= Empirical process
{title2=$\mathbb G_n$}
Given <independent and identically distributed random variables> $X_1,\ldots,X_n$ with distribution $P$, the <empirical process> indexed by test functions $f\in\mathcal F$ is
$$
\mathbb G_n(f)=\sqrt n\left(\frac1n\sum_{i=1}^nf(X_i)-\mathbb E f(X_1)\right).
$$
It studies sampling fluctuations for a whole family of questions at once, rather than one fixed average. Its covariance is $\operatorname{Cov}(\mathbb G_n(f),\mathbb G_n(g))=P(fg)-Pf\,Pg$ when the second moments exist.
Taking $f_x(t)=1_{\{t\le x\}}$ gives $\mathbb G_n(f_x)=\sqrt n(F_n(x)-F(x))$, where $F_n$ is the <empirical distribution function>. At a point with $F(x)=1/2$, the variance is $1/4$ regardless of $n$ after this normalization. Uniform bounds over $x$ underlie the <Kolmogorov-Smirnov statistic>. Convergence at every fixed test function is weaker than convergence uniformly over the class; <chaining>, entropy and measurability conditions address that extra difficulty.<Measurability Woes by David Pollard>[Indexed processes, separable versions and common exceptional sets]{ref}
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