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We introduce a method to construct general multivariate positive definitekernels on a nonempty set $X$ that employs a prescribed bounded completelymonotone function and special multivariate functions on $X$.\ The method isconsistent with a generalized version of Aitken's integral formula forGaussians.\ In the case where $X$ is a cartesian product, the method producesnonseparable positive definite kernels that may be useful in multivariateinterpolation.\ In addition, it can be interpreted as an abstract multivariategeneralization of the well-established Gneiting's model for constructingspace-time covariances commonly cited in the literature.\ Many parametricmodels discussed in statistics can be interpreted as particular cases of themethod.

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