Stein-Malliavin Approximations for Nonlinear Functionals of Random Eigenfunctions on ${\mathbb{S}}^{d}$

TL;DR

This paper develops Stein-Malliavin approximation techniques for nonlinear functionals of Gaussian eigenfunctions on high-dimensional spheres, providing new asymptotic results and quantitative CLTs, especially for the excursion area.

Contribution

It introduces novel asymptotic analysis and convergence rates for nonlinear functionals of eigenfunctions on spheres, extending and improving previous results for the 2D case.

Findings

Asymptotic variance analysis of eigenfunctions

Quantitative CLT for excursion areas

Improved bounds for the 2D case

Abstract

We investigate Stein-Malliavin approximations for nonlinear functionals of geometric interest of Gaussian random eigenfunctions on the unit $d$ -dimensional sphere ${\mathbb{S}}^{d},$ $d\geq 2.$ All our results are established in the high energy limit, i.e. for eigenfunctions corresponding to growing eigenvalues. More precisely, we provide an asymptotic analysis for the variance of random eigenfunctions, and also establish rates of convergence for various probability metrics for Hermite subordinated processes, arbitrary polynomials of finite order and square integral nonlinear transforms; the latter, for instance, allows to prove a quantitative Central Limit Theorem for the excursion area. Some related issues were already considered in the literature for the $2$-dimensional case ${\mathbb{S}}^{2}$; our results are new or improve the existing bounds even for this special case. Proofs are based on the asymptotic analysis of moments of all order for Gegenbauer polynomials, and make extensive use of the recent literature on so-called fourth-moment theorems by Nourdin and Peccati.