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

TL;DR

This paper studies the asymptotic behavior of the largest eigenvalues of large sample covariance matrices from spiked populations with diagonal covariance, showing they behave similarly to Gaussian cases.

Contribution

It provides new theoretical results on the limiting distribution of largest eigenvalues in the diagonal spiked population model for large matrices.

Findings

Largest eigenvalues follow the same fluctuations as Gaussian models.

Results hold for both complex and real sample covariance matrices.

Conditions on moments ensure the universality of the asymptotic behavior.

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.