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

TL;DR

This paper proves that the largest eigenvalue of high-dimensional sample covariance matrices converges to the Tracy--Widom law at a rate nearly N^{-1/3}, improving previous estimates and using advanced probabilistic techniques.

Contribution

It provides a nearly optimal convergence rate to the Tracy--Widom law for sample covariance matrices, enhancing previous bounds with a new proof technique.

Findings

Convergence rate to Tracy--Widom law is nearly N^{-1/3}.

Improves previous estimate of N^{-2/9}.

Uses Green function comparison and cumulant expansions.

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 $X^*X$ converge to its Tracy--Widom limit at a rate nearly $N^{-1/3}$, where $X$ is an $M \times N$ 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 $N^{-2/9}$ obtained by Wang [73]. Our proof relies on a Green function comparison method [27] using iterative cumulant expansions, the local laws for the Green function and asymptotic properties of the correlation kernel of the white Wishart ensemble.