# zbMATH — the first resource for mathematics

High-frequency asymptotics for subordinated stationary fields on an abelian compact group. (English) Zbl 1143.60007
Let $$G$$ be a connected compact second countable abelian group, $$\widehat G$$ be its character group, $$T=\{T(g):g\in G\}$$ be a real valued stationary Gaussian field. Put $$F[T](g)=F(T(g)), \;g\in G$$, where $$F\in L^2(\mathbb{R}, \exp({-x^2}/2)dx)$$, and $$\widetilde a_\chi(F)=\int_G F[T](g)\chi(g^{-1})dg, \;\chi\in \widehat G$$. The aim of the article is to establish sufficient (and in many cases, also necessary) conditions when the following CLT holds:
$\mathbb{E}[| \tilde a_\chi(F)| ^2]^{1/2}\tilde a_\chi(F)\overset{law}{\longrightarrow} N+iN',$ where $$N$$ and $$N'$$ are two independent centered Gaussian random variables with common variance equal to $$1/2$$. The convergence means that an infinite sequence elements $$\{\chi_l\}$$ of $$\widehat G$$ is taken and the limit is taken as $$l\rightarrow +\infty$$. The proofs of the main results involved recently established criteria for the weak convergence of multiple Winer–Itô integrals.

##### MSC:
 60B15 Probability measures on groups or semigroups, Fourier transforms, factorization 60F05 Central limit and other weak theorems 60G60 Random fields
Full Text:
##### References:
 [1] P. Baldi, G. Kerkyacharian, D. Marinucci, D. Picard, High-frequency asymptotics for wavelet-based tests for Gaussianity and isotropy on the torus, Journal of Multivariate Analysis (2006) (in press) · Zbl 1333.62223 [2] 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 [3] Bartolo, N.; Komatsu, E.; Matarrese, S.; Riotto, A., Non-gaussianity from inflation: theory and observations, Physical reports, 402, 103-266, (2004) [4] Bartolo, N.; Matarrese, S.; Riotto, A., Non-gaussianity in the curvaton scenario, Physical review D, 69, (2004), id. 043503 [5] Blanc-Lapierre, A.; Fortet, R., Theory of random functions. vol. I and II, (1968), Gordon and Breach Science Publishers New York, London, Paris · Zbl 0185.44502 [6] Breuer, P.; Major, P., Central limit theorem for non-linear functionals of Gaussian fields, Journal of multivariate analysis, 13, 425-441, (1983) · Zbl 0518.60023 [7] 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 $$f_{\operatorname{NL}}$$ with WMAP data, Monthly notices of the royal astronomical society, 369, 819-824, (2006) [8] Chiang, L.-Y., Spawning and merging of Fourier modes and phase coupling in the cosmological density bispectrum, Monthly notices of the royal astronomical society, 350, 1310-1321, (2004) [9] Coles, P.; Dineen, P.; Earl, J.; Wright, D., Phase correlations in cosmic microwave background temperature maps, Monthly notices of the royal astronomical society, 350, 989-1004, (2004) [10] P. Diaconis, Group Representations in Probability and Statistics, in: IMS Lecture Notes — Monograph Series, vol. 11, Hayward, CA, 1988 [11] Dodelson, S., Modern cosmology, (2003), Academic Press [12] Duistermaat, J.J.; Kolk, J.A.C., Lie groups, (1997), Springer-Verlag [13] Giraitis, L.; Surgailis, D., CLT and other limit theorems for functionals of Gaussian processes, Zeitschrift fur wahrscheinlichkeitstheorie und verwandte gebiete, 70, 191-212, (1985) · Zbl 0575.60024 [14] Guyon, X.; Leon, J., Convergence en loi des $$H$$-variations d’un processus gaussien stationnaire sur $$\mathbb{R}$$, Annales de l’institut Poincaré (PS), 25, 265-282, (1989) · Zbl 0691.60017 [15] Hall, P.; Heyde, C., Martingale limit theory and its applications, (1980), Academic Press · Zbl 0462.60045 [16] Hu, Y.; Nualart, D., Renormalized self-intersection local time for fractional Brownian motion, The annals of probability, 33, 3, 948-983, (2005) · Zbl 1093.60017 [17] Janson, S., Gaussian Hilbert spaces, (1997), Cambridge University Press · Zbl 0887.60009 [18] Kogo, N.; Komatsu, E., Angular trispectrum of CMB temperature anisotropy from primordial non-gaussianity with the full radiation transfer function, Physical review D, 73, (2006), id. 083007 [19] Maldacena, J., Non-Gaussian features of primordial fluctuations in single field inflationary models, Journal of high energy physics, 5, 0-13, (2003) [20] Marinucci, D., High resolution asymptotics for the angular bispectrum of spherical random fields, The annals of statistics, 34, 1-41, (2006) · Zbl 1104.60020 [21] D. Marinucci, G. Peccati, Group representations and high-resolution central limit theorems for subordinated spherical random fields. math.PR/0706.2851v2, 2007. Preprint · Zbl 1284.60099 [22] Muciaccia, P.F.; Natoli, P.; Vittorio, N., Fast spherical harmonic analysis. A quick algorithm for generating and/or inverting full-sky, high-resolution CMB anisotropy maps, The astrophysical journal, 488, 2, L63-L66, (1997) [23] Niu, X.F.; Tiao, G.C., Modelling satellitar ozone data, Journal of the American statistical association, 90, 869-893, (1995) [24] Nualart, D.; Peccati, G., Central limit theorems for sequences of multiple stochastic integrals, The annals of probability, 33, 177-193, (2005) · Zbl 1097.60007 [25] Peccati, G., On the convergence of multiple random integrals, Studia scientiarum mathematicarum hungarica, 37, 429-470, (2001) · Zbl 0997.60032 [26] G. Peccati, J.-R. Pycke, Decompositions of stochastic processes based on irreducible group representations. math.PR/0509569, 2005. Preprint · Zbl 1229.60039 [27] Peccati, G.; Taqqu, M.S., Stable convergence of $$L^2$$ generalized stochastic integrals and the principle of conditioning, The electronic journal of probability, 447-480, (2007) · Zbl 1139.60024 [28] Peccati, G.; Tudor, C.A., Gaussian limits for vector-valued multiple stochastic integrals, (), 247-262 · Zbl 1063.60027 [29] Peccati, G.; Yor, M., Identities in law between quadratic functionals of bivariate Gaussian processes, through Fubini theorems and symmetric projections, (), 235-250 · Zbl 1116.60013 [30] Peebles, P.J.E., Principles of physical cosmology, (1993), Princeton University Press [31] Pycke, J.-R., A decomposition for invariant tests of uniformity on the sphere, Proceedings of the American mathematical society, 135, 9, 2983-2993, (2007) · Zbl 1112.62051 [32] Rudin, W., Fourier analysis on groups, (1962), Wiley · Zbl 0107.09603 [33] Shorack, G.R.; Wellner, J.A., Empirical processes with applications to statistics, (1986), Wiley New York · Zbl 1170.62365 [34] Surgailis, D., CLTs for polynomials of linear sequences: diagram formula with illustrations, (), 111-128 · Zbl 1032.60017 [35] Taqqu, M.S., Weak convergence to fractional Brownian motion and to the Rosenblatt process, Zeitschrift fur wahrscheinlichkeitstheorie und verwandte gebiete, 31, 287-302, (1975) · Zbl 0303.60033 [36] Taqqu, M.S., Convergence of integrated processes of arbitrary Hermite rank, Zeitschrift fur wahrscheinlichkeitstheorie und verwandte gebiete, 50, 53-83, (1979) · Zbl 0397.60028 [37] Varshalovich, D.A.; Moskalev, A.N.; Khersonskii, V.K., Quantum theory of angular momentum, (1988), World Scientific Press [38] Wu, W.B., Fourier transforms of stationary processes, Proceedings of the American mathematical society, 133, 1, 285-293, (2005) · Zbl 1055.60016
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.