  We consider $n$ independent $p$-dimensional Gaussian vectors with covariance
matrix having Toeplitz structure. We test that these vectors have independent
components against a stationary distribution with sparse Toeplitz covariance
matrix, and also select the support of non-zero entries. We assume that the
non-zero values can occur in the recent past (time-lag less than $p/2$). We
build test procedures that combine a sum and a scan-type procedures, 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.
