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January 2002 On the Distribution of the Square Integral of the Brownian Bridge
Leonid Tolmatz
Ann. Probab. 30(1): 253-269 (January 2002). DOI: 10.1214/aop/1020107767


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.


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Leonid Tolmatz. "On the Distribution of the Square Integral of the Brownian Bridge." Ann. Probab. 30 (1) 253 - 269, January 2002.


Published: January 2002
First available in Project Euclid: 29 April 2002

zbMATH: 1018.60039
Digital Object Identifier: 10.1214/aop/1020107767

Primary: 60G15
Secondary: 60J65

Keywords: $\omega^2$-criterion , asymptotics , Brownian bridge , distribution , goodness of fit , parabolic cylinder functions

Rights: Copyright © 2002 Institute of Mathematical Statistics


Vol.30 • No. 1 • January 2002
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