A matrix-valued Schoenberg's problem and its applications

Authors: Pavel Ievlev, Svyatoslav Novikov
Published in: Electronic Communications in Probability (2023)
Date:
doi

Keywords

Matrix-valued positive definite functions, Positive definiteness, Multivariate processes, Gaussian processes, Multivariate Ornstein-Uhlenbeck process, Multivariate fractional Brownian motion, Cross-variogram, Stationary time-series

In this paper we present a criterion for positive definiteness of the matrix-valued function $$f(t) := \exp(-|t|^\alpha[B^+ + B^- \text{sign}(t)]),$$ where $\alpha \in (0, 2]$ and $B^\pm$ are real symmetric and antisymmetric $d \times d$ matrices. We also find a criterion for positive definiteness of its multidimensional generalization $$f(t) := \exp(- \int_{\mathbb{S}^{d-1}} |t^\top s|^\alpha[B^+ + B^- \text{sign}(t^\top s)]d\Lambda(s))$$ where $\Lambda$ is a finite measure on the unit sphere $\mathbb{S}^{d-1} \subset \mathbb{R}^d$ under a more restrictive assumption that $B^\pm$ commute and are normal. The associated stationary Gaussian random field may be viewed as a generalization of the univariate fractional Ornstein-Uhlenbeck process. This generalization turns out to be particularly useful for the asymptotic analysis of $\mathbb{R}^d$-valued Gaussian random fields. Another possible application of these findings may concern variogram modelling and general stationary time series analysis.

BibTeX

@article {MR4684061,
    AUTHOR = {Ievlev, Pavel and Novikov, Svyatoslav},
    TITLE = {A matrix-valued {S}choenberg's problem and its applications},   
    JOURNAL = {Electron. Commun. Probab.},
    FJOURNAL = {Electronic Communications in Probability},
    VOLUME = {28},
    YEAR = {2023},
    PAGES = {Paper No. 48, 12},
    ISSN = {1083-589X},
    MRCLASS = {42A82 (47A56 60G15 60G60)},
    MRNUMBER = {4684061},
    MRREVIEWER = {Ana\ Paula\ Peron},
    DOI = {10.1214/23-ecp562},
    URL = {https://doi.org/10.1214/23-ecp562}
}