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  • Cited by 173
Publisher:
Cambridge University Press
Online publication date:
October 2011
Print publication year:
2011
Online ISBN:
9780511751677

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

References
[1] Abramowitz, M., Stegun, I. (1964) Handbook of Mathematical Functions, Dover.
[2] Adler, R.J. (1981) The Geometry of Random Fields, J. Wiley.
[3] Adler, R.J., Taylor, J.E. (2007) Random Fields and Geometry, Springer-Verlag.
[4] Anderes, E., Chatterjee, S. (2009) Consistent estimates of deformed isotropic Gaussian random fields on the plane, Annals of Statistics, 37, 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, 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, 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, 308–321.
[8] Babich, D., Creminelli, P., Zaldarriaga, M. (2004) The shape of non-Gaussianities, Journal of Cosmology and Astroparticle Physics, 8, 009.
[9] Balbi, A. (2007), The Music of the Big Bang, Springer-Verlag.
[10] Baldi, P., Marinucci, D. (2007). Some characterizations of the spherical harmonics coefficients for isotropic random fields, Statistics & Probability Letters, 77(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, 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(4), 606–636.
[13] Baldi, P., Kerkyacharian, G., Marinucci, D., Picard, D. (2009) Asymptotics for Spherical Needlets, Annals of Statistics, 37(3), 1150–1171, arxiv:math/0606599.
[14] Baldi, P., Kerkyacharian, G.Marinucci, D., Picard, D. (2009) Subsampling Needlet Coefficients on the Sphere, Bernoulli, 15(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(6A), 3362–3395.
[16] Baldi, P., Kerkyacharian, G., Marinucci, D., Picard, D. (2009) 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, 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, 103–266.
[19] Bartolo, N., Matarrese, S., Riotto, A. (2010) Non-Gaussianity and the Cosmic Microwave Background anisotropies, Advances in Astronomy, 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, 1008:08, 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, 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) 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, Addison-Wesley.
[24] Billingsley, P. (1968) Convergence of Probability Measures, J. Wiley.
[25] Bishop, R.L., Goldberg, S. (1980) Tensor Analysis on Manifolds, Dover.
[26] Blei, R. (2001) Analysis in Integer and Fractional Dimensions, Cambridge University Press.
[27] Breuer, P., Major, P. (1983) Central limit theorems for nonlinear functionals of Gaussian fields, Journal of Multivariate Analysis, 13, 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, No. 1, 6–37.
[29] Brillinger, D. W. (1975) Time series. Data Analysis and Theory, Holt, Rinehart and Winston.
[30] Brockwell, P.J., Davis, R.A. (1991) Time Series: Theory and Methods, 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, Springer-Verlag.
[32] Bump, D. (2005) Lie Groups, Graduate Texts in Mathematics, 225, Springer-Verlag.
[33] Cabella, P., Kamionkowskii, M. (2005) Theory of Cosmic Microwave Background Polarization, 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, 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, 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, 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(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(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(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, 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, 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, 387–404.
[43] Davidson, J. (1994), Stochastic Limit Theory, 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, 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, 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, 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, 9487–9500.
[48] Dennis, M. (2005) Correlations between Maxwell's multipoles for Gaussian random functions on the sphere, Journal of Physics A, 38, 1653–1658.
[49] Diaconis, P. (1988) Group Representations in Probability and Statistics, 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(2), 397–423.
[51] Dodelson, S. (2003) Modern Cosmology, 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, 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, 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, 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, no. 6, 425–430.
[56] Dudley, R.M. (2002) Real Analysis and Probability, 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, Springer-Verlag.
[58] Duffin, R.J., Schaeffer, A.C. (1952) A class of nonharmonic Fourier series, Transactions of the American Mathematical Society, 72, 341–366.
[59] Durastanti, C., Geller, D., Marinucci, D. (2010) Nonparametric Regression on Spin fiber Bundles, under revision for the Journal of Multivariate Analysis, arXiv preprint 1009.4345.
[60] Durrer, R. (2008) The Cosmic Microwave Background, 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, 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, 14–20.
[63] Faraut, J. (2006) Analyse sur le Groupes de Lie, 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, 78:083013.
[65] Faÿ, G., Guilloux, F. (2008) 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, 2nd Edition J. Wiley.
[67] Fergusson, J.R., Liguori, M., Shellard, E.P.S. (2009) 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) The CMB Bispectrum, arXiv: 1006.1642.
[69] Foulds, L.R. (1992) Graph Theory and Applications, 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, 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, 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, D78:123533, arXiv:0811.2881.
[73] Geller, D., Lan, X., Marinucci, D. (2009) Spin needlets spectral estimation, Electronic Journal of Statistics, Vol. 3, 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, Issue 6, pages 840–884, arXiv: 0811.2835.
[75] Geller, D., Marinucci, D. (2011) Mixed needlets, Journal of Mathematical Analysis and Applications, Vol. 375, n.2, pp. 610–630, arXiv: 1006.3835.
[76] Geller, D., Mayeli, A. (2009) Continuous wavelets on manifolds, Mathematische Zeitschrift, Vol. 262, 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), 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, 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, 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, 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(6A), 3236–3271.
[82] Genovese, C.R., Perone-Pacifico, M., Verdinelli, I., Wasserman, L. (2010) Non-parametric filament estimation, arXiv:1003.5536.
[83] Ghosh, T., Delabrouille, J., Remazeilles, M., Cardoso, J.-F., Souradeep, T. (2010) 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(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, 759–771.
[86] Gradshteyn, I. S., Ryzhik, I. M. (1980) Table of Integrals, Series, and Products, Academic Press.
[87] Guilloux, F., Fay, G., Cardoso, J.-F. (2008) Practical wavelet design on the sphere, Applied and Computational Harmonic Analysis, 26, no. 2, 143–160.
[88] Guionnet, A. (2009) 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, 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, 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, Issue 1, L5–L9.
[92] Hannan, E.J. (1970) Multiple Time Series. 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, L67–L70.
[94] Hardle, W., Kerkyacharian, G., Picard, D. and Tsybakov, A. (1998) Wavelets, Approximation, and Statistical Applications, Springer Lecture Notes in Statistics, 129.
[95] Hausman, J.A. (1978) Specification tests in econometrics, Econometrica, 6, 1251–1271.
[96] Havin, V. and Joricke, B. (1994) The Uncertainty Principle in Harmonic Analysis, Springer-Verlag.
[97] Hernandez, E., Weiss, G. (1996) A First Course on Wavelets, 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, 389:1439–1446.
[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, 180:225–245.
[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, Issue 1, pp. 2–17.
[101] Holschneider, M., Iglewska-Nowak., I. (2007) Poisson wavelets on the sphereJournal of Fourier Analysis and its Applications, 13, 405–420.
[102] Hu, W. (2001) The angular trispectrum of the Cosmic Microwave Background, Physical Review D, Volume 64, Issue 8, id.083005.
[103] Hu, Y., Nualart, D. (2005) Renormalized self-intersection local time for fractional Brownian motion, The Annals of Probability, 33(3), 948–983.
[104] Ivanov, A.V., Leonenko, N.N. (1989), Statistical Analysis of Random Fields, Kluwer.
[105] Jansson, S. (1997) Gaussian Hilbert Spaces, Cambridge University Press.
[106] Johnson, N.L., Kotz, S.J. (1972) Distributions in Statistics: Continuous Multivariate Distributions, J. Wiley.
[107] Kagan, A.M., Linnik, Y.V., Rao, C.R. (1973) Characterization Problems in Mathematical Statistics, J. Wiley.
[108] Kamionkowski, M., Kosowski, A., Stebbins, A. (1997) Statistics of Cosmic Microwave Background Polarization, Physical Review D, 55, 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, 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, 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, in press, arXiv:1102.2450.
[112] Kerkyacharian, G., Pham Ngoc, T.M., Picard, D. (2009) Localized spherical deconvolution, Annals of Statistics, in press, arXiv: 0908.1952.
[113] Kim, P.T., Koo, J.-Y. (2002) Optimal spherical deconvolution, Journal of Multivariate Analysis, 80, 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, 1962–1978.
[115] Kitching, T. et al. (2010) Gravitational lensing accuracy testing 2010 (GREAT10) challenge handbook, preprint, arXiv: 1009.0779.
[116] Kolb, E., Turner, M. (1994), The Early Universe, Cambridge University Press.
[117] Komatsu, E., Spergel, D.N. (2001) Acoustic signatures in the primary Microwave Background bispectrum, Physycal Review D, 63, 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, 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, 664:680–686.
[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 678:578.
[121] Komatsu, et al. (2009) Five-Year Wilkinson Microwave Anisotropy Probe observations: cosmological interpretation, Astrophysical Journal Supplement Series, 180, 2, 330–376.
[122] Koornwinder, T.H. (2008), 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, 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, 3749–3766.
[125] Leonenko, N. (1999) Limit Theorems for Random Fields with Singular Spectrum, Kluwer.
[126] Leonenko, N., Sakhno, L. (2009) On spectral representations of tensor random fields on the sphere, arXiv:0912.3389.
[127] Liboff, R.L. (1999) Introductory Quantum Mechanics, Addison-Wesley.
[128] Magnus, J.R., Neudecker, H. (1988) Matrix Differential Calculus with Applications to Statistics and Econometrics, J. Wiley.
[129] Malyarenko, A. (2009) 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, 294–307.
[131] Marinucci, D. (2006) High-resolution asymptotics for the angular bispectrum of spherical random fields, Annals of Statistics, 34, 1–41.
[132] Marinucci, D. (2008) A central limit theorem and higher order results for the angular bispectrum, Probability Theory and Related Fields, 141(3–4), 389–409.
[133] Marinucci, D., Piccioni, M. (2004) The empirical process on Gaussian spherical harmonics, Annals of Statistics, 32, 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 (4), 585–613.
[135] Marinucci, D., Peccati, G. (2010) Group representations and high-resolution Central Limit Theorems for subordinated spherical random fields, Bernoulli, 16, 798–824.
[136] Marinucci, D.; Peccati, G. (2010) Representations of SO(3) and angular polyspectra, Journal of Multivariate Analysis, 101, 77–100.
[137] Marinucci, D.; Peccati, G. (2010) Ergodicity and Gaussianity for spherical random fields, Journal of Mathematical Physics, 51, n. 4, 043301, 23 pp.
[138] Marinucci, D., Wigman, I. (2010) 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, 539–545, arXiv: 0707.0844.
[140] Mayeli, A. (2010) Asymptotic uncorrelation for Mexican needlets, Journal of Mathematical Analysis and Applications, Vol. 363, 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, 495–510.
[142] Miller, W.Topics 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, 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, 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, pp. 1100–1112.
[146] Newman, E. T., Penrose, R. (1966) Note on the Bondi-Metzner-Sachs group, Journal of Mathematical Physics, 7, 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(5), 1947–1985.
[148] Nourdin, I., Peccati, G. (2009). Stein's method on Wiener chaos, Probability Theory and Related Fields, 145(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, 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(1), 45–58.
[151] Nualart, D. (2006) The Malliavin Calculus and Related Topics. Second edition, Springer-Verlag.
[152] Nualart, D., Peccati, G. (2005) Central limit theorems for sequences of multiple stochastic integrals, Annals of Probability, 33, 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, pp. 1185–1194.
[154] Peacock, J.A. (1999) Cosmological Physics, Cambridge University Press.
[155] Peccati, G. (2001) On the convergence of multiple random integrals, Studia Sc. Math. Hungarica, 37, 429–470.
[156] Peccati, G. (2007) Gaussian approximations of multiple integrals, Electronic Communications in Probability 12, 350–364.
[157] Peccati, G., Pycke, J.-R. (2010) Decompositions of stochastic processes based on irreducible group representations, Theory of Probability and Applications, 54(2), 217–245.
[158] Peccati, G., Taqqu, M.S. (2008) Stable convergence of multiple Wiener-Itô integrals, Journal of Theoretical Probability, 21(3), 527–570.
[159] Peccati, G., M.S., Taqqu (2010) Wiener Chaos: Moments, Cumulants and Diagrams. A Survey with Computer Implementation, Springer-Verlag.
[160] Peccati, G., Tudor, C.A. (2005) Gaussian limits for vector-valued multiple stochastic integrals. In: Séminaire de Probabilités XXXVIII, 247–262, Springer Verlag.
[161] Peebles, J. (1993), Principles of Cosmology, 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, 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, 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, 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, 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, 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, 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, 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, Springer-Verlag.
[170] Robinson, P.M. (1995) Log-periodogram regression of time series with long range dependence, Annals of Statistics, 23, 1048–1072.
[171] Robinson, P.M. (1995) Gaussian semiparametric estimation of long range dependence, Annals of Statistics, 23, 1630–1661.
[172] Rosca, D. (2007) Wavelet bases on the sphere obtained by radial projection, Journal of Fourier Analysis and its Applications, 13, 421–434.
[173] Rudin, W. (1962) Fourier Analysis on Groups, Wiley Classics Library, Wiley.
[174] Rudin, W. (1975) Real and Complex Analysis, 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, 701:369–376, 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, 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, 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, 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, in press, arXiv: 1004.5576.
[180] Simon, B. (1996) Representations of Finite and Compact Groups, 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, 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, 1001:028.
[183] Serre, J.P. (1977) Linear Representation of Finite Groups, Springer-Verlag.
[184] Regan, D.M., Shellard, E.P.S. (2009), 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, 191–202.
[186] Shyraev, A.N. (1984) Probability, 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, 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, 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, 2, 377–408.
[190] Stein, E.M., Weiss, G. (1971) Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press.
[191] Sternberg, S.Group Theory and Physics, 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, 111–128, Birkhäuser.
[193] Szego, G. (1975) Orthogonal Polynomials, American Mathematical Society Colloquium Publications, Volume 23 Reprinted version of the 1939 original.
[194] Varadarajan, V.S. (1999) An Introduction to Harmonic Analysis on Semisimple Lie Groups, 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, 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, Issue 1, 89–104.
[197] Vilenkin, N.Ja. and Klimyk, A.U. (1991) Representation of Lie Groups and Special Functions, 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, 664:680–686.
[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, Issue 18, id. 181301.
[200] Yadav, A.P.S. and Wandelt, B.D. (2010) 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, 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, 477–494.
[203] Wiaux, Y., Jacques, L., Vandergheynst, P. (2005) Correspondence principle between spherical and Euclidean wavelets, The Astrophysical Journal, Vol. 632, 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, 226:2359–2371.
[205] Wiener, N. (1938), The homogeneous chaos, American Journal of Mathematics, 60, 879–936.
[206] Wigman, I. (2009) On the distribution of the nodal sets of random spherical harmonics, Journal of Mathematical Physics, 50, no. 1, 013521, 44 pp.
[207] Wigman, I. (2010) Fluctuations of the nodal length of random spherical harmonics, Communications in Mathematical Physics, Vol. 298, n. 3, pp. 787–831, arXiv: 0907.1648.
[208] Zaldarriaga, M., Seljak, U. (2000) CMBFAST for spatially closed Universes, Astrophysical Journal Supplements Series, 129, 431–434.

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