# Deviation inequalities for eigenvalues of deformed random matrices

## Abstract/Contents

- Abstract
- We consider two models of deformed random matrices. The first model is the deformed GOE. Let G be a n by n matrix in the Gaussian Orthogonal Ensemble (GOE) with off diagonal entries having variance $\frac{\sigma^2}{n}$ and let P be a n by n real symmetric matrix whose non zero eigenvalues are $\theta_1 \geq \cdots \geq \theta_r > 0$. We establish non-asymptotic deviation inequalities for the extreme eigenvalues of A = P + G of the following type: $P(\lambda_i (A) - \lambda_{\theta_i}[vertical line] \geq t) \leq C_1 n^{r-i+1} e^{-C_2 nt^2 / \sigma^2}, 1\leq i \leq r$, where $\lambda_{\theta_i} = \theta_i + \frac{\sigma^2}{\theta_i}$ or $2\sigma$, depending on $\theta_i > \sigma$ or $0 < \theta_i \leq\sigma$, with $C_1, C_2$ positive constants independent of n, r and t. The second model studied is the spiked population model, which is sometimes called Laguerre Orthogonal Ensemble (LOE). We establish similar deviation inequalities for extreme eigenvalues of matrices in this model. Unlike classical approach in the study of extreme eigenvalues of large dimensional random matrices, which relies on the moment method and the eigenvalue density formulae, our approach is based on the minimax characterization of eigenvalues and comparison inequalities for Gaussian processes.

## Description

Type of resource | text |
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Form | electronic; electronic resource; remote |

Extent | 1 online resource. |

Publication date | 2013 |

Issuance | monographic |

Language | English |

## Creators/Contributors

Associated with | Peng, Minyu | |
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Associated with | Stanford University, Department of Mathematics. | |

Primary advisor | Papanicolaou, George | |

Thesis advisor | Papanicolaou, George | |

Thesis advisor | Dembo, Amir | |

Thesis advisor | Johnstone, Iain | |

Advisor | Dembo, Amir | |

Advisor | Johnstone, Iain |

## Subjects

Genre | Theses |
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## Bibliographic information

Statement of responsibility | Minyu Peng. |
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Note | Submitted to the Department of Mathematics. |

Thesis | Thesis (Ph.D.)--Stanford University, 2013. |

Location | electronic resource |

## Access conditions

- Copyright
- © 2013 by Minyu Peng
- License
- This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).

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