On Limit of the Largest Eigenvalue of the Large Dimensional Sample Covariance Matrix.

Yin, Yanqing; Bai, Zijian; Krishnaiah, P. R.

doi:10.21236/ada150589

Cited by 134 publications

(175 citation statements)

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“…Such results apply to more general samples other than Gaussian (see e.g. [31,32,4]). One piece of information that can be extracted from this is that if there are non-zero eigenvalues of the sample covariance matrix well separated from the rest of the sample eigenvalues, one can infer that the samples are not i.i.d.…”

Section: Introductionmentioning

confidence: 66%

Eigenvalues of large sample covariance matrices of spiked population models

Baik

Silverstein

2006

Journal of Multivariate Analysis

612

567

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We consider a spiked population model, proposed by Johnstone, in which all the population eigenvalues are one except for a few fixed eigenvalues. The question is to determine how the sample eigenvalues depend on the non-unit population ones when both sample size and population size become large. This paper completely determines the almost sure limits of the sample eigenvalues in a spiked model for a general class of samples.

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Section: Introductionmentioning

confidence: 66%

Eigenvalues of large sample covariance matrices of spiked population models

Baik

Silverstein

2006

Journal of Multivariate Analysis

612

567

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“…The meaning of (1.1) in the limit regime is that, for a family of matrices as above whose dimensions m and n increase to infinity and whose aspect ratio m/n converges to a constant, the ratio W /( √ n + √ m) converges to 1 almost surely [32], see also [7]. In the non-limit regime, i.e.…”

Section: Matrices With Independent Entriesmentioning

confidence: 99%

Spectral norm of products of random and deterministic matrices

Vershynin

2010

Probab. Theory Relat. Fields

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Abstract. We study the spectral norm of matrices W that can be factored as W = BA, where A is a random matrix with independent mean zero entries and B is a fixed matrix. Under the (4 + ε)-th moment assumption on the entries of A, we show that the spectral norm of such an m× n matrix W is bounded by √ m + √ n, which is sharp. In other words, in regard to the spectral norm, products of random and deterministic matrices behave similarly to random matrices with independent entries. This result along with the previous work of M. Rudelson and the author implies that the smallest singular value of a random m × n matrix with i.i.d. mean zero entries and bounded (4 + ε)-th moment is bounded below by √ m − √ n − 1 with high probability.

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“…standard normals as entries converges was first proved by Geman [31]. Later Bai et al [7], Yin et al [71] generalized it to the matrices with entries being i.i.d. (arbitrary) random variables with the finite fourth moment.…”

Section: T Jiangmentioning

confidence: 99%

“…For a q × q square matrix A, write λ 1 (A) ≥ λ 2 (A) ≥ · · · ≥ λ q (A) for all the eigenvalues of A. By Theorem 2.16 from [3] (see also [7] and [71]) that…”

Section: Proof Of Theoremmentioning

confidence: 99%

Approximation of Haar distributed matrices and limiting distributions of eigenvalues of Jacobi ensembles

Jiang

2008

Probab. Theory Relat. Fields

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We develop a tool to approximate the entries of a large dimensional complex Jacobi ensemble with independent complex Gaussian random variables. Based on this and the author's earlier work in this direction, we obtain the TracyWidom law of the largest singular values of the Jacobi emsemble. Moreover, the circular law, the Marchenko-Pastur law, the central limit theorem, and the laws of large numbers for the spectral norms are also obtained.

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On Limit of the Largest Eigenvalue of the Large Dimensional Sample Covariance Matrix.

Cited by 134 publications

References 0 publications

Eigenvalues of large sample covariance matrices of spiked population models

Eigenvalues of large sample covariance matrices of spiked population models

Spectral norm of products of random and deterministic matrices

Approximation of Haar distributed matrices and limiting distributions of eigenvalues of Jacobi ensembles

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