Rate of Convergence of the Empirical Spectral Distribution Function to the Semi-Circular Law

Abstract

Let $\mathbf{X}=(X_{jk}{)}_{j,k=1}^n$ denote a Hermitian random matrix with entries $X_{jk}$, which are independent for $1\le j\le k\le n$. We consider the rate of convergence of the empirical spectral distribution function of the matrix $\mathbf{W}=\frac{1}{\sqrt{n}}\mathbf{X}$ to the semi-circular law assuming that $\mathbf{E}{X}_{jk}=0$, $\mathbf{E}{X}_{jk}^2=1$ and uniformly bounded eight moments. By means of a recursion argument it is shown that the Kolmogorov distance between the empirical spectral distribution of the Wigner matrix $\mathbf{W}$ and the semi--circular law is of order $O(n^{-1}{\log }^{5}n)$ with high probability.

Version 2 got mixed up with Version 1 in the last submission. Here are now the intended changes: On p. 10 definition of ${\tilde{\Lambda }}_n^{(j)}$ has been changed, followed by appropriate changes in (6.29)-(6.58) etc. arXiv admin note: substantial text overlap with arXiv:1405.7820

Cited by in corpus (2)

• Local semicircle law under moment conditions. Part I: The Stieltjes transform
• Local semicircle law under moment conditions. Part II: Localization and delocalization