Statistics Papers

Document Type

Journal Article

Date of this Version

2011

Publication Source

The Annals of Statistics

Volume

39

Issue

3

Start Page

1496

Last Page

1525

DOI

10.1214/11-AOS879

Abstract

Testing covariance structure is of significant interest in many areas of statistical analysis and construction of compressed sensing matrices is an important problem in signal processing. Motivated by these applications, we study in this paper the limiting laws of the coherence of an n × p random matrix in the high-dimensional setting where p can be much larger than n. Both the law of large numbers and the limiting distribution are derived. We then consider testing the bandedness of the covariance matrix of a high-dimensional Gaussian distribution which includes testing for independence as a special case. The limiting laws of the coherence of the data matrix play a critical role in the construction of the test. We also apply the asymptotic results to the construction of compressed sensing matrices.

Keywords

Chen–Stein method, coherence, compressed sensing matrix, covariance structure, law of large numbers, limiting distribution, maxima, moderate deviations, mutual incoherence property, random matrix, sample correlation matrix

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Date Posted: 27 November 2017

This document has been peer reviewed.