Extremal eigenvalues of sample covariance matrices with general population

Abstract

We consider the eigenvalues of sample covariance matrices of the form $\mathcal{Q}=(\Sigma^{1/2}X)(\Sigma^{1/2}X)^*$. The sample $X$ is an $M\times N$ rectangular random matrix with real independent entries and the population covariance matrix $\Sigma$ is a positive definite diagonal matrix independent of $X$. Assuming that the limiting spectral density of $\Sigma$ exhibits convex decay at the right edge of the spectrum, in the limit $M, N \to \infty$ with $N/M \to d\in(0,\infty)$, we find a certain threshold $d_+$ such that for $d>d_+$ the limiting spectral distribution of $\mathcal{Q}$ also exhibits convex decay at the right edge of the spectrum. In this case, the largest eigenvalues of $\mathcal{Q}$ are determined by the order statistics of the eigenvalues of $\Sigma$, and in particular, the limiting distribution of the largest eigenvalue of $\mathcal{Q}$ is given by a Weibull distribution. In case $d<d_+$, we also prove that the limiting distribution of the largest eigenvalue of $\caQ$ is Gaussian if the entries of $\Sigma$ are i.i.d. random variables. While $\Sigma$ is considered to be random mostly, the results also hold for deterministic $\Sigma$ with some additional assumptions.