Fast nonasymptotic testing and support recovery for large sparse Toeplitz covariance matrices

We consider n independent p-dimensional Gaussian vectors with covariance matrix having Toeplitz structure. The aim is two-fold: to test that these vectors have independent components against a stationary distribution with sparse Toeplitz covariance matrix, and also to select the support of non-zero entries under the alternative hypothesis. Our model assumes that the non-zero values occur in the recent past (time-lag less than p/2). We build test procedures that combine a sum and a scan-type procedure, but are computationally fast, and show their non-asymptotic behaviour in both one-sided (only positive correlations) and two-sided alternatives, respectively. We also exhibit a selector of significant lags and bound the Hamming-loss risk of the estimated support. These results can be extended to the case of nearly Toeplitz covariance structure and to sub-Gaussian vectors. Numerical results illustrate the excellent behaviour of both test procedures and support selectors - larger the dimension p, faster are the rates. (c) 2021 Elsevier Inc. All rights reserved.

Automated summary

This study examines n independent p-dimensional Gaussian vectors with covariance matrix having a Toeplitz structure, aiming to test the independence of the vectors and select the support of non-zero entries. Test procedures are developed, showing non-asymptotic behavior under one-sided and two-sided alternatives. Results also extend to nearly Toeplitz covariance structure and sub-Gaussian vectors, with numerical results indicating excellent performance of the test procedures and support selectors, particularly as dimension p increases.