Gneiting Class, Semi-Metric Spaces and Isometric Embeddings

Abstract

This paper revisits the Gneiting class of positive definite kernels originally proposed as a class of covariance functions for space-time processes. Under the framework of quasi-metric spaces and isometric embeddings,the paper proposes a general and unifying framework that encompasses results provided by earlier literature. Ourresults allow to study the positive definiteness of the Gneiting class over products of either Euclidean spaces or highdimensional spheres and quasi-metric spaces. In turn, Gneiting’s theorem is proved here by a direct construction,eluding Fourier inversion (the so-called Gneiting’s lemma) and convergence arguments that are required by Gneitingto preserve an integrability assumption.