Extremal eigenvalues of sample covariance matrices with general population

TL;DR

This paper analyzes the extremal eigenvalues of sample covariance matrices with general population covariance, revealing phase transitions and distribution types depending on the aspect ratio and spectral decay.

Contribution

It establishes a threshold for the aspect ratio determining whether the largest eigenvalue follows a Weibull or Gaussian distribution, extending understanding of eigenvalue behavior in high-dimensional covariance matrices.

Findings

Largest eigenvalues follow Weibull distribution when spectral decay is convex and aspect ratio exceeds threshold.

For smaller aspect ratios, the largest eigenvalue is Gaussian distributed under certain conditions.

Results apply to both random and deterministic population covariance matrices.

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.