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Wang Zhou - One of the best experts on this subject based on the ideXlab platform.
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Tracy-Widom limit for the Largest Eigenvalue of high-dimensional covariance matrices in elliptical distributions
arXiv: Statistics Theory, 2019Co-Authors: Jun Wen, Wang ZhouAbstract:Let $X$ be an $M\times N$ random matrices consisting of independent $M$-variate elliptically distributed column vectors $\mathbf{x}_{1},\dots,\mathbf{x}_{N}$ with general population covariance matrix $\Sigma$. In the literature, the quantity $XX^{*}$ is referred to as the sample covariance matrix, where $X^{*}$ is the transpose of $X$. In this article, we show that the limiting behavior of the scaled Largest Eigenvalue of $XX^{*}$ is universal for a wide class of elliptical distributions, namely, the scaled Largest Eigenvalue converges weakly to the same limit as $M,N\to\infty$ with $M/N\to\phi>0$ regardless of the distributions that $\mathbf{x}_{1},\dots,\mathbf{x}_{N}$ follow. In particular, via comparing the Green function with that of the sample covariance matrix of multivariate normally distributed data, we conclude that the limiting distribution of the scaled Largest Eigenvalue is the celebrated Tracy-Widom law. Applications of our results to the statistical signal detection problems have also been discussed.
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universality for the Largest Eigenvalue of sample covariance matrices with general population
Annals of Statistics, 2015Co-Authors: Zhigang Bao, Guangming Pan, Wang ZhouAbstract:This paper is aimed at deriving the universality of the Largest Eigenvalue of a class of high-dimensional real or complex sample covariance matrices of the form $\mathcal{W}_{N}=\Sigma^{1/2}XX^{*}\Sigma^{1/2}$. Here, $X=(x_{ij})_{M,N}$ is an $M\times N$ random matrix with independent entries $x_{ij}$, $1\leq i\leq M$, $1\leq j\leq N$ such that $\mathbb{E}x_{ij}=0$, $\mathbb{E}|x_{ij}|^{2}=1/N$. On dimensionality, we assume that $M=M(N)$ and $N/M\rightarrow d\in(0,\infty)$ as $N\rightarrow\infty$. For a class of general deterministic positive-definite $M\times M$ matrices $\Sigma$, under some additional assumptions on the distribution of $x_{ij}$’s, we show that the limiting behavior of the Largest Eigenvalue of $\mathcal{W}_{N}$ is universal, via pursuing a Green function comparison strategy raised in [Probab. Theory Related Fields154 (2012) 341–407, Adv. Math.229 (2012) 1435–1515] by Erdős, Yau and Yin for Wigner matrices and extended by Pillai and Yin [Ann. Appl. Probab.24 (2014) 935–1001] to sample covariance matrices in the null case ($\Sigma=I$). Consequently, in the standard complex case ($\mathbb{E}x_{ij}^{2}=0$), combing this universality property and the results known for Gaussian matrices obtained by El Karoui in [Ann. Probab.35 (2007) 663–714] (nonsingular case) and Onatski in [Ann. Appl. Probab.18 (2008) 470–490] (singular case), we show that after an appropriate normalization the Largest Eigenvalue of $\mathcal{W}_{N}$ converges weakly to the type 2 Tracy–Widom distribution $\mathrm{TW}_{2}$. Moreover, in the real case, we show that when $\Sigma$ is spiked with a fixed number of subcritical spikes, the type 1 Tracy–Widom limit $\mathrm{TW}_{1}$ holds for the normalized Largest Eigenvalue of $\mathcal{W}_{N}$, which extends a result of Feral and Peche in [J. Math. Phys.50 (2009) 073302] to the scenario of nondiagonal $\Sigma$ and more generally distributed $X$. In summary, we establish the Tracy–Widom type universality for the Largest Eigenvalue of generally distributed sample covariance matrices under quite light assumptions on $\Sigma$. Applications of these limiting results to statistical signal detection and structure recognition of separable covariance matrices are also discussed.
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Universality for the Largest Eigenvalue of sample covariance matrices with general population
The Annals of Statistics, 2015Co-Authors: Zhigang Bao, Guangming Pan, Wang ZhouAbstract:In this paper, we will derive the universality of the Largest Eigenvalue of a class of large dimensional real or complex sample covariance matrices in the form of WN = Σ1/2XX∗Σ1/2. Here X = (xij)M,N is an M × N random matrix with independent entries xij , 1 ≤ i ≤ M, 1 ≤ j ≤ N such that Exij = 0, E|xij | = 1/N . We say WN is a classical complex sample covariance matrix if there also exists Exij = 0, 1 ≤ i ≤ M, 1 ≤ j ≤ N . Moreover, on dimensions we assume that M = M(N) and N/M → d ∈ (0,∞) as N → ∞. For a class of highly general deterministic positive definite M ×M matrices Σ, we show that the limiting behavior of the Largest Eigenvalue of WN is universal under some additional assumptions on the distribution of (xij) ′s via pursuing a Green function comparison strategy raised in [18, 19] by Erdos, Yau and Yin for Wigner matrices and extended by Pillai and Yin [31] to sample covariance matrices in the null case (Σ = I). Consequently, in the classical complex case, combing this universality property and the results known for Gaussian matrices derived by El Karoui in [9] (nonsingular case) and Onatski in [27] (singular case) we show that after appropriate normalization the Largest Eigenvalue of WN converges weakly to the type 2 TracyWidom distribution TW2. Moreover, in the real case, we show that when Σ is spiked with fixed number of sub-critical spikes, the type 1 Tracy-Widom distribution TW1 holds for the Largest Eigenvalue of WN , which extends a result of Feral and Peche in [21] to the scenario of nondiagonal Σ and more generally distributed X. In other words, this result establishes the Tracy-Widom type universality property for the Largest Eigenvalue of generally distributed sample covariance matrices under nearly lightest assumptions on Σ. The proof is based on our former results on the square root behavior and local MP type law on the right edge of the spectrum of WN in [5] and a new bound on the entries of some congruent transformation of the resolvent matrix of WN .
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universality for the Largest Eigenvalue of sample covariance matrices with general population
arXiv: Probability, 2013Co-Authors: Zhigang Bao, Guangming Pan, Wang ZhouAbstract:This paper is aimed at deriving the universality of the Largest Eigenvalue of a class of high-dimensional real or complex sample covariance matrices of the form $\mathcal{W}_N=\Sigma^{1/2}XX^*\Sigma ^{1/2}$. Here, $X=(x_{ij})_{M,N}$ is an $M\times N$ random matrix with independent entries $x_{ij},1\leq i\leq M,1\leq j\leq N$ such that $\mathbb{E}x_{ij}=0$, $\mathbb{E}|x_{ij}|^2=1/N$. On dimensionality, we assume that $M=M(N)$ and $N/M\rightarrow d\in(0,\infty)$ as $N\rightarrow\infty$. For a class of general deterministic positive-definite $M\times M$ matrices $\Sigma$, under some additional assumptions on the distribution of $x_{ij}$'s, we show that the limiting behavior of the Largest Eigenvalue of $\mathcal{W}_N$ is universal, via pursuing a Green function comparison strategy raised in [Probab. Theory Related Fields 154 (2012) 341-407, Adv. Math. 229 (2012) 1435-1515] by Erd\H{o}s, Yau and Yin for Wigner matrices and extended by Pillai and Yin [Ann. Appl. Probab. 24 (2014) 935-1001] to sample covariance matrices in the null case ($\Sigma=I$). Consequently, in the standard complex case ($\mathbb{E}x_{ij}^2=0$), combing this universality property and the results known for Gaussian matrices obtained by El Karoui in [Ann. Probab. 35 (2007) 663-714] (nonsingular case) and Onatski in [Ann. Appl. Probab. 18 (2008) 470-490] (singular case), we show that after an appropriate normalization the Largest Eigenvalue of $\mathcal{W}_N$ converges weakly to the type 2 Tracy-Widom distribution $\mathrm{TW}_2$. Moreover, in the real case, we show that when $\Sigma$ is spiked with a fixed number of subcritical spikes, the type 1 Tracy-Widom limit $\mathrm{TW}_1$ holds for the normalized Largest Eigenvalue of $\mathcal {W}_N$, which extends a result of F\'{e}ral and P\'{e}ch\'{e} in [J. Math. Phys. 50 (2009) 073302] to the scenario of nondiagonal $\Sigma$ and more generally distributed $X$.
Sandrine Peche - One of the best experts on this subject based on the ideXlab platform.
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the Largest Eigenvalue of rank one deformation of large wigner matrices
arXiv: Probability, 2006Co-Authors: Delphine Feral, Sandrine PecheAbstract:The purpose of this paper is to establish universality of the fluctuations of the Largest Eigenvalue of some non necessarily Gaussian complex Deformed Wigner Ensembles. The real model is also considered. Our approach is close to the one used by A. Soshnikov in the investigations of classical real or complex Wigner Ensembles. It is based on the computation of moments of traces of high powers of the random matrices under consideration.
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The Largest Eigenvalue of small rank perturbations of Hermitian random matrices
Probability Theory and Related Fields, 2006Co-Authors: Sandrine PecheAbstract:We compute the limiting Eigenvalue statistics at the edge of the spectrum of large Hermitian random matrices perturbed by the addition of small rank deterministic matrices. To be more precise, we consider random Hermitian matrices with independent Gaussian entries $M_{ij}, i\leq j$ with various expectations. We prove that the Largest Eigenvalue of such random matrices exhibits, in the large $N$ limit, various limiting distributions depending on both the Eigenvalues of the matrix $(\mathbb{E}M_{ij})_{i,j=1}^N$ and its rank.
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phase transition of the Largest Eigenvalue for nonnull complex sample covariance matrices
Annals of Probability, 2005Co-Authors: Jinho Baik, Gerard Ben Arous, Sandrine PecheAbstract:AbstractWe compute the limiting distributions of the Largest Eigenvalue of a complex Gaussian samplecovariance matrix when both the number of samples and the number of variables in each samplebecome large. When all but finitely many, say r, Eigenvalues of the covariance matrix arethe same, the dependence of the limiting distribution of the Largest Eigenvalue of the samplecovariance matrix on those distinguished r Eigenvalues of the covariance matrix is completelycharacterized in terms of an infinite sequence of new distribution functions that generalizethe Tracy-Widom distributions of the random matrix theory. Especially a phase transitionphenomena is observed. Our results also apply to a last passage percolation model and aqueuing model. 1 Introduction Consider M independent, identically distributed samples y 1 ,...,~y M , all of which are N ×1 columnvectors. We further assume that the sample vectors ~y k are Gaussian with mean µ and covarianceΣ, where Σ is a fixed N ×N positive matrix; the density of a sample ~y isp(~y) =1(2π)
Guanglu Zhou - One of the best experts on this subject based on the ideXlab platform.
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on the Largest Eigenvalue of a symmetric nonnegative tensor
Numerical Linear Algebra With Applications, 2013Co-Authors: Guanglu ZhouAbstract:SUMMARY In this paper, some important spectral characterizations of symmetric nonnegative tensors are analyzed. In particular, it is shown that a symmetric nonnegative tensor has the following properties: (i) its spectral radius is zero if and only if it is a zero tensor; (ii) it is weakly irreducible (respectively, irreducible) if and only if it has a unique positive (respectively, nonnegative) Eigenvalue–eigenvector; (iii) the minimax theorem is satisfied without requiring the weak irreducibility condition; and (iv) if it is weakly reducible, then it can be decomposed into some weakly irreducible tensors. In addition, the problem of finding the Largest Eigenvalue of a symmetric nonnegative tensor is shown to be equivalent to finding the global solution of a convex optimization problem. Subsequently, algorithmic aspects for computing the Largest Eigenvalue of symmetric nonnegative tensors are discussed. Copyright © 2013 John Wiley & Sons, Ltd.
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On the Largest Eigenvalue of a symmetric nonnegative tensor
Numerical Linear Algebra with Applications, 2013Co-Authors: Guanglu ZhouAbstract:In this paper, some important spectral characterizations of symmetric nonnegative tensors are analyzed. In particular, it is shown that a symmetric nonnegative tensor has the following properties: (i) its spectral radius is zero if and only if it is a zero tensor; (ii) it is weakly irreducible (respectively, irreducible) if and only if it has a unique positive (respectively, nonnegative) Eigenvalue-eigenvector; (iii) the minimax theorem is satisfied without requiring the weak irreducibility condition; and (iv) if it is weakly reducible, then it can be decomposed into some weakly irreducible tensors. In addition, the problem of finding the Largest Eigenvalue of a symmetric nonnegative tensor is shown to be equivalent to finding the global solution of a convex optimization problem. Subsequently, algorithmic aspects for computing the Largest Eigenvalue of symmetric nonnegative tensors are discussed.Department of Applied Mathematic
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efficient algorithms for computing the Largest Eigenvalue of a nonnegative tensor
Frontiers of Mathematics in China, 2013Co-Authors: Guanglu ZhouAbstract:Consider the problem of computing the Largest Eigenvalue for nonnegative tensors. In this paper, we establish the Q-linear convergence of a power type algorithm for this problem under a weak irreducibility condition. Moreover, we present a convergent algorithm for calculating the Largest Eigenvalue for any nonnegative tensors.
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Efficient algorithms for computing the Largest Eigenvalue of a nonnegative tensor
Frontiers of Mathematics in China, 2013Co-Authors: Guanglu ZhouAbstract:Consider the problem of computing the Largest Eigenvalue for nonnegative tensors. In this paper, we establish the Q-linear convergence of a power type algorithm for this problem under a weak irreducibility condition. Moreover, we present a convergent algorithm for calculating the Largest Eigenvalue for any nonnegative tensors.Department of Applied Mathematic
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Finding the Largest Eigenvalue of a Nonnegative Tensor
SIAM Journal on Matrix Analysis and Applications, 2010Co-Authors: Guanglu ZhouAbstract:In this paper we propose an iterative method for calculating the Largest Eigenvalue of an irreducible nonnegative tensor. This method is an extension of a method of Collatz (1942) for calculating the spectral radius of an irreducible nonnegative matrix. Numerical results show that our proposed method is promising. We also apply the method to studying higher-order Markov chains.Department of Applied Mathematic
Zhigang Bao - One of the best experts on this subject based on the ideXlab platform.
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universality for the Largest Eigenvalue of sample covariance matrices with general population
Annals of Statistics, 2015Co-Authors: Zhigang Bao, Guangming Pan, Wang ZhouAbstract:This paper is aimed at deriving the universality of the Largest Eigenvalue of a class of high-dimensional real or complex sample covariance matrices of the form $\mathcal{W}_{N}=\Sigma^{1/2}XX^{*}\Sigma^{1/2}$. Here, $X=(x_{ij})_{M,N}$ is an $M\times N$ random matrix with independent entries $x_{ij}$, $1\leq i\leq M$, $1\leq j\leq N$ such that $\mathbb{E}x_{ij}=0$, $\mathbb{E}|x_{ij}|^{2}=1/N$. On dimensionality, we assume that $M=M(N)$ and $N/M\rightarrow d\in(0,\infty)$ as $N\rightarrow\infty$. For a class of general deterministic positive-definite $M\times M$ matrices $\Sigma$, under some additional assumptions on the distribution of $x_{ij}$’s, we show that the limiting behavior of the Largest Eigenvalue of $\mathcal{W}_{N}$ is universal, via pursuing a Green function comparison strategy raised in [Probab. Theory Related Fields154 (2012) 341–407, Adv. Math.229 (2012) 1435–1515] by Erdős, Yau and Yin for Wigner matrices and extended by Pillai and Yin [Ann. Appl. Probab.24 (2014) 935–1001] to sample covariance matrices in the null case ($\Sigma=I$). Consequently, in the standard complex case ($\mathbb{E}x_{ij}^{2}=0$), combing this universality property and the results known for Gaussian matrices obtained by El Karoui in [Ann. Probab.35 (2007) 663–714] (nonsingular case) and Onatski in [Ann. Appl. Probab.18 (2008) 470–490] (singular case), we show that after an appropriate normalization the Largest Eigenvalue of $\mathcal{W}_{N}$ converges weakly to the type 2 Tracy–Widom distribution $\mathrm{TW}_{2}$. Moreover, in the real case, we show that when $\Sigma$ is spiked with a fixed number of subcritical spikes, the type 1 Tracy–Widom limit $\mathrm{TW}_{1}$ holds for the normalized Largest Eigenvalue of $\mathcal{W}_{N}$, which extends a result of Feral and Peche in [J. Math. Phys.50 (2009) 073302] to the scenario of nondiagonal $\Sigma$ and more generally distributed $X$. In summary, we establish the Tracy–Widom type universality for the Largest Eigenvalue of generally distributed sample covariance matrices under quite light assumptions on $\Sigma$. Applications of these limiting results to statistical signal detection and structure recognition of separable covariance matrices are also discussed.
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Universality for the Largest Eigenvalue of sample covariance matrices with general population
The Annals of Statistics, 2015Co-Authors: Zhigang Bao, Guangming Pan, Wang ZhouAbstract:In this paper, we will derive the universality of the Largest Eigenvalue of a class of large dimensional real or complex sample covariance matrices in the form of WN = Σ1/2XX∗Σ1/2. Here X = (xij)M,N is an M × N random matrix with independent entries xij , 1 ≤ i ≤ M, 1 ≤ j ≤ N such that Exij = 0, E|xij | = 1/N . We say WN is a classical complex sample covariance matrix if there also exists Exij = 0, 1 ≤ i ≤ M, 1 ≤ j ≤ N . Moreover, on dimensions we assume that M = M(N) and N/M → d ∈ (0,∞) as N → ∞. For a class of highly general deterministic positive definite M ×M matrices Σ, we show that the limiting behavior of the Largest Eigenvalue of WN is universal under some additional assumptions on the distribution of (xij) ′s via pursuing a Green function comparison strategy raised in [18, 19] by Erdos, Yau and Yin for Wigner matrices and extended by Pillai and Yin [31] to sample covariance matrices in the null case (Σ = I). Consequently, in the classical complex case, combing this universality property and the results known for Gaussian matrices derived by El Karoui in [9] (nonsingular case) and Onatski in [27] (singular case) we show that after appropriate normalization the Largest Eigenvalue of WN converges weakly to the type 2 TracyWidom distribution TW2. Moreover, in the real case, we show that when Σ is spiked with fixed number of sub-critical spikes, the type 1 Tracy-Widom distribution TW1 holds for the Largest Eigenvalue of WN , which extends a result of Feral and Peche in [21] to the scenario of nondiagonal Σ and more generally distributed X. In other words, this result establishes the Tracy-Widom type universality property for the Largest Eigenvalue of generally distributed sample covariance matrices under nearly lightest assumptions on Σ. The proof is based on our former results on the square root behavior and local MP type law on the right edge of the spectrum of WN in [5] and a new bound on the entries of some congruent transformation of the resolvent matrix of WN .
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universality for the Largest Eigenvalue of sample covariance matrices with general population
arXiv: Probability, 2013Co-Authors: Zhigang Bao, Guangming Pan, Wang ZhouAbstract:This paper is aimed at deriving the universality of the Largest Eigenvalue of a class of high-dimensional real or complex sample covariance matrices of the form $\mathcal{W}_N=\Sigma^{1/2}XX^*\Sigma ^{1/2}$. Here, $X=(x_{ij})_{M,N}$ is an $M\times N$ random matrix with independent entries $x_{ij},1\leq i\leq M,1\leq j\leq N$ such that $\mathbb{E}x_{ij}=0$, $\mathbb{E}|x_{ij}|^2=1/N$. On dimensionality, we assume that $M=M(N)$ and $N/M\rightarrow d\in(0,\infty)$ as $N\rightarrow\infty$. For a class of general deterministic positive-definite $M\times M$ matrices $\Sigma$, under some additional assumptions on the distribution of $x_{ij}$'s, we show that the limiting behavior of the Largest Eigenvalue of $\mathcal{W}_N$ is universal, via pursuing a Green function comparison strategy raised in [Probab. Theory Related Fields 154 (2012) 341-407, Adv. Math. 229 (2012) 1435-1515] by Erd\H{o}s, Yau and Yin for Wigner matrices and extended by Pillai and Yin [Ann. Appl. Probab. 24 (2014) 935-1001] to sample covariance matrices in the null case ($\Sigma=I$). Consequently, in the standard complex case ($\mathbb{E}x_{ij}^2=0$), combing this universality property and the results known for Gaussian matrices obtained by El Karoui in [Ann. Probab. 35 (2007) 663-714] (nonsingular case) and Onatski in [Ann. Appl. Probab. 18 (2008) 470-490] (singular case), we show that after an appropriate normalization the Largest Eigenvalue of $\mathcal{W}_N$ converges weakly to the type 2 Tracy-Widom distribution $\mathrm{TW}_2$. Moreover, in the real case, we show that when $\Sigma$ is spiked with a fixed number of subcritical spikes, the type 1 Tracy-Widom limit $\mathrm{TW}_1$ holds for the normalized Largest Eigenvalue of $\mathcal {W}_N$, which extends a result of F\'{e}ral and P\'{e}ch\'{e} in [J. Math. Phys. 50 (2009) 073302] to the scenario of nondiagonal $\Sigma$ and more generally distributed $X$.
Noureddine El Karoui - One of the best experts on this subject based on the ideXlab platform.
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tracy widom limit for the Largest Eigenvalue of a large class of complex sample covariance matrices
Annals of Probability, 2007Co-Authors: Noureddine El KarouiAbstract: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.
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tracy widom limit for the Largest Eigenvalue of a large class of complex sample covariance matrices
arXiv: Probability, 2005Co-Authors: Noureddine El KarouiAbstract: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 $n\times p$ matrix, and let its rows be i.i.d. complex normal vectors with mean 0 and covariance $\Sigma_p$. We show that for a large class of covariance matrices $\Sigma_p$, 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.
