On concentration of empirical measures and convergence to the semi-circle law (Q1960238)

Let \((\xi_{i,j})_{1\leq i\leq j\leq n}\) a family of independent centered random variables with variance 1, and consider the eigenvalues \(X_1\leq X_2\leq\ldots\leq X_n\) of the associated random symmetric matrices. It is well known that under mild additional conditions, the empirical means of the \(X_i\) tend to Wigner's semicircle law for \(n\to\infty\) where the rate of convergence is of order \(1/n\) w.r.t. the Kolmogorov distance for Gaussian unitary ensembles. In the non-Gaussian case, no results about order of convergence were known. In the paper under review it is shown that the rate of convergence above is of order at least \(n^{-2/3}\) whenever the distributions of \(\xi_{i,j}\) satisfy a certain Poincaré-type inequality. Moreover, under this condition, a similar rate of convergence is derived for the rate of fluctuations.