Moments of the Kolmogorov distribution Up to what order do the moments of the Kolmogorov distribution exist? References would be appreciated.
 A: The Kolmogorov distribution is defined by the distribution of the random variable $K:=\sup_{0\leqslant t\leqslant 1}|B(t)|$, where $B(t)$ is the Brownian Bridge. 
The problem of existence of moments for $K$ is actually the same as the study of moments of $K':=\sup_{0\leqslant t\leqslant 1}|W(t)|$, where $W(t)$ is a standard Brownian motion. An application of Doob's (sub)martingale inequality gives that for all $C>0$,
$$P(K'\geqslant C)\leqslant \exp\left(-\frac{C^2}2\right).$$
Using the fact that for a non-negative random variable $X$ and $p>1$, we have $$E(X^p)=\int_0^{+\infty}pt^{p-1}P(X\geqslant t)dt,$$
we conclude that Kolmogorov distribution admits moments of any order.
As for $p<0$, we have $K^p\leqslant B(1)^p$, we can say that $K$ admits moments of order $p>-1$ for all $p$.
A: Looks like this paper could help you: Evaluating Kolmogorov's Distribution
Check the 3rd header, Limiting Forms, for a mention of how the moments are found.
Not only do all moments exist, but they are all simply expressible in analytic form:
$\left<x^m\right> = \frac{\Gamma(m/2 + 1) \, \eta(m)}{2^{m/2 - 1}} $
You can obtain this formula by using the form of the series definition containing factors $e^{-2 k^2 x^2}$ and integrating term-by-term.
(Sorry to have to add this information via edit, but "protecting" this question from answers by people who have not used the site previously prevented me from putting this in a separate answer.)
