On the rate of convergence to the semi-circular law

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 $$\mathrm{Pr}\left\{|X_{jk}|\;>\;t \right\}\leq\;x^{-1}\;\exp\left\{-t^x\right\}$$ 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 \(b\;>\;0\)