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

Year 2021, , 384 - 399, 13.12.2021

Valdir Menegatto , Claudemir Oliveira

https://doi.org/10.33205/cma.964096

Cited By: 2

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 in which $X$ 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 version of the well-established Gneiting's model for constructing space-time covariances commonly highly cited in the literature. Many parametric models discussed in statistics can be interpreted as particular cases of the method.

Keywords

positive definite kernels, conditionally negative definite functions, Aitken's integral, Schur exponential, Oppenheim's inequality, Gneiting's model

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