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

This paper addresses the problem of finding parametric constraints that ensure the validity of the multivariate Mat{\é}rn covariance for modeling the spatial correlation structure of coregionalized variables defined in an Euclidean space. To date, much attention has been given to the bivariate setting, while the multivariate setting has 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{\é}rn model appealing for the case of weak spatial cross-dependence only. We provide a collection of sufficient validity conditions for the multivariate Mat{\é}rn covariance that allows for more flexible parameterizations than those currently available, and prove that one can attain considerably higher upper bounds for the collocated correlation coefficients in comparison with our competitors. We conclude with an illustration on a trivariate geochemical data set and show that our enlarged parametric space yields better fitting performances.