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Random Fields on the Sphere : Representation, Limit Theorems and Cosmological Applications.

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Reviews recent developments in the analysis of isotropic spherical random fields, with a view towards applications in cosmology.

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Bibliographic Details
Online Access: Full Text (via ProQuest)
Main Author: Marinucci, Domenico
Other Authors: Peccati, Giovanni, 1975-
Format: eBook
Language:English
Published: Cambridge : Cambridge University Press, 2011.
Series:London Mathematical Society Lecture Note Series, 389.
Subjects:
Compact groups.
Cosmology > Statistical methods.
Random fields.
Spherical harmonics.
Table of Contents:
  • Cover; Title; Copyright; Contents; Dedication; Preface; 1 Introduction; 1.1 Overview; 1.2 Cosmological motivations; 1.3 Mathematical framework; 1.4 Plan of the book; 2 Background Results in Representation Theory; 2.1 Introduction; 2.2 Preliminary remarks; 2.3 Groups: basic definitions; 2.3.1 First definitions and examples; 2.3.2 Cosets and quotients; 2.3.3 Actions; 2.4 Representations of compact groups; 2.4.1 Basic definitions; 2.4.2 Group representations and Schur Lemma; 2.4.3 Direct sum and tensor product representations; 2.4.4 Orthogonality relations; 2.5 The Peter-Weyl Theorem.
  • 3 Representations of SO(3) and Harmonic Analysis on S23.1 Introduction; 3.2 Euler angles; 3.2.1 Euler angles for SU(2); 3.2.2 Euler angles for SO(3); 3.3 Wigner's D matrices; 3.3.1 A family of unitary representations of SU(2); 3.3.2 Expressions in terms of Euler angles and irreducibility; 3.3.3 Further properties; 3.3.4 The dual of SO(3); 3.4 Spherical harmonics and Fourier analysis on S2; 3.4.1 Spherical harmonics and Wigner's Dl matrices; 3.4.2 Some properties of spherical harmonics; 3.4.3 An alternative characterization of spherical harmonics; 3.5 The Clebsch-Gordan coefficients.
  • 3.5.1 Clebsch-Gordan matrices3.5.2 Integrals of multiple spherical harmonics; 3.5.3 Wigner 3 j coefficients; 4 Background Results in Probability and Graphical Methods; 4.1 Introduction; 4.2 Brownian motion and stochastic calculus; 4.3 Moments, cumulants and diagram formulae; 4.4 The simplified method of moments on Wiener chaos; 4.4.1 Real kernels; 4.4.2 Further results on complex kernels; 4.5 The graphical method for Wigner coefficients; 4.5.1 From diagrams to graphs; 4.5.2 Further notation; 4.5.3 First example: sums of squares; 4.5.4 Cliques and Wigner 6 j coefficients.
  • 4.5.5 Rule n. 1: loops are zero4.5.6 Rule n. 2: paired sums are one; 4.5.7 Rule n. 3: 2-loops can be cut, and leave a factor; 4.5.8 Rule n. 4: three-loops can be cut, and leave a clique; 5 Spectral Representations; 5.1 Introduction; 5.2 The Stochastic Peter-Weyl Theorem; 5.2.1 General statements; 5.2.2 Decompositions on the sphere; 5.3 Weakly stationary random fields in Rm; 5.4 Stationarity and weak isotropy in R3; 6 Characterizations of Isotropy; 6.1 Introduction; 6.2 First example: the cyclic group; 6.3 The spherical harmonics coefficients; 6.4 Group representations and polyspectra.
  • 6.5 Angular polyspectra and the structure of?l1 ... ln6.5.1 Spectra of strongly isotropic fields; 6.5.2 The structure of?l1 ... ln; 6.6 Reduced polyspectra of arbitrary orders; 6.7 Some examples; 7 Limit Theorems for Gaussian Subordinated Random Fields; 7.1 Introduction; 7.2 First example: the circle; 7.3 Preliminaries on Gaussian-subordinated fields; 7.4 High-frequency CLTs; 7.4.1 Hermite subordination; 7.5 Convolutions and random walks; 7.5.1 Convolutions on?SO (3); 7.5.2 The cases q = 2 and q = 3; 7.5.3 The case of a general q: results and conjectures; 7.6 Further remarks.
  • Machine generated contents note: 1. Introduction
  • 1.1. Overview
  • 1.2. Cosmological motivations
  • 1.3. Mathematical framework
  • 1.4. Plan of the book
  • 2. Background Results in Representation Theory
  • 2.1. Introduction
  • 2.2. Preliminary remarks
  • 2.3. Groups: basic definitions
  • 2.4. Representations of compact groups
  • 2.5. Peter-Weyl Theorem
  • 3. Representations of SO(3) and Harmonic Analysis on S2
  • 3.1. Introduction
  • 3.2. Euler angles
  • 3.3. Wigner's D matrices
  • 3.4. Spherical harmonics and Fourier analysis on S2
  • 3.5. Clebsch-Gordan coefficients
  • 4. Background Results in Probability and Graphical Methods
  • 4.1. Introduction
  • 4.2. Brownian motion and stochastic calculus
  • 4.3. Moments, cumulants and diagram formulae
  • 4.4. simplified method of moments on Wiener chaos
  • 4.5. graphical method for Wigner coefficients
  • 5. Spectral Representations
  • 5.1. Introduction
  • 5.2. Stochastic Peter-Weyl Theorem
  • 5.3. Weakly stationary random fields in Rm
  • 5.4. Stationarity and weak isotropy in R3
  • 6. Characterizations of Isotropy
  • 6.1. Introduction
  • 6.2. First example: the cyclic group
  • 6.3. spherical harmonics coefficients
  • 6.4. Group representations and polyspectra
  • 6.5. Angular polyspectra and the structure of δl1 ... l1
  • 6.6. Reduced polyspectra of arbitrary orders
  • 6.7. Some examples
  • 7. Limit Theorems for Gaussian Subordinated Random Fields
  • 7.1. Introduction
  • 7.2. First example: the circle
  • 7.3. Preliminaries on Gaussian-subordinated fields
  • 7.4. High-frequency CLTs
  • 7.5. Convolutions and random walks
  • 7.6. Further remarks
  • 7.7. Application: algebraic/exponential dualities
  • 8. Asymptotics for the Sample Power Spectrum
  • 8.1. Introduction
  • 8.2. Angular power spectrum estimation
  • 8.3. Interlude: some practical issues
  • 8.4. Asymptotics in the non-Gaussian case
  • 8.5. quadratic case
  • 8.6. Discussion
  • 9. Asymptotics for Sample Bispectra
  • 9.1. Introduction
  • 9.2. Sample bispectra
  • 9.3. central limit theorem
  • 9.4. Limit theorems under random normalizations
  • 9.5. Testing for non-Gaussianity
  • 10. Spherical Needlets and their Asymptotic Properties
  • 10.1. Introduction
  • 10.2. construction of spherical needlets
  • 10.3. Properties of spherical needlets
  • 10.4. Stochastic properties of needlet coefficients
  • 10.5. Missing observations
  • 10.6. Mexican needlets
  • 11. Needlets Estimation of Power Spectrum and Bispectrum
  • 11.1. Introduction
  • 11.2. general convergence result
  • 11.3. Estimation of the angular power spectrum
  • 11.4. functional central limit theorem
  • 11.5. central limit theorem for the needlets bispectrum
  • 12. Spin Random Fields
  • 12.1. Introduction
  • 12.2. Motivations
  • 12.3. Geometric background
  • 12.4. Spin needlets and spin random fields
  • 12.5. Spin needlets spectral estimator
  • 12.6. Detection of asymmetries
  • 12.7. Estimation with noise
  • 13. Appendix
  • 13.1. Orthogonal polynomials
  • 13.2. Spherical harmonics and their analytic properties
  • 13.3. proof of needlets' localization.

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