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March 2014 Necessary and sufficient conditions for the asymptotic distributions of coherence of ultra-high dimensional random matrices
Qi-Man Shao, Wen-Xin Zhou
Ann. Probab. 42(2): 623-648 (March 2014). DOI: 10.1214/13-AOP837

Abstract

Let $\mathbf{x} _{1},\ldots,\mathbf{x}_{n}$ be a random sample from a $p$-dimensional population distribution, where $p=p_{n}\to\infty$ and $\log p=o(n^{\beta})$ for some $0<\beta\leq1$, and let $L_{n}$ be the coherence of the sample correlation matrix. In this paper it is proved that $\sqrt{n/\log p}L_{n}\to2$ in probability if and only if $Ee^{t_{0}|x_{11}|^{\alpha}}<\infty$ 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.

Citation

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Qi-Man Shao. Wen-Xin Zhou. "Necessary and sufficient conditions for the asymptotic distributions of coherence of ultra-high dimensional random matrices." Ann. Probab. 42 (2) 623 - 648, March 2014. https://doi.org/10.1214/13-AOP837

Information

Published: March 2014
First available in Project Euclid: 24 February 2014

zbMATH: 1354.60020
MathSciNet: MR3178469
Digital Object Identifier: 10.1214/13-AOP837

Subjects:
Primary: 60F05, 62E20

Rights: Copyright © 2014 Institute of Mathematical Statistics

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Vol.42 • No. 2 • March 2014
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