Correlation tests and sample spectral coherence matrix in the high-dimensional regime

It is established that the linear spectral statistics (LSS) of the smoothed periodogram estimate of the spectral coherence matrix of a complex Gaussian high-dimensional times series (yn)n∈Z with independent components satisfy at each frequency a central limit theorem (CLT) in the asymptotic regime where the sample size N, the dimension M of the observation, and the smoothing span B both converge towards +∞ in such a way that M=O(Nα) for α<1 and M B→c, c∈(0,1). It is deduced that two recentered and renormalized versions of the LSS, one based on an average in the frequency domain and the other one based on a sum of squares also in the frequency domain, and both evaluated over a well-chosen frequency grid, also verify a central limit theorem. These two statistics are proposed to test with controlled asymptotic level the hypothesis that the components of y are independent. Numerical simulations assess the performance of the two tests.