THE TRACY-WIDOM LAW FOR THE LARGEST EIGENVALUE OF F TYPE MATRICES

• 成果类型: Article
• 署名作者: Han, Xiao; Pan, Guangming; Zhang, Bo
• 署名单位: Nanyang Technological University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/15-AOS1427
• 发表日期: 2016
• 页码: 1564-1592
• 关键词: sample covariance matrices; MULTIVARIATE-ANALYSIS; UNIVERSALITY; statistics; fluctuations; population; ensembles; limit

摘要

Let A(p) = YY*/m and B-p = XX*/n be two independent random matrices where X = (X-ij)(pxn) and Y = (Y-ij)(pxm) respectively consist of real (or complex) independent random variables with EXij = EYij = 0, E vertical bar X-ij vertical bar(2) = E vertical bar Y-ij vertical bar(2) = 1. Denote by lambda(1) the largest root of the determinantal equation det(lambda A(p) - B-p) = 0. We establish the Tracy-Widom type universality for lambda(1) under some moment conditions on X-ij and Y-ij when p/m and p/n approach positive constants as p -> infinity.