Quantitative universality for the largest eigenvalue of sample covariance matrices

Abstract

We prove the first explicit rate of convergence to the Tracy–Widom distribution for the fluctuation of the largest eigenvalue of sample covariance matrices that are not integrable. Our primary focus is matrices of type ${\mathit{X}}^{\ast }\mathit{X}$ and the proof follows the Erdös–Schlein–Yau dynamical method. We use a recent approach to the analysis of the Dyson Brownian motion from (J. Eur. Math. Soc. (JEMS) 24 (2022) 2823–2873) to obtain a quantitative error estimate for the local relaxation flow at the edge. Together with a quantitative version of the Green function comparison theorem, this gives the rate of convergence.

Combined with a result of Lee–Schnelli (Ann. Appl. Probab. 26 (2016) 3786–3839), some quantitative estimates also hold for more general separable sample covariance matrices ${\mathit{X}}^{\ast }\mathrm{\Sigma }\mathit{X}$ with general diagonal population Σ.