Abstract
In this paper we present a criterion for positive definiteness of the matrix-valued function , where and are real symmetric and antisymmetric matrices. We also find a criterion for positive definiteness of its multidimensional generalization where Λ is a finite measure on the unit sphere under a more restrictive assumption that commute and are normal. The associated stationary Gaussian random field may be viewed as as a generalization of the univariate fractional Ornstein-Uhlenbeck process. This generalization turns out to be particularly useful for the asymptotic analysis of -valued Gaussian random fields. Another possible application of these findings may concern variogram modelling and general stationary time series analysis.
Funding Statement
Supported by SNSF Grant 200021-196888.
Acknowledgments
The authors kindly acknowledge the financial support by SNSF Grant 200021-196888. We also are in debt to the referee and the handling Editor for numerous suggestions that improved this manuscript significantly.
Citation
Pavel Ievlev. Svyatoslav Novikov. "A matrix-valued Schoenberg’s problem and its applications." Electron. Commun. Probab. 28 1 - 12, 2023. https://doi.org/10.1214/23-ECP562
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