On the rate of convergence to the semi-circular law

Let X = (Xjk) denote a Hermitian random matrix with entries Xjk, which are independent for 1 ≤ j ≤ k. We consider the rate of convergence of the empirical spectral distribution function of the matrix X to the semi-circular law assuming that EXjk = 0, EX 2 jk = 1 and that the distributions of the matrix elements Xjk have a uniform sub exponential decay in the sense that there exists a constant κ > 0 such that for any 1 ≤ j ≤ k ≤ n and any t ≥ 1 we have Pr{|Xjk| > t} ≤ κ−1 exp{−t}. By means of a short recursion argument it is shown that the Kolmogorov distance between the empirical spectral distribution of the Wigner matrix W = 1 √ n X and the semicircular law is of order O(n−1 log n) with some positive constant b > 0.