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.
Publié le : 2002-01-14
Classification:
Brownian bridge,
distribution,
$\omega^2$-criterion,
goodness of fit,
asymptotics,
parabolic cylinder functions,
60G15,
60J65
@article{1020107767,
author = {Tolmatz, Leonid},
title = {On the Distribution of the Square Integral of the Brownian
Bridge},
journal = {Ann. Probab.},
volume = {30},
number = {1},
year = {2002},
pages = { 253-269},
language = {en},
url = {http://dml.mathdoc.fr/item/1020107767}
}
Tolmatz, Leonid. On the Distribution of the Square Integral of the Brownian
Bridge. Ann. Probab., Tome 30 (2002) no. 1, pp. 253-269. http://gdmltest.u-ga.fr/item/1020107767/