Spherical Poisson Waves

TL;DR

This paper introduces a model of Poisson random waves on the sphere and analyzes their asymptotic behavior as both the process intensity and wave frequency increase, providing quantitative CLTs and convergence results.

Contribution

It develops a new probabilistic model for spherical Poisson waves and studies their limit theorems under diverging parameters, highlighting the interplay between eigenvalues and Poisson measures.

Findings

Quantitative CLTs for finite-dimensional distributions

Convergence in law in functional spaces

Analysis of harmonic coefficients behavior

Abstract

We introduce a model of Poisson random waves in $\mathbb{S}^{2}$ and we study Quantitative Central Limit Theorems when both the rate of the Poisson process and the energy (i.e., frequency) of the waves (eigenfunctions) diverge to infinity. We consider finite-dimensional distributions, harmonic coefficients and convergence in law in functional spaces, and we investigate carefully the interplay between the rates of divergence of eigenvalues and Poisson governing measures.