On the distribution of the square integral of the Brownian bridge

Abstract

Smirnov obtained the distribution F for his ω2-test in the form of a certain series. F is identical to the distribution of the the Brownian bridge in the L2 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.