1 - Introduction

Summary

Overview

The purpose of this monograph is to discuss recent developments in the analysis of isotropic spherical random fields, with a view towards applications in cosmology. We shall be concerned in particular with the interplay among three leading themes, namely:

• the connection between isotropy, representation of compact groups and spectral analysis for random fields, including the characterization of polyspectra and their statistical estimation;

• the interplay between Gaussianity, Gaussian subordination, nonlinear statistics, and recent developments in the methods of moments and diagram formulae to establish weak convergence results;

• the various facets of high-resolution asymptotics, including the high-frequency behaviour of Gaussian subordinated random fields and asymptotic statistics in the high-frequency sense.

These basic themes will be exploited in a number of different applications, some with a probabilistic flavour and others with a more statistical focus.

On the probabilistic side, we mention, for instance, a systematic study of the connections between Gaussianity, independence of Fourier coefficients, ergodicity and high-frequency asymptotics of Gaussian subordinated fields. We will also discuss at length the role of isotropy in constraining the behaviour of angular power spectra and polyspectra, thus providing a characterization of the dependence structure of random fields, as well as a sound mathematical background for establishing meaningful estimation procedures.

Among the statistical applications, we mention the estimation of angular power spectra and polyspectra, and their use to implement tests for Gaussianity, isotropy and asymmetry.