Marinucci D., Peccati G. Random Fields on the Sphere: Representation, Limit Theorems and Cosmological Applications

Cambridge, UK: Cambridge University Press, 2011. — 354 p. — (London Mathematical Society Lecture Note Series 389). — ISBN: 0521175615.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 analyze 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 waveletsBackground Results in Representation Theory
Representations of SO(3) and Harmonic Analysis on S²
Background Results in Probability and Graphical Methods
Spectral Representations
Characterizations of Isotropy
Limit Theorems for Gaussian Subordinated Random Fields
Asymptotics for the Sample Power Spectrum
Asymptotics for Sample Bispectra
Spherical Needlets and their Asymptotic Properties
Needlets Estimation of Power Spectrum and Bispectrum
Spin Random Fields