## 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.

Original language | English (US) |
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Pages (from-to) | 1496-1525 |

Number of pages | 30 |

Journal | Annals of Statistics |

Volume | 39 |

Issue number | 3 |

DOIs | |

State | Published - Jun 2011 |

### Bibliographical note

Copyright:Copyright 2011 Elsevier B.V., All rights reserved.

## Keywords

- Chen
- Coherence
- Compressed sensing matrix
- Covariance structure
- Law of large numbers
- Limiting distribution
- Maxima
- Moderate deviations
- Mutual incoherence property
- Random matrix
- Sample correlation matrix
- Stein method