STRONG LIMIT OF THE EXTREME EIGENVALUES OF A SYMMETRIZED AUTO-CROSS COVARIANCE MATRIX

• 成果类型: Article
• 署名作者: Wang, Chen; Jin, Baisuo; Bai, Z. D.; Nair, K. Krishnan; Harding, Matthew
• 署名单位: National University of Singapore; Chinese Academy of Sciences; University of Science & Technology of China, CAS; Northeast Normal University - China; Northeast Normal University - China; Stanford University; Duke University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/14-AAP1092
• 发表日期: 2015
• 页码: 3624-3683
• 关键词: spectral distribution; no eigenvalues; support

摘要

The auto-cross covariance matrix is defined as M-n = 1/2T Sigma(T)(j=1) (e(j)e(j+pi)* + e(j+tau)e(j)*), where e(j)'s are n-dimensional vectors of independent standard complex components with a common mean 0, variance sigma(2), and uniformly bounded 2 + eta th moments and tau is the lag. Jin et al. [Ann. AppL Probab. 24 (2014) 119912251 has proved that the LSD of M-n exists uniquely and nonrandomly, and independent of tau for all tau >= 1. And in addition they gave an analytic expression of the LSD. As a continuation of Jin et al. [Ann. AppL Probab. 24 (2014) 1199-1225], this paper proved that under the condition of uniformly bounded fourth moments, in any closed interval outside the support of the LSD, with probability 1 there will be no eigenvalues of M-n for all large n. As a consequence of the main theorem, the limits of the largest and smallest eigenvalue of M-n are also obtained.