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.
Citation
Leonid Tolmatz. "On the Distribution of the Square Integral of the Brownian Bridge." Ann. Probab. 30 (1) 253 - 269, January 2002. https://doi.org/10.1214/aop/1020107767
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