Ergodicity and Gaussianity for spherical random fields

We investigate the relationship between ergodicity and asymptotic Gaussianity of isotropic spherical random fields in the high-resolution (or high-frequency) limit. In particular, our results suggest that under a wide variety of circumstances the two conditions are equivalent, i.e., the sample angular power spectrum may converge to the population value if and only if the underlying field is asymptotically Gaussian in the high-frequency sense. These findings may shed some light on the role of cosmic variance in cosmic microwave background radiation data analysis.