### Abstract

We study the statistics of the largest eigenvalues of p×p sample covariance matrices Σ_{p,n}=M_{p,n}M_{p,n}^{∗} when the entries of the p×n matrix M_{p,n} are sparse and have a distribution with tail t^{−α}, α>0. On average the number of nonzero entries of M_{p,n} is of order n^{μ+1}, 0≤μ≤1. We prove that in the large n limit, the largest eigenvalues are Poissonian if α<2(1+μ^{−1}) and converge to a constant in the case α>2(1+μ^{−1}). We also extend the results of Benaych-Georges and Péché (2014) in the Hermitian case, removing restrictions on the number of nonzero entries of the matrix.

Original language | English (US) |
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Pages (from-to) | 3310-3330 |

Number of pages | 21 |

Journal | Stochastic Processes and their Applications |

Volume | 126 |

Issue number | 11 |

DOIs | |

State | Published - Nov 1 2016 |

### Keywords

- Eigenvalue distribution
- Heavy tail
- Random matrices
- Sparse

### ASJC Scopus subject areas

- Statistics and Probability
- Modeling and Simulation
- Applied Mathematics

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## Cite this

Auffinger, A., & Tang, S. (2016). Extreme eigenvalues of sparse, heavy tailed random matrices.

*Stochastic Processes and their Applications*,*126*(11), 3310-3330. https://doi.org/10.1016/j.spa.2016.04.029