  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.
