  In this paper we study the asymptotic behavior of the angular bispectrum of
spherical random fields. Here, the asymptotic theory is developed in the
framework of fixed-radius fields, which are observed with increasing resolution
as the sample size grows. The results we present are then exploited in a set of
procedures aimed at testing non-Gaussianity; for these statistics, we are able
to show convergence to functionals of standard Brownian motion under the null
hypothesis. Analytic results are also presented on the behavior of the tests in
the presence of a broad class of non-Gaussian alternatives. The issue of
testing for non-Gaussianity on spherical random fields has recently gained
enormous empirical importance, especially in connection with the statistical
analysis of cosmic microwave background radiation.
