Asymptotic regime for impropriety tests of complex random vectors

TL;DR

This paper analyzes the asymptotic behavior of impropriety tests for complex random vectors, deriving limiting distributions and identifying phase transitions, which enhance detection capabilities in large-scale applications.

Contribution

It provides new asymptotic results for impropriety tests, including distributions of test statistics and phase transition phenomena, advancing understanding in high-dimensional settings.

Findings

Derived limiting distributions for impropriety coefficients and test statistics.

Identified phase transition in Roy's test for low-rank subspace detection.

Simulations confirm the accuracy of asymptotic approximations.

Abstract

Impropriety testing for complex-valued vector has been considered lately due to potential applications ranging from digital communications to complex media imaging. This paper provides new results for such tests in the asymptotic regime, i.e. when the vector dimension and sample size grow commensurately to infinity. The studied tests are based on invariant statistics named impropriety 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 subspace corrupted by proper noise in large datasets. Simulations illustrate the accuracy of the proposed asymptotic approximations.