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This paper addresses the problem of finding parametric constraints thatensure the validity of the multivariate Mat{\'e}rn covariance for modeling thespatial correlation structure of coregionalized variables defined in anEuclidean space. To date, much attention has been given to the bivariatesetting, while the multivariate setting has been explored to a limited extentonly. The existing conditions often imply severe restrictions on the upperbounds for the collocated correlation coefficients, which makes themultivariate Mat{\'e}rn model appealing for the case of weak spatialcross-dependence only. We provide a collection of sufficient validityconditions for the multivariate Mat{\'e}rn covariance that allows for moreflexible parameterizations than those currently available, and prove that onecan attain considerably higher upper bounds for the collocated correlationcoefficients in comparison with our competitors. We conclude with anillustration on a trivariate geochemical data set and show that our enlargedparametric space yields better fitting performances.

Flexible Validity Conditions for the Multivariate Mat\'ern Covariance in any Spatial Dimension and for any Number of Components