= Empirical distribution process
{title2=$\alpha_n(x)=\sqrt n\,(G_n(x)-G(x))$}
The empirical distribution process is the centred, scaled <empirical distribution function> of an independent sample with <cumulative distribution function> $G$. Its mean is zero and its <covariance> is $G(\min(x,y))-G(x)G(y)$. This follows by expanding the two indicator averages: different sample indices have zero covariance, while the same index contributes the covariance of the two nested events. The scaling by $\sqrt n$ keeps these covariances independent of $n$.
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