Open AccessOn optimal bounds in the local semicircle law under four moment condition
Author(s) -
F. Götze,
A. A. Naumov,
A. N. Tikhomirov
Publication year - 2019
Publication title -
doklady akademii nauk. rossijskaâ akademiâ nauk
Language(s) - English
Resource type - Journals
ISSN - 0869-5652
DOI - 10.31857/s0869-56524843265-268
Subject(s) - mathematics , moment (physics) , distribution (mathematics) , spectral power distribution , zero (linguistics) , random matrix , triangular matrix , mathematical analysis , convergence (economics) , second moment of area , circular law , order (exchange) , matrix (chemical analysis) , function (biology) , rate of convergence , probability distribution , law , random variable , geometry , convergence of random variables , pure mathematics , statistics , eigenvalues and eigenvectors , key (lock) , physics , philosophy , materials science , economic growth , ecology , linguistics , sum of normally distributed random variables , optics , composite material , biology , classical mechanics , quantum mechanics , evolutionary biology , finance , economics , invertible matrix , political science
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.