Needlet analysis of spherical random fields

This thesis is a collection of essays on spherical wavelets and their applications on statistical models. In particular, its aim is to investigate a form of second generation wavelets on the sphere called needlets, and their statistical applications to the analysis of isotropic spherical random fields. These methods are strongly motivated by many applications, especially from Cosmology and Astrophysics; in particular, the analysis of so-called Cosmic Microwave Background (CMB) radiation. Chapter 1 is to introduce the physical background and motivations of this thesis. We will emphasize on the nature of CMB, and the statistical challenges in this field. The concept of needlet is introduced in Chapter 2. This chapter is also preliminary to all the remaining work we have done. It includes the introduction of continuous and discrete spherical wavelets, isotropic spherical random fields, and the diagram formulae, i.e. the basic ingredients that will be needed in the following chapters. Chapter 3 is devoted to the investigation of non-Gaussianity testing, focusing on a new statistical procedure which we label the needlets bispectrum. We also establish a central limit theorem and multivariate procedures for these statistics and investigate their power properties against non-Gaussian alternatives. In Chapter 4, we consider an extension of the needlet ideas, leading to a new class of spherical wavelets, called Mexican needlets. In particular, we investigate the extent in which such Mexican needlets enjoy the same stochastic properties as the 2standard construction. By means of these, we go on to establish some limit results for related statistics. Some auxiliary material is collected in Appendices A-C.