CONVERGENCE RATE TO THE TRACY-WIDOM LAWS FOR THE LARGEST EIGENVALUE OF SAMPLE COVARIANCE MATRICES

• 成果类型: Article
• 署名作者: Schnelli, Kevin; Xu, Yuanyuan
• 署名单位: Royal Institute of Technology; Institute of Science & Technology - Austria
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/22-AAP1826
• 发表日期: 2023
• 页码: 677-725
• 关键词: fixed-energy universality; edge universality; MULTIVARIATE-ANALYSIS; Sufficient condition; limit-theorem; statistics; fluctuations; RIGIDITY; spectrum

摘要

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 x N random matrix whose entries are independent real or complex random variables, assuming that both M and N tend to in-finity at a constant rate. This result improves the previous estimate 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 expan-sions, the local laws for the Green function and asymptotic properties of the correlation kernel of the white Wishart ensemble.