Up to what order do the moments of the [Kolmogorov distribution](http://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Smirnov_test#Kolmogorov_distribution) exist? References would be appreciated.



Moments of the Kolmogorov distribution
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](http://en.wikipedia.org/wiki/Doob%2527s_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.


