TRACY-WIDOM DISTRIBUTION FOR THE LARGEST EIGENVALUE OF REAL SAMPLE COVARIANCE MATRICES WITH GENERAL POPULATION

• 成果类型: Article
• 署名作者: Lee, Ji Oon; Schnelli, Kevin
• 署名单位: Korea Advanced Institute of Science & Technology (KAIST); Institute of Science & Technology - Austria
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/16-AAP1193
• 发表日期: 2016
• 页码: 3786-3839
• 关键词: wigner matrices; UNIVERSALITY; statistics; limit; ensembles; fluctuations; signals; edge

摘要

We consider sample covariance matrices of the form Q = (Sigma X-1/2)((EX)-X-1/2)*, where the sample X is an M x N random matrix whose entries are real independent random variables with variance 1/N and where Sigma is an M x M positive-definite deterministic matrix. We analyze the asymptotic fluctuations of the largest resealed eigenvalue of Q when both M and N tend to infinity with N/M -> d is an element of (0, infinity). For a large class of populations Sigma in the sub-critical regime, we show that the distribution of the largest rescaled eigenvalue of Q is given by the type-1 Tracy-Widom distribution under the additional assumptions that (1) either the entries of X are i.i.d. Gaussians or (2) that Sigma is diagonal and that the entries of X have a sub-exponential decay.