Bivariate covariance functions of Pólya type

We provide sufficient conditions of Pólya type which guarantee the positive definiteness an isotropic 2 × 2-matrix-valued function in R and R3. Several bivariate covariance models have been proposed literature, where all components matrix are same parametric family, such as Matérn model. Based on conditions, we introduce two novel this class, powered exponential (or stable) model generalized Cauchy Both allow for flexible smoothness, variance, scale, cross-correlation parameters. The smoothness parameters (0,1]. Additionally, allows distinct long range also show that univariate spherical can be to case within above class only a trivial way.