Strict positive definiteness of multivariate covariance functions on compact two-point homogeneous spaces

The authors provide a characterization of the continuous and isotropic multivariate covariance functions associated to a Gaussian random field with index set varying over a compact two-point homogeneous space. Sufficient conditions for the strict positive definiteness based on this characterization are presented. Under the assumption that the space is not a sphere, a necessary and sufficient condition is given for the continuous and isotropic multivariate covariance function to be strictly positive definite. Under the same assumption, an alternative necessary and sufficient condition is also provided for the strict positive definiteness of a continuous and isotropic bivariate covariance function based on the main diagonal entries in the matrix representation for the covariance function.