Strict Positive Definiteness of a Product of Covariance Functions

Abstract

Positive definiteness represents an admissibility condition for a function to be a covariance. Nevertheless, the more restricted condition of strict positive definiteness has received attention in literature, especially in spatial statistics, since it ensures that the kriging system has a unique solution. Most known covariance functions are isotropic but there are applications where isotropy is not appropriate, e.g., space-time covariance functions. One way to construct non-isotropic covariance functions is to use a product or a product-sum. In this article, it is given a necessary as well as a sufficient condition for a product of two covariance functions to be strictly positive definite. This result is extended to the well-known product-sum covariance model.

Keywords:

• Kronecker product
• Product model
• Product-sum model
• Separability
• Strict positive definiteness

Mathematics Subject Classification:

• 62M30
• 62M40
• 86A32
• 41A15
• 41A63