Multivariate and multiradial Schoenberg measures with their dimension walks

The paper fixes some important properties of matrix-valued correlation functions associated to Multivariate Gaussian fields in a Euclidean space RdRd. In particular, we focus (a) on the isotropic (radially symmetric) case and (b) on anisotropy obtained through isotropy between components of the lag vector. This second case includes, as special case, space–time and fully symmetric correlation functions.Starting from the multivariate analogue of the Schoenberg integral representation of isotropic correlation functions, we characterize their associated measures, which are called here mm-Schoenberg measures. We also propose a new dimension walk for the componentwise isotropic case. Finally, we obtain examples where dimension walks for multivariate correlations are not well defined.