Open Access
2023 A matrix-valued Schoenberg’s problem and its applications
Pavel Ievlev, Svyatoslav Novikov
Author Affiliations +
Electron. Commun. Probab. 28: 1-12 (2023). DOI: 10.1214/23-ECP562

Abstract

In this paper we present a criterion for positive definiteness of the matrix-valued function f(t):=exp(|t|α[B++Bsign(t)]), where α(0,2] and B± are real symmetric and antisymmetric d×d matrices. We also find a criterion for positive definiteness of its multidimensional generalization f(t):=exp(Sd1|ts|α[B++Bsign(ts)]dΛ(s)) where Λ is a finite measure on the unit sphere Sd1Rd under a more restrictive assumption that B± 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 Rd-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

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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

Information

Received: 21 May 2023; Accepted: 15 October 2023; Published: 2023
First available in Project Euclid: 14 November 2023

Digital Object Identifier: 10.1214/23-ECP562

Subjects:
Primary: 42A82 , 47A56
Secondary: 60G15 , 60G60

Keywords: cross-variogram , Gaussian processes , matrix-valued positive definite kernels , multivariate fractional Brownian motion , multivariate Ornstein-Uhlenbeck process , multivariate processes , positive definite function , stationary time-series

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