Asymptotic analysis of linear spectral statistics of the sample coherence matrix

Correlation tests of multiple Gaussian signals are typically formulated as linear spectral statistics on the eigenvalues of the sample coherence matrix. This is the case of the Generalized Likelihood Ratio Test (GLRT), which is formulated as the determinant of the sample coherence matrix, or the locally most powerful invariant test (LMPIT), which is formulated as the Frobenius norm of this matrix. In this paper, the asymptotic behavior of general linear spectral statistics is analyzed assuming that both the sample size and the observation dimension increase without bound at the same rate. More specifically, almost sure convergence of a general class of linear spectral statistics is established, and an associated central limit theorem is formulated. These asymptotic results are shown to provide an accurate statistical description of the behavior of the GLRT and the LMPIT in situations where the sample size and the observation dimension are both large but comparable in magnitude.