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.
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{AIHPB_2015__51_3_1159_0, author = {Cammarota, Valentina and Marinucci, Domenico}, title = {On the limiting behaviour of needlets polyspectra}, journal = {Annales de l'I.H.P. Probabilit\'es et statistiques}, pages = {1159--1189}, publisher = {Gauthier-Villars}, volume = {51}, number = {3}, year = {2015}, doi = {10.1214/14-AIHP609}, mrnumber = {3365977}, zbl = {1325.60014}, language = {en}, url = {http://www.numdam.org/articles/10.1214/14-AIHP609/} }
TY - JOUR AU - Cammarota, Valentina AU - Marinucci, Domenico TI - On the limiting behaviour of needlets polyspectra JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2015 SP - 1159 EP - 1189 VL - 51 IS - 3 PB - Gauthier-Villars UR - http://www.numdam.org/articles/10.1214/14-AIHP609/ DO - 10.1214/14-AIHP609 LA - en ID - AIHPB_2015__51_3_1159_0 ER -
%0 Journal Article %A Cammarota, Valentina %A Marinucci, Domenico %T On the limiting behaviour of needlets polyspectra %J Annales de l'I.H.P. Probabilités et statistiques %D 2015 %P 1159-1189 %V 51 %N 3 %I Gauthier-Villars %U http://www.numdam.org/articles/10.1214/14-AIHP609/ %R 10.1214/14-AIHP609 %G en %F AIHPB_2015__51_3_1159_0
Cammarota, Valentina; Marinucci, Domenico. On the limiting behaviour of needlets polyspectra. Annales de l'I.H.P. Probabilités et statistiques, Volume 51 (2015) no. 3, pp. 1159-1189. doi : 10.1214/14-AIHP609. http://www.numdam.org/articles/10.1214/14-AIHP609/
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