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$\in$Z with independent components satisfy at each frequency a central limit theorem in the asymptotic regime where the sample size N , the dimension M of the observation, and the smoothing span B both converge towards +$\infty$ in such a way that M = O(N $\alpha$ ) for $\alpha$ < 1 and M B $\rightarrow$ c, c $\in$ (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.