CLT for linear spectral statistics of large-dimensional sample covariance matrices

• 成果类型: Article
• 署名作者: Bai, ZD; Silverstein, JW
• 署名单位: Northeast Normal University - China; North Carolina State University; National University of Singapore
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 2004
• 页码: 553-605
• 关键词: limit-theorem; eigenvalues

摘要

Let B-n = (1/N)T-n(1/2) XnX*T-n(n)1/2 where X-n = (X-ij) is n x N with i.i.d. complex standardized entries having finite fourth moment, and T-n(1/2) is a Hermitian square root of the nonnegative definite Hermitian matrix T-n. The limiting behavior, as n --> infinity with n/N approaching a positive constant, of functionals of the eigenvalues of B-n, where each is given equal weight, is studied. Due to the limiting behavior of the empirical spectral distribution of B,,, it is known that these linear spectral statistics converges a.s. to a nonrandom quantity. This paper shows their rate of convergence to be 1/n by proving, after proper scaling, that they form a tight sequence. Morc-over, if EX2=0 and E\X (11)\(4) = 2, or if X-11 and T-n are real and EX114=3, they are shown to have Gaussian limits.