Weighted Multivariate Cram´er-von Mises-type Statistics

In this paper, we consider weighted quadratic functionals of the multivariate uniform empirical process. By deriving the Karhunen-Lo`eve expansion of the corresponding limiting Gaussian process, we obtain the asymptotic distribution of these statistics. Our results have direct applications to tests of goodness of fit and tests of independence by Cram´er-von Mises-type statistics. Key words and phrases: Cram´er-von Mises tests, tests of goodness of fit, tests of independence, weak laws, empirical processes, Karhunen-Lo‘eve decompositions, Gaussian processes, Bessel functions.