Normal Approximations for Wavelet Coefficients on Spherical Poisson Fields

TL;DR

This paper derives explicit bounds on how closely wavelet coefficients of spherical Poisson fields approximate a Gaussian distribution, aiding astrophysical data analysis.

Contribution

It extends Malliavin calculus and Stein's method to assess Gaussian convergence rates for needlet coefficients on spherical Poisson fields.

Findings

Explicit upper bounds on distributional distance to Gaussian

Assessment of convergence rates for needlet coefficients

Applications to astrophysical point source detection

Abstract

We compute explicit upper bounds on the distance between the law of a multivariate Gaussian distribution and the joint law of wavelets/needlets coefficients based on a homogeneous spherical Poisson field. In particular, we develop some results from Peccati and Zheng (2011), based on Malliavin calculus and Stein's methods, to assess the rate of convergence to Gaussianity for a triangular array of needlet coefficients with growing dimensions. Our results are motivated by astrophysical and cosmological applications, in particular related to the search for point sources in Cosmic Rays data.