Multivariate Gaussian Random Fields over Generalized Product Spaces involving the Hypertorus

TL;DR

This paper characterizes and constructs multivariate Gaussian random fields over complex product spaces involving the hypertorus, extending covariance function theory beyond isotropic assumptions.

Contribution

It introduces spectral representations and new construction methods for matrix-valued covariance functions on generalized product spaces, including non-isotropic cases.

Findings

Spectral representation of covariance functions

Methods for constructing matrix-valued covariance functions

Representation theorems for non-isotropic Gaussian fields

Abstract

The paper deals with multivariate Gaussian random fields defined over generalized product spaces that involve the hypertorus. The assumption of Gaussianity implies the finite dimensional distributions to be completely specified by the covariance functions, being in this case matrix valued mappings. We start by considering the spectral representations that in turn allow for a characterization of such covariance functions. We then provide some methods for the construction of these matrix valued mappings. Finally, we consider strategies to evade radial symmetry (called isotropy in spatial statistics) and provide representation theorems for such a more general case.