We investigate the asymptotic distribution of the maximum of a frequency smoothed estimate of the spectral coherence of a M-variate complex Gaussian time series with mutually independent components when the dimension M and the number of samples N both converge to infinity. If B denotes the smoothing span of the underlying smoothed periodogram estimator, a type I extreme value limiting distribution is obtained under the rate assumptions M N $\rightarrow$ 0 and M B $\rightarrow$ c $\in$ (0, +$\infty$). This result is then exploited to build a statistic with controlled asymptotic level for testing independence between the M components of the observed time series. Numerical simulations support our results.

翻译：当尺寸M和样本N数量都接近无限时,我们调查一个带相互独立的M-变异复合高斯时间序列的光谱一致性最大平滑估计值的无线分布。如果B表示基本平滑周期测算仪的平滑范围,则根据MN N$\rightrow$ 0和MB$\rightrowr$ c\in$(0,+$\infty$)的假设,获得一种限制分布的I型极端值。然后,利用这一结果建立一个有控制的无线统计,测试所观测时间序列M部分之间的独立。数字模拟支持我们的结果。