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.2010 Mathematics Subject Classification. 60B05,60F05,60G57,62E20.

A Berry–Esseén theorem for partial sums of functionals of heavy-tailed moving averages

“…In particular, quantitative bounds for normal approximations were obtained by combining Malliavin calculus on the Poisson space with Stein's method. The resulting bounds have successfully been applied in various contexts such as stochastic geometry (see, e.g., [7,13,14,16,19,36]), the theory of U-statistics (see, e.g., [7,8,9,36]), non-parametric Bayesian survival analysis (see [26,27]) or statistics of spherical point fields (see, e.g., [3,4]). We refer the reader also to [24], which contains a representative collection of survey articles.…”

Nonparametric needlet estimation for partial derivatives of a probability density function on the $d$-torus

“…As already mentioned in Section 1, needlets have been originally introduced on the d-dimensional sphere in [NPW06b,NPW06a], and then generalized to compact manifolds (see [BD17,GM09,KNP12]). Needlet-like wavelets on T d have been already used in [BD18] in the framework of the two-sample problem.…”

“…As far as the d-dimensional torus is concerned, toroidal needlets have already been discussed and applied in the framework of the two sample problem, in [BD18]. As already remarked in [BD17], the choice of T d as the support of the probability density function is quite general, in view of the fact that R d and T d are locally homeomorphic and, thus, the spatial localization property of the toroidal needlets ensures the validity of results here achieved for any local approximation of R d by T d . 1.2.…”

Flexible-bandwidth Needlets

“…Needlets on one hand represent a tightframe system and hence satisfy classical requirements of approximation theory; on the other hand under some regularity conditions needlet coefficients have been shown to enjoy asymptotic uncorrelation properties (in the high-resolution sense) which makes their application to random fields extremely powerful. Extensions of the needlet construction to more general homogeneous spaces of compact groups were given for instance by [10,16,20]; statistical applications are currently too many to be recalled in any reasonable completeness: we refer for instance to [18,19] or more recently [6,9,11,12,22,36,38,14,23]. Applications in Cosmology and Astrophysics are discussed for instance in [7,27,32,35,39,40]) and the references therein.…”