Abstract
Let X:= (X jk ) denote a Hermitian random matrix with entries X jk which are independent for all 1 ≤ j ≤ k. We study the rate of convergence of the expected spectral distribution function of the matrix X to the semi-circular law under the conditions E X jk = 0, E X 2 jk = 1, and E|X jk |2+η ≤ M η < ∞, 0 < η ≤ 2. The bounds of order \( O(n^{ - \frac{\eta } {{2 + \eta }}} ) \) for 1 ≤ η ≤ 2, and those of order \( O(n^{ - \frac{{2\eta }} {{(2 + \eta )(3 - \eta )}}} ) \) for 0 < η ≤ 1, are obtained.
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References
Z. D. Bai, B. Miao, and J. Tsay, “Convergence rates of the spectral distributions of large Wigner matrices,” Int.Math. J. 1, 65–90 (2002).
S. Bobkov, F. Götze, and A. N. Tikhomirov, On Concentration of EmpiricalMeasures and Convergence to the Semi-circle Law, Preprint 08-110 (Universität Bielefeld, SFB701, 2008).
V. L. Girko, “Extended proof of the statement: Convergence rate of expected spectral functions of symmetric random matrices Σn is equal O(n −½) and the method of critical stepest descent,” Random Oper. Stoch. Equ. 10, 253–300 (2002).
F. Götze and A. N. Tikhomirov, “The rate of convergence for spectra of GUE and LUE matrix ensembles,” Cent. Eur. J. Math. 3(4), 666–704 (2005).
F. Götze and A. N. Tikhomirov, “Rate of convergence to the semi-circular law,” Probab. Theory Relat. Fields 127, 228–276 (2003).
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Tikhomirov, A.N. On the rate of convergence of the expected spectral distribution function of a Wigner matrix to the semi-circular law. Sib. Adv. Math. 19, 211–223 (2009). https://doi.org/10.3103/S1055134409030067
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DOI: https://doi.org/10.3103/S1055134409030067