Characterization Theorems for Pseudo-Variograms

TL;DR

This paper provides a comprehensive characterization of pseudo-variograms, establishing criteria and connections to correlation functions, and extends univariate space-time covariance models to multivariate contexts.

Contribution

It introduces necessary and sufficient conditions for matrix-valued functions to be pseudo-variograms and extends Gneiting's univariate models to multivariate space-time covariance functions.

Findings

Established criteria for pseudo-variograms.

Connected pseudo-variograms with multivariate correlation functions.

Extended Gneiting's covariance model to multivariate space-time cases.

Abstract

Pseudo-variograms appear naturally in the context of multivariate Brown-Resnick processes, and are a useful tool for analysis and prediction of multivariate random fields. We give a necessary and sufficient criterion for a matrix-valued function to be a pseudo-variogram, and further provide a Schoenberg-type result connecting pseudo-variograms and multivariate correlation functions. By means of these characterizations, we provide extensions of the popular univariate space-time covariance model of Gneiting to the multivariate case.