## Illinois Journal of Mathematics

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

#### Article information

Source
Illinois J. Math., Volume 61, Number 1-2 (2017), 97-125.

Dates
Revised: 15 November 2017
First available in Project Euclid: 3 March 2018

https://projecteuclid.org/euclid.ijm/1520046211

Digital Object Identifier
doi:10.1215/ijm/1520046211

Mathematical Reviews number (MathSciNet)
MR3770838

Zentralblatt MATH identifier
1392.60006

#### Citation

Bourguin, Solesne; Durastanti, Claudio. On high-frequency limits of $U$-statistics in Besov spaces over compact manifolds. Illinois J. Math. 61 (2017), no. 1-2, 97--125. doi:10.1215/ijm/1520046211. https://projecteuclid.org/euclid.ijm/1520046211

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