Extending the Gneiting class for modeling spatially isotropic and temporally symmetric vector random fields

We provide new classes of nonseparable univariate and multivariate space-time covariance functions that extend the Gneiting class. In particular, we prove that the spatial generator of the Gneiting class does not need to be completely monotone and can be replaced with the radial profile of a continuous positive semidefinite function in a finite dimensional Euclidean space, and we weaken the assumptions on the temporal margins so as to include more general behaviors than that of the original Gneiting construction. Our findings allow for covariance functions that are compactly-supported in space and/or nonmonotone in time, i.e., they offer versatility to model complex features commonly observed on data indexed with space-time coordinates.