Phase transition for the smallest eigenvalue of covariance matrices

Abstract

In this paper, we study the smallest non-zero eigenvalue of the sample covariance matrices $\mathcal{S}(Y)=Y{Y}^{\ast }$, where $Y=(y_{ij})$ is an $M\times N$ matrix with iid mean $0$ variance $N^{-1}$ entries. We prove a phase transition for its distribution, induced by the fatness of the tail of $y_{ij}$'s. More specifically, we assume that $y_{ij}$ is symmetrically distributed with tail probability $\mathbb{P}(|\sqrt{N}{y}_{ij}|\ge x)\sim x^{-\alpha }$ when $x\to \infty$, for some $\alpha \in (2,4)$. We show the following conclusions: (i). When $\alpha >\frac{8}{3}$, the smallest eigenvalue follows the Tracy-Widom law on scale $N^{-\frac{2}{3}}$; (ii). When $2<\alpha <\frac{8}{3}$, the smallest eigenvalue follows the Gaussian law on scale $N^{-\frac{\alpha }{4}}$; (iii). When $\alpha =\frac{8}{3}$, the distribution is given by an interpolation between Tracy-Widom and Gaussian; (iv). In case $\alpha \le \frac{10}{3}$, in addition to the left edge of the MP law, a deterministic shift of order $N^{1-\frac{\alpha }{2}}$ shall be subtracted from the smallest eigenvalue, in both the Tracy-Widom law and the Gaussian law. Overall speaking, our proof strategy is inspired by \cite{ALY} which is originally done for the bulk regime of the Lévy Wigner matrices. In addition to various technical complications arising from the bulk-to-edge extension, two ingredients are needed for our derivation: an intermediate left edge local law based on a simple but effective matrix minor argument, and a mesoscopic CLT for the linear spectral statistic with asymptotic expansion for its expectation.

Typos in equations (1.13) and (2.3) have been corrected