The rate of convergence for spectra of GUE and LUE matrix ensembles
Friedrich Götze ; Alexander Tikhomirov
Open Mathematics, Tome 3 (2005), p. 666-704 / Harvested from The Polish Digital Mathematics Library

We obtain optimal bounds of order O(n −1) for the rate of convergence to the semicircle law and to the Marchenko-Pastur law for the expected spectral distribution functions of random matrices from the GUE and LUE, respectively.

Publié le : 2005-01-01
EUDML-ID : urn:eudml:doc:268703
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     author = {Friedrich G\"otze and Alexander Tikhomirov},
     title = {The rate of convergence for spectra of GUE and LUE matrix ensembles},
     journal = {Open Mathematics},
     volume = {3},
     year = {2005},
     pages = {666-704},
     zbl = {1108.60014},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_BF02475626}
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Friedrich Götze; Alexander Tikhomirov. The rate of convergence for spectra of GUE and LUE matrix ensembles. Open Mathematics, Tome 3 (2005) pp. 666-704. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_BF02475626/

[1] R. Askey and S. Wainger: “Mean convergence of expansion in Laguerre and Hermitean series”, American Journal of Mathematics, Vol. 87, (1965), pp. 695–707. http://dx.doi.org/10.2307/2373069 | Zbl 0125.31301

[2] Z.D. Bai: “Convergence rate of expected spectral distributions of large random matrices. Part I. Wigner matrices”, Ann. Probab., Vol. 21, (1993), pp. 625–648. | Zbl 0779.60024

[3] Z.D. Bai: “Convergence rate of expected spectral distributions of large random matrices. II. Sample covariance matrices”, Ann. Probab., Vol. 21, (1993), pp. 649–672. | Zbl 0779.60025

[4] Z.D. Bai: “Methodologies in spectral analysis of large dimensional random matrices: a review”, Statistica Sinica, Vol. 9, (1999), pp. 611–661. | Zbl 0949.60077

[5] Z.D. Bai, B. Miao and J. Tsay: “Convergence rates of the spectral distributions of large Wigner matrices”, Int. Math. J., Vol. 1 (2002), pp. 65–90. | Zbl 0987.60050

[6] Z.D. Bai, B. Miao and J.-F. Yao: “Convergence rate of spectral distributions of large sample covariance matrices”, SIAM J. Matrix Anal. Appl., Vol. 25, (2003), pp. 105–127. http://dx.doi.org/10.1137/S0895479801385116 | Zbl 1059.60036

[7] P. Deift: Orthogonal Polynomials and Random Matrices: A Rieman-Hilbert Approach, Courant Lectures Notes, Vol. 3, Amer. Math. Soc., 2000.

[8] P. Deift, T. Kriecherbauer, K.D.T.-R. McLaughlin, S. Venakides and X. Zhou: “Strong asymptotics of orthogonal polynomials with respect to exponential weights”, Comm. Pure and Applied Math., Vol. LII, (1999), pp. 1491–1552. http://dx.doi.org/10.1002/(SICI)1097-0312(199912)52:12<1491::AID-CPA2>3.0.CO;2-# | Zbl 1026.42024

[9] N.M. Ercolani and K.D.T.-R. McLaughlin: “Asymptotics of the partition function for random matrices via Riemann-Hilbert techniques and applications to grafical enumeration”, Int. Math. Res. Not., Vol. 14, (2003), pp. 755–820. http://dx.doi.org/10.1155/S1073792803211089 | Zbl 1140.82307

[10] A. Erdelyi: “Asymptotic solutions of differencial equations with transition points or singularities”, J. Math. Phys., Vol. 1, (1960), pp. 16–26. http://dx.doi.org/10.1063/1.1703631 | Zbl 0125.04802

[11] P. Forrester: Log-gases and Random Matrices, Book Manuscript: www.ms.unimelb.edu.au/∼matpjf/matpjf.html | Zbl 1217.82003

[12] V.L. Girko: “Asymptotics distribution of the spectrum of random matrices”, Russian Math. Surveys., Vol. 44, (1989), pp. 3–36. http://dx.doi.org/10.1070/RM1989v044n04ABEH002143 | Zbl 0717.60033

[13] V.L. Girko: “Convergence rate of the expected spectral functions of symmetric random matrices equals to O(n −1/2)”, Random Oper. and Stoch. Equ., Vol. 6, (1998), pp. 359–406. http://dx.doi.org/10.1515/rose.1998.6.4.359 | Zbl 0912.60004

[14] V.L. Girko: “Extended proof of the statement: Convergence rate of the expected spectral functions of symmetric random matrices Ξn is equal to O(n −1/2) and the method of critical steepest descent”, Random Oper. and Stoch. Equ., Vol. 10, (2002), pp. 253–300. | Zbl 1010.62041

[15] N.R. Goodman: “Statistical analysis based on a certain multivariate complex Gaussian distribution (an introduction)”, Ann. Math. Statistics, Vol. 34, (1963), pp. 152–177. | Zbl 0122.36903

[16] F. Götze and A.N. Tikhomirov: “Rate of convergence to the semi-circular law for the Gaussian Unitary Ensemble”, Theory Probab. Appl., Vol. 47, (2002), pp. 381–388. | Zbl 1041.60033

[17] F. Götze and A.N. Tikhomirov: “Rate of convergence to the semi-circular law”, Probab. Theory Relat. Fields, Vol. 127, (2003), pp. 228–276. http://dx.doi.org/10.1007/s00440-003-0285-z | Zbl 1031.60019

[18] F. Götze and A.N. Tikhomirov: “Rate of Convergence in Probability to the Marchenko-Pastur Law”, Bernoulli, Vol. 10(1), (2004), 1–46. http://dx.doi.org/10.3150/bj/1077544601 | Zbl 1049.60018

[19] F. Götze and A.N. Tikhomirov: Limit theorems for spectra of random matrices with martingale structure, Bielefeld University, Preprint 03-018 2003, www.mathemathik.uni-bielefeld.de/fgweb/preserv.html

[20] I.S. Gradstein and I.M. Ryzhik: Table of Integrals, Series, and Products, Academic Press, Inc. New York, 1994.

[21] J. Gustavsson: Gaussian fluctuations of eigenvalues in the GUE, arXiv: math. PR/0401076 v1, 1–27, (2004).

[22] U. Haagerup and S. Thorbjørnsen: Random matrices with complex Gaussian entries, Expo. Math., Vol. 21, (2003), pp. 293–337. | Zbl 1041.15018

[23] R. Janik and M. Nowak: “Wishart and anti-Wishart random matrices”, J. of Phys. A: Math. Gen., Vol. 36, (2003), pp. 3629–3637. http://dx.doi.org/10.1088/0305-4470/36/12/343

[24] M. Ledoux: Differential operators and spectral distribution functions of invariant ensembles from the classical orthogonal polynomials. The continuous case, Preprint, University of Toulouse, 2002, pp. 1–31.

[25] V.M. Marchenko and L.A. Pastur: “The eigenvalue distribution in some ensembles of random matrices”, Math. USSR Sbornik, Vol. 1, (1967), pp. 457–483. http://dx.doi.org/10.1070/SM1967v001n04ABEH001994 | Zbl 0162.22501

[26] M.L. Mehta: Random matrices, 2nd ed., Academic Press, San Diego, 1991.

[27] B. Muckenhaupt: “Mean convergence of Hermitian and Laguerre series I, II”, Trans. American. Math. Soc., Vol. 147, (1970), pp. 419–460. http://dx.doi.org/10.2307/1995204

[28] G. Szegö: Orthogonal Polynomials, American Math. Soc., New York, 1967.