On strict positive definiteness of product and product–sum covariance models

Although positive definiteness is a sufficient condition for a function to be a covariance, the stronger strict positive definiteness is important for many applications, especially in spatial statistics, since it ensures that the kriging equations have a unique solution. In particular, spatial–temporal prediction has received a lot of attention, hence strictly positive definite spatial–temporal covariance models (or equivalently strictly conditionally negative definite variogram models) are needed. In this paper the necessary and sufficient condition for the product and the product–sum space–time covariance models to be strictly positive definite (or the variogram function to be strictly conditionally negative definite) is given. In addition it is shown that an example appeared in the recent literature which purports to show that product–sum covariance functions may be only semi-definite is itself invalid. Strict positive definiteness of the sum of products model is also discussed.