Gaussian approximations of nonlinear statistics on the sphere

TL;DR

This paper develops a method to quantify how quickly nonlinear statistics based on wavelet coefficients on the sphere converge to a Gaussian distribution, with applications in non-parametric density estimation and uniformity testing.

Contribution

It introduces a Stein-Malliavin approach to assess Gaussian approximation rates for U-statistics on the sphere, extending previous methods to spherical Poisson fields of arbitrary dimension.

Findings

Provides explicit convergence rates for Gaussian approximations

Applies to variance estimation in non-parametric density estimation

Enables Sobolev tests for uniformity on the sphere

Abstract

We show how it is possible to assess the rate of convergence in the Gaussian approximation of triangular arrays of $U$-statistics, built from wavelets coefficients evaluated on a homogeneous spherical Poisson field of arbitrary dimension. For this purpose, we exploit the Stein-Malliavin approach introduced in the seminal paper by Peccati, Sol\'e, Taqqu and Utzet (2011); we focus in particular on statistical applications covering evaluation of variance in non-parametric density estimation and Sobolev tests for uniformity.