Matrix valued positive definite kernels related to the generalized Aitken's integral for Gaussians

TL;DR

This paper presents a novel method for constructing multivariate positive definite kernels on arbitrary sets, generalizing Aitken's integral for Gaussians and extending Gneiting's space-time covariance models.

Contribution

It introduces a unified approach to generate nonseparable multivariate kernels using completely monotone functions, applicable to various statistical models and multivariate interpolation.

Findings

Method produces positive definite kernels consistent with Aitken's integral.

Applicable to nonseparable kernels on Cartesian product spaces.

Encompasses many existing statistical models as special cases.

Abstract

We introduce a method to construct general multivariate positive definite kernels on a nonempty set $X$ that employs a prescribed bounded completely monotone function and special multivariate functions on $X$.\ The method is consistent with a generalized version of Aitken's integral formula for Gaussians.\ In the case where $X$ is a cartesian product, the method produces nonseparable positive definite kernels that may be useful in multivariate interpolation.\ In addition, it can be interpreted as an abstract multivariate generalization of the well-established Gneiting's model for constructing space-time covariances commonly cited in the literature.\ Many parametric models discussed in statistics can be interpreted as particular cases of the method.