Quantitative Universality for the Largest Eigenvalue of Sample Covariance Matrices

Haoyu Wang

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arXiv · 1912.05473

Quantitative Universality for the Largest Eigenvalue of Sample Covariance Matrices

Haoyu Wang

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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^*X $ and the proof follows the Erd\"{o}s-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^* \Sigma X $ with general diagonal population $ \Sigma $.

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Haoyu Wang. 2019-12-11. Quantitative Universality for the Largest Eigenvalue of Sample Covariance Matrices. https://arxiv.org/abs/1912.05473

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