Random Fields on the Sphere : Representation, Limit Theorems and Cosmological Applications - Domenico Marinucci

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Random Fields on the Sphere

Representation, Limit Theorems and Cosmological Applications

By: Domenico Marinucci, Giovanni Peccati

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Paperback | 25 August 2011 | Edition Number 1

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354 Pages

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22.86 x 15.24 x 2.01

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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.

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"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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Preface	p. xi
Introduction	p. 1
Overview	p. 1
Cosmological motivations	p. 2
Mathematical framework	p. 5
Plan of the book	p. 7
Background Results in Representation Theory	p. 12
Introduction	p. 12
Preliminary remarks	p. 13
Groups: basic definitions	p. 15
Representations of compact groups	p. 19
The Peter-Weyl Theorem	p. 39
Representations of SO(3) and Harmonic Analysis on S²	p. 45
Introduction	p. 45
Euler angles	p. 46
Wigner's D matrices	p. 51
Spherical harmonics and Fourier analysis on S²	p. 63
The Clebsch-Gordan coefficients	p. 77
Background Results in Probability and Graphical Methods	p. 85
Introduction	p. 85
Brownian motion and stochastic calculus	p. 86
Moments, cumulants and diagram formulae	p. 94
The simplified method of moments on Wiener chaos	p. 99
The graphical method for Wigner coefficients	p. 104
Spectral Representations	p. 114
Introduction	p. 114
The Stochastic Peter-Weyl Theorem	p. 114
Weakly stationary random fields in R^m	p. 126
Stanonarity and weak isotropy in R³	p. 130
Characterizations of Isotropy	p. 134
Introduction	p. 134
First example: the cyclic group	p. 135
The spherical harmonics coefficients	p. 137
Group representations and polyspectra	p. 147
Angular polyspectra and the structure of _{l₁…l_n}	p. 151
Reduced polyspectra of arbitrary orders	p. 156
Some examples	p. 160
Limit Theorems for Gaussian Subordinated Random Fields	p. 169
Introduction	p. 169
First example: the circle	p. 171
Preliminaries on Gaussian-subordinated fields	p. 173
High-frequency CLTs	p. 176
Convolutions and random walks	p. 181
Further remarks	p. 188
Application: algebraic/exponential dualities	p. 190
Asymptotics for the Sample Power Spectrum	p. 194
Introduction	p. 194
Angular power spectrum estimation	p. 196
Interlude: some practical issues	p. 201
Asymptotics in the non-Gaussian case	p. 205
The quadratic case	p. 210
Discussion	p. 219
Asymptotics for Sample Bispectra	p. 223
Introduction	p. 223
Sample bispectra	p. 224
A central limit theorem	p. 230
Limit theorems under random normalizations	p. 234
Testing for non-Gaussianity	p. 238
Spherical Needlets and their Asymptotic Properties	p. 245
Introduction	p. 245
The construction of spherical needlets	p. 248
Properties of spherical needlets	p. 252
Stochastic properties of needlet coefficients	p. 257
Missing observations	p. 259
Mexican needlets	p. 261
Needlets Estimation of Power Spectrum and Bispectrum	p. 265
Introduction	p. 265
A general convergence result	p. 265
Estimation of the angular power spectrum	p. 272
A functional central limit theorem	p. 273
A central limit theorem for the needlets bispectrum	p. 275
Spin Random Fields	p. 282
Introduction	p. 282
Motivations	p. 284
Geometric background	p. 286
Spin needlets and spin random fields	p. 290
Spin needlets spectral estimator	p. 294
Detection of asymmetries	p. 302
Estimation with noise	p. 305
Appendix	p. 312
Orthogonal polynomials	p. 312
Spherical harmonics and their analytic properties	p. 317
The proof of needlets' localization.	p. 320
References	p. 326
Index	p. 338
Table of Contents provided by Ingram. All Rights Reserved.

Random Fields on the Sphere

Representation, Limit Theorems and Cosmological Applications

At a Glance

Paperback

Industry Reviews

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Defective items

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