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


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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.

作者简介

F. Götze

University of Bielefeld

Email: anaumov@hse.ru
德国, Bielefeld

A. Naumov

National Research University Higher School of Economics; Kharkevich Institute for Information Transmission Problems, Russian Academy of Sciences

编辑信件的主要联系方式.
Email: anaumov@hse.ru
俄罗斯联邦, Moscow, 101000; Moscow, 127994

A. Tikhomirov

National Research University Higher School of Economics; Komi Center of Science, Ural Branch, Russian Academy
of Sciences

Email: anaumov@hse.ru
俄罗斯联邦, Moscow, 101000; Syktyvkar

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