May 2024 Inverse covariance operators of multivariate nonstationary time series
Jonas Krampe, Suhasini Subba Rao
Author Affiliations +
Bernoulli 30(2): 1177-1196 (May 2024). DOI: 10.3150/23-BEJ1628


For multivariate stationary time series many important properties, such as partial correlation, graphical models and autoregressive representations are encoded in the inverse of its spectral density matrix. This is not true for nonstationary time series, where the pertinent information lies in the inverse infinite dimensional covariance matrix operator associated with the multivariate time series. This necessitates the study of the covariance of a multivariate nonstationary time series and its relationship to its inverse. We show that if the rows/columns of the infinite dimensional covariance matrix decay at a certain rate then the rate (up to a factor) transfers to the rows/columns of the inverse covariance matrix. This is used to obtain a nonstationary autoregressive representation of the time series and a Baxter-type bound between the parameters of the autoregressive infinite representation and the corresponding finite autoregressive projection. The aforementioned results lay the foundation for the subsequent analysis of locally stationary time series. In particular, we show that smoothness properties on the covariance matrix transfer to (i) the inverse covariance (ii) the parameters of the vector autoregressive representation and (iii) the partial covariances. All results are set up in such a way that the constants involved depend only on the eigenvalue of the covariance matrix and can be applied in the high-dimensional settings with non-diverging eigenvalues.

Funding Statement

The first author was supported by the Research Center (SFB) 884 “Political Economy of Reforms”(Project B6), funded by the German Research Foundation (DFG), and acknowledges the partial support of DFG (travel grant 493207657) and National Institute of Health (grants R01GM135926 and R21NS120227). The second author was partially supported by the National Science Foundation (grant DMS-1812128 and DMS-2210726).


The authors would like to thank two anonymous referees for their comments and suggestions, which greatly improved all aspects of the paper.


Download Citation

Jonas Krampe. Suhasini Subba Rao. "Inverse covariance operators of multivariate nonstationary time series." Bernoulli 30 (2) 1177 - 1196, May 2024.


Received: 1 February 2022; Published: May 2024
First available in Project Euclid: 31 January 2024

MathSciNet: MR4699549
Digital Object Identifier: 10.3150/23-BEJ1628

Keywords: Autoregressive parameters , Baxter’s inequality , High dimensional time series , local stationarity , partial covariance


This article is only available to subscribers.
It is not available for individual sale.

Vol.30 • No. 2 • May 2024
Back to Top