Convergence rate to the Tracy–Widom laws for the largest eigenvalue of sample covariance matrices

Abstract

We establish a quantitative version of the Tracy–Widom law for the largest eigenvalue of high-dimensional sample covariance matrices. To be precise, we show that the fluctuations of the largest eigenvalue of a sample covariance matrix ${\mathit{X}}^{\ast }\mathit{X}$ converge to its Tracy–Widom limit at a rate nearly ${\mathit{N}}^{-1/3}$, where X is an $\mathit{M}\times \mathit{N}$ random matrix whose entries are independent real or complex random variables, assuming that both M and N tend to infinity at a constant rate. This result improves the previous estimate ${\mathit{N}}^{-2/9}$ obtained by Wang (2019). Our proof relies on a Green function comparison method (Adv. Math. 229 (2012) 1435–1515) using iterative cumulant expansions, the local laws for the Green function and asymptotic properties of the correlation kernel of the white Wishart ensemble.