NECESSARY AND SUFFICIENT CONDITIONS FOR THE ASYMPTOTIC DISTRIBUTIONS OF COHERENCE OF ULTRA-HIGH DIMENSIONAL RANDOM MATRICES

• 成果类型: Article
• 署名作者: Shao, Qi-Man; Zhou, Wen-Xin
• 署名单位: Chinese University of Hong Kong; Hong Kong University of Science & Technology
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/13-AOP837
• 发表日期: 2014
• 页码: 623-648
• 关键词: sample correlation-matrices; large deviations; stable recovery; largest entries; REPRESENTATIONS

摘要

Let x(1),...,x(n) be a random sample from a p-dimensional population distribution, where p = p(n) -> infinity and log p = o(n(beta)) for some 0 < beta <= 1, and let L-n be the coherence of the sample correlation matrix. In this paper it is proved that root n/log pL(n) -> 2 in probability if and only if Ee(t0 vertical bar x11 vertical bar alpha) < infinity for some t(0) > 0, where alpha satisfies beta = alpha/(4 - alpha). Asymptotic distributions of L-n are also proved under the same sufficient condition. Similar results remain valid for m-coherence when the variables of the population are m dependent. The proofs are based on self-normalized moderate deviations, the Stein-Chen method and a newly developed randomized concentration inequality.