On the limiting empirical measure of eigenvalues of the sum of rank one matrices with log-concave distribution

Pajor, Alain; Pastur, L. А.

doi:10.4064/sm195-1-2

Cited by 58 publications

(60 citation statements)

References 15 publications

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“…Various authors have then extended this result to more general settings, see e.g. Auburn [3], Yin and Krisnaiah [40], Silverstein [31], Götze and Tikhomirov [19,20], El Karoui [15,17], Adamczak [1], Pajor and Pastur [28], Bordenave et al [7], Chafai [10], Chatterjee et al [12], and O'Rourke [27]. In particular, the result holds for X i 's drawn independently from isotrophic log-concave distributions, this is a result of Pajor and Pastur [28].…”

Section: Introductionmentioning

confidence: 99%

“…Auburn [3], Yin and Krisnaiah [40], Silverstein [31], Götze and Tikhomirov [19,20], El Karoui [15,17], Adamczak [1], Pajor and Pastur [28], Bordenave et al [7], Chafai [10], Chatterjee et al [12], and O'Rourke [27]. In particular, the result holds for X i 's drawn independently from isotrophic log-concave distributions, this is a result of Pajor and Pastur [28]. Extensions to settings with some martingale-type assumptions were carried out in [19,20,1] , and extensions to settings with some concentration conditions on the distributions of X i 's were done in [15].…”

Section: Introductionmentioning

confidence: 99%

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The Spectrum of Random Kernel Matrices: Universality Results for Rough and Varying Kernels

2013

Random Matrices: Theory Appl.

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Abstract. We consider random matrices whose entries are f (X T i X j ) or f ( X i − X j 2 ) for iid vectors X i ∈ R p with normalized distribution. Assuming that f is sufficiently smooth and the distribution of X i 's is sufficiently nice, El Karoui [17] showed that the spectral distributions of these matrices behave as if f is linear in the Marčhenko-Pastur limit. When X i 's are Gaussian vectors, variants of this phenomenon were recently proved for varying kernels, i.e. when f may depend on p, by Cheng-Singer [13]. Two results are shown in this paper: first it is shown that for a large class of distributions the regularity assumptions on f in El Karoui's results can be reduced to minimal; and secondly it is shown that the Gaussian assumptions in Cheng-Singer's result can be removed, answering a question posed in [13] about the universality of the limiting spectral distribution.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The Spectrum of Random Kernel Matrices: Universality Results for Rough and Varying Kernels

2013

Random Matrices: Theory Appl.

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“…In particular, if all entries {A (n) jk } |j|,|k|≤m are non-vanishing, then we get the quarter-circle law, and this fact was proved long time ago [12] (see also [14,15]). …”

Section: )mentioning

confidence: 99%

“…The limiting measure N is uniquely defined via its Stieltjes transform f (see [1,15]) 14) by the formula…”

Section: Definition 12 [Good Vectors]mentioning

confidence: 99%

On a Limiting Distribution of Singular Values of Random Band Matrices

Lytova¹,

Pastur²

2015

Z. mat. fiz. anal. geom.

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An equation is obtained for the Stieltjes transform of the normalized distribution of singular values of non-symmetric band random matrices in the limit when the band width and rank of the matrix simultaneously tend to infinity. Conditions under which this limit agrees with the quarter-circle law are found. An interesting particular case of lower triangular random matrices is also considered and certain properties of the corresponding limiting singular value distribution are given.

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“…The convergence of the eigenvalue distribution of Σ N Σ * N towards MP(2σ 4 , c * ) follows immediately from the general results of [9]. The convergence of the extreme eigenvalues of Σ N Σ * N is more demanding, and follows an approach developed in a different context in [5] and [11].…”

mentioning

confidence: 92%

On the Behaviour of the Estimated Fourth-Order Cumulants Matrix of a High-Dimensional Gaussian White Noise

Gouédard¹,

Loubaton²

2017

Latent Variable Analysis and Signal Separation

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Abstract. This paper is devoted to the study of the traditional estimator of the fourth-order cumulants matrix of a high-dimensional multivariate Gaussian white noise. If M represents the dimension of the noise and N the number of available observations, it is first established that this M 2 × M 2 matrix converges towards 0 in the spectral norm sensThe behaviour of the estimated fourth-order cumulants matrix is then evaluated in the asymptotic regime where M and N converge towards +∞ in such a way thatconverges towards a constant. In this context, it is proved that the matrix does not converge towards 0 in the spectral norm sense, and that its empirical eigenvalue distribution converges towards a shifted Marcenko-Pastur distribution. It is finally claimed that the largest and the smallest eigenvalue of the cumulant matrix converges almost surely towards the rightend and the leftend points of the support of the Marcenko-Pastur distribution.Keywords: Estimated joint fourth-order cumulants matrices, large random matrices, blind source separation in the high-dimensional context.

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On the limiting empirical measure of eigenvalues of the sum of rank one matrices with log-concave distribution

Cited by 58 publications

References 15 publications

The Spectrum of Random Kernel Matrices: Universality Results for Rough and Varying Kernels

The Spectrum of Random Kernel Matrices: Universality Results for Rough and Varying Kernels

On a Limiting Distribution of Singular Values of Random Band Matrices

On the Behaviour of the Estimated Fourth-Order Cumulants Matrix of a High-Dimensional Gaussian White Noise

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