• Bernoulli
  • Volume 16, Number 3 (2010), 798-824.

Group representations and high-resolution central limit theorems for subordinated spherical random fields

Domenico Marinucci and Giovanni Peccati

Full-text: Open access


We study the weak convergence (in the high-frequency limit) of the frequency components associated with Gaussian-subordinated, spherical and isotropic random fields. In particular, we provide conditions for asymptotic Gaussianity and establish a new connection with random walks on the hypergroup $\widehat{\mathit{SO}(3)}$ (the dual of the group of rotations $SO(3)$), which mirrors analogous results previously established for fields defined on Abelian groups (see Marinucci and Peccati [Stochastic Process. Appl. 118 (2008) 585–613]). Our work is motivated by applications to cosmological data analysis.

Article information

Bernoulli, Volume 16, Number 3 (2010), 798-824.

First available in Project Euclid: 6 August 2010

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Clebsch–Gordan coefficients cosmic microwave background Gaussian subordination group representations high resolution asymptotics spectral representation spherical random fields


Marinucci, Domenico; Peccati, Giovanni. Group representations and high-resolution central limit theorems for subordinated spherical random fields. Bernoulli 16 (2010), no. 3, 798--824. doi:10.3150/09-BEJ230.

Export citation


  • [1] Adler, R.J. and Taylor, J. (2007)., Random Fields and Geometry. New York: Springer.
  • [2] Baldi, P. and Marinucci, D. (2007). Some characterizations of the spherical harmonics coefficients for isotropic random fields., Statist. Probab. Lett. 77 490–496.
  • [3] Baldi, P., Marinucci, D. and Varadarajan, V.S. (2007). On the characterization of isotropic random fields on homogeneous spaces of compact groups., Electron. Comm. Probab. 12 291–302.
  • [4] Bartolo, N., Komatsu, E., Matarrese, S. and Riotto, A. (2004). Non-Gaussianity from inflation: Theory and observations., Phys. Rep. 402 103–266.
  • [5] Bloom, W.R. and Heyer, H. (1995)., The Harmonic Analysis of Probability Measures on Hypergroups. Berlin: de Gruyter.
  • [6] Cabella, P. and Marinucci, D. (2008). Statistical challenges in the analysis of Cosmic Microwave Background radiation., Ann. Appl. Statist. 3 61–95.
  • [7] Diaconis, P. (1988)., Group Representations in Probability and Statistics. IMS Lecture Notes – Monograph Series 11. Hayward, CA: IMS.
  • [8] Dodelson, S. (2003)., Modern Cosmology. Academic Press.
  • [9] Efstathiou, G. (2004). Myths and truths concerning estimation of power spectra: The case for a hybrid estimator., Monthly Notices of the Royal Astronomical Society 349 603–626.
  • [10] Faraut, J. (2006)., Analyse sur le groupes de Lie. Calvage et Mounet.
  • [11] Guivarc’h, Y., Keane, M. and Roynette, B. (1977)., Marches Aléatoires sur les Groupes de Lie. Lecture Notes in Math. 624. Berlin: Springer.
  • [12] Hu, Y. and Nualart, D. (2005). Renormalized self-intersection local time for fractional Brownian motion., Ann. Probab. 33 948–983.
  • [13] Jansson, S. (1997)., Gaussian Hilbert Spaces. Cambridge Univ. Press.
  • [14] Leonenko, N. (1999)., Limit Theorems for Random Fields with Singular Spectrum. Dordrecht: Kluwer.
  • [15] Liboff, R.L. (1999)., Introductory Quantum Mechanics. Addison-Wesley.
  • [16] Marinucci, D. (2006). High-resolution asymptotics for the angular bispectrum of spherical random fields., Ann. Statist. 34 1–41.
  • [17] Marinucci, D. (2008). A central limit theorem and higher order results for the angular bispectrum., Probab. Theory Related Fields 141 389–409.
  • [18] Marinucci, D. and Piccioni, M. (2004). The empirical process on Gaussian spherical harmonics., Ann. Statist. 32 1261–1288.
  • [19] Marinucci, D. and Peccati, G. (2008). High-frequency asymptotics for subordinated stationary fields on an Abelian compact group., Stochastic Process. Appl. 118 585–613.
  • [20] Marinucci, D. and Peccati, G. (2009). Group representations and angular polyspectra., J. Multivariate Anal. To appear.
  • [21] Marinucci, D. and Peccati, G. (2007). Group representations and high-resolution central limit theorems for subordinated spherical random fields. Preprint. Available at, arXiv0706.2851v3.
  • [22] Nourdin, I. and Peccati, G. (2009). Stein’s method on Wiener chaos., Probab. Theory Related Fields 145 75–118.
  • [23] Nourdin, I., Peccati, G. and Réveillac, A. (2009). Multidimensional normal approximation via Stein’s method and Malliavin calculus., Ann. Inst. H. Poincaré Probab. Statist. To appear.
  • [24] Nualart, D. (2006)., The Malliavin Calculus and Related Topics, 2nd ed. Berlin: Springer.
  • [25] Nualart, D. and Peccati, G. (2005). Central limit theorems for sequences of multiple stochastic integrals., Ann. Probab. 33 177–193.
  • [26] Peccati, G. (2007). Gaussian approximations of multiple integrals., Electron. Comm. Probab. 12 350–364.
  • [27] Peccati, G. and Pycke, J.-R. (2005). Decompositions of stochastic processes based on irreducible group representations., Theory Probab. Appl. To appear.
  • [28] Peccati, G. and Taqqu, M.S. (2008). Stable convergence of multiple Wiener–Itô integrals., J. Theoret. Probab. 21 527–570.
  • [29] Peccati, G. and Tudor, C.A. (2005). Gaussian limits for vector-valued multiple stochastic integrals. In, Séminaire de Probabilités XXXVIII 247–262. Berlin: Springer.
  • [30] Pycke, J.-R. (2007). A decomposition for invariant tests of uniformity on the sphere., Proc. Amer. Math. Soc. To appear.
  • [31] Stein, E.M. and Weiss, G. (1971)., Introduction to Fourier Analysis on Euclidean Spaces 32. Princeton, NJ: Princeton Univ. Press.
  • [32] Surgailis, D. (2003). CLTs for polynomials of linear sequences: Diagram formula with illustrations. In, Theory and Applications of Long Range Dependence 111–128. Boston, MA: Birkhäuser.
  • [33] Varadarajan, V.S. (1999)., An Introduction to Harmonic Analysis on Semisimple Lie Groups. Cambridge Univ. Press. Corrected reprint of the 1989 original.
  • [34] Varshalovich, D.A., Moskalev, A.N. and Khersonskii, V.K. (1988)., Quantum Theory of Angular Momentum. Teaneck, NJ: World Scientific Press.
  • [35] Vilenkin, N.J. and Klimyk, A.U. (1991)., Representation of Lie Groups and Special Functions. Dordrecht: Kluwer.