From Schoenberg to Pick-Nevanlinna: Toward a complete picture of the variogram class

We show that a large subclass of variograms is closed under products and that some desirable stability properties, such as the product of special compositions, can be obtained within the proposed setting. We introduce new classes of kernels of Schoenberg-L{e}vy type and demonstrate some important properties of rotationally invariant variograms.

Space-time covariance functions with compact support

We characterize completely the Gneiting class of space-time covariance functions and give more relaxed conditions on the involved functions. We then show necessary conditions for the construction of compactly supported functions of the Gneiting type. These conditions are very general since they do not depend on the Euclidean norm. Finally, we discuss a general class of positive definite functions, used for multivariate Gaussian random fields. For this class, we show necessary criteria for its generator to be compactly supported.

The Dagum family of isotropic correlation functions

A function $rho:[0,infty)to(0,1]$ is a completely monotonic function if and only if $rho(Vertmathbf{x}Vert^2)$ is positive definite on $mathbb{R}^d$ for all $d$ and thus it represents the correlation function of a weakly stationary and isotropic Gaussian random field. Radial positive definite functions are also of importance as they represent characteristic functions of spherically symmetric probability distributions. In this paper, we analyze the function [rho(beta ,gamma)(x)=1-biggl(frac{x^{beta}}{1+x^{beta}}biggr )^{gamma},qquad xge 0, beta,gamma>0,] called the Dagum function, and show those ranges for which this function is completely monotonic, that is, positive definite, on any $d$-dimensional Euclidean space. Important relations arise with other families of completely monotonic and logarithmically completely monotonic functions.