  We establish weak convergence of the empirical process on the spherical
harmonics of a Gaussian random field in the presence of an unknown angular
power spectrum. This result suggests various Gaussianity tests with an
asymptotic justification. The issue of testing for Gaussianity on isotropic
spherical random fields has recently received strong empirical attention in the
cosmological literature, in connection with the statistical analysis of cosmic
microwave background radiation.
