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June 2016 Gaussian fluctuations for linear spectral statistics of large random covariance matrices
Jamal Najim, Jianfeng Yao
Ann. Appl. Probab. 26(3): 1837-1887 (June 2016). DOI: 10.1214/15-AAP1135

## 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.

## Citation

Jamal Najim. Jianfeng Yao. "Gaussian fluctuations for linear spectral statistics of large random covariance matrices." Ann. Appl. Probab. 26 (3) 1837 - 1887, June 2016. https://doi.org/10.1214/15-AAP1135

## Information

Received: 1 October 2014; Revised: 1 July 2015; Published: June 2016
First available in Project Euclid: 14 June 2016

zbMATH: 06618844
MathSciNet: MR3513608
Digital Object Identifier: 10.1214/15-AAP1135

Subjects:
Primary: 15A52
Secondary: 15A18‎ , 60F15

Keywords: central limit theorem , Large random matrix , linear statistics of the eigenvalues  