Consider a sample from a multiple time series that is stationary and Gaussian. A test is presented for independence among the multivariate observations that comprise this sample. The test is a generalization of the Kolmogorov-Smirnov test for serial correlation in a single time series. In the test, pairs of spectral-matrix estimates are compared using the largest-root statistic. The comparisons, which are tested simultaneously, are between estimates obtained from upper and lower parts of the frequency band. Under the null hypothesis, the joint distribution of the largest roots is obtained in a form suitable for computation of significance levels.
"A Test for Serial Correlation in Multivariate Data." Ann. Statist. 5 (2) 408 - 413, March, 1977. https://doi.org/10.1214/aos/1176343808