A Universality Result for the Smallest Eigenvalues of Certain Sample Covariance Matrices

Feldheim, Ohad N.; Sodin, Sasha

doi:10.1007/s00039-010-0055-x

Cited by 132 publications

(206 citation statements)

References 25 publications

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“…Note that both edges of the spectrum are "soft" when γ ∞ ∈ (0; 1). These results are in consistence with results by Ben Arous and Péché [4], Tao and Vu [24], Soshnikov [21], Péché [20], and Feldheim and Sodin [11], who obtained similar results for the (more relevant) correlation function of the eigenvalues, yet under stronger assumptions on the underlying distributions. Ben Arous and Péché [4] proved the occurrence of the sine kernel in the bulk of the spectrum for a certain class of complex sample covariance matrices with γ ∞ = 1.…”

Section: Introductionsupporting

confidence: 89%

“…Soshnikov [21] and Péché [20] established the occurrence of the Airy kernel at the upper edge of the spectrum for real and complex sample covariance matrices whose underlying distributions are symmetric with exponential tails (or at least finite 36th moments). Recently, Feldheim and Sodin [11] proposed another approach to obtain these results, which also works for the lower edge of the spectrum. Thus, our results add some support to the wide-spread expectation that correlation functions in random matrix theory are universal, and subject to weak moment conditions only.…”

Section: Introductionmentioning

confidence: 99%

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Characteristic polynomials of sample covariance matrices: The non-square case

Kösters

2010

Open Mathematics

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Abstract:We consider the sample covariance matrices of large data matrices which have i.i.d. complex matrix entries and which are non-square in the sense that the difference between the number of rows and the number of columns tends to infinity. We show that the second-order correlation function of the characteristic polynomial of the sample covariance matrix is asymptotically given by the sine kernel in the bulk of the spectrum and by the Airy kernel at the edge of the spectrum. Similar results are given for real sample covariance matrices. MSC:15A52

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Section: Introductionsupporting

confidence: 89%

Section: Introductionmentioning

confidence: 99%

Characteristic polynomials of sample covariance matrices: The non-square case

Kösters

2010

Open Mathematics

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“…gave a detailed overview of these techniques and also extended the universality results to so-called generalized Wigner matrices, where the entries are independent but have different variances, and to certain classes of banded Hermitian random matrices. Universality at the edge of the spectrum of sample covariance and correlation matrices has been studied by Feldheim and Sodin (2010), Bao et al (2012) and Pillai and Jin (2012). Lee and Yin (2012) established necessary and sufficient conditions for edge universality of a Wigner matrix.…”

Section: Universalitymentioning

confidence: 99%

Random matrix theory in statistics: A review

Paul

Aue

2014

Journal of Statistical Planning and Inference

165

115

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b s t r a c tWe give an overview of random matrix theory (RMT) with the objective of highlighting the results and concepts that have a growing impact in the formulation and inference of statistical models and methodologies. This paper focuses on a number of application areas especially within the field of high-dimensional statistics and describes how the development of the theory and practice in high-dimensional statistical inference has been influenced by the corresponding developments in the field of RMT.

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“…Theorem 2 [30]. The probability that the largest eigenvalue of X is larger than [(1 + √ y) 2 + δ]σ 2 is no greater than C exp (−dδ 3/2 /C), where C is a constant.…”

Section: Quantum Data Locking From Phase Modulationmentioning

confidence: 96%

Robust quantum data locking from phase modulation

2014

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Quantum data locking is a uniquely quantum phenomenon that allows a relatively short key of constant size to (un)lock an arbitrarily long message encoded in a quantum state, in such a way that an eavesdropper who measures the state but does not know the key has essentially no information about the message. The application of quantum data locking in cryptography would allow one to overcome the limitations of the one-time pad encryption, which requires the key to have the same length as the message. However, it is known that the strength of quantum data locking is also its Achilles heel, as the leakage of a few bits of the key or the message may in principle allow the eavesdropper to unlock a disproportionate amount of information. In this paper we show that there exist quantum data locking schemes that can be made robust against information leakage by increasing the length of the key by a proportionate amount. This implies that a constant size key can still lock an arbitrarily long message as long as a fraction of it remains secret to the eavesdropper. Moreover, we greatly simplify the structure of the protocol by proving that phase modulation suffices to generate strong locking schemes, paving the way to optical experimental realizations. Also, we show that successful data locking protocols can be constructed using random code words, which very well could be helpful in discovering random codes for data locking over noisy quantum channels.

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A Universality Result for the Smallest Eigenvalues of Certain Sample Covariance Matrices

Abstract: After proper rescaling and under some technical assumptions, the smallest eigenvalue of a sample covariance matrix with aspect ratio bounded away from 1 converges to the Tracy-Widom distribution. This complements the results on the largest eigenvalue, due to Soshnikov and Péché.

Cited by 132 publications

References 25 publications

Characteristic polynomials of sample covariance matrices: The non-square case

Characteristic polynomials of sample covariance matrices: The non-square case

Random matrix theory in statistics: A review

Robust quantum data locking from phase modulation

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