Tracy–Widom limit for the largest eigenvalue of a large class of complex sample covariance matrices

By Noureddine El Karoui. 2005

240 citations recorded by the source corpus.

We consider the asymptotic fluctuation behavior of the largest eigenvalue of certain sample covariance matrices in the asymptotic regime where both dimensions of the corresponding data matrix go to infinity. More precisely, let X be an n x p matrix, and let its rows be i.i.d. complex normal vectors with mean 0 and covariance Σ p . We show that for a large class of covariance matrices £ p, the largest eigenvalue of X*X is asymptotically distributed (after recentering and rescaling) as the Tracy-Widom distribution that appears in the study of the Gaussian unitary ensemble. We give explicit formulas for the centering and scaling sequences that are easy to implement and involve only the spectral distribution of the population covariance, n and p. The main theorem applies to a number of covariance models found in applications. For example, well-behaved Toeplitz matrices as well as covariance matrices whose spectral distribution is a sum of atoms (under some conditions on the mass of the atoms) are among the models the theorem can handle. Generalizations of the theorem to certain spiked versions of our models and a.s. results about the largest eigenvalue are given. We also discuss a simple corollary that does not require normality of the entries of the data matrix and some consequences for applications in multivariate statistics.

Subfields: Applied Mathematics, Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/c7111c1cd2ff9782462d9ba7245cc836accf423c

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Citations in the corpus

Universality for the largest eigenvalue of sample covariance matrices with general population (2013, methodology): “Corresponding results analogous to those in [9] and [27] are still open for the real Wishart ensemble.”
Asymptotic theory for stationary processes (2011, background): “Geman (1980) and Yin, Bai and Krishnaiah (1988) considered the convergence problem of largest eigenvalues of sample covariance matrices of iid random vectors which has independent entries; see also Johnstone (2001), El Karoui (2007) and Bai and Silverstein (2010).”
Large deviations for the largest eigenvalue of rank one deformations of Gaussian ensembles (2006, background): “log |x − y|dσβ(y) := Sθ(x), so that from (10) and (11), we get that F β θ (x) = 1 2 ∫ x √ 2β √ z2 − 2βdz − θx + 1 4 x(2) + β 4 − β 2 log β 2 + β 2 log θ.”
A necessary and sufficient condition for edge universality at the largest singular values of covariance matrices (2016, background): “The scaling factor γ0 is given by [11] 1 γ3 0 = 1 d ∫ ( x 1 +m2c(λr)x 3 πN (dx) − 1 m2c(λr)(3) , and Assumption 2.”
Two Sample Tests for High Dimensional Covariance Matrices (2012, background): “probability limits and the limiting distributions of extreme eigenvalues of the sample covariance matrix based on the random matrix theory are developed in Bai (1993), Bai and Yin (1993), Tracy and Widom (1996), Johnstone (2001) and El Karoui (2007), Johnstone and Lu (2009), Bai and Silverstein (201”
Tracy-Widom Distribution for the Largest Eigenvalue of Real Sample Covariance Matrices with General Population (2014, background): “The scaling factor γ0 ≡ γ0(N) is given by [12] 1 γ3 0 = 1 d ∫ ( t 1− tξ+ )3 dρ̂(t) + 1 ξ3 + .”
Anisotropic local laws for random matrices (2014, background): “1214/009117906000000575 [11] Karoui, N, Tracy-Widom limit for the largest eigenvalue of a large class of complex sample covariance matrices, Ann.”
Rank 1 real Wishart spiked model (2011, result): “This has led to the results in [10] and results on the largest eigenvalue distributions for more general complex Wishart ensembles [38], [57].”
High Dimensional Statistical Inference and Random Matrices (2006, background): “For example, El Karoui [39] establishes that (15) holds, along with explicit formulas for μnp(Σ) and σnp(Σ), if enough eigenvalues accumulate near the largest eigenvalue, or if a small number of eigenvalues are not too isolated, as we describe below in a specific setting below.”
Spectrum estimation for large dimensional covariance matrices using random matrix theory (2006, background): “One thing is, however, clear from the most recent theoretical results: if we wish to de-bias extreme sample eigenvalues, we need an accurate estimate of the so-called population spectral distribution, a probability measure that characterizes the population eigenvalues (see [17]).”
Finite sample approximation results for principal component analysis: a matrix perturbation approach (2009, methodology): “24) was recently derived by El-Karoui [12], who also studied the finite p, n fluctuations around this mean.”
Analysis of the limiting spectral measure of large random matrices of the separable covariance type (2013, background): “Finally, it has been noticed in the large random matrix community that there is an intimate connection between the square root behavior of the density of the limit spectral measure at the edges of the support and the Tracy-Widom fluctuations of the eigenvalues near those edges (see [6] dealing with ”
Covariance regularization by thresholding (2009, methodology): “is reprint diﬀers from the original in pagination and typographic detail. 1 2 P. J. BICKEL AND E. LEVINA to the more recent work of Johnstone and his students on the theory of the largest eigenvalues [12, 23, 30] and associated eigenvectors [24]. However, with the exception of a method for estimatin”
Random matrix theory in statistics: A review (2014, methodology): “A modified form of the asymptotics of the weighted Laguerre polynomials was used by El Karoui (2003) to extend the domain of Theorem 3.6 to the setting when p=n-0 as p;n-1. Paul (2011) used the technique of El Karoui (2003) to prove that under the latter setting, the scaling limits of the normalized”
BANDING SAMPLE AUTOCOVARIANCE MATRICES OF STATIONARY PROCESSES (2009, background): “For other contributions to inconsistency of the largest eigenvalues of sample covariance matrices see Johnstone (2001) and El Karoui (2007), and references therein.”
Concentration of measure and spectra of random matrices: Applications to correlation matrices, elliptical distributions and beyond (2009, background): “I would also like to thank an anonymous referee and the Editor for constructive comments that led to a substantial improvement of the paper.”
The eigenvalues and eigenvectors of finite, low rank perturbations of large random matrices (2009, background): “This trick cannot be applied for Wishart matrices since Wishart matrices do not exhibit the symmetries of Wigner matrices.”
Global Testing and Large-Scale Multiple Testing for High-Dimensional Covariance Structures (2017, background): “See also Baik et al. (2005); El Karoui (2007); Onatski et al. (2013) for other optimality results.”