Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

On the limiting behaviour of needlets polyspectra

Valentina Cammarota and Domenico Marinucci

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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 non-Gaussianity analysis in the astrophysical community, and on the area of excursion sets. Our results are based on Stein–Malliavin approximations for nonlinear transforms of Gaussian fields, and on an explicit derivation on the high-frequency limit of their variances, which may have some independent interest.

Résumé

Dans cet article on prouve un TCL pour des fonctionnelles nonlinéaires de champs aléatoires sur la sphère avec bornes en variation totale dans le sens de la limite en haute fréquence. Les suites de champs aléatoires que l’on considère sont des moyennes régularisées de fonctions propres gaussiennes sur la sphère qui peuvent être vues comme des coefficients aléatoires d’ondelettes/needlets continues. En particulier on se concentre sur le polyspectre en needlets lequel est un outil couramment utilisé dans l’analyse de la nongaussianité en astrophysique et dans le domaine des ensembles de niveau. Nos résultats sont basés sur des approximations de type Stein–Malliavin pour des fonctionnelles nonlinéaires de champs gaussiens ainsi que sur le calcul explicite de la limite en haute fréquence de leur variance, ce qui pourrait constituer un résultat ayant un interêt en lui même.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 51, Number 3 (2015), 1159-1189.

Dates
Received: 18 July 2013
Revised: 3 February 2014
Accepted: 12 February 2014
First available in Project Euclid: 1 July 2015

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1435759244

Digital Object Identifier
doi:10.1214/14-AIHP609

Mathematical Reviews number (MathSciNet)
MR3365977

Zentralblatt MATH identifier
1325.60014

Subjects
Primary: 60G60: Random fields 62M15: Spectral analysis 42C15: General harmonic expansions, frames

Keywords
Spherical random fields Stein–Malliavin approximations Polyspectra Excursion sets Wavelets Needlets

Citation

Cammarota, Valentina; Marinucci, Domenico. On the limiting behaviour of needlets polyspectra. Ann. Inst. H. Poincaré Probab. Statist. 51 (2015), no. 3, 1159--1189. doi:10.1214/14-AIHP609. https://projecteuclid.org/euclid.aihp/1435759244


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