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

F. Götze; A. N. Tikhomirov

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arXiv · 1407.2780

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

F. Götze·A. N. 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 $\mathbf W=\frac1{\sqrt n}\mathbf X$ 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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BibTeX RIS

F. Götze, A. N. Tikhomirov. 2015-02-08. Rate of Convergence of the Empirical Spectral Distribution Function to the Semi-Circular Law. https://arxiv.org/abs/1407.2780

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