The Schoenberg kernel and more flexible multivariate covariance models in Euclidean spaces

The J-Bessel univariate kernel Ω _d introduced by Schoenberg plays a central role in the characterization of stationary isotropic covariance models defined in a d-dimensional Euclidean space. In the multivariate setting, a matrix-valued isotropic covariance is a scale mixture of the kernel Ω _d against a matrix-valued measure that is nondecreasing with respect to matrix inequality. We prove that constructions based on a p-variate kernel [Ωdij]i,j=1p are feasible for different dimensions d_ij, at the expense of some parametric restrictions. We illustrate how multivariate covariance models inherit such restrictions and provide new classes of hypergeometric, Matérn, Cauchy and compactly-supported models to illustrate our findings.