On Nonlinear Functionals of Random Spherical Eigenfunctions

TL;DR

This paper establishes CLTs and bounds for nonlinear functionals of spherical Gaussian eigenfunctions, combining asymptotic moment analysis, Malliavin calculus, and applications to geometric and invariant statistics relevant in astrophysics.

Contribution

It introduces new CLTs and bounds for nonlinear spherical eigenfunction functionals using advanced probabilistic and analytical techniques.

Findings

Proved CLTs for nonlinear spherical eigenfunctionals

Derived Stein-like bounds for asymptotic distributions

Applied results to astrophysical data analysis

Abstract

We prove Central Limit Theorems and Stein-like bounds for the asymptotic behaviour of nonlinear functionals of spherical Gaussian eigenfunctions. Our investigation combine asymptotic analysis of higher order moments for Legendre polynomials and, in addition, recent results on Malliavin calculus and Total Variation bounds for Gaussian subordinated fields. We discuss application to geometric functionals like the Defect and invariant statistics, e.g. polyspectra of isotropic spherical random fields. Both of these have relevance for applications, especially in an astrophysical environment.