Kiefer considered the asymptotics of g-sample Cramer-Von Mises statist
ics for a fixed q and sample sizes tending to infinity. For univariate
observations, McDonald proved the asymptotic normality of these stati
stics when q goes to infinity while the sample sizes stay fixed, Here
we define a class of multivariate randomness statistics that generaliz
es the class considered by McDonald. We also prove the asymptotic norm
ality of such statistics when the sample sizes stay fixed while q tend
s to infinity. (C) 1995 Academic Press, Inc.