On the Distribution of the Square Integral of the Brownian Bridge

Abstract

Smirnov obtained the distribution $F$ for his $ \omega ^2$-test in the form of a certain series. $F$ is identical to the distribution of the the Brownian bridge in the $L^2$ norm. Smirnov, Kac and Shepp determined the Laplace--Stieltjes transform of $F$. Anderson and Darling expressed $F$ in terms of Bessel functions. In the present paper we compute the moments of $F$ and their asymptotics, obtain expansions of $F$ and its density $f$ in terms of the parabolic cylinder functions and Laguerre functions, and determine their asymptotics for the small and large values of the argument. A novel derivation of expansions of Smirnov and of Anderson and Darling is obtained.