On Optimal Bounds in the Local Semicircle Law under Four Moment Condition

We consider symmetric random matrices , whose upper triangular entries are independent random variables with zero mean and unit variance. Under the assumption , j, k = 1, 2, ..., n, it is shown that the fluctuations of the Stieltjes transform m(n)(z), of the empirical spectral distribution function of the matrix about the Stieltjes transform of Wigner's semicircle law are of order (n). An application of the result obtained to the convergence rate in probability of the empirical spectral distribution function of to Wigner's semicircle law in the uniform metric is discussed.