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 m_n(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${v}$)$^{{ - 1}}lnn$. 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.