A test for independence of two sets of variables when the number of variables is large relative to the sample size

A simple statistic is proposed for testing the independence of two subvectors of a random vector having a multivariate normal distribution. The asymptotic null distribution of this statistic, as both the sample size and the number of variables in the random vector go to infinity, is shown to be normal. Some simulation results are obtained so as to assess the adequacy of the normal approximation and to compare the performance of this new test to that of the likelihood ratio test.