It is shown that the Kolmogorov distance between the expected spectral distribution function of an n × n matrix from the Deformed Gaussian Ensemble and the distribution function of the semi-circle law is of order O(n −2/3+v ).
@article{bwmeta1.element.doi-10_2478_s11533-007-0006-4,
author = {Friedrich G\"otze and Alexander Tikhomirov and Dmitry Timushev},
title = {Rate of convergence to the semi-circle law for the Deformed Gaussian Unitary Ensemble},
journal = {Open Mathematics},
volume = {5},
year = {2007},
pages = {305-334},
zbl = {1155.15027},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-007-0006-4}
}
Friedrich Götze; Alexander Tikhomirov; Dmitry Timushev. Rate of convergence to the semi-circle law for the Deformed Gaussian Unitary Ensemble. Open Mathematics, Tome 5 (2007) pp. 305-334. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-007-0006-4/
[1] Z. D. Bai: “Convergence rate of expected spectral distributions of large random matrices. I. Wigner matrices”, Ann. Probab., Vol. 21, (1993), pp. 625–648. | Zbl 0779.60024
[2] 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
[3] Z. D. Bai: “Remarks on the convergence rate of the spectral distributions of Wigner matrices”, J. Theoret. Probab., Vol. 12, (1999), pp. 301–311. http://dx.doi.org/10.1023/A:1021617825555
[4] Z. D. Bai, B. Miao, J. Tsay: “Convergence rate of the spectral distributions of large Wigner matrices”, Int. Math. J., Vol. 1, (2002), pp. 65–90. | Zbl 0987.60050
[5] P. Deift, T. Kriecherbauer, K. D. T.-R. McLaughlin, S. Venakides, X. Zhou: “Strong asymptotics of orthogonal polynomials with respect to exponential weights”, Comm. Pure Appl. Math., Vol. 52, (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
[6] N. M. Ercolani, K. D. T.-R. McLaughlin: “Asymptotics of the partition function for random matrices via Riemann-Hilbert techniques and applications to graphical enumeration”, Int. Math. Res. Not., Vol. 14, (2003), pp. 755–820. http://dx.doi.org/10.1155/S1073792803211089 | Zbl 1140.82307
[7] V. L. Girko: “Convergence rate of the expected spectral functions of symmetric random matrices equals to O(n −1/2 )”, Random Oper. Stochastic Equations, Vol. 6, (1998), pp. 359–406. | Zbl 0912.60004
[8] 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. Stochastic Equations, Vol. 10, (2002), pp. 253–300. http://dx.doi.org/10.1515/rose.2002.10.3.253 | Zbl 1010.62041
[9] F. Götze, E. F. Kushmanova, A. N. Tikhomirov: “Rate of convergence to the semicircular law almost surely”, In preparation.
[10] F. Götze, A. N. Tikhomirov: “Rate of convergence in probability to the Marchenko-Pastur law”, Bernuolii, Vol. 10(1), (2004), pp. 1–46. http://dx.doi.org/10.3150/bj/1077544601 | Zbl 1049.60018
[11] F. Götze, 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
[12] F. Götze, A. N. Tikhomirov: “Rate of convergence to the semi-circular law for the Gaussian unitary ensemble”, Teor. Veroyatnost. i Primenen., Vol. 47, (2002), pp. 381–387. | Zbl 1041.60033
[13] F. Götze, A. N. Tikhomirov: “The rate of convergence for the spectra of GUE and LUE matrix ensembles”, Cent. Eur. J. Math., Vol. 3, (2005), pp. 666–704. http://dx.doi.org/10.2478/BF02475626 | Zbl 1108.60014
[14] K. Johansson: “Universality of the local spacing distribution in certain ensembles of Hermitian Wigner matrices”, Comm. Math. Phys., Vol. 215, (2001), pp. 683–705. http://dx.doi.org/10.1007/s002200000328 | Zbl 0978.15020
[15] A. I. Markushevich: Theory of Functions of a Complex Variable, 2nd ed., Chelsea Publishing Company, New York, 1977.
[16] M. L. Mehta: Random Matrices, 2nd ed., Academic Press, San Diego, 1991.
[17] L. A. Pastur: “Random matrices as paradigm”, In: Mathematical physics 2000, Imp. Coll. Press, London, 2000, pp. 216–265. | Zbl 1017.82023
[18] L. A. Pastur: “Spectra of random self-adjoint operators”, Russian Math. Surveys, Vol. 28, (1973), pp. 1–67. http://dx.doi.org/10.1070/rm1973v028n01ABEH001396 | Zbl 0277.60049
[19] E. P. Wigner: “On the characteristic vectors of bordered matrices with infinite dimensions”, Ann. of Math., Vol. 62, (1955), pp. 548–564. http://dx.doi.org/10.2307/1970079 | Zbl 0067.08403