On Valid Multivariate Generalizations of the Confluent Hypergeometric Covariance Function

Modeling of multivariate random fields through Gaussian processes calls for the construction of valid cross-covariance functions describing the dependence between any two component processes at different spatial locations. The required validity conditions often present challenges that lead to complicated restrictions on the parameter space. The purpose of this paper is to present a simplified techniques for establishing multivariate validity for the recently-introduced Confluent Hypergeometric (CH) class of covariance functions. Specifically, we use multivariate mixtures to present both simplified and comprehensive conditions for validity, based on results on conditionally negative semidefinite matrices and the Schur product theorem. In addition, we establish the spectral density of the CH covariance and use this to construct valid multivariate models as well as propose new cross-covariances. We show that our proposed approach leads to valid multivariate cross-covariance models that inherit the desired marginal properties of the CH model and outperform the multivariate Mat\'ern model in out-of-sample prediction under slowly-decaying correlation of the underlying multivariate random field. We also establish properties of multivariate CH models, including equivalence of Gaussian measures, and demonstrate their use in modeling a multivariate oceanography data set consisting of temperature, salinity and oxygen, as measured by autonomous floats in the Southern Ocean.