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Rate of Convergence of the Empirical Spectral Distribution Function to the Semi-Circular Law (1407.2780v3)

Published 10 Jul 2014 in math.PR, math-ph, and math.MP

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 $\mathbf W=\frac1{\sqrt n}\mathbf X$ 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.

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