Abstract
Let \(X\;=\;(X_{jk})^n_{j,k=1}\) denote a Hermitian random matrix with entries X jk, which are independent for \(1\;\leq\;j\;\leq\;k\;\leq\;n\). We consider the rate of convergence of the empirical spectral distribution function of the matrix X to the semi-circular law assuming that \(\mathbf{E}X_{jk}\;=\;0,\;\mathbf{E}X^2_{jk}\;=\;1\) and that the distributions of the matrix elements X jk have a uniform sub exponential decay in the sense that there exists a constant ϰ> 0 such that for any \(1\;\leq\;j\;\leq\;k\;\leq\;n\) and any \(t\;\geq\;1\) we have
By means of a short recursion argument it is shown that the Kolmogorov distance between the empirical spectral distribution of the Wigner matrix \(\mathbf{W}\;=\;\frac{1}{\sqrt{n}}\mathbf{X}\) and the semicircular law is of order \(O(n^{-1}\;\log^b\;n)\) with some positive constant \(b\;>\;0\)
Mathematics Subject Classification (2010). Primary 60F99; Secondary 60B20.
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Götze, F., Tikhomirov, A. (2013). On the Rate of Convergence to the Semi-circular Law. In: Houdré, C., Mason, D., Rosiński, J., Wellner, J. (eds) High Dimensional Probability VI. Progress in Probability, vol 66. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0490-5_10
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DOI: https://doi.org/10.1007/978-3-0348-0490-5_10
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