On optimal bounds in the local semicircle law under four moment condition

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Abstract


We consider symmetric random matrices with independent mean zero and unit variance entries in the upper triangular part. Assuming that the distributions of matrix entries have finite moment of order four, we prove optimal bounds for the distance between the Stieltjes transforms of the empirical spectral distribution function and the semicircle law. Application concerning the convergence rate in probability of the empirical spectral distribution to the semicircle law is discussed as well.


F. Götze

Билефельдский университет

Author for correspondence.
Email: anaumov@hse.ru

Germany

A. A. Naumov

Национальный исследовательский университет «Высшая школа экономики»; Институт проблем передачи информации Российской Академии наук

Email: anaumov@hse.ru

Russian Federation, Москва

A. N. Tikhomirov

Национальный исследовательский университет «Высшая школа экономики»; Коми научный центр Уральского отделения Российской Академии наук

Email: anaumov@hse.ru

Russian Federation, Москва; Сыктывкар

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