POINT PROCESS CONVERGENCE FOR THE OFF-DIAGONAL ENTRIES OF SAMPLE COVARIANCE MATRICES

• 成果类型: Article
• 署名作者: Heiny, Johannes; Mikosch, Thomas; Yslas, Jorge
• 署名单位: Ruhr University Bochum; University of Copenhagen
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/20-AAP1597
• 发表日期: 2021
• 页码: 538-560
• 关键词: asymptotic-distribution; LARGEST EIGENVALUES; large deviations; poisson statistics; LIMIT-THEOREMS; distributions; coherence

摘要

We study point process convergence for sequences of i.i.d. random walks. The objective is to derive asymptotic theory for the extremes of these random walks. We show convergence of the maximum random walk to the Gumbel distribution under the existence of a (2 + delta)th moment. We make heavy use of precise large deviation results for sums of i.i.d. random variables. As a consequence, we derive the joint convergence of the off-diagonal entries in sample covariance and correlation matrices of a high-dimensional sample whose dimension increases with the sample size. This generalizes known results on the asymptotic Gumbel property of the largest entry.