Classes of compactly supported covariance functions for multivariate random fields

The paper combines simple general methodologies to obtain new classes of matrix-valued covariance functions that have two important properties: (i) the domains of the compact support of the several components of the matrix-valued functions can vary between components; and (ii) the overall differentiability at the origin can also vary. These models exploit a class of functions called here the Wendland-Gneiting class; their use is illustrated via both a simulation study and an application to a North American bivariate dataset of precipitation and temperature. Because for this dataset, as for others, the empirical covariances exhibit a hole effect, the turning bands operator is extended to matrix-valued covariance functions so as to obtain matrix-valued covariance models with negative covariances.