Let $X_1,X_2,\ldots$ be i.i.d random variables whose common distribution has the CDF $F$. The empirical cumulative distribution function $\hat{F}_n$ (computed from the first $n$ samples) is $$ \hat{F}_n (t) = \frac{\#\Big\{i\in\{1,2,\ldots,n\}\,|\, X_i \leq t \Big\}}{n}. $$
The Kolmogoriv Statistic $D_n = \sup_t \Big|\hat{F}_n(t)-F(t) \Big|$ measures the maximum distance between the empirical CDF and the true CDF. For large $n$, $D_n \to 0$ a.s., but with suitable rescaling it converges to the supremum of a standard brownian bridge, i.e. $$ \lim_{n\to\infty} \mathbb{P}\big(\sqrt{n}D_n \leq x\big) = \mathbb{P}\big(\sup_{t\in{0,1}} |B(t)| \leq x\big) $$ for a standard brownian bridge $B$.
There are various precise and well as approximate methods to compute $\mathbb{P}\big(\sup_{t\in{0,1}} |B(t)| \leq x\big)$, and also to compute the finite-sample probabilities $ \mathbb{P}\big(\sqrt{n}D_n \leq x\big)$. However, I could find very little on other properties of these distributions, in particular in their moments.
What I would ideally want is mean and variance of $D_n$, but since my $n$ are generally large, Id be happy with mean and variance of $\sup_{t\in{0,1}} |B(t)|$. Since brownian bridges seem to be quite well-understood, I'd be surprised if these moments weren't known. I, however, wasn't able to find them in the literature, so I'd be happy for any pointers.