On the limiting behaviour of needlets polyspectra
Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 3, pp. 1159-1189.

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.

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.

DOI : 10.1214/14-AIHP609
Mots clés : spherical random fields, Stein–Malliavin approximations, polyspectra, excursion sets, wavelets, needlets
@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, Tome 51 (2015) no. 3, pp. 1159-1189. doi : 10.1214/14-AIHP609. http://www.numdam.org/articles/10.1214/14-AIHP609/

[1] E. Anderes. On the consistent separation of scale and variance for Gaussian random fields. Ann. Statist. 38 (2) (2010) 870–893. | MR | Zbl

[2] P. Baldi, G. Kerkyacharian, D. Marinucci and D. Picard. Asymptotics for spherical needlets. Ann. Statist. 37 (3) (2009) 1150–1171. | MR | Zbl

[3] P. Baldi, G. Kerkyacharian, D. Marinucci and D. Picard. Subsampling needlet coefficients on the sphere. Bernoulli 15 (2009) 438–463. | DOI | MR | Zbl

[4] G. Blum, S. Gnutzmann and U. Smilansky. Nodal domains statistics: A criterion for quantum chaos. Phys. Rev. Lett. 88 (2002) 114101.

[5] H. Dehling and M. S. Taqqu. The empirical process of some long-range dependent sequences with applications to U-statistics. Ann. Statist. 17 (4) (1989) 1767–1783. | MR | Zbl

[6] S. Dodelson. Modern Cosmology. Academic Press, San Diego, 2003.

[7] S. Donzelli, F. K. Hansen, M. Liguori, D. Marinucci and S. Matarrese. On the linear term correction for needlets/wavelets non-Gaussianity estimators. Astrophysical Journal 755 (2012) 19.

[8] C. Durastanti, X. Lan and D. Marinucci. Needlet–Whittle estimates on the unit sphere. Electron. J. Stat. 7 (2013) 597–646. | DOI | MR | Zbl

[9] R. Durrer. The Cosmic Microwave Background. Cambridge Univ. Press, Cambridge, 2008. | DOI

[10] I. S. Gradshteyn and I. M. Ryzhik. Tables of Integrals, Series and Products, 7th edition. Academic Press, Elsevier, 2007. | MR | Zbl

[11] X. Lan and D. Marinucci. The needlets bispectrum. Electron. J. Stat. 2 (2008) 332–367. | MR | Zbl

[12] N. N. Lebedev. Special Functions and Their Applications. Prentice-Hall Inc., Englewood Cliffs, NJ, 1965. | DOI | MR | Zbl

[13] N. Leonenko. Limit Theorems for Random Fields with Singular Spectrum. Mathematics and Its Applications 465. Kluwer Academic, Dordrecht, 1999. | DOI | MR | Zbl

[14] W.-L. Loh. Fixed-domain asymptotics for a subclass of matérn-type Gaussian random fields. Ann. Statist. 33 (5) (2005) 2344–2394. | MR | Zbl

[15] A. Malyarenko. Invariant Random Fields on Spaces with a Group Action. Probability and Its Applications. Springer, Heidelberg, 2012. | MR | Zbl

[16] A. Malyarenko. Invariant random fields in vector bundles and application to cosmology. Ann. Inst. Henri Poincaré Probab. Stat. 47 (4) (2011) 1068–1095. | Numdam | MR | Zbl

[17] D. Marinucci and G. Peccati. Random Fields on the Sphere. Representation, Limit Theorem and Cosmological Applications. Cambridge Univ. Press, Cambridge, 2011. | DOI | MR | Zbl

[18] D. Marinucci and G. Peccati. Mean square continuity on homogeneous spaces of compact groups. Electron. Commun. Probab. 18 (2013) no. 37, 10. | MR | Zbl

[19] D. Marinucci, D. Pietrobon, A. Balbi, P. Baldi, P. Cabella, G. Kerkyacharian, P. Natoli, D. Picard and N. Vittorio. Spherical needlets for CMB data analysis. Monthly Notices of the Royal Astronomical Society 383 (2) (2008) 539–545.

[20] D. Marinucci and S. Vadlamani. High-frequency asymptotics for Lipschitz–Killing curvatures of excursion sets on the sphere, 2013. Available at arXiv:1303.2456. | MR | Zbl

[21] D. Marinucci and I. Wigman. On the area of excursion sets of spherical Gaussian eigenfunctions. J. Math. Phys. 52 (2011) 093301. | MR | Zbl

[22] D. Marinucci and I. Wigman. The defect variance of random spherical harmonics. J. Phys. A 44 (2011) 355206. | Zbl

[23] D. Marinucci and I. Wigman. On nonlinear functionals of random spherical eigenfunctions. Comm. Math. Phys. 327 (3) (2014) 849–872. | MR | Zbl

[24] J. D. Mcewen, P. Vielva, Y. Wiaux, R. B. Barreiro, I. Cayón, M. P. Hobson, A. N. Lasenby, E. Martínez-González and J. L. Sanz. Cosmological applications of a wavelet analysis on the sphere. J. Fourier Anal. Appl. 13 (4) (2007) 495–510. | MR | Zbl

[25] F. J. Narcowich, P. Petrushev and J. D. Ward. Localized tight frames on spheres. SIAM J. Math. Anal. 38 (2006) 574–594. | DOI | MR | Zbl

[26] I. Nourdin and G. Peccati. Stein’s method on Wiener chaos. Probab. Theory Related Fields 145 (1–2) (2009) 75–118. | MR | Zbl

[27] I. Nourdin and G. Peccati. Normal Approximations Using Malliavin Calculus: From Stein’s Method to Universality. Cambridge Univ. Press, Cambridge, 2012. | MR | Zbl

[28] G. Peccati and M. S. Taqqu. Wiener Chaos: Moments, Cumulants and Diagrams. A Survey with Computer Implementations. Springer, Milan, 2008. | MR | Zbl

[29] D. Pietrobon, A. Balbi and D. Marinucci. Integrated Sachs–Wolfe effect from the cross correlation of WMAP3 year and the NRAO VLA sky survey data: New results and constraints on dark energy. Phys. Rev. D 74 (2006) 043524.

[30] Planck Collaboration. Planck 2013 results. XXIV. Constraints on primordial non-Gaussianity, 2013. Available at arXiv:1303.5084.

[31] Planck Collaboration. Planck 2013 results. XXII. Isotropy and statistics of the CMB, 2013. Available at arXiv:1303.5083.

[32] O. Rudjord, F. K. Hansen, X. Lan, M. Liguori, D. Marinucci and S. Matarrese. An estimate of the primordial non-Gaussianity parameter f NL using the needlet bispectrum from WMAP. Astrophysical Journal 701 (1) (2009) 369–376.

[33] O. Rudjord, F. K. Hansen, X. Lan, M. Liguori, D. Marinucci and S. Matarrese. Directional variations of the non-Gaussianity parameter f NL . Astrophysical Journal 708 (2) (2010) 1321–1325.

[34] J.-L. Starck, F. Murtagh and J. Fadili. Sparse Image and Signal Processing: Wavelets, Curvelets, Morphological Diversity. Cambridge Univ. Press, Cambridge, 2010. | MR | Zbl

[35] M. L. Stein. Interpolation of Spatial Data. Some Theory for Kriging. Springer Series in Statistics. Springer, New York, 1999. | MR | Zbl

[36] G. Szego. Orthogonal Polynomials, 4th edition. Colloquium Publications. American Mathematical Society, Providence, RI, 1975. | MR

[37] D. A. Varshalovich, A. N. Moskalev and V. K. Khersonskii. Quantum Theory of Angular Momentum. World Scientific, Teaneck, NJ, 1988. | DOI | MR | Zbl

[38] V.-H. Pham. On the rate of convergence for central limit theorems of sojourn times of Gaussian fields. Stochastic Process. Appl. 123 (6) (2013) 2158–2174. | MR | Zbl

[39] D. Wang and W.-L. Loh. On fixed-domain asymptotics and covariance tapering in Gaussian random field models. Electron. J. Stat. 5 (2011) 238–269. | DOI | MR | Zbl

[40] I. Wigman. On the distribution of the nodal sets of random spherical harmonics. J. Math. Phys. 50 (1) (2009) 013521. | DOI | MR | Zbl

[41] I. Wigman. Fluctuation of the nodal length of random spherical harmonics. Comm. Math. Phys. 298 (3) (2010) 787–831. | MR | Zbl

Cité par Sources :