A NECESSARY AND SUFFICIENT CONDITION FOR EDGE UNIVERSALITY AT THE LARGEST SINGULAR VALUES OF COVARIANCE MATRICES

• 成果类型: Article
• 署名作者: Ding, Xiucai; Yang, Fan
• 署名单位: University of Toronto; University of Wisconsin System; University of Wisconsin Madison
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/17-AAP1341
• 发表日期: 2018
• 页码: 1679-1738
• 关键词: limiting spectral distribution; eigenvalue based detection; tracy-widom limit; statistics; signals; tests

摘要

In this paper, we prove a necessary and sufficient condition for the edge universality of sample covariance matrices with general population. We consider sample covariance matrices of the form Q = TX(TX)*, where X is an M-2 x N random matrix with X-ij = N-1/2 qij such that qij are i.i.d. random variables with zero mean and unit variance, and T is an M-1 x M-2 deterministic matrix such that T*T is diagonal. We study the asymptotic behavior of the largest eigenvalues of Q when M := min{M-1 x M-2} and N tend to infinity with lim(N ->infinity) N/M = d is an element of (0, infinity). We prove that the Tracy-Widom law holds for the largest eigenvalue of Q if and only if lim(s ->infinity) s(4)P(vertical bar qij vertical bar >= s) = 0 under mild assumptions of T. The necessity and sufficiency of this condition for the edge universality was first proved for Wigner matrices by Lee and Yin [Duke Math. J. 163 (2014) 117-173].