Large random matrix approach for testing independence of a large number of Gaussian time series

The asymptotic behaviour of Linear Spectral Statistics (LSS) of the smoothed periodogram estimator of the spectral coherency matrix of a complex Gaussian high-dimensional time series (yn) n$\in$Z with independent components is studied under the asymptotic regime where both the dimension M of y and the smoothing span of the estimator grow to infinity at the same rate. It is established that the estimated spectral coherency matrix is close from the sample covariance matrix of an independent identically N C (0, I M) distributed sequence, and that its empirical eigenvalue distribution converges towards the Marcenko-Pastur distribution. This allows to conclude that each LSS has a deterministic behaviour that can be evaluated explicitely. Using concentration inequalities, it is shown that the order of magnitude of the deviation of each LSS from its deterministic approximation is of the order of M N where N is the sample size. Numerical simulations suggest that these results can be used to test whether a large number of time series are uncorrelated or not. MSC 2010 subject classifications: Primary 60B20, 62H15; secondary 62M15.