Spherical Poisson Needlets with Shrinking Bandwidth

TL;DR

This paper develops a multiscale analysis framework for spherical Poisson random fields using needlets with shrinking bandwidth, providing quantitative probabilistic guarantees for high-resolution spherical data modeling.

Contribution

It establishes Central Limit Theorems and functional convergence results for spherical Poisson needlets with shrinking bandwidth, using Stein-Malliavin techniques.

Findings

Quantitative CLTs for needlet coefficients

Explicit rates of normal approximation

Functional convergence of needlet-based fields

Abstract

Flexible bandwidth needlets provide a localized multiscale framework with scale-adaptive frequency resolution, enabling effective analysis of spherical Poisson random fields exhibiting spatial inhomogeneity and scale variation. We establish here quantitative Central Limit Theorems for finite-dimensional distributions of spherical Poisson needlets and for the related Poisson needlet coefficients constructed via needlets with shrinking bandwidth on the sphere, and using Stein-Malliavin techniques, we derive explicit rates of normal approximation. In addition, we study the functional convergence of the associated needlet-based random fields. Indeed, our framework provides quantitative control on the limiting behavior in appropriate function spaces. Together, these results offer rigorous probabilistic guarantees for high-resolution spherical data modeling under Poisson sampling.