QUANTITATIVE UNIVERSALITY FOR THE LARGEST EIGENVALUE OF SAMPLE COVARIANCE MATRICES

• 成果类型: Article
• 署名作者: Wang, Haoyu
• 署名单位: Yale University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/22-AAP1910
• 发表日期: 2024
• 页码: 2539-2565
• 关键词: fixed-energy universality; generalized wigner; principal components; edge universality; local statistics; limit; beta

摘要

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 Erdos-Schlein-Yau dynamical method. We use a recent approach to the analysis of the Dyson Brownian motion from (J. Eur. Math. Soc. (JEMS) 24 (2022) 2823-2873) 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 (Ann. Appl. Probab. 26 (2016) 3786-3839), some quantitative estimates also hold for more general separable sample covariance matrices X*Sigma X with general diagonal population Sigma.