## Bernoulli

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

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

#### Abstract

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

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

Dates
First available in Project Euclid: 6 August 2010

https://projecteuclid.org/euclid.bj/1281099885

Digital Object Identifier
doi:10.3150/09-BEJ230

Mathematical Reviews number (MathSciNet)
MR2730649

Zentralblatt MATH identifier
1284.60099

#### Citation

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. https://projecteuclid.org/euclid.bj/1281099885

#### References

• [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.