Vector Random Fields with Second-Order Moments or Second-Order Increments

This article is concerned with vector (multivariate, or multidimensional) random fields with second-order moments or second-order increments. Two crucial questions for such a random field are what kind of the square matrix function can be employed as its covariance matrix or variogram matrix, and, in particular, what type of the functions can be employed as its cross covariances or cross variograms. We attempt to explore the relationships between the direct covariance and the cross covariance in a covariance matrix and the relationships between the direct variogram and the cross variogram in a variogram matrix. Necessary and sufficient conditions are obtained for a given square matrix function to be the covariance matrix or variogram matrix of a vector Gaussian or elliptically contoured random field, and some parametric or nonparametric examples are given for stationary and nonstationary cases in a temporal, spatial, or spatio-temporal domain.