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

Abstract

We consider symmetric random matrices $${{{\mathbf{X}}}_{n}} = [{{X}_{{jk}}}]_{{j,k = 1}}^{n},n \geqslant 1$$ , whose upper triangular entries are independent random variables with zero mean and unit variance. Under the assumption $$\mathbb{E}{\text{|}}{{X}_{{jk}}}{{{\text{|}}}^{4}} < C$$ , j, k = 1, 2, ..., n, it is shown that the fluctuations of the Stieltjes transform mn(z), $$z = u + i{v},{v} > 0,$$ of the empirical spectral distribution function of the matrix $${{{\mathbf{X}}}_{n}}{\text{/}}\sqrt n $$ about the Stieltjes transform $${{m}_{{{\text{sc}}}}}(z)$$ of Wigner’s semicircle law are of order (n $${v}$$ ) $$^{{ - 1}}\text{ln}n$$ . An application of the result obtained to the convergence rate in probability of the empirical spectral distribution function of $${{{\mathbf{X}}}_{n}}{\text{/}}\sqrt n $$ to Wigner’s semicircle law in the uniform metric is discussed.