all $i$, the variables are not independent, $F_n(x) = \mathbf{1}\{X_1 \le x\}$ for all $n$, and $\sqrt{n}(F_n - F)$ does not have the behaviour above. More generally, for a probability measure $P$ and a class $\mathcal{F}$ of functions, the empirical process indexed by $\mathcal{F}$ is $f \mapsto \sqrt{n}\bigl(\frac{1}{n}\sum_{i=1}^n f(X_i) - \int f \, dP\bigr)$; the case above takes the indicator functions $\mathbf{1}_{(-\infty, x]}$.