A class of valid Matérn cross-covariance functions for multivariate spatio-temporal random fields

We extend a current result in the literature by introducing a parametric family of matrix-valued cross-covariance functions, which would help describe multivariate space-time dependence structures for any number of variables. All the direct and cross-covariance functions belong to the Matérn class. The smoothness, space-time separability and scale parameters are allowed to be different for each variable.