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 $X^{\ast }X$ and the proof follows the Erdös-Schlein-Yau dynamical method. We use a recent approach to the analysis of the Dyson Brownian motion from [5] 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 [26], some quantitative estimates also hold for more general separable sample covariance matrices $X^{\ast }\Sigma X$ with general diagonal population $\Sigma$.

References in corpus (6)

• Universality of covariance matrices
• Universality for the largest eigenvalue of sample covariance matrices with general population
• A rate of convergence result for the largest eigenvalue of complex white Wishart matrices
• Edge Universality for Deformed Wigner Matrices
• Edge universality of correlation matrices
• A necessary and sufficient condition for edge universality at the largest singular values of covariance matrices

Cited by in corpus (1)

• Convergence rate to the Tracy-Widom laws for the largest eigenvalue of Wigner matrices