The largest eigenvalues of sample covariance matrices for a spiked population: diagonal case

Abstract

We consider large complex random sample covariance matrices obtained from "spiked populations", that is when the true covariance matrix is diagonal with all but finitely many eigenvalues equal to one. We investigate the limiting behavior of the largest eigenvalues when the population and the sample sizes both become large. Under some conditions on moments of the sample distribution, we prove that the asymptotic fluctuations of the largest eigenvalues are the same as for a complex Gaussian sample with the same true covariance. The real setting is also considered.

References in corpus (5)

• The largest eigenvalue of rank one deformation of large Wigner matrices
• Central limit theorems for eigenvalues in a spiked population model
• On the top eigenvalue of heavy-tailed random matrices
• Instabilities in complex mixtures with a large number of components
• The Tracy--Widom limit for the largest eigenvalues of singular complex Wishart matrices

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