## Annals of Applied Probability

### Gaussian fluctuations for linear spectral statistics of large random covariance matrices

#### Abstract

Consider a $N\times n$ matrix $\Sigma_{n}=\frac{1}{\sqrt{n}}R_{n}^{1/2}X_{n}$, where $R_{n}$ is a nonnegative definite Hermitian matrix and $X_{n}$ is a random matrix with i.i.d. real or complex standardized entries. The fluctuations of the linear statistics of the eigenvalues

$\operatorname{Trace}f(\Sigma_{n}\Sigma_{n}^{*})=\sum_{i=1}^{N}f(\lambda_{i}),\qquad (\lambda_{i})\mbox{ eigenvalues of }\Sigma_{n}\Sigma_{n}^{*},$ are shown to be Gaussian, in the regime where both dimensions of matrix $\Sigma_{n}$ go to infinity at the same pace and in the case where $f$ is of class $C^{3}$, that is, has three continuous derivatives. The main improvements with respect to Bai and Silverstein’s CLT [Ann. Probab. 32 (2004) 553–605] are twofold: First, we consider general entries with finite fourth moment, but whose fourth cumulant is nonnull, that is, whose fourth moment may differ from the moment of a (real or complex) Gaussian random variable. As a consequence, extra terms proportional to

$|\mathcal{V}|^{2}=|\mathbb{E}(X_{11}^{n})^{2}|^{2}\quad\mbox{and}\quad\kappa=\mathbb{E}\vert X_{11}^{n}\vert^{4}-\vert\mathcal{V} \vert^{2}-2$ appear in the limiting variance and in the limiting bias, which not only depend on the spectrum of matrix $R_{n}$ but also on its eigenvectors. Second, we relax the analyticity assumption over $f$ by representing the linear statistics with the help of Helffer–Sjöstrand’s formula.

The CLT is expressed in terms of vanishing Lévy–Prohorov distance between the linear statistics’ distribution and a Gaussian probability distribution, the mean and the variance of which depend upon $N$ and $n$ and may not converge.

#### Article information

Source
Ann. Appl. Probab., Volume 26, Number 3 (2016), 1837-1887.

Dates
Revised: July 2015
First available in Project Euclid: 14 June 2016

https://projecteuclid.org/euclid.aoap/1465905021

Digital Object Identifier
doi:10.1214/15-AAP1135

Mathematical Reviews number (MathSciNet)
MR3513608

Zentralblatt MATH identifier
06618844

Subjects
Primary: 15A52
Secondary: 15A18: Eigenvalues, singular values, and eigenvectors 60F15: Strong theorems

#### Citation

Najim, Jamal; Yao, Jianfeng. Gaussian fluctuations for linear spectral statistics of large random covariance matrices. Ann. Appl. Probab. 26 (2016), no. 3, 1837--1887. doi:10.1214/15-AAP1135. https://projecteuclid.org/euclid.aoap/1465905021