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 \[\operatorname {Trace}f \bigl(Σ_nΣ_n^*\bigr)=\sum_{i=1}^Nf(λ_i),\qquad (λ_i)\ eigenvalues\ of\ Σ_nΣ_n^*,\] 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 $|\mathcal{V}{|}^2=|\mathbb{E}(X_{11}^n{)}^2{|}^2$ and $\kappa =\mathbb{E}|X_{11}^n{|}^4-|\mathcal{V}{|}^2-2$ 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.

Published at http://dx.doi.org/10.1214/15-AAP1135 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (4)

• Central limit theorem for linear eigenvalue statistics of random matrices with independent entries
• Central Limit Theorem for linear eigenvalue statistics of the Wigner and sample covariance random matrices
• Central limit theorem for signal-to-interference ratio of reduced rank linear receiver
• Functional CLT for sample covariance matrices

Cited by in corpus (16)

• Designing for Situation Awareness of Future Power Grids: An Indicator System Based on Linear Eigenvalue Statistics of Large Random Matrices
• Beyond universality in random matrix theory
• Non-Hermitian random matrices with a variance profile (I): Deterministic equivalents and limiting ESDs
• On fluctuations of eigenvalues of random band matrices
• Large complex correlated Wishart matrices: Fluctuations and asymptotic independence at the edges
• Unbounded Largest Eigenvalue of Large Sample Covariance Matrices: Asymptotics, Fluctuations and Applications
• Central limit theorem for mesoscopic eigenvalue statistics of deformed Wigner matrices and sample covariance matrices
• Joint CLT for top eigenvalues of sample covariance matrices of separable high dimensional long memory processes
• Second order statistics of robust estimators of scatter. Application to GLRT detection for elliptical signals
• Bias for the Trace of the Resolvent and Its Application on Non-Gaussian and Non-centered MIMO Channels
• Linear Eigenvalue Statistics of $X{X}^{\prime }$ matrices
• On the asymptotic behaviour of the eigenvalue distribution of block correlation matrices of high-dimensional time series
• Linear spectral statistics of sequential sample covariance matrices
• Gaussian fluctuations for linear spectral statistics of deformed Wigner matrices
• CLT for linear eigenvalue statistics for a tensor product version of sample covariance matrices
• CLT for LSS of sample covariance matrices with unbounded dispersions