In this thesis we investigate random fields taking values in Hilbert spaces, mostly based on the properties of their covariance functions, which are operator-valued in this setting. First, we generalize familiar results regarding the representation of stationary covariances in terms of positive operator valued measures starting with a functional version of Bochner’s Theorem. Then, we investigate some particular cases of radially symmetric covariances and other related classes, such as exponentially convex which can be written in terms of Laplace transforms and isotropic covariances on the sphere written as a series in terms of Gegenbauer polynomials. Our main results provide conditions for equivalence of probability distributions for these functional fields. We generalize standard results in finding criteria involving the covariance functions for stationary and spherical processes, which can find applications in the growing field of functional data analysis.
- Functional random field
- Hilbert spaces
- Equivalence of Gaussian measures
- Positive definite
- Exponentially convex
Operator Methods in Statistics and Data Analysis
Ferreira, V. (Author). 20 Dec 2024
Student thesis: Doctoral Thesis