On the rate of convergence of the expected spectral distribution function of a Wigner matrix to the semi-circular law |
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Authors: | A N Tikhomirov |
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Institution: | 1. Komi Research Center of Ural Branch of the Russian Academy of Sciences, Syktyvkar, 167001, Russia
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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
jk
2 = 1, and E|X
jk
|2+η
≤ M
η
< ∞, 0 < η ≤ 2. The bounds of order $
O(n^{ - \frac{\eta }
{{2 + \eta }}} )
$
O(n^{ - \frac{\eta }
{{2 + \eta }}} )
for 1 ≤ η ≤ 2, and those of order $
O(n^{ - \frac{{2\eta }}
{{(2 + \eta )(3 - \eta )}}} )
$
O(n^{ - \frac{{2\eta }}
{{(2 + \eta )(3 - \eta )}}} )
for 0 < η ≤ 1, are obtained. |
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Keywords: | |
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