Asymptotic regime for improperness tests of complex random vectors

Improperness testing for complex-valued vectors and signals has been considered lately due to potential applications in complex-valued time series analysisencountered in many applications from communications to oceanography. This paper provides new results for such tests in the asymptotic regime, i.e. when the vector and sample sizes grow commensurately to infinity. The studied tests are based on invariant statistics named canonical correlation coefficients. Limiting distributions for these statistics are derived, together with those of the Generalized Likelihood Ratio Test (GLRT) and Roy's test, in the Gaussian case. This characterization in the asymptotic regime allows also to identify a phase transition in Roy's test with potential application in detection of complex-valued low-rank signals corrupted by proper noise in large datasets. Simulations illustrate the accuracy of the proposed asymptotic approximations.