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• W1639175092 abstract "Consider a $Ntimes n$ 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(Sigma_nSigma_n^*bigr)=sum_{i=1}^Nf(lambda_i),qquad (lambda_i) eigenvalues of Sigma_nSigma_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 $ vert mathcal{V}vert ^2=bigl|mathbb{E}bigl(X_{11}^nbigr) ^2bigr|^2$ and $kappa=mathbb{E}bigl vert X_{11}^nbigr vert ^4-vert {mathcal{V}}vert ^2-2$ 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-Sj{o}strand's formula. The CLT is expressed in terms of vanishing L'{e}vy-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." @default.
• W1639175092 created "2016-06-24" @default.
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• W1639175092 date "2016-06-01" @default.
• W1639175092 modified "2023-10-10" @default.
• W1639175092 title "Gaussian fluctuations for linear spectral statistics of large random covariance matrices" @default.
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• W1639175092 doi "https://doi.org/10.1214/15-aap1135" @default.
• W1639175092 hasPublicationYear "2016" @default.
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