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Regular version of the site

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

Doklady Mathematics. 2019. Vol. 99. No. 1. P. 40-43.
Naumov A., Tikhomirov A., Гётце Ф.

We consider symmetric random matrices whose upper triangular entries are independent random variables with zero mean and unit variance. Under the assumption of finite fourth moment 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 about the Stieltjes transform m_{sc}(𝑧) of Wigner’s semicircle law are of order (nv)^{-1} log n. An application of the result obtained to the convergence rate in probability of the empirical spectral distribution function to Wigner’s semicircle law in the uniform metric is discussed.