Necessary and sufficient conditions for the asymptotic distributions of coherence of ultrahigh dimensional random matrices
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 $Ee^{t_0x_{11}^{\alpha}}<\infty$ 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 selfnormalized moderate deviations, the SteinChen method and a newly developed randomized concentration inequality.
 Publication:

arXiv eprints
 Pub Date:
 February 2014
 arXiv:
 arXiv:1402.6173
 Bibcode:
 2014arXiv1402.6173S
 Keywords:

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