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We investigate the asymptotic distribution of the maximum of a frequencysmoothed estimate of the spectral coherence of a M-variate complex Gaussiantime series with mutually independent components when the dimension M and thenumber of samples N both converge to infinity. If B denotes the smoothing spanof the underlying smoothed periodogram estimator, a type I extreme valuelimiting distribution is obtained under the rate assumptions M N $\rightarrow$0 and M B $\rightarrow$ c $\in$ (0, +$\infty$). This result is then exploitedto build a statistic with controlled asymptotic level for testing independencebetween the M components of the observed time series. Numerical simulationssupport our results.

On the asymptotic distribution of the maximum sample spectral coherence of Gaussian time series in the high dimensional regime