Random Fields on the Sphere : Representation, Limit Theorems and Cosmological Applications.
Reviews recent developments in the analysis of isotropic spherical random fields, with a view towards applications in cosmology.
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Full Text (via ProQuest) |
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| Format: | eBook |
| Language: | English |
| Published: |
Cambridge :
Cambridge University Press,
2011.
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| Series: | London Mathematical Society Lecture Note Series, 389.
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| Subjects: |
MARC
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| 100 | 1 | |a Marinucci, Domenico. | |
| 245 | 1 | 0 | |a Random Fields on the Sphere : |b Representation, Limit Theorems and Cosmological Applications. |
| 260 | |a Cambridge : |b Cambridge University Press, |c 2011. | ||
| 300 | |a 1 online resource (355 pages) | ||
| 336 | |a text |b txt |2 rdacontent. | ||
| 337 | |a computer |b c |2 rdamedia. | ||
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| 490 | 1 | |a London Mathematical Society Lecture Note Series, 389 ; |v v. 389. | |
| 505 | 0 | |6 880-01 |a 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. | |
| 505 | 8 | |a 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. | |
| 505 | 8 | |a 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. | |
| 505 | 8 | |a 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. | |
| 505 | 8 | |a 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. | |
| 500 | |a 7.6.1 Convolutions as mixed states. | ||
| 504 | |a Includes bibliographical references (pages 326-337) and index. | ||
| 520 | |a Reviews recent developments in the analysis of isotropic spherical random fields, with a view towards applications in cosmology. | ||
| 588 | 0 | |a Print version record. | |
| 650 | 0 | |a Compact groups. | |
| 650 | 0 | |a Cosmology |x Statistical methods. | |
| 650 | 0 | |a Random fields. | |
| 650 | 0 | |a Spherical harmonics. | |
| 650 | 7 | |a Compact groups. |2 fast |0 (OCoLC)fst00871287. | |
| 650 | 7 | |a Random fields. |2 fast |0 (OCoLC)fst01089800. | |
| 650 | 7 | |a Spherical harmonics. |2 fast |0 (OCoLC)fst01129683. | |
| 700 | 1 | |a Peccati, Giovanni, |d 1975- | |
| 776 | 0 | 8 | |i Print version: |a Marinucci, Domenico. |t Random Fields on the Sphere : Representation, Limit Theorems and Cosmological Applications. |d Cambridge : Cambridge University Press, ©2011 |z 9780521175616. |
| 830 | 0 | |a London Mathematical Society Lecture Note Series, 389. | |
| 856 | 4 | 0 | |u https://ebookcentral.proquest.com/lib/ucb/detail.action?docID=775025 |z Full Text (via ProQuest) |
| 880 | 0 | 0 | |6 505-01/(S |g Machine generated contents note: |g 1. |t Introduction -- |g 1.1. |t Overview -- |g 1.2. |t Cosmological motivations -- |g 1.3. |t Mathematical framework -- |g 1.4. |t Plan of the book -- |g 2. |t Background Results in Representation Theory -- |g 2.1. |t Introduction -- |g 2.2. |t Preliminary remarks -- |g 2.3. |t Groups: basic definitions -- |g 2.4. |t Representations of compact groups -- |g 2.5. |t Peter-Weyl Theorem -- |g 3. |t Representations of SO(3) and Harmonic Analysis on S2 -- |g 3.1. |t Introduction -- |g 3.2. |t Euler angles -- |g 3.3. |t Wigner's D matrices -- |g 3.4. |t Spherical harmonics and Fourier analysis on S2 -- |g 3.5. |t Clebsch-Gordan coefficients -- |g 4. |t Background Results in Probability and Graphical Methods -- |g 4.1. |t Introduction -- |g 4.2. |t Brownian motion and stochastic calculus -- |g 4.3. |t Moments, cumulants and diagram formulae -- |g 4.4. |t simplified method of moments on Wiener chaos -- |g 4.5. |t graphical method for Wigner coefficients -- |g 5. |t Spectral Representations -- |g 5.1. |t Introduction -- |g 5.2. |t Stochastic Peter-Weyl Theorem -- |g 5.3. |t Weakly stationary random fields in Rm -- |g 5.4. |t Stationarity and weak isotropy in R3 -- |g 6. |t Characterizations of Isotropy -- |g 6.1. |t Introduction -- |g 6.2. |t First example: the cyclic group -- |g 6.3. |t spherical harmonics coefficients -- |g 6.4. |t Group representations and polyspectra -- |g 6.5. |t Angular polyspectra and the structure of δl1 ... l1 -- |g 6.6. |t Reduced polyspectra of arbitrary orders -- |g 6.7. |t Some examples -- |g 7. |t Limit Theorems for Gaussian Subordinated Random Fields -- |g 7.1. |t Introduction -- |g 7.2. |t First example: the circle -- |g 7.3. |t Preliminaries on Gaussian-subordinated fields -- |g 7.4. |t High-frequency CLTs -- |g 7.5. |t Convolutions and random walks -- |g 7.6. |t Further remarks -- |g 7.7. |t Application: algebraic/exponential dualities -- |g 8. |t Asymptotics for the Sample Power Spectrum -- |g 8.1. |t Introduction -- |g 8.2. |t Angular power spectrum estimation -- |g 8.3. |t Interlude: some practical issues -- |g 8.4. |t Asymptotics in the non-Gaussian case -- |g 8.5. |t quadratic case -- |g 8.6. |t Discussion -- |g 9. |t Asymptotics for Sample Bispectra -- |g 9.1. |t Introduction -- |g 9.2. |t Sample bispectra -- |g 9.3. |t central limit theorem -- |g 9.4. |t Limit theorems under random normalizations -- |g 9.5. |t Testing for non-Gaussianity -- |g 10. |t Spherical Needlets and their Asymptotic Properties -- |g 10.1. |t Introduction -- |g 10.2. |t construction of spherical needlets -- |g 10.3. |t Properties of spherical needlets -- |g 10.4. |t Stochastic properties of needlet coefficients -- |g 10.5. |t Missing observations -- |g 10.6. |t Mexican needlets -- |g 11. |t Needlets Estimation of Power Spectrum and Bispectrum -- |g 11.1. |t Introduction -- |g 11.2. |t general convergence result -- |g 11.3. |t Estimation of the angular power spectrum -- |g 11.4. |t functional central limit theorem -- |g 11.5. |t central limit theorem for the needlets bispectrum -- |g 12. |t Spin Random Fields -- |g 12.1. |t Introduction -- |g 12.2. |t Motivations -- |g 12.3. |t Geometric background -- |g 12.4. |t Spin needlets and spin random fields -- |g 12.5. |t Spin needlets spectral estimator -- |g 12.6. |t Detection of asymmetries -- |g 12.7. |t Estimation with noise -- |g 13. |t Appendix -- |g 13.1. |t Orthogonal polynomials -- |g 13.2. |t Spherical harmonics and their analytic properties -- |g 13.3. |t proof of needlets' localization. |
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