## Abstract

Consider a matrix [Formula presented] where σ>0 and X_{n}=(x_{ij} ^{n}) is a N×n random matrix with i.i.d. real or complex standardized entries and A_{n} is a N×n deterministic matrix with bounded spectral norm. The fluctuations of the linear spectral statistics of the eigenvalues: Tracef(Y_{n}Y_{n} ^{∗})=∑i=1Nf(λ_{i}),(λ_{i})eigenvalues ofY_{n}Y_{n} ^{∗},are shown to be Gaussian, in the case where f is a smooth function of class C^{3} with bounded support, and in the regime where both dimensions of matrix Y_{n} go to infinity at the same pace. The CLT is expressed in terms of vanishing Lévy–Prokhorov distance between the linear statistics’ distribution and a centered Gaussian probability distribution, the variance of which depends upon N and n and may not converge. The proof combines ideas from [2,18] and [32].

Original language | English (US) |
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Pages (from-to) | 2250-2281 |

Number of pages | 32 |

Journal | Stochastic Processes and their Applications |

Volume | 130 |

Issue number | 4 |

DOIs | |

State | Published - Apr 2020 |

## Keywords

- Central limit theorem
- Large random matrices
- Linear statistics of the eigenvalues

## ASJC Scopus subject areas

- Statistics and Probability
- Modeling and Simulation
- Applied Mathematics