LIMITING SPECTRAL DISTRIBUTION OF A SYMMETRIZED AUTO-CROSS COVARIANCE MATRIX

• 成果类型: Article
• 署名作者: Jin, Baisuo; Wang, Chen; Bai, Z. D.; Nair, K. Krishnan; Harding, Matthew
• 署名单位: Chinese Academy of Sciences; University of Science & Technology of China, CAS; National University of Singapore; Northeast Normal University - China; Northeast Normal University - China; Stanford University; Stanford University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/13-AAP945
• 发表日期: 2014
• 页码: 1199-1225
• 关键词: empirical distribution; eigenvalues; THEOREMS; product; statistics; number

摘要

This paper studies the limiting spectral distribution (LSD) of a symmetrized auto-cross covariance matrix. The auto-cross covariance matrix is defined as M-tau = 1/2T Sigma(T)(j=1) (e(j) e(j+tau)*+ e(j+tau)e(j)*) where e(j) is an N dimensional vectors of independent standard complex components with properties stated in Theorem 1.1, and tau is the lag. M-0 is well studied in the literature whose LSD is the Mareenko-Pastur (MP) Law. The contribution of this paper is in determining the LSD of M-tau where tau >= 1. It should be noted that the LSD of the M-tau does not depend on tau. This study arose from the investigation of and plays an key role in the model selection of any large dimensional model with a lagged time series structure, which is central to large dimensional factor models and singular spectrum analysis.