Electronic Journal of Statistics

The needlets bispectrum

Xiaohong Lan and Domenico Marinucci

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The purpose of this paper is to join two different threads of the recent literature on random fields on the sphere, namely the statistical analysis of higher order angular power spectra on one hand, and the construction of second-generation wavelets on the sphere on the other. To this aim, we introduce the needlets bispectrum and we derive a number of convergence results. Here, the limit theory is developed in the high resolution sense. The leading motivation of these results is the need for statistical procedures for searching non-Gaussianity in Cosmic Microwave Background radiation.

Article information

Electron. J. Statist., Volume 2 (2008), 332-367.

First available in Project Euclid: 20 May 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G20: Asymptotic properties
Secondary: 62M15: Spectral analysis 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization 60G60: Random fields

Bispectrum Needlets Spherical Random Fields Cosmic Microwave Background Radiation High Resolution Asymptotics


Lan, Xiaohong; Marinucci, Domenico. The needlets bispectrum. Electron. J. Statist. 2 (2008), 332--367. doi:10.1214/08-EJS197. https://projecteuclid.org/euclid.ejs/1211317529

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  • [1], Adler, R.J. and Taylor, J.E. (2007) Random Fields and Geometry,Springer.
  • [2], Antoine, J.-P. and Vandergheynst, P. (2007) Wavelets on the Sphere and Other Conic Sections, Journal of Fourier Analysis and its Applications, 13, 369–386.
  • [3], Babich, D., Creminelli, P., Zaldarriaga, M. (2004) The Shape of non-Gaussianities, Journal of Cosmology and Astroparticle Physics 8, 009.
  • [4], Baldi, P., Marinucci, D. (2007), Some Characterizations of the Spherical Harmonics Coefficients for Isotropic Random Fields, Statistics and Probability Letters 77, 490–496.
  • [5], Baldi, P., Kerkyacharian, G., Marinucci, D. and Picard, D. (2008) High Frequency Asymptotics for Wavelet-Based Tests for Gaussianity and Isotropy on the Torus, Journal of Multivariate Analysis, 99, 606–636, arxiv:math/0606154
  • [6], Baldi, P., Kerkyacharian, G., Marinucci, D. and Picard, D. (2006) Asymptotics for Spherical Needlets, Annals of Statistics, in press, arxiv:math/0606599
  • [7], Baldi, P., Kerkyacharian, G. Marinucci, D. and Picard, D. (2007) Subsampling Needlet Coefficients on the Sphere, preprint,arxiv 0706.4169
  • [8], Bartolo, N., Komatsu, E., Matarrese, S., Riotto, A. (2004) Non-Gaussianity from Inflation: Theory and Observations, Phys.Rept. 402, 103–266.
  • [9], Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.
  • [10], Cabella, P., Hansen, F.K., Marinucci, D., Pagano, D. and Vittorio, N. (2004) Search for non-Gaussianity in Pixel, Harmonic, and Wavelet Space: Compared and Combined. Physical Review D 69 063007.
  • [11], Cabella, P., Hansen, F.K., Liguori, M., Marinucci, D., Matarrese, S., Moscardini, L., and Vittorio, N. (2006) The Integrated Bispectrum as a Test of CMB non-Gaussianity: Detection Power and Limits on f_NL with WMAP Data, Monthly Notices of the Royal Astronomical Society 369, 819–824, arxiv:astro-ph/0512112
  • [12], Cruz, M., Cayon, L., Martinez-Gonzalez, E., Vielva, P., Jin, J., (2007) The non-Gaussian Cold Spot in the 3-year WMAP data, Astrophysical Journal 655, 11–20.
  • [13], Cruz, M., Cayon, L., Martinez-Gonzalez, E., Vielva, P., (2006) The non-Gaussian Cold Spot in WMAP: Significance, Morphology and Foreground Contribution, Monthly Notices of the Royal Astronomical Society 369, 57–67.
  • [14], DeJong, P. (1990) A Central Limit Theorem for Generalized Multilinear Forms. Journal of Multivariate Analysis 34, 275–289.
  • [15], S. Dodelson (2003) Modern Cosmology, Academic Press.
  • [16], Doroshkevich, A.G., Naselsky, P.D., Verkhodanov, O.V., Novikov, D.I., Turchaninov, V.I., Novikov, I.D., Christensen, P.R., Chiang, L.-Y. (2005) Gauss-Legendre Sky Pixelization (GLESP) for CMB Maps, International Journal of Modern Physics D 14, 275, arxiv:astro-ph/0305537
  • [17], Guilloux, F., Fay, G., Cardoso, J.-F. (2007) Practical Wavelet Design on the Sphere, arxiv 0706.2598
  • [18], Gorski, K. M., Hivon, E., Banday, A. J., Wandelt, B. D., Hansen, F. K., Reinecke, M., Bartelman, M., (2005) HEALPix – a Framework for High Resolution Discretization, and Fast Analysis of Data Distributed on the Sphere, Astrophysical Journal 622, 759–771, arxiv:astro-ph/0409513
  • [19], Kerkyacharian, G., Petrushev, P., Picard, D., Willer, T. (2007) Needlet Algorithms for Estimation in Inverse Problems, Electronic Journal of Statistics, 1, 30–76.
  • [20], Komatsu, E. and Spergel, D.N. (2001) Acoustic Signatures in the Primary Microwave Background Bispectrum. Physycal Review D 63 063002, arXiv:astro-ph/0005036
  • [21], Yadav, A.P.S., Komatsu, E., Wandelt, B. D. (2007) Fast Estimator of Primordial Non-Gaussianity from Temperature and Polarization Anisotropies in the Cosmic Microwave Background, The Astrophysical Journal, Volume 664, Issue 2, pp. 680–686.
  • [22], Yadav, A. P. S., Komatsu, E., Wandelt, B. D., Liguori, M., Hansen, F. K., Matarrese, S. (2007) Fast Estimator of Primordial Non-Gaussianity from Temperature and Polarization Anisotropies in the Cosmic Microwave Background II: Partial Sky Coverage and Inhomogeneous Noise, preprint arXiv:0711.4933
  • [23], Marinucci, D. (2006), High-Resolution Asymptotics for the Angular Bispectrum of Spherical Random Fields, The Annals of Statistics 34, 1–41, arxiv;math/0502434
  • [24], Marinucci, D. (2007), A Central Limit Theorem and Higher Order Results for the Angular Bispectrum, Probability Theory and Related Fields, in press, arxiv:math/0509430
  • [25], Marinucci, D. and Peccati, G. (2007), Group Representations and High-Resolution Central Limit Theorems for Subordinated Spherical Random Fields, arXiv:0706.2851
  • [26], Marinucci, D., Pietrobon, D., Balbi, A., Baldi, P., Cabella, P., Kerkyacharian, G., Natoli, P., Picard, D., Vittorio, N. (2008) Spherical Needlets for CMB Data Analysis, Monthly Notices of the Royal Astronomical Society, Vol. 383, 539–545, arxiv:0707/0844
  • [27], McEwen, J.D., Vielva, P., Wiaux, Y., Barreiro, R.B., Cayon, L., Hobson, M.P., Lasenby, A.N., Martinez-Gonzalez, E., Sanz, J. (2007) Cosmological Applications of a Wavelet Analysis on the Sphere, Journal of Fourier Analysis and its Applications, 13, 495–510.
  • [28], Martinez-Gonzalez, E., Forni, O. (2006) Report on the non-Gaussian Working Group activities, internal communication within the Planck collaboration.
  • [29], Mhaskar, H.N, Narcowich, F.J and Ward, J.D. (2000) Spherical Marcinkiewicz-Zygmund Inequalities and Positive Quadrature, Math. Comp. 70 (2001), no. 235, 1113–1130. 41A55 (42C10 65D30).
  • [30], Narcowich, F.J., Petrushev, P. and Ward, J.D. (2006a) Localized Tight Frames on Spheres, SIAM Journal of Mathematical Analysis 38, 2, 574–594.
  • [31], Narcowich, F.J., Petrushev, P. and Ward, J.D. (2006b) Decomposition of Besov and Triebel-Lizorkin Spaces on the Sphere, Journal of Functional Analysis 238, 2, 530–564.
  • [32], Nualart, D., Peccati, G. (2005) Central Limit Theorems For Sequences of Multiple Stochastic Integrals, Annals of Probability 33, 1, 177–193.
  • [33], Pietrobon, D., Balbi, A., Marinucci, D. (2006) 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, Physical Review D, 74, 043524.
  • [34], Surgailis, D. CLTs for Polynomials of Linear Sequences: Diagram Formula with Illustrations,(English summary) Theory and applications of long-range dependence, 111–127, Birkhäuser Boston, Boston, MA, 2003. 60F05 (60G15).
  • [35], Straf, M.L. (1972) Weak Convergence of Stochastic Processes with Several Parameters, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, II, 187–221.
  • [36], Varshalovich, D.A., Moskalev, A.N. and Khersonskii, V.K. (1988).Quantum Theory of Angular Momentum. World Scientific, Singapore.
  • [37], Yadav, A.P.S. and Wandelt, B.D. (2007) Detection of Primordial non-Gaussianity (fNL) in the WMAP 3-Year Data at above 99.5% Confidence, arxiv: 0712.1148
  • [38], Wiaux, Y., McEwen, J.D., Vielva, P., (2007) Complex Data Processing: Fast Wavelet Analysis on the Sphere, Journal of Fourier Analysis and its Applications, 13, 477–494.