On the limiting behaviour of needlets polyspectra

Abstract

This paper provides quantitative Central Limit Theorems for nonlinear transforms of spherical random fields, in the high frequency limit. The sequences of fields that we consider are represented as smoothed averages of spherical Gaussian eigenfunctions and can be viewed as random coefficients from continuous wavelets/needlets; as such, they are of immediate interest for spherical data analysis. In particular, we focus on so-called needlets polyspectra, which are popular tools for nonGaussianity analysis in the astrophysical community, and on the area of excursion sets. Our results are based on Stein-Maliavin approximations for nonlinear transforms of Gaussian fields, and on an explicit derivation on the high-frequency limit of the fields' variances, which may have some independent interest.• where P ℓ are the usual Legendre polynomials defined dy Rodrigues' formula P ℓ (t) := 1 2 ℓ ℓ!

Adaptive global thresholding on the sphere

“…In particular, nonparametric estimation results have been achieved on S d by using a second generation wavelet system, namely, the spherical needlets. Needlets were introduced by Narcowich et al [39,40], while their stochastic properties dealing with various applications to spherical random fields were examined in [2,6,34,35,36]. Needlet-like constructions were also established over more general manifolds by Geller and Mayeli [18,19,20,21], Kerkyacharian et al [25] and Pesenson [41] among others, and over spin fiber bundles by Geller and Marinucci [16,17].…”

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

“…The concentration property in the spatial domain of the needlets (see (9)) allows one to focus on any subset of a given manifold without having to take into account the entire structure. Some of the most remarkable statistical applications are discussed and studied in [BKMP09b,BKMP09a,CM15,DLM14].…”

A note on global suprema of band-limited spherical random functions

“…which have a clear interpretation as wavelet components of the original field, and as such lend themselves to a number of statistical applications, see for instance [4], [8], [17], [18]. Band-limited spherical fields have also been widely studied in other context of mathematical physics, although in such cases b(.)…”