Up to what order do the moments of the Kolmogorov distribution exist? References would be appreciated.

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 nonnegative random variable $X$ and $p>1$, we have $$E(X^p)=\int_0^{+\infty}pt^{p1}P(X\geqslant t)dt,$$ we conclude that Kolmogorov distribution admits moments of any order. 

