Flexible Validity Conditions for the Multivariate Matérn Covariance in any Spatial Dimension and for any Number of Components

Flexible multivariate covariance models for spatial data are on demand. This paper addresses the problem of parametric constraints for positive semidefiniteness of the multivariate Mat{\'e}rn model. Much attention has been given to the bivariate case, while highly multivariate cases have been explored to a limited extent only. The existing conditions often imply severe restrictions on the upper bounds for the collocated correlation coefficients, which makes the multivariate Mat{\'e}rn model appealing for the case of weak spatial cross-dependence only. We provide a collection of validity conditions for the multivariate Mat{\'e}rn covariance model that allows for more flexible parameterizations than those currently available. We also prove that, in several cases, we can attain much higher upper bounds for the collocated correlation coefficients in comparison with our competitors. We conclude with a simple illustration on a trivariate geochemical dataset and show that our enlarged parametric space allows for better fitting performance with respect to our competitors.