On Testing for Impropriety of Complex-Valued Gaussian Vectors

We consider the problem of testing whether a complex-valued random vector is proper, i.e., is uncorrelated with its complex conjugate. We formulate the testing problem in terms of real-valued Gaussian random vectors, so we can make use of some useful existing results which enable us to study the null distributions of two test statistics. The tests depend only on the sample-size $n $ and the dimensionality of the vector $p$ . The basic behaviors of the distributions of the test statistics are derived and critical values (thresholds) are calculated and presented for certain $(n,p)$ values. For one of these tests we derive a distributional approximation for a transform of the statistic, potentially very useful in practice for rapid and simple testing. We also study the power (detection probability) of the tests. Our results mean that testing for propriety can be a practical and undaunting procedure.