On high-frequency limits of $U$-statistics in Besov spaces over compact manifolds

TL;DR

This paper establishes quantitative bounds for high-frequency central limit theorems involving Poisson-based U-statistics on compact Riemannian manifolds, highlighting how regularity affects convergence rates.

Contribution

It introduces new convergence rates for U-statistics built from needlet wavelets on manifolds, linking regularity of the Poisson process control measure to the speed of convergence.

Findings

Derived new convergence rates depending on regularity

Connected Besov space regularity to CLT convergence speed

Provided bounds for U-statistics on Riemannian manifolds

Abstract

In this paper, quantitative bounds in high-frequency central limit theorems are derived for Poisson based $U$-statistics of arbitrary degree built by means of wavelet coefficients over compact Riemannian manifolds. The wavelets considered here are the so-called needlets, characterized by strong concentration properties and by an exact reconstruction formula. Furthermore, we consider Poisson point processes over the manifold such that the density function associated to its control measure lives in a Besov space. The main findings of this paper include new rates of convergence that depend strongly on the degree of regularity of the control measure of the underlying Poisson point process, providing a refined understanding of the connection between regularity and speed of convergence in this framework.