JOINT CONVERGENCE OF SAMPLE AUTOCOVARIANCE MATRICES WHEN p/n → 0 WITH APPLICATION

• 成果类型: Article
• 署名作者: Bhattacharjee, Monika; Bose, Arup
• 署名单位: State University System of Florida; University of Florida; Indian Statistical Institute; Indian Statistical Institute Kolkata
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/18-AOS1785
• 发表日期: 2019
• 页码: 3470-3503
• 关键词: linear spectral statistics; time-series; Covariance matrices; clt

摘要

Consider a high-dimensional linear time series model where the dimen- sion p and the sample size n grow in such a way that p/n -> 0. Let (Gamma) over cap (u) be the uth order sample autocovariance matrix. We first show that the LSD of any symmetric polynomial in {(Gamma) over cap (u) , (Gamma) over cap (u)*, u >= 0} exists under independence and moment assumptions on the driving sequence together with weak assumptions on the coefficient matrices. This LSD result, with some additional effort, implies the asymptotic normality of the trace of any polynomial in {(Gamma) over cap (u) , (Gamma) over cap (u)*, u >= 0}. We also study similar results for several independent MA processes. We show applications of the above results to statistical inference problems such as in estimation of the unknown order of a high-dimensional MA process and in graphical and significance tests for hypotheses on coefficient matrices of one or several such independent processes.