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∈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.

In this paper, we address the problem of detection, in the frequency domain, of a M-dimensional time series modeled as the output of a M × K MIMO filter driven by a K-dimensional Gaussian white noise, and disturbed by an additive M-dimensional Gaussian colored noise. We consider the study of test statistics based of the Spectral Coherence Matrix (SCM) obtained as renormalization of the smoothed periodogram matrix of the observed time series over N samples, and with smoothing span B. To that purpose, we consider the asymptotic regime in which M, B, N all converge to infinity at certain specific rates, while K remains fixed. We prove that the SCM may be approximated in operator norm by a correlated Wishart matrix, for which Random Matrix Theory (RMT) provides a precise description of the asymptotic behaviour of the eigenvalues. These results are then exploited to study the consistency of a test based on the largest eigenvalue of the SCM, and provide some numerical illustrations to evaluate the statistical performance of such a test.

This paper analyzes the detection of a M -dimensional useful signal modeled as the output of a M ×K MIMO filter driven by a K-dimensional white Gaussian noise, and corrupted by a M -dimensional Gaussian noise with mutually uncorrelated components. The study is focused on frequency domain test statistics based on the eigenvalues of an estimate of the spectral coherence matrix (SCM), obtained as a renormalization of the frequency-smoothed periodogram of the observed signal. If N denotes the sample size and B the smoothing span, it is proved that in the high-dimensional regime where M, B, N converge to infinity while K remains fixed, the SCM behaves as a certain correlated Wishart matrix. Exploiting well-known results on the behaviour of the eigenvalues of such matrices, it is deduced that the standard tests based on linear spectral statistics of the SCM fail to detect the presence of the useful signal in the high-dimensional regime. A new test based on the SCM, which is proved to be consistent, is also proposed, and its statistical performance is evaluated through numerical simulations.

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