Convergence Rates of the Spectral Distributions of Large Random Quaternion Self-Dual Hermitian Matrices

Abstract

In this paper, convergence rates of the spectral distributions of quaternion self-dual Hermitian matrices are investigated. We show that under conditions of finite 6th moments, the expected spectral distribution of a large quaternion self-dual Hermitian matrix converges to the semicircular law in a rate of \(O(n^{-1/2})\) and the spectral distribution itself converges to the semicircular law in rates \(O_p(n^{-2/5})\) and \(O_{a.s.}(n^{-2/5+\eta })\). Those results include GSE as a special case.