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This paper considers multivariate Gaussian fields with their associated matrix valued covariance functions. In particular, we characterize the class of stationary-isotropic matrix valued covariance functions on dd-dimensional Euclidean spaces, as being the scale mixture of the characteristic function of a dd dimensional random vector being uniformly distributed on the spherical shell of Rd, with a uniquely determined matrix valued and signed measure. This result is the analogue of celebrated Schoenberg theorem, which characterizes stationary and isotropic covariance functions associated to an univariate Gaussian fields.The elements C, being matrix valued, radially symmetric and positive definite on Rd, have a matrix valued generator φ such that C(τ)=φ(‖τ‖), ∀τ∈Rd, and where ‖⋅‖ is the Euclidean norm. This fact is the crux, together with our analogue of Schoenberg’s theorem, to show the existence of operators that, applied to the generators φ of a matrix valued mapping C being positive definite on Rd, allow to obtain generators associated to other matrix valued mappings, say C̃, being positive definite on Euclidean spaces of different dimensions.
Author(s): Alonso-Malaver CE, Porcu E, Henao RG
Publication type: Article
Publication status: Published
Journal: Brazilian Journal of Probability and Statistics
Year: 2017
Volume: 31
Issue: 1
Pages: 144-159
Online publication date: 25/01/2017
Acceptance date: 01/12/2015
ISSN (print): 0103-0752
Publisher: Brazilian Statistical Association
URL: https://doi.org/10.1214/15-BJPS306
DOI: 10.1214/15-BJPS306
Notes:
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