  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 work is to present
techniques using multivariate mixtures for establishing validity that are
simultaneously simplified and comprehensive. This is accomplished using results
on conditionally negative semidefinite matrices and the Schur product theorem.
For illustration, we use the recently-introduced Confluent Hypergeometric (CH)
class of covariance functions. In addition, we establish the spectral density
of the Confluent Hypergeometric covariance and use this to construct valid
multivariate models as well as propose new cross-covariances. Our approach
leads to valid multivariate cross-covariance models that inherit the desired
marginal properties of the Confluent Hypergeometric 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 the new models, including results on equivalence of Gaussian
measures. We demonstrate the new model's use for multivariate oceanography
dataset consisting of temperature, salinity and oxygen, as measured by
autonomous floats in the Southern Ocean.
