On the Rate of Convergence to the Semi-circular Law

Abstract

Let $X\;=\;(X_{jk})^n_{j,k=1}$ denote a Hermitian random matrix with entries X _jk, which are independent for $1\;\leq\;j\;\leq\;k\;\leq\;n$. We consider the rate of convergence of the empirical spectral distribution function of the matrix X to the semi-circular law assuming that $\mathbf{E}X_{jk}\;=\;0,\;\mathbf{E}X^2_{jk}\;=\;1$ and that the distributions of the matrix elements X _jk have a uniform sub exponential decay in the sense that there exists a constant ϰ> 0 such that for any $1\;\leq\;j\;\leq\;k\;\leq\;n$ and any $t\;\geq\;1$ we have

$$\mathrm{Pr}\left\{|X_{jk}|\;>\;t \right\}\leq\;x^{-1}\;\exp\left\{-t^x\right\}$$

By means of a short recursion argument it is shown that the Kolmogorov distance between the empirical spectral distribution of the Wigner matrix $\mathbf{W}\;=\;\frac{1}{\sqrt{n}}\mathbf{X}$ and the semicircular law is of order $O(n^{-1}\;\log^b\;n)$ with some positive constant $b\;>\;0$

Mathematics Subject Classification (2010). Primary 60F99; Secondary 60B20.