From Schoenberg to Pick-Nevanlinna: towards a complete picture of the variogram class

We show that a large subclass of variograms is closed under Schur products and that some desirable stability properties, like the Schur product of \emph{ad hoc} compositions, can be obtained under the proposed setting. We introduce new classes of kernels of Schoenberg-L\'{e}vy type and show some important properties of eventually constant, radially symmetric variograms. In particular, we characterize eventually constant variograms in terms of their permissibility in Euclidean spaces of arbitrary high dimension.