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  3. Random Fields on the Sphere
  • Cited by 179
Publisher:
Cambridge University Press
Online publication date:
October 2011
Print publication year:
2011
Online ISBN:
9780511751677
DOI:
https://doi.org/10.1017/CBO9780511751677
Subjects:
Mathematics (general), Physics and Astronomy, Cosmology, Relativity and Gravitation, Probability Theory and Stochastic Processes, Statistics and Probability
Series:
London Mathematical Society Lecture Note Series (389)
Information

Book description

Random Fields on the Sphere presents a comprehensive analysis of isotropic spherical random fields. The main emphasis is on tools from harmonic analysis, beginning with the representation theory for the group of rotations SO(3). Many recent developments on the method of moments and cumulants for the analysis of Gaussian subordinated fields are reviewed. This background material is used to analyse spectral representations of isotropic spherical random fields and then to investigate in depth the properties of associated harmonic coefficients. Properties and statistical estimation of angular power spectra and polyspectra are addressed in full. The authors are strongly motivated by cosmological applications, especially the analysis of cosmic microwave background (CMB) radiation data, which has initiated a challenging new field of mathematical and statistical research. Ideal for mathematicians and statisticians interested in applications to cosmology, it will also interest cosmologists and mathematicians working in group representations, stochastic calculus and spherical wavelets.

Reviews

"The methods described in the book shed light on extremely important issues in astrophysics, cosmology, and fundamental physics. Most of the results of the book were first proved by the authors. Rigourous mathematical proofs of other results appear here for the first time in a monograph form. ...the material is very accessible, both technically interesting and a pleasure to read. The presentation is very clear. The book is a must for mathematicians and for graduate and postgraduate students who would like to work in the area of statistical analysis of cosmological data."
Anatoliy Malyarenko, Mathematical Reviews

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Contents

Contents
References
References
[1] Abramowitz, M., Stegun, I. (1964) Handbook of Mathematical Functions Google Scholar, Dover.
[2] Adler, R.J. (1981) The Geometry of Random Fields Google Scholar, J. Wiley.
[3] Adler, R.J., Taylor, J.E. (2007) Random Fields and Geometry Google Scholar, Springer-Verlag.
[4] Anderes, E., Chatterjee, S. (2009) Consistent estimates of deformed isotropic Gaussian random fields on the plane, Annals of Statistics, 37 CrossRef | Google Scholar, No. 5A, 2324–2350.
[5] Antoine, J.-P., Vandergheynst, P. (1999) Wavelets on the sphere: a group-theoretic approach, Applied and Computational Harmonic Analysis, 7 CrossRef | Google Scholar, 262–291.
[6] Antoine, J.-P., Vandergheynst, P. (2007) Wavelets on the sphere and other conic sections, Journal of Fourier Analysis and its Applications, 13 CrossRef | Google Scholar, 369–386.
[7] Arjunwadkar, M., Genovese, C.R., Miller, C.J., Nichol, R.C., Wasserman, L. (2004) Nonparametric inference for the Cosmic Microwave Background, Statistical Science, 19 Google Scholar, 308–321.
[8] Babich, D., Creminelli, P., Zaldarriaga, M. (2004) The shape of non-Gaussianities, Journal of Cosmology and Astroparticle Physics, 8 CrossRef | Google Scholar, 009.
[9] Balbi, A. (2007), The Music of the Big Bang Google Scholar, Springer-Verlag.
[10] Baldi, P., Marinucci, D. (2007). Some characterizations of the spherical harmonics coefficients for isotropic random fields, Statistics & Probability Letters, 77 CrossRef | Google Scholar(5), 490–496.
[11] Baldi, P., Marinucci, D., Varadarajan, V.S. (2007) On the characterization of isotropic random fields on homogeneous spaces of compact groups, Electronic Communications in Probability, 12 CrossRef | Google Scholar, 291–302.
[12] Baldi, P., Kerkyacharian, G., Marinucci, D., Picard, D. (2008) High frequency asymptotics for wavelet-based tests for Gaussianity and isotropy on the torus, Journal of Multivariate Analysis, 99 CrossRef | Google Scholar(4), 606–636.
[13] Baldi, P., Kerkyacharian, G., Marinucci, D., Picard, D. (2009) Asymptotics for Spherical Needlets, Annals of Statistics, 37 CrossRef | Google Scholar(3), 1150–1171, arxiv:math/0606599.
[14] Baldi, P., Kerkyacharian, G.Marinucci, D., Picard, D. (2009) Subsampling Needlet Coefficients on the Sphere, Bernoulli, 15 CrossRef | Google Scholar(2), 438–463, arxiv 0706.4169.
[15] Baldi, P., Kerkyacharian, G., Marinucci, D., Picard, D. (2009) Density estimation for directional data using needlets, Annals of Statistics, 37 CrossRef | Google Scholar(6A), 3362–3395.
[16] Baldi, P., Kerkyacharian, G., Marinucci, D., Picard, D. (2009 Google Scholar) Besov spaces for sections of spin fiber bundles on the sphere, preprint.
[17] Balkar, E., Lovesey, S.W. (2009), Introduction to the Graphical Theory of Angular Momentum CrossRef | Google Scholar, Springer Tracts on Modern Physics, Springer.
[18] Bartolo, N., Komatsu, E., Matarrese, S., Riotto, A. (2004). Non-Gaussianity from inflation: theory and observations, Physical Reports, 402 CrossRef | Google Scholar, 103–266.
[19] Bartolo, N., Matarrese, S., Riotto, A. (2010) Non-Gaussianity and the Cosmic Microwave Background anisotropies, Advances in Astronomy CrossRef | Google Scholar, in press, arXiv: 1001.3957.
[20] Bartolo, N., Fasiello, M., Matarrese, S., Riotto, A. (2010) Large non-Gaussianities in the effective field theory approach to single-field inflation: the bispectrum, Journal of Cosmology and Astroparticle Physics, 1008 CrossRef | Google Scholar:08, arXiv: 1004.0893.
[21] Bennett, C. L., Halpern, M., Hinshaw, G., Jarosik, N., Kogut, A., Limon, M., Meyer, S. S., Page, L., Spergel, D. N., Tucker, G. S., Wollack, E., Wright, E. L., Barnes, C., Greason, M. R., Hill, R. S., Komatsu, E., Nolta, M. R., Odegard, N., Peiris, H. V., Verde, L., Weiland, J. L. (2003) First -Year Wilkinson Microwave Anisotropy Probe (WMAP) observations: preliminary maps and basic results, Astrophysical Journal Supplement Series, Volume 148 CrossRef | Google Scholar, Issue 1, pp. 1–27.
[22] Bennett, C. L., Hill, S., Hinshaw, G., Larson, D., Smith, K. M., Dunkley, J., Gold, B., Halpern, M., Jarosik, N., Kogut, A., Komatsu, E., Limon, M., Meyer, S. S., Nolta, M. R., Odegard, N., Page, L., Spergel, D. N., Tucker, G. S., Weiland, J. L., Wollack, E., Wright, E. L. (2010 Google Scholar) Seven-year Wilkinson Microwave Anisotropy Probe (WMAP) observations: are there Cosmic Microwave Background anomalies?, arXiv: 1001.4758.
[23] Biedenharn, L.C., Louck, J.D. (1981) The Racah-Wigner Algebra in Quantum Theory, Encyclopedia of Mathematics and its Applications, Volume 9 Google Scholar, Addison-Wesley.
[24] Billingsley, P. (1968) Convergence of Probability Measures Google Scholar, J. Wiley.
[25] Bishop, R.L., Goldberg, S. (1980) Tensor Analysis on Manifolds Google Scholar, Dover.
[26] Blei, R. (2001) Analysis in Integer and Fractional Dimensions CrossRef | Google Scholar, Cambridge University Press.
[27] Breuer, P., Major, P. (1983) Central limit theorems for nonlinear functionals of Gaussian fields, Journal of Multivariate Analysis, 13 CrossRef | Google Scholar, no. 3, 425–441.
[28] Bridles, S. et al. (2009) Handbook for the GREAT08 Challenge: An image analysis competition for cosmological lensing, Annals of Applied Statistics, Vol. 3 CrossRef | Google Scholar, No. 1, 6–37.
[29] Brillinger, D. W. (1975) Time series. Data Analysis and Theory Google Scholar, Holt, Rinehart and Winston.
[30] Brockwell, P.J., Davis, R.A. (1991) Time Series: Theory and Methods CrossRef | Google Scholar, Second edition, Springer Series in Statistics, Springer-Verlag.
[31] Brocker, T., tom Dieck, T. (1985) Representations of Compact Lie Groups, Graduate Texts in Mathematics, 98 CrossRef | Google Scholar, Springer-Verlag.
[32] Bump, D. (2005) Lie Groups Google Scholar, Graduate Texts in Mathematics, 225, Springer-Verlag.
[33] Cabella, P., Kamionkowskii, M. (2005) Theory of Cosmic Microwave Background Polarization Google Scholar, Lectures given at the 2003 Villa Mondragone School of Gravitation and Cosmology: “The Polarization of the Cosmic Microwave Background,” Rome, arxiv: astro.ph/0403392.
[34] Cabella, P., Hansen, F.K., Marinucci, D., Pagano, D., Vittorio, N. (2004) Search for non-Gaussianity in pixel, harmonic, and wavelet space: compared and combined, Physical Review D, 69 CrossRef | Google Scholar, 063007.
[35] Cabella, P., Hansen, F.K., Liguori, M., Marinucci, D., Matarrese, S., Moscardini, L., Vittorio, N. (2005) Primordial non-Gaussianity: local curvature method and statistical significance of constraints on fNL from WMAP data, Monthly Notices of the Royal Astronomical Society, Vol. 358 CrossRef | Google Scholar, pp. 684–692.
[36] Cabella, P., Hansen, F.K., Liguori, M., Marinucci, D., Matarrese, S., Moscardini, L., Vittorio, N. (2006) The integrated bispectrum as a test of CMB non-Gaussianity: detection power and limits on fNL with WMAP data, Monthly Notices of the Royal Astronomical Society, 369 CrossRef | Google Scholar, 819–824, arxiv: astro-ph/0512112.
[37] Cabella, P., Marinucci, D. (2009) Statistical challenges in the analysis of Cosmic Microwave Background radiation, Annals of Applied Statistics, 3 CrossRef | Google Scholar(1), 61–95.
[38] Chambers, D., Slud, E. (1989) Necessary conditions for nonlinear functionals of Gaussian processes to satisfy central limit theorems, Stochastic Processes and their Applications, 32 CrossRef | Google Scholar(1), 93–107.
[39] Chambers, D., Slud, E. (1989) Central Limit Theorems for nonlinear functionals of stationary Gaussian processes, Probability Theory and Related Fields, 80 CrossRef | Google Scholar(3), 323–346.
[40] 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 CrossRef | Google Scholar, 11–20.
[41] 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 CrossRef | Google Scholar, 57–67.
[42] Dahlke, S., Steidtl, G., Teschke, G. (2007) Frames and coorbit theory on homogeneous spaces with a special guidance on the sphere, Journal of Fourier Analysis and its Applications, 13 CrossRef | Google Scholar, 387–404.
[43] Davidson, J. (1994), Stochastic Limit Theory CrossRef | Google Scholar, Oxford University Press.
[44] De Bernardis, P. et al. (2000) A flat Universe from high-resolution maps of the Cosmic Microwave Background radiation, Nature, Vol. 404 CrossRef | Google Scholar | PubMed, Issue 6781, pp. 955–959.
[45] de Gasperis, G., Balbi, A., Cabella, P., Natoli, P., Vittorio, N. (2005) ROMA: A map-making algorithm for polarised CMB data sets, Astronomy and Astrophysics, Vol. 436 CrossRef | Google Scholar, Issue 3, pp. 1159–1165.
[46] Delabrouille, J., Cardoso, J.-F., Le Jeune, M., Betoule, M., Fay, G., Guilloux, F. (2009) A full sky, low foreground, high resolution CMB map from WMAP, Astronomy and Astrophysics, Vol. 493 CrossRef | Google Scholar, Issue 3, pp. 835–857, arXiv:0807.0773.
[47] Dennis, M. (2004), Canonical representation of spherical functions: Sylvester's theorem, Maxwell's multipoles and Majorana's sphere, Journal of Physics A, 37 CrossRef | Google Scholar, 9487–9500.
[48] Dennis, M. (2005) Correlations between Maxwell's multipoles for Gaussian random functions on the sphere, Journal of Physics A, 38 CrossRef | Google Scholar, 1653–1658.
[49] Diaconis, P. (1988) Group Representations in Probability and Statistics Google Scholar, IMS Lecture Notes – Monograph Series, 11, Hayward.
[50] Diaconis, P., Freedman, D. (1987) A dozen de Finetti-style results in search of a theory, Annales Institute Henri Poincaré Probabilités et Statistiques, 23 Google Scholar(2), 397–423.
[51] Dodelson, S. (2003) Modern Cosmology Google Scholar, Academic Press.
[52] Doré, O., Colombi, S., Bouchet, F.R. (2003) Probing non-Gaussianity using local curvature, Monthly Notices of the Royal Astronomical Society, 344 CrossRef | Google Scholar, 905–916.
[53] 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 CrossRef | Google Scholar, 275.
[54] Doukhan, P. (1988) Formes de Toeplitz associées à une analyse multi-échelle, (French) [Toeplitz forms associated to a multiscale analysis]Comptes Rendus de l'Académie des Sciences. Série I. Mathématique, 306 Google Scholar, no. 15, 663–666.
[55] Doukhan, P., Leon, J. R. (1990) Formes quadratique d'estimateurs de densité par projections orthogonales. (French) [Quadratic deviation of projection density estimates]Comptes Rendus de l'Académie des Sciences. Série I. Mathématique, 310 Google Scholar, no. 6, 425–430.
[56] Dudley, R.M. (2002) Real Analysis and Probability CrossRef | Google Scholar, revised reprint of the 1989 original, Cambridge Studies in Advanced Mathematics, 74, Cambridge University Press.
[57] Duistermaat, J.J., Kolk, J.A.C. (1997) Lie Groups Google Scholar, Springer-Verlag.
[58] Duffin, R.J., Schaeffer, A.C. (1952) A class of nonharmonic Fourier series, Transactions of the American Mathematical Society, 72 CrossRef | Google Scholar, 341–366.
[59] Durastanti, C., Geller, D., Marinucci, D. (2010) Nonparametric Regression on Spin fiber Bundles, under revision for the Journal of Multivariate Analysis Google Scholar, arXiv preprint 1009.4345.
[60] Durrer, R. (2008) The Cosmic Microwave Background CrossRef | Google Scholar, Cambridge University Press.
[61] Efstathiou, G. (2004) Myths and truths concerning estimation of power spectra: the case for a hybrid estimator, Monthly Notices of the Royal Astronomical Society, 349 CrossRef | Google Scholar, Issue 2, pp. 603–626.
[62] Eriksen, H.K., Hansen, F.K., Banday, A.J., Gorski, K.M., Lilje, P.B. (2004) Asymmetries in the CMB anisotropy field, Astrophysical Journal, 605 CrossRef | Google Scholar, 14–20.
[63] Faraut, J. (2006) Analyse sur le Groupes de Lie Google Scholar, Calvage et Mounet.
[64] Faÿ, G., Guilloux, F., Betoule, M., Cardoso, J.-F., Delabrouille, J., Le Jeune, M. (2008) CMB power spectrum estimation using waveletsPhysical Review D, 78 CrossRef | Google Scholar:083013.
[65] Faÿ, G., Guilloux, F. (2008 Google Scholar) Consistency of a Needlet Spectral Estimator on the Sphere, arXiv:0807.2162.
[66] Feller, W. (1970) An Introduction to Probability Theory and its Applications, Volume II Google Scholar, 2nd Edition J. Wiley.
[67] Fergusson, J.R., Liguori, M., Shellard, E.P.S. (2009 Google Scholar) General CMB and Primordial Bispectrum Estimation I: Mode Expansion, Map-Making and Measures of fNL, arXiv: 0912.5516.
[68] Fergusson, J.R., Liguori, M., Shellard, E.P.S. (2010 Google Scholar) The CMB Bispectrum, arXiv: 1006.1642.
[69] Foulds, L.R. (1992) Graph Theory and Applications CrossRef | Google Scholar, Universitext, Springer-Verlag.
[70] Freeden, W., Schreiner, M. (1998) Orthogonal and nonorthogonal multiresolution analysis, scale discrete and exact fully discrete wavelet transform on the sphere. Constructive Approximations, 14 CrossRef | Google Scholar, 4, 493–515.
[71] Gangolli, R. (1967) Positive definite kernels on homogeneous spaces and certain stochastic processes related to Lévy's Brownian motion of several parameters. Annales de l'Institut H. Poincaré Sect. B, Vol. 3 Google Scholar, 121–226.
[72] Geller, D., Hansen, F.K., Marinucci, D., Kerkyacharian, G., Picard, D. (2008), Spin needlets for Cosmic Microwave Background Polarization data analysis, Physical Review D, D78 Google Scholar:123533, arXiv:0811.2881.
[73] Geller, D., Lan, X., Marinucci, D. (2009) Spin needlets spectral estimation, Electronic Journal of Statistics, Vol. 3 CrossRef | Google Scholar, 1497–1530, arXiv:0907.3369.
[74] Geller, D., Marinucci, D. (2008) Spin wavelets on the sphere, Journal of Fourier Analysis and its Applications, Vol. 16 CrossRef | Google Scholar, Issue 6, pages 840–884, arXiv: 0811.2835.
[75] Geller, D., Marinucci, D. (2011) Mixed needlets, Journal of Mathematical Analysis and Applications, Vol. 375 CrossRef | Google Scholar, n.2, pp. 610–630, arXiv: 1006.3835.
[76] Geller, D., Mayeli, A. (2009) Continuous wavelets on manifolds, Mathematische Zeitschrift, Vol. 262 CrossRef | Google Scholar, pp. 895–927, arXiv: math/0602201.
[77] Geller, D., Mayeli, A. (2009) Nearly Tight frames and space-frequency analysis on compact manifolds, Mathematische Zeitschrift, Vol. 263 (2009 CrossRef | Google Scholar), pp. 235–264, arXiv: 0706.3642.
[78] Geller, D., Mayeli, A. (2009) Besov spaces and frames on compact manifolds, Indiana University Mathematics Journal, Vol. 58 CrossRef | Google Scholar, pp. 2003–2042, arXiv:0709.2452.
[79] Geller, D., Mayeli, A. (2009) Nearly tight frames of spin wavelets on the sphere, Sampling Theory in Signal and Image Processing Google Scholar, in press, arXiv:0907.3164.
[80] Geller, D., Pesenson, I. (2010), Band-limited localized Parseval frames and Besov spaces on compact homogeneous manifolds, Journal of Geometric Analysis Google Scholar, in press, arXiv:1002.3841.
[81] Genovese, C.R., Perone-Pacifico, M., Verdinelli, I., Wasserman, L. (2009) On the path density of a gradient field. Annals of Statistics, 37 CrossRef | Google Scholar(6A), 3236–3271.
[82] Genovese, C.R., Perone-Pacifico, M., Verdinelli, I., Wasserman, L. (2010) Non-parametric filament estimation, arXiv:1003.5536 Google Scholar.
[83] Ghosh, T., Delabrouille, J., Remazeilles, M., Cardoso, J.-F., Souradeep, T. (2010 Google Scholar) Foreground maps in WMAP frequency bands, arxiv: 1006.0916.
[84] Goldberg, J.N., Newman, E.T., (1967) Spin-s Spherical Harmonics and ð, Journal of Mathematical Physics, 8 CrossRef | Google Scholar(11), 2155–2166.
[85] 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 CrossRef | Google Scholar, 759–771.
[86] Gradshteyn, I. S., Ryzhik, I. M. (1980) Table of Integrals, Series, and Products Google Scholar, Academic Press.
[87] Guilloux, F., Fay, G., Cardoso, J.-F. (2008) Practical wavelet design on the sphere, Applied and Computational Harmonic Analysis, 26 CrossRef | Google Scholar, no. 2, 143–160.
[88] Guionnet, A. (2009 Google Scholar) Large random matrices: lectures on macroscopic asymptotics. Lecture Notes in Mathematics, Vol. 1957, Springer-Verlag.
[89] Guivarc'h, Y., Keane, M. and Roynette, B. (1977) Marches Aléatoires sul les Groupes de Lie, Lecture Notes in Mathematics, Vol. 624 CrossRef | Google Scholar, Springer-Verlag.
[90] Hamann, Jan, Wong, Yvonne, Y. Y. (2008) The effects of Cosmic Microwave Background (CMB) temperature uncertainties on cosmological parameter estimation, Journal of Cosmology and Astroparticle Physics CrossRef | Google Scholar, Issue 03, pp. 025.
[91] Hanany, S., Ade, P., Balbi, A., Bock, J., Borrill, J., Boscaleri, A., de Bernardis, P., Ferreira, P. G., Hristov, V. V., Jaffe, A. H., Lange, A. E., Lee, A. T., Mauskopf, P. D., Netterfield, C. B., Oh, S., Pascale, E., Rabii, B., Richards, P. L., Smoot, G. F., Stompor, R., Winant, C. D., Wu, J. H. P. (2000) MAXIMA-1: A measurement of the Cosmic Microwave Background anisotropy on angular scales of 10'-5°, The Astrophysical Journal, Vol. 545 CrossRef | Google Scholar, Issue 1, L5–L9.
[92] Hannan, E.J. (1970) Multiple Time Series CrossRef | Google Scholar. J. Wiley.
[93] Hansen, F.K., Cabella, P., Marinucci, D., Vittorio, N. (2004) Asymmetries in the local curvature of the WMAP data, Astrophysical Journal Letters CrossRef | Google Scholar, L67–L70.
[94] Hardle, W., Kerkyacharian, G., Picard, D. and Tsybakov, A. (1998) Wavelets, Approximation, and Statistical Applications CrossRef | Google Scholar, Springer Lecture Notes in Statistics, 129.
[95] Hausman, J.A. (1978) Specification tests in econometrics, Econometrica, 6 CrossRef | Google Scholar, 1251–1271.
[96] Havin, V. and Joricke, B. (1994) The Uncertainty Principle in Harmonic Analysis CrossRef | Google Scholar, Springer-Verlag.
[97] Hernandez, E., Weiss, G. (1996) A First Course on Wavelets CrossRef | Google Scholar, Studies in Advanced Mathematics, CRC Press.
[98] Hikage, C., Matsubara, T., Coles, P., Liguori, M., Hansen, F.K., Matarrese, S. (2008) Primordial non-Gaussianity from Minkowski functionals of the WMAP temperature anisotropies, Monthly Notices Royal Astronomical Society, 389 CrossRef | Google Scholar:1439–1446.
[99] Hinshaw, G., Weiland, J. L., Hill, R. S., Odegard, N., Larson, D., Bennett, C. L., Dunkley, J., Gold, B., Greason, M. R., Jarosik, N., Komatsu, E., Nolta, M. R., Page, L., Spergel, D. N., Wollack, E., Halpern, M., Kogut, A., Limon, M., Meyer, S. S., Tucker, G. S., Wright, E. L. (2009) Five-Year Wilkinson Microwave Anisotropy Probe (WMAP) observations: data processing, sky maps, and basic results, Astrophysical Journal Supplement Series, 180 CrossRef | Google Scholar:225–245.
[100] Hivon, E., Gorski, K.M., Netterfield, C.B., Crill, B.P., Prunet, S., Hansen, F.K. (2002) MASTER of the Cosmic Microwave Background anisotropy power spectrum: a fast method for statistical analysis of large and complex Cosmic Microwave Background data sets, Astrophysical Journal, Volume 567 CrossRef | Google Scholar, Issue 1, pp. 2–17.
[101] Holschneider, M., Iglewska-Nowak., I. (2007) Poisson wavelets on the sphereJournal of Fourier Analysis and its Applications, 13 CrossRef | Google Scholar, 405–420.
[102] Hu, W. (2001) The angular trispectrum of the Cosmic Microwave Background, Physical Review D, Volume 64 CrossRef | Google Scholar, Issue 8, id.083005.
[103] Hu, Y., Nualart, D. (2005) Renormalized self-intersection local time for fractional Brownian motion, The Annals of Probability, 33 CrossRef | Google Scholar(3), 948–983.
[104] Ivanov, A.V., Leonenko, N.N. (1989), Statistical Analysis of Random Fields CrossRef | Google Scholar, Kluwer.
[105] Jansson, S. (1997) Gaussian Hilbert Spaces CrossRef | Google Scholar, Cambridge University Press.
[106] Johnson, N.L., Kotz, S.J. (1972) Distributions in Statistics: Continuous Multivariate Distributions Google Scholar, J. Wiley.
[107] Kagan, A.M., Linnik, Y.V., Rao, C.R. (1973) Characterization Problems in Mathematical Statistics Google Scholar, J. Wiley.
[108] Kamionkowski, M., Kosowski, A., Stebbins, A. (1997) Statistics of Cosmic Microwave Background Polarization, Physical Review D, 55 CrossRef | Google Scholar, 7368–7388.
[109] Keihänen, E., Kurki-Suonio, H., Poutanen, T. (2005) MADAM- a map-making method for CMB experiments, Monthly Notices of the Royal Astronomical Society, Vol. 360 CrossRef | Google Scholar, Issue 1, pp. 390–400.
[110] Kerkyacharian, G., Petrushev, P., Picard, D., Willer, T. (2007) Needlet algorithms for estimation in inverse problems, Electronic Journal of Statistics, 1 CrossRef | Google Scholar, 30–76.
[111] Kerkyacharian, G., Nickl, R., Picard, D. (2010) Concentration inequalities and confidence bands for needlet density estimators on compact homogeneous manifolds, Probability Theory and Related Fields Google Scholar, in press, arXiv:1102.2450.
[112] Kerkyacharian, G., Pham Ngoc, T.M., Picard, D. (2009) Localized spherical deconvolution, Annals of Statistics Google Scholar, in press, arXiv: 0908.1952.
[113] Kim, P.T., Koo, J.-Y. (2002) Optimal spherical deconvolution, Journal of Multivariate Analysis, 80 CrossRef | Google Scholar, 21–42.
[114] Kim, P.T., Koo, J.-Y., Luo, Z.-M. (2009) Weyl eigenvalue asymptotics and sharp adaptation on vector bundles, Journal of Multivariate Analysis, 100 CrossRef | Google Scholar, 1962–1978.
[115] Kitching, T. et al. (2010 Google Scholar) Gravitational lensing accuracy testing 2010 (GREAT10) challenge handbook, preprint, arXiv: 1009.0779.
[116] Kolb, E., Turner, M. (1994), The Early Universe Google Scholar, Cambridge University Press.
[117] Komatsu, E., Spergel, D.N. (2001) Acoustic signatures in the primary Microwave Background bispectrum, Physycal Review D, 63 Google Scholar, 063002.
[118] Komatsu, E.Wandelt, B.D., Spergel, D.N., Banday, A.J., Gorski, K.M. (2002), Measurement of the Cosmic Microwave Backgroun bispectrum on the COBE DMR sky maps, Astrophysical Journal, 566 CrossRef | Google Scholar, 19–29.
[119] Komatsu, E., Yadav, A., Wandelt, B. (2007) Fast estimator of primordial non-Gaussianity from temperature and polarization anisotropies in the Cosmic Microwave Background, Astrophysical Journal, 664 Google Scholar:680–686.
[120] Komatsu, E., Yadav, A., Wandelt, B., Liguori, M., Hansen, F.K., Matarrese, S. (2008) Fast estimator of primordial non-Gaussianity from temperature and polarization anisotropies in the Cosmic Microwave Background II: partial sky coverage and inhomogeneous noise, Astrophysical Journal 678 Google Scholar:578.
[121] Komatsu, et al. (2009) Five-Year Wilkinson Microwave Anisotropy Probe observations: cosmological interpretation, Astrophysical Journal Supplement Series, 180 CrossRef | Google Scholar, 2, 330–376.
[122] Koornwinder, T.H. (2008 Google Scholar), Representations of SU(2) and Jacobi polynomials, preprint, available online http://staff.science.uva.nl/ thk/edu/orthopoly.pdf.
[123] Lan, X., Marinucci, D. (2008) The needlets bispectrum, Electronic Journal of Statistics, 2 CrossRef | Google Scholar, 332–367.
[124] Lan, X., Marinucci, D. (2009) On the dependence structure of wavelet coefficients for spherical random fields, Stochastic Processes and their Applications, 119 CrossRef | Google Scholar, 3749–3766.
[125] Leonenko, N. (1999) Limit Theorems for Random Fields with Singular Spectrum CrossRef | Google Scholar, Kluwer.
[126] Leonenko, N., Sakhno, L. (2009 Google Scholar) On spectral representations of tensor random fields on the sphere, arXiv:0912.3389.
[127] Liboff, R.L. (1999) Introductory Quantum Mechanics Google Scholar, Addison-Wesley.
[128] Magnus, J.R., Neudecker, H. (1988) Matrix Differential Calculus with Applications to Statistics and Econometrics Google Scholar, J. Wiley.
[129] Malyarenko, A. (2009 Google Scholar) Invariant random fields in vector bundles and application to cosmology, preprint arXiv: 0907.4620.
[130] Marinucci, D. (2004) Testing for non-Gaussianity on Cosmic Microwave Background radiation: a review, Statistical Science, 19 CrossRef | Google Scholar, 294–307.
[131] Marinucci, D. (2006) High-resolution asymptotics for the angular bispectrum of spherical random fields, Annals of Statistics, 34 CrossRef | Google Scholar, 1–41.
[132] Marinucci, D. (2008) A central limit theorem and higher order results for the angular bispectrum, Probability Theory and Related Fields, 141 CrossRef | Google Scholar(3–4), 389–409.
[133] Marinucci, D., Piccioni, M. (2004) The empirical process on Gaussian spherical harmonics, Annals of Statistics, 32 Google Scholar, 1261–1288.
[134] Marinucci, D., Peccati, G. (2008) High-frequency asymptotics for subordinated stationary fields on an Abelian compact group, Stochastic Processes and their Applications, 118 CrossRef | Google Scholar (4), 585–613.
[135] Marinucci, D., Peccati, G. (2010) Group representations and high-resolution Central Limit Theorems for subordinated spherical random fields, Bernoulli, 16 CrossRef | Google Scholar, 798–824.
[136] Marinucci, D.; Peccati, G. (2010) Representations of SO(3) and angular polyspectra, Journal of Multivariate Analysis, 101 CrossRef | Google Scholar, 77–100.
[137] Marinucci, D.; Peccati, G. (2010) Ergodicity and Gaussianity for spherical random fields, Journal of Mathematical Physics, 51 CrossRef | Google Scholar, n. 4, 043301, 23 pp.
[138] Marinucci, D., Wigman, I. (2010 Google Scholar) On the excursion sets of spherical Gaussian eigenfunctions, preprint, arXiv: 1009.4367.
[139] 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 CrossRef | Google Scholar, 539–545, arXiv: 0707.0844.
[140] Mayeli, A. (2010) Asymptotic uncorrelation for Mexican needlets, Journal of Mathematical Analysis and Applications, Vol. 363 CrossRef | Google Scholar, Issue 1, pp. 336–344, arXiv: 0806.3009.
[141] 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 CrossRef | Google Scholar, 495–510.
[142] Miller, W. Google ScholarTopics in Harmonic Analysis with Applications to Radar and Sonar, preprint, available online http://www.ima.umn.edu/ miller/radarla.pdf.
[143] Narcowich, F.J., Petrushev, P., Ward, J.D. (2006) Localized tight frames on spheres, SIAM Journal of Mathematical Analysis, 38 CrossRef | Google Scholar, 2, 574–594.
[144] Narcowich, F.J., Petrushev, P., Ward, J.D. (2006) Decomposition of Besov and Triebel-Lizorkin spaces on the sphere, Journal of Functional Analysis, 238 CrossRef | Google Scholar, 2, 530–564.
[145] Natoli, P., Degasperis, G., Marinucci, D., Vittorio, N. (2002) Non-iterative methods to estimate the in-flight noise properties of CMB detectors, Astronomy and Astrophysics, 383 CrossRef | Google Scholar, pp. 1100–1112.
[146] Newman, E. T., Penrose, R. (1966) Note on the Bondi-Metzner-Sachs group, Journal of Mathematical Physics, 7 CrossRef | Google Scholar, 863–870.
[147] Nourdin, I., Peccati, G. and Reinert, G. (2010) Invariance principles for homogeneous sums: universality of the Gaussian Wiener chaos, Annals of Probability, 38 CrossRef | Google Scholar(5), 1947–1985.
[148] Nourdin, I., Peccati, G. (2009). Stein's method on Wiener chaos, Probability Theory and Related Fields, 145 CrossRef | Google Scholar(1), 75–118.
[149] Nourdin, I., Peccati, G. (2009) Stein's method meets Malliavin calculus: a short survey with new estimates. In the volume: Recent Advances in Stochastic Dynamics and Stochastic Analysis Google Scholar, World Scientific.
[150] Nourdin, I., Peccati, G., Réveillac, A. (2008). Multivariate normal approximation using Stein's method and Malliavin calculus, Annales de l'Institut H. Poincaré (B), 46 CrossRef | Google Scholar(1), 45–58.
[151] Nualart, D. (2006) The Malliavin Calculus and Related Topics Google Scholar. Second edition, Springer-Verlag.
[152] Nualart, D., Peccati, G. (2005) Central limit theorems for sequences of multiple stochastic integrals, Annals of Probability, 33 CrossRef | Google Scholar, 177–193.
[153] Patanchon, G., Delabrouille, J., Cardoso, J.-F., Vielva, P. (2005) CMB and foreground in WMAP first-year data, Monthly Notices of the Royal Astronomical Society, 364 CrossRef | Google Scholar, pp. 1185–1194.
[154] Peacock, J.A. (1999) Cosmological Physics Google Scholar, Cambridge University Press.
[155] Peccati, G. (2001) On the convergence of multiple random integrals, Studia Sc. Math. Hungarica, 37 Google Scholar, 429–470.
[156] Peccati, G. (2007) Gaussian approximations of multiple integrals, Electronic Communications in Probability 12 CrossRef | Google Scholar, 350–364.
[157] Peccati, G., Pycke, J.-R. (2010) Decompositions of stochastic processes based on irreducible group representations, Theory of Probability and Applications, 54 CrossRef | Google Scholar(2), 217–245.
[158] Peccati, G., Taqqu, M.S. (2008) Stable convergence of multiple Wiener-Itô integrals, Journal of Theoretical Probability, 21 CrossRef | Google Scholar(3), 527–570.
[159] Peccati, G., M.S., Taqqu (2010) Wiener Chaos: Moments, Cumulants and Diagrams. A Survey with Computer Implementation Google Scholar, Springer-Verlag.
[160] Peccati, G., Tudor, C.A. (2005) Gaussian limits for vector-valued multiple stochastic integrals. In: Séminaire de Probabilités XXXVIII CrossRef | Google Scholar, 247–262, Springer Verlag.
[161] Peebles, J. (1993), Principles of Cosmology Google Scholar, Princeton University Press.
[162] Pietrobon, D., Balbi, A., Marinucci, D. (2006) Integrated Sachs-Wolfe effect from the cross correlation of WMAP 3-Year and the NRAO VLA Sky Survey Data: new results and constraints on dark energy, Physical Review D, 74 CrossRef | Google Scholar, 043524.
[163] Pietrobon, D.Amblard, A., Balbi, A., Cabella, P., Cooray, A., Marinucci, D. (2008) Needlet detection of features in the WMAP CMB sky and the impact on anisotropies and hemispherical asymmetries, Physical Review D, Vol. 78 CrossRef | Google Scholar, Issue 10, id. 103504.
[164] Pietrobon, D.Amblard, A., Balbi, A., Cabella, P., Cooray, A., Vittorio, N. (2009) Constraints on primordial non-Gaussianity from a needlet analysis of the WMAP-5 data, Monthly Notices of the Royal Astronomical Society, Volume 396 CrossRef | Google Scholar, Issue 3, pp. 1682–1688.
[165] Pietrobon, D.Amblard, A., Balbi, A., Cabella, P., Cooray, A., Vittorio, N. (2009) Needlet bispectrum asymmetries in the WMAP 5-year Data, Monthly Notices of the Royal Astronomical Society, L367 Google Scholar, arXiv: 0905.3702.
[166] Pietrobon, D., Balbi, A., Cabella, P.Gorski, K. M. (2010) Needatool: A Needlet Analysis Tool for Cosmological Data Processing, Astrophysical Journal, 723 CrossRef | Google Scholar, 1.
[167] Polenta, G., Marinucci, D., Balbi, A., De Bernardis, P., Hivon, E., Masi, S., Natoli, P., Vittorio, N. (2005) Unbiased estimation of angular power spectra, Journal of Cosmology and Astroparticle Physics CrossRef | Google Scholar, Issue 11, n.1, pp.1–17.
[168] Pycke, J.-R. (2007) A decomposition for invariant tests of uniformity on the sphere, Proceedings of the American Mathematical Society, 135 CrossRef | Google Scholar, 2983–2993.
[169] Revuz, D., Yor, M. (1999) Continuous Martingales and Brownian motion, Third edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 293 CrossRef | Google Scholar, Springer-Verlag.
[170] Robinson, P.M. (1995) Log-periodogram regression of time series with long range dependence, Annals of Statistics, 23 CrossRef | Google Scholar, 1048–1072.
[171] Robinson, P.M. (1995) Gaussian semiparametric estimation of long range dependence, Annals of Statistics, 23 CrossRef | Google Scholar, 1630–1661.
[172] Rosca, D. (2007) Wavelet bases on the sphere obtained by radial projection, Journal of Fourier Analysis and its Applications, 13 CrossRef | Google Scholar, 421–434.
[173] Rudin, W. (1962) Fourier Analysis on Groups Google Scholar, Wiley Classics Library, Wiley.
[174] Rudin, W. (1975) Real and Complex Analysis Google Scholar, McGraw-Hill.
[175] Rudjord, O., Hansen, F.K.Lan, X., Liguori, M., Marinucci, D., Matarrese, S. (2009) An estimate of the primordial non-Gaussianity parameter fNL using the needlet bispectrum from WMAP, The Astrophysical Journal, 701 CrossRef | Google Scholar:369–376, arXiv:0901.3154.
[176] Rudjord, O., Hansen, F.K.Lan, X., Liguori, M., Marinucci, D., Matarrese, S. (2010), Directional variations of the non-Gaussianity parameter fNL, Astrophysical Journal, Vol. 708 CrossRef | Google Scholar, 2, 1321–1325.
[177] Schreiber, M. (1969) Fermeture en probabilité de certains sous-espaces d'un espace L2, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 14 CrossRef | Google Scholar, 36–48.
[178] Schwartzman, A., Mascarenhas, W.F. and Taylor, J.E.T. (2008) Inference for eigenvalues and eigenvectors of Gaussian symmetric matrices, Annals of Statistics, 36 CrossRef | Google Scholar, no. 6, 2886–2919.
[179] Scodeller, S., Rudjord, O., Hansen, F.K., Marinucci, D., Geller, D., Mayeli, A. (2010) Introducing Mexican needlets for CMB analysis: Issues for practical applications and comparison with standard needlets, Astrophysical Journal Google Scholar, in press, arXiv: 1004.5576.
[180] Simon, B. (1996) Representations of Finite and Compact Groups Google Scholar, Graduate Studies in Mathematics, 10, American Mathematical Society.
[181] Seljak, U., Zaldarriaga, M. (1996) Line-of-Sight integration approach to Cosmic Microwave Background anisotropies, Astrophysical Journal, Vol.469 CrossRef | Google Scholar, p.437.
[182] Senatore, L., Smith, K.M., Zaldarriaga, M. (2010) Non-Gaussianities in single field inflation and their optimal limits from the WMAP 5-year data, Journal of Cosmology and Astroparticle Physics, 1001 CrossRef | Google Scholar:028.
[183] Serre, J.P. (1977) Linear Representation of Finite Groups CrossRef | Google Scholar, Springer-Verlag.
[184] Regan, D.M., Shellard, E.P.S. (2009 Google Scholar), Cosmic string power spectrum, bispectrum and trispectrum, arXiv:0911.2491.
[185] Shigekawa, I. (1986) De Rham–Hodge–Kodaira's decomposition on an abstract Wiener space, Journal of Mathematics of the Kyoto University, 26 CrossRef | Google Scholar, 191–202.
[186] Shyraev, A.N. (1984) Probability CrossRef | Google Scholar, Springer-Verlag.
[187] Smoot, G. F., Bennett, C. L., Kogut, A., Wright, E. L., Aymon, J., Boggess, N. W., Cheng, E. S., de Amici, G., Gulkis, S., Hauser, M. G., Hinshaw, G., Jackson, P. D., Janssen, M., Kaita, E., Kelsall, T., Keegstra, P., Lineweaver, C., Loewenstein, K., Lubin, P., Mather, J., Meyer, S. S., Moseley, S. H., Murdock, T., Rokke, L., Silverberg, R. F., Tenorio, L., Weiss, R., Wilkinson, D. T. (1992) Structure in the COBE differential microwave radiometer first-year maps, Astrophysical Journal, Part 2 - Letters, Vol. 396 CrossRef | Google Scholar, no. 1, pp. L1–L5.
[188] Spergel, D.N. et al. (2003) First-Year Wilkinson Microwave Anisotropy Probe (WMAP) observations: determination of cosmological parameters, Astrophysical Journal Supplement Series, 148 CrossRef | Google Scholar, 1, pp. 175–194.
[189] Spergel, D.N. et al. (2007) Three-Year Wilkinson Microwave Anisotropy Probe (WMAP) observations: implications for cosmology, Astrophysical Journal Supplement Series, 170 CrossRef | Google Scholar, 2, 377–408.
[190] Stein, E.M., Weiss, G. (1971) Introduction to Fourier Analysis on Euclidean Spaces Google Scholar, Princeton University Press.
[191] Sternberg, S.Group Theory and Physics Google Scholar, Cambridge University Press.
[192] Surgailis, D. (2003) CLTs for polynomials of linear sequences: Diagram formula with illustrations. In Theory and Applications of Long Range Dependence Google Scholar, 111–128, Birkhäuser.
[193] Szego, G. (1975) Orthogonal Polynomials, American Mathematical Society Colloquium Publications, Volume 23 Google Scholar Reprinted version of the 1939 original.
[194] Varadarajan, V.S. (1999) An Introduction to Harmonic Analysis on Semisimple Lie Groups Google Scholar, Corrected reprint of the 1989 original, Cambridge University Press.
[195] Varshalovich, D.A., Moskalev, A.N., Khersonskii, V.K. (1988). Quantum Theory of Angular Momentum CrossRef | Google Scholar, World Scientific Press.
[196] Vielva, P., Martínez-González, E., Gallegos, J. E., Toffolatti, L., Sanz, J. L. (2003) Point source detection using the spherical Mexican hat wavelet on simulated all-sky Planck maps, Monthly Notice of the Royal Astronomical Society, Vol. 344 CrossRef | Google Scholar, Issue 1, 89–104.
[197] Vilenkin, N.Ja. and Klimyk, A.U. (1991) Representation of Lie Groups and Special Functions CrossRef | Google Scholar, Kluwer.
[198] 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, Astrophysical Journal, 664 CrossRef | Google Scholar:680–686.
[199] Yadav, A.P.S. and Wandelt, B.D. (2008) Evidence of primordial non-Gaussianity (fNL) in the Wilkinson Microwave Anisotropy Probe 3-Year Data at 2.8 sigma, Physical Review Letters, Vol. 100 CrossRef | Google Scholar, Issue 18, id. 181301.
[200] Yadav, A.P.S. and Wandelt, B.D. (2010 Google Scholar) Primordial non-Gaussianity in the Cosmic Microwave Background, Advances in Astronomy, in press, arXiv: 1006.0275.
[201] Yadrenko, M.I. (1983) Spectral Theory of Random Fields Google Scholar, Translated from the Russian, Translation Series in Mathematics and Engineering, Optimization Software, Inc., Publications Division.
[202] 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 CrossRef | Google Scholar, 477–494.
[203] Wiaux, Y., Jacques, L., Vandergheynst, P. (2005) Correspondence principle between spherical and Euclidean wavelets, The Astrophysical Journal, Vol. 632 CrossRef | Google Scholar, Issue 1, pp. 15–28.
[204] Wiaux, Y., Jacques, L., Vandergheynst, P. (2007) Fast spin +-2 spherical harmonics transforms and application in cosmology, Journal of Computational Physics, 226 CrossRef | Google Scholar:2359–2371.
[205] Wiener, N. (1938), The homogeneous chaos, American Journal of Mathematics, 60 CrossRef | Google Scholar, 879–936.
[206] Wigman, I. (2009) On the distribution of the nodal sets of random spherical harmonics, Journal of Mathematical Physics, 50, no. 1 CrossRef | Google Scholar, 013521, 44 pp.
[207] Wigman, I. (2010) Fluctuations of the nodal length of random spherical harmonics, Communications in Mathematical Physics, Vol. 298 CrossRef | Google Scholar, n. 3, pp. 787–831, arXiv: 0907.1648.
[208] Zaldarriaga, M., Seljak, U. (2000) CMBFAST for spatially closed Universes, Astrophysical Journal Supplements Series, 129 CrossRef | Google Scholar, 431–434.

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