On the Limiting Behaviour of Needlets Polyspectra

TL;DR

This paper establishes quantitative Central Limit Theorems for needlets polyspectra, which are important for analyzing non-Gaussian features in spherical data, especially in astrophysics, using advanced probabilistic techniques.

Contribution

It provides the first explicit high-frequency variance limits and CLTs for needlets polyspectra of spherical Gaussian fields, enhancing understanding of their asymptotic behavior.

Findings

Quantitative CLTs for needlets polyspectra in high frequency limit.

Explicit variance asymptotics for nonlinear transforms of spherical Gaussian fields.

Application relevance to non-Gaussianity analysis in astrophysics.

Abstract

This paper provides quantitative Central Limit Theorems for nonlinear transforms of spherical random fields, in the high frequency limit. The sequences of fields that we consider are represented as smoothed averages of spherical Gaussian eigenfunctions and can be viewed as random coefficients from continuous wavelets/needlets; as such, they are of immediate interest for spherical data analysis. In particular, we focus on so-called needlets polyspectra, which are popular tools for nonGaussianity analysis in the astrophysical community, and on the area of excursion sets. Our results are based on Stein-Maliavin approximations for nonlinear transforms of Gaussian fields, and on an explicit derivation on the high-frequency limit of their variances, which may have some independent interest.