# Ergodicity and Gaussianity for Spherical Random Fields

## TL;DR

This paper explores the connection between ergodicity and Gaussianity in isotropic spherical random fields, showing they are often equivalent in the high-frequency limit, with implications for CMB data analysis.

## Contribution

It demonstrates the equivalence of ergodicity and asymptotic Gaussianity for high-frequency spherical fields under broad conditions.

## Key findings

- Sample angular power spectrum converges to the true value if and only if the field is asymptotically Gaussian.
- Results provide insights into the role of Cosmic Variance in CMB analysis.
- Establishes conditions linking ergodicity and Gaussianity in spherical random fields.

## Abstract

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 (CMB) radiation data analysis.