A necessary and sufficient condition for edge universality at the largest singular values of covariance matrices

Abstract

In this paper, we prove a necessary and sufficient condition for the edge universality of sample covariance matrices with general population. We consider sample covariance matrices of the form $\mathcal{Q}=TX(TX{)}^{\ast }$, where the sample $X$ is an $M_2\times N$ random matrix with $i.i.d.$ entries with mean zero and variance $N^{-1}$, and $T$ is an $M_1\times M_2$ deterministic matrix satisfying $T^{\ast }T$ is diagonal. We study the asymptotic behavior of the largest eigenvalues of $\mathcal{Q}$ when $M:=\min \{M_1,M_2\}$ and $N$ tends to infinity with ${\lim }_{N\to \infty }N/M=d\in (0,\infty )$. Under mild assumptions of $T$, we prove that the Tracy-Widom law holds for the largest eigenvalue of $\mathcal{Q}$ if and only if ${\lim }_{s\to \infty }{s}^4\mathbb{P}(|\sqrt{N}{x}_{ij}|\ge s)=0$. This condition was first proposed for Wigner matrices by Lee and Yin.

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• Spiked sample covariance matrices with possibly multiple bulk components
• Tracy-Widom limit for Kendall's tau
• On the spectral property of kernel-based sensor fusion algorithms of high dimensional data
• Deformed Fréchet law for Wigner and sample covariance matrices with tail in crossover regime