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arXiv · 1405.7820

Optimal Bounds for Convergence of Expected Spectral Distributions to the Semi-Circular Law

F. Götze·A. Tikhomirov
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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 $$ \sup_{n\ge1}\sup_{1\le j,k\le n}{\mathbf E}|X_{jk}|^4=:\mu_4<\infty \quad \text{and} \sup_{1\le j,k\le n}|X_{jk}|\le D_0n^{\frac14}. $$ By means of a recursion argument it is shown that the Kolmogorov distance between the expected spectral distribution of the Wigner matrix $\mathbf W=\frac1{\sqrt n}\mathbf X$ and the semicircular law is of order $O(n^{-1})$.

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F. Götze, A. Tikhomirov. 2014-05-30. Optimal Bounds for Convergence of Expected Spectral Distributions to the Semi-Circular Law. https://arxiv.org/abs/1405.7820

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