Distances, Gegenbauer expansions, curls, and dimples: On dependence measures for random fields

Abstract

The aim of this thesis is to analyze several aspects of dependence structures for stochastic processes. To this end, new dependence measures for spatial stochastic processes will be introduced. Further, different analytical properties of correlation functions and their relation to properties of the realizations of the corresponding stochastic processes will be studied. The newly introduced dependence measures are based on the distance correlation. Thus, they are defined for larger classes of processes than the well-known Pearson correlation function. In addition, the new dependence measures allow to quantify non-linear dependencies, as well as dependencies in multivariate stochastic processes. On the one hand, the investigation of analytical properties of correlation functions considers isotropic positive definite functions on spheres of different dimensions. These functions are characterized by Gegenbauer expansions involving so-called Schoenberg coefficients. Relationships between those Schoenberg coefficients will be given. On the other hand, characterizations of correlation functions of random vector fields with almost surely divergence-free and irrotational realizations will be considered. Further, the dimple property of spatio-temporal random fields will be analyzed. This dimple corresponds to a non-monotonic temporal behaviour of the correlation function, which can be generated by so-called transport fields. These transport fields and their relation to the dimple property will be investigated.