publication . Article . Preprint . 2009

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

Kösters, Holger;
Open Access
  • Published: 15 Dec 2009 Journal: Open Mathematics, volume 8 (eissn: 2391-5455, Copyright policy)
  • Publisher: Walter de Gruyter GmbH
Abstract
<jats:title>Abstract</jats:title><jats:p>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.</jats:p>
Subjects
free text keywords: Estimation of covariance matrices, Matrix (mathematics), Integer matrix, Mathematics, Covariance function, Rational quadratic covariance function, Covariance, Matrix analysis, Law of total covariance, Mathematical analysis, Mathematics - Probability, 15A52, 60B99, 62E20
25 references, page 1 of 2

S(a) := = S(a) ; w(z) = 2 N A(a) := Ai ( z) = [AS] Abramowitz, M.; Stegun, I. (1965): Handbook of Mathematical Functions. Dover Publications, New York.

[AF] Akemann, G.; Fyodorov, Y.V. (2003): Universal random matrix correlations of ratios of characteristic polynomials at the spectral edges. Nuclear Physics B, 664, 457{476. [OpenAIRE]

[BDS] Baik, J.; Deift, P.; Strahov, E. (2003): Products and ratios of characteristic polynomials of random hermitian matrices. J. Math. Phys., 44, 3657{3670. [OpenAIRE]

[BP] Ben Arous, G.; Peche, S. (2005): Universality of local eigenvalue statistics for some sample covariance matrices. Comm. Pure Appl. Math., 58, 1316{1357. [OpenAIRE]

[BS] Borodin, A.; Strahov, E. (2006): Averages of characteristic polynomials in random matrix theory. Comm. Pure Appl. Math., 59, 161{253.

[BH1] Brezin, E.; Hikami, S. (2000): Characteristic polynomials of random matrices. Comm. Math. Phys., 214, 111{135. [OpenAIRE]

[BH2] Brezin, E.; Hikami, S. (2001): Characteristic polynomials of real symmetric random matrices. Comm. Math. Phys., 223, 363{382. [OpenAIRE]

[BH3] Brezin, E.; Hikami, S. (2003): New correlation functions for random matrices and integrals over supergroups. J. Phys. A: Math. Gen., 36, 711{751. [OpenAIRE]

[De] Deift, P.A. (1999): Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach. Courant Lecture Notes in Mathematics, vol. 3, Courant Institute of Mathematical Sciences, New York.

[Er] Erdelyi, A.; Magnus, W.; Oberhettinger, F.; Tricomi, F.G. (1954): Tables of Integral Transforms, volume I. McGraw-Hill Book Company, New York.

[FS-1] Feldheim, O.N.; Sodin, S. (2008): A universality result for the smallest eigenvalues of certain sample covariance matrices. Preprint.

[Fo] Forrester, P.J. (2008+): Log Gases and Random Matrices. Book in preparation, www.ms.unimelb.edu.au/~matpjf/matpjf.html

[FS-2] Fyodorov, Y.V.; Strahov, E (2003): An exact formula for general spectral correlation function of random Hermitian matrices. J. Phys. A: Math. Gen., 36, 3202{3213.

[GK] Go&#x7f;tze, F.; Ko&#x7f;sters, H. (2009): On the second-order correlation function of the characteristic polynomial of a Hermitian Wigner matrix. Comm. Math. Phys., 285, 1183{1205.

[Ko&#x7f;1] Ko&#x7f;sters, H. (2008): On the second-order correlation function of the characteristic polynomial of a real-symmetric Wigner matrix. Electron. Comm. Prob., 13, 435{447.

25 references, page 1 of 2
Powered by OpenAIRE Open Research Graph
Any information missing or wrong?Report an Issue
publication . Article . Preprint . 2009

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

Kösters, Holger;