Optimal Bounds for Convergence of Expected Spectral Distributions 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 $\mathbf{X}$ to the semi-circular law assuming that $\mathbf{E}{X}_{jk}=0$, $\mathbf{E}{X}_{jk}^2=1$ and that $$\underset{n\ge 1}{\sup }\underset{1\le j,k\le n}{\sup }\mathbf{E}|X_{jk}{|}^4=:{\mu }_4<\infty \text{and}\underset{1\le j,k\le n}{\sup }|X_{jk}|\le D_0{n}^{\frac{1}{4}}.$$ By means of a recursion argument it is shown that the Kolmogorov distance between the expected spectral distribution of the Wigner matrix $\mathbf{W}=\frac{1}{\sqrt{n}}\mathbf{X}$ and the semicircular law is of order $O(n^{-1})$.

Some typos and signs e.g. in (3.2), (5.32) and other locations corrected

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