### Abstract

For a sample of n independent identically distributed p-dimensional centered random vectors with covariance matrix σ_{n} let S~n denote the usual sample covariance (centered by the mean) and S_{n} the non-centered sample covariance matrix (i.e. the matrix of second moment estimates), where p>n. In this paper, we provide the limiting spectral distribution and central limit theorem for linear spectral statistics of the Moore-Penrose inverse of S_{n} and S~n. We consider the large dimensional asymptotics when the number of variables p→∞ and the sample size n→∞ such that p/n→c∈(1, +∞). We present a Marchenko-Pastur law for both types of matrices, which shows that the limiting spectral distributions for both sample covariance matrices are the same. On the other hand, we demonstrate that the asymptotic distribution of linear spectral statistics of the Moore-Penrose inverse of S~n differs in the mean from that of S_{n}.

Original language | English |
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Pages (from-to) | 160-172 |

Number of pages | 13 |

Journal | Journal of Multivariate Analysis |

Volume | 148 |

DOIs | |

Publication status | Published - 1 Jun 2016 |

Externally published | Yes |

### Keywords

- CLT
- Large-dimensional asymptotics
- Moore-Penrose inverse
- Random matrix theory

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## Cite this

*Journal of Multivariate Analysis*,

*148*, 160-172. https://doi.org/10.1016/j.jmva.2016.03.001