Rate of Convergence of the Empirical Spectral Distribution Function to the Semi-Circular Law

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Let X=(Xjk)ⱼ,ₖ₌₁ⁿ denote a Hermitian random matrix with entries Xjk, which are independent for 1≤ j≤ k≤ n. We consider the rate of convergence of the empirical spectral distribution function of the matrix W=1√ nX to the semi-circular law assuming that E Xjk=0, E Xjk²=1 and uniformly bounded eight moments. By means of a recursion argument it is shown that the Kolmogorov distance between the empirical spectral distribution of the Wigner matrix W and the semi--circular law is of order O(n⁻¹⁵n) with high probability.