## 6 Conclusion 

This paper studies the asymptotic power of the signal detection tests in complex- valued Gaussian data as both the number of observations and data dimensionality go to infinity. Contrary to the conventional wisdom that detection of signals be- comes nearly impossible when their strength, measured by the size of the covariance spikes, is below the phase transition threshold, we find that detection of such sig- nals may be possible with high probability. The detection power lies not in the different behavior of a few of the largest sample covariance eigenvalues under the null and the alternative, which is exploited by the popular signal detection tests, but in small deviations of the empirical distribution of all the eigenvalues from the Marchenko-Pastur limit. 

To derive our results, we consider the ratio of the densities of the sample co- variance eigenvalues under the null and under the alternative hypothesis. We establish a contour integral representation of this likelihood ratio, and use the Laplace approximation to derive its asymptotic limit. Our analysis of the limiting log-likelihood ratio process shows that the sub-critical region, where the sizes of the covariance spikes are below the phase transition threshold, is the region of mutual contiguity of the joint densities of the sample covariance eigenvalues under the null and the alternative. We use the derived limiting log-likelihood process along with Le Cam’s third lemma and the Neyman-Pearson lemma to obtain the asymptotic power envelope for the signal detection tests. Preliminary analysis indicates that the asymptotic power of the likelihood ratio test based on the sample covariance eigenvalues is close to the asymptotic power envelope. 

Our technical analysis is based on what we believe to be a novel representation of the Harish-Chandra/Itzykson-Zuber integral with one of the p × p matrices being of reduced rank r in the form of an r × r matrix of contour integrals. We obtain 

