Mathematical Developments on Isotropic Positive Definite Functions on Spheres

Abstract

Abstract:In this thesis, we investigate some problems related to positive definite functions on the sphere. As known, positive definite functions play an important role in representation theory, probability theory, stochastic processes, harmonic analysis and machine learning. This role is emphasized a long the thesis.First, equivalence of Gaussian measures represents a fundamental tool to establish the properties of maximum likelihood estimators as well as kriging predictions under fixed domain asymptotics. Classical results for the equivalence and orthogonality of measures associated to Gaussian fields on bounded sets of R d are available in the literature. The present work considers Gaussian fields defined over spheres of R d+1 , with covariance functions depending on the great circle distance. We provide necessary and sufficient conditions for the equivalence of two Gaussian measures with two different covariance models with associated d-Schoenberg sequences. As an example we study equivalence of Gaussian measures for some parametric families of covariance functions valid on spheres. A simulation study explores the consistency of the maximum likelihood estimator associated to the covariance parameters of some covariance models on the sphere. We face Problems 1 and 3 proposed in the essay by Gneiting (2013b) and related to the d-Schoenberg coefficients in the series expansion of members of Ψ d for a given d. Such problems have precise implications for the simulation of Gaussian fields on spheres as well as for applications to geostatistical data, with special emphasis to atmospheric sciences. We also show how to deduce the 2-Schoenberg coefficients of given parametric families of members of the class Ψ d , called respectively exponential and Askey families, for which the 1-coefficients were available but the 2-dimensional case was still elusive.Third, we propose and define a family of marked point processes in a noncompact semisimple Lie groups. We first generate Lévy processes via marked point processes by using jump-diffusion processes. We then build a family of Markov processes in a maximal compact subgroup of a given semisimple Lie group. With all my heart, thank to you all.