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 have components against a stationary distribution sparse matrix, and also select the support of non-zero entries under alternative hypothesis. Our model assumes values occur in recent past (time-lag less than p/2). build procedures combine sum scan-type procedure, but are computationally fast, show their non-asymptotic behaviour both one-sided (only positive correlations) two-sided alternatives, respectively. exhibit selector significant lags bound Hamming-loss risk estimated support. These results can be extended case nearly structure sub-Gaussian vectors. Numerical illustrate excellent selectors — larger dimension p, faster rates.