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Bourguin, Solesne; Durastanti, Claudio

On high-frequency limits of \(U\)-statistics in Besov spaces over compact manifolds. (English) Zbl 1392.60006

Ill. J. Math. 61, No. 1-2, 97-125 (2017).
Authors’ 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.”
The paper is very large structured in 5 chapters: 1. Introduction – 1.1. Motivations and overview of the literature – 1.2. Framework and main results – 1.2.1. Main results: Poissonized case – 1.2.2. Main results: De-Poissonized case – 1.3. Applications to cosmology – 1.3.1. Global thresholding – 1.3.2. Point sources detection – 1.4. Plan of paper. 2. Preliminaries – 2.1. Poisson random measures and \(U\)-statistics – 2.2. Stein-Malliavin bounds – 2.3. Needlets on compact manifolds – 2.4. Besov spaces on compact manifolds. 3. Main results – 4. Some interpretations and comparisons with other results – 5. Auxiliary results.
Reviewer: Ludwig Paditz (Dresden)

Cited in 3 Documents

MSC:

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

Keywords:

Poisson based \(U\)-statistics; Poisson random measures; Malliavin calculus; Stein-Malliavin method; wavelet methods; high-frequency limit; quantitative bounds; Besov space; compact Riemannian manifolds, needlets; astrostatistics
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Full Text: arXiv Euclid

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