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Representations of SO(3) and angular polyspectra. (English) Zbl 1216.60027
Summary: We characterize the angular polyspectra, of arbitrary order, associated with isotropic fields defined on the sphere \(S^{2}=\{(x,y,z): x^{2}+ y^{2}+ z^{2}=1\}\). Our techniques rely heavily on group representation theory, and specifically on the properties of Wigner matrices and Clebsch-Gordan coefficients. The findings of the present paper constitute a basis upon which one can build formal procedures for the statistical analysis and the probabilistic modelization of the cosmic microwave background radiation, which is currently a crucial topic of investigation in cosmology. We also outline an application to random data compression and “simulation” of Clebsch-Gordan coefficients.

60G10 Stationary stochastic processes
60G35 Signal detection and filtering (aspects of stochastic processes)
20C12 Integral representations of infinite groups
20C35 Applications of group representations to physics and other areas of science
Full Text: DOI
[1] Diaconis, P., ()
[2] Bump, D., Lie groups, (2004), Springer-Verlag Berlin, Heidelberg, New York · Zbl 1053.22001
[3] Diaconis, P.; Mayer-Wolf, E.; Zeitouni, O.; Zerner, M., The poisson – dirichlet law is the unique invariant distribution for uniform split-merge transformations, The annals of probability, 32, 1B, 915-938, (2004) · Zbl 1049.60088
[4] Fulman, J., Convergence rates of random walk on irreducible representations of finite groups, Journal of theoretical probability, 21, 193-211, (2008) · Zbl 1138.60010
[5] Guivarc’h, Y.; Keane, M.; Roynette, B., ()
[6] J.-R. Pycke, A decomposition for invariant tests of uniformity on the sphere, Proceedings of the American Mathematical Society (2007) (in press) · Zbl 1112.62051
[7] Raimond, O., Flots browniens isotropes sur la sph ère, Annales de l’institut Henri Poincaré (B) probabilité s et statistiques, 35, 3, 313-354, (1999) · Zbl 0924.60028
[8] Yadrenko, M.I., Spectral theory of random fields, (1983), Optimization Software, Inc. New York · Zbl 0539.60048
[9] G. Peccati, J.-R. Pycke, Decompositions of stochastic processes based on irreducible group representations, Theory of Probability and its Applications (2005) (in press) · Zbl 1229.60039
[10] Marinucci, D.; Peccati, G., High-frequency asymptotics for subordinated stationary fields on an abelian compact group, Stochastic processes and their applications, 118, 585-613, (2008) · Zbl 1143.60007
[11] D. Marinucci, G. Peccati, Group representations and high-resolution central limit theorems for subordinated spherical random fields, 2008, Preprint. arXiv:0706.2851 · Zbl 1284.60099
[12] Varshalovich, D.A.; Moskalev, A.N.; Khersonskii, V.K., Quantum theory of angular momentum, (1988), World Scientific Press
[13] Vilenkin, N.Ja.; Klimyk, A.U., Representation of Lie groups and special functions, (1991), Kluwer Dordrecht · Zbl 0826.22001
[14] Cabella, P.; Marinucci, D., Statistical challenges in the analysis of cosmic microwave background, The annals of applied statistics, 3, 1, (2009) · Zbl 1160.62097
[15] Dodelson, S., Modern cosmology, (2003), Academic Press
[16] Hu, W., The angular trispectrum of the CMB, Physical review D, 64, (2001), id. 083005
[17] Komatsu, E.; Spergel, D.N., Acoustic signatures in the primary microwave background bispectrum, Physical review D, 63, (2001), id. 063002
[18] Marinucci, D., High-resolution asymptotics for the angular bispectrum of spherical random fields, The annals of statistics, 34, 1-41, (2006) · Zbl 1104.60020
[19] Marinucci, D., A central limit theorem and higher order results for the angular bispectrum, Probability theory and related fields, 141, 389-409, (2007) · Zbl 1141.60028
[20] Cabella, P.; Hansen, F.K.; Liguori, M.; Marinucci, D.; Matarrese, S.; Moscardini, L.; Vittorio, N., The integrated bispectrum as a test of cosmic microwave background non-gaussianity: detection power and limits on \(f_{N L}\) with WMAP data, Monthly notices of the royal astronomical society, 369, 819-824, (2006)
[21] A.P.S. Yadav, B.D. Wandelt, Detection of primordial non-Gaussianity \((f_{N L})\) in the WMAP 3-year data at above 99.5% confidence, 2007. arxiv:0712.1148
[22] A.P.S. Yadav, E. Komatsu, B.D. Wandelt, M. Liguori, F.K. Hansen, S. Matarrese, Fast estimator of primordial Non-Gaussianity from temperature and polarization anisotropies in the cosmic microwave background II: Partial sky coverage and inhomogeneous noise, 2007, Preprint. arXiv:0711.4933
[23] Babich, D.; Creminelli, P.; Zaldarriaga, M., The shape of non-gaussianities, Journal of cosmology and astroparticle physics, 8, 009, (2004)
[24] Bartolo, N.; Komatsu, E.; Matarrese, S.; Riotto, A., Non-gaussianity from inflation: theory and observations, Physical reports, 402, 103-266, (2004)
[25] Maldacena, J., Non-Gaussian features of primordial fluctuations in single field inflationary models, Journal of high energy physics, 5, 0-13, (2003)
[26] Chung, M.K.; Dalton, K.M.; Evans, A.C.; Davidson, R.J., Tensor-based cortical surface morphometry via weighted spherical harmonics representation, IEEE transactions on medical imaging, (2007)
[27] Simons, F.J.; Dahlen, F.A.; Wieczorek, M.A., Spatiospectral concentration on a sphere, SIAM review, 48, 3, 504-536, (2006) · Zbl 1117.42003
[28] Wieczorek, M.A.; Simons, F.J., Minimum-variance multitaper spectral estimation on the sphere, The journal of Fourier analysis and applications, 13, 6, 665-692, (2007) · Zbl 1257.33018
[29] Adler, R.J.; Taylor, J.E., Random fields and geometry, (2007), Springer · Zbl 1149.60003
[30] Baldi, P.; Marinucci, D., Some characterizations of the spherical harmonics coefficients for isotropic random fields, Statistics and probability letters, 77, 5, 490-496, (2007) · Zbl 1117.60053
[31] Baldi, P.; Marinucci, D.; Varadarajan, V.S., On the characterization of isotropic random fields on homogeneous spaces of compact groups, Electronic communications in probability, 12, 291-302, (2007) · Zbl 1128.60039
[32] D. Geller, D. Marinucci, Spin wavelets on the sphere, 2008, Preprint. arXiv:0811.2935 · Zbl 1206.42039
[33] Liboff, R.L., Introductory quantum mechanics, (1999), Addison-Wesley
[34] Stanley, R.P., Enumerative combinatorics, vol. I, (1997), Cambridge University Press · Zbl 0889.05001
[35] Biedenharn, L.C.; Louck, J.D., ()
[36] Surgailis, D., CLTs for polynomials of linear sequences: diagram formula with illustrations, (), 111-128 · Zbl 1032.60017
[37] Sagan, B.E., The symmetric group. representations, combinatorial algorithms and symmetric functions, (2001), Springer-Verlag · Zbl 0964.05070
[38] Sternberg, S., Group theory and physics, (1999), Cambridge University Press · Zbl 0829.53001
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