Extremal eigenvalues of sample covariance matrices with general population

Abstract

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