QUANTITATIVE CLT FOR LINEAR EIGENVALUE STATISTICS OF WIGNER MATRICES

• 成果类型: Article
• 署名作者: Bao, Zhigang; He, Yukun
• 署名单位: Hong Kong University of Science & Technology; City University of Hong Kong
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/23-AAP1945
• 发表日期: 2023
• 页码: 5171-5207
• 关键词: spectral statistics

摘要

In this article, we establish a near-optimal convergence rate for the CLT of linear eigenvalue statistics of N x N Wigner matrices, in Kolmogorov- Smirnov distance. For all test functions f is an element of C5(R), we show that the conver-gence rate is either N-1/2+epsilon or N-1+epsilon, depending on the first Chebyshev coefficient of f and the third moment of the diagonal matrix entries. The condition that distinguishes these two rates is necessary and sufficient. For a general class of test functions, we further identify matching lower bounds for the convergence rates. In addition, we identify an explicit, nonuniversal con-tribution in the linear eigenvalue statistics, which is responsible for the slow rate N-1/2+epsilon for non-Gaussian ensembles. By removing this nonuniversal part, we show that the shifted linear eigenvalue statistics have the unified convergence rate N-1+epsilon for all test functions.