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contributor.advisorMarinucci, Domenico-
contributor.authorXiaohong, Lan-
description.abstractThis 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.en
description.tableofcontents1 Motivations and Overview - 2 Preliminaries - 2.1 Introduction - 2.2 Fourier Analysis on S2 - 2.3 Connection with Group Representation Properties for SO(3) - 2.4 The Spectral Representation, Isotropy and Joint Moments - 2.5 Some Background on the Existing Literature for Wavelet Analysis on the Sphere - 2.5.1 A Quick Review on Continuous Wavelet Transforms - 2.5.2 Some Examples of Spherical Wavelets - 2.6 Second-generation Spherical Wavelets: Needlets - 2.6.1 Frames - 2.6.2 Voronoi Cells - 2.6.3 The Cubature Points and Cubature Weights - 2.6.4 Construction of Needlets - 2.7 Diagram Formula - 3 The Needlets Bispectrum - 3.1 A Central Limit Theorem for the Needlets Bispectrum - 3.1.1 Bispectrum - 3.1.2 The Needlets Bispectrum - 3.1.3 Unknown Angular Power Spectrum - 3.2 Convergence to Multiparameter Gaussian Processes - 3.3 Behaviour under non-Gaussianity. - 4 On the Dependence Structure of Wavelet Coefficients for Spherical Random Fields - 4.1 Spherical Mexican Needlets - 4.2 Stochastic Properties of Mexican Needlet Coefficients - 4.3 Correlation Across Different Frequencies - 4.4 Statistical Applications References - A The 3-j Symbols - B Construction of b() - C A proof for the bound of dPl(cos )- den
format.extent659140 bytes-
subjectspherical random fieldsen
subjecthigh frequency asymptoticsen
subjectcosmic microwave background radiationen
subject.classificationMAT/06 Probabilità e statistica matematicaen
titleNeedlet analysis of spherical random fieldsen
typeDoctoral thesisen
degree.disciplineFacoltà di Scienze Matematiche Fisiche e Naturalien
degree.grantorUniversità degli studi di Roma Tor Vergataen
date.dateofdefenseA.A. 2008/2009en
Appears in Collections:Tesi di dottorato in scienze matematiche e fisiche

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