Necessary and sufficient conditions for the asymptotic distributions of coherence of ultra-high dimensional random matrices

Abstract

Let ${\mathbf{x}}_1,\dots ,{\mathbf{x}}_n$ be a random sample from a $p$-dimensional population distribution, where $p=p_n\to \infty$ and $\log p=o(n^{\beta })$ for some $0<\beta \le 1$, and let $L_n$ be the coherence of the sample correlation matrix. In this paper it is proved that $\sqrt{n/\log p}{L}_n\to 2$ in probability if and only if $E{e}^{t_0|x_{11}{|}^{\alpha }}<\infty$ 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.

Published in at http://dx.doi.org/10.1214/13-AOP837 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (2)

• The asymptotic distribution and Berry--Esseen bound of a new test for independence in high dimension with an application to stochastic optimization
• Some strong limit theorems for the largest entries of sample correlation matrices

Cited by in corpus (4)

• Are Discoveries Spurious? Distributions of Maximum Spurious Correlations and Their Applications
• Cramér type moderate deviation theorems for self-normalized processes
• Asymptotically Independent U-Statistics in High-Dimensional Testing
• Coherence of high-dimensional random matrices in a Gaussian case : application of the Chen-Stein method