Illinois Journal of Mathematics

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

Solesne Bourguin and Claudio Durastanti

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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
Received: 17 March 2017
Revised: 15 November 2017
First available in Project Euclid: 3 March 2018

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1520046211

Digital Object Identifier
doi:10.1215/ijm/1520046211

Mathematical Reviews number (MathSciNet)
MR3770838

Zentralblatt MATH identifier
1392.60006

Subjects
Primary: 60B05: Probability measures on topological spaces 60F05: Central limit and other weak theorems 60G57: Random measures 62E20: Asymptotic distribution theory

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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