 Empirical distribution process

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 $n$.