## Abstract

Let Sn = 1/n X_{n}X_{n} where X_{n} = {X _{ij}} is a p × n matrix with i.i.d. complex standardized entries having finite fourth moments. Let Y_{n}(t_{1}, t _{2},σ)=√p(x_{n}(t_{1}) ^{*}(Sn +σI)^{-1}x_{n}(t_{2})-x _{n}(t_{1})*x_{n}(t_{2})m _{n}(σ)) in which σ > 0 and m_{n}(σ)= ∫dF_{yn}(x)/x+σ where F_{yn}(x) is the Marčenko-Pastur law with parameter y_{n} = p/n; which converges to a positive constant as n → ∞ and x_{n}(t_{1}) and x_{n}(t_{2}) are unit vectors in ℂp, having indices t _{1} and t_{2}, ranging in a compact subset of a finite-dimensional Euclidean space. In this paper, we prove that the sequence Y_{n}(t_{1}, t_{2}, σ) converges weakly to a (2m + 1)-dimensional Gaussian process. This result provides further evidence in support of the conjecture that the distribution of the eigenmatrix of S _{n} is asymptotically close to that of a Haar-distributed unitary matrix.

Original language | English |
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Pages (from-to) | 1994-2015 |

Number of pages | 22 |

Journal | Annals of Applied Probability |

Volume | 21 |

Issue number | 5 |

DOIs | |

Publication status | Published - Oct 2011 |

## Scopus Subject Areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

## User-Defined Keywords

- Central limit theorems
- Haar distribution
- Linear spectral statistics
- Marčenko-Pastur law
- Random matrix
- Sample covariance matrix
- Semicircular law