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