On the rate of convergence to the semi-circular law
By Friedrich Götze, A. Tikhomirov. 2011
47 citations recorded by the source corpus.
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 \(b\;>\;0\)
Subfields: Mathematical Physics, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/b03bebd877970cf57c1b54de60f7955ec923d34b
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In Mathematical Physics
In Statistics and Probability
Citations in the corpus
- Random covariance matrices: Universality of local statistics of eigenvalues (2009, background): “ing, that lemma was phrased for the semi-circular distribution rather than the Marchenko– Pastur distribution, but an inspection of the proof shows the proof can be modified without difficulty. See also [15] and [7], Corollary 4.2, for closely related lemmas. In view of this lemma, we see that to show ”
- Random matrices: Sharp concentration of eigenvalues (2012, background): “mes non-trivial when T ≫ logC n for some sufficiently large C. In that regime, we see that this result removes the double-logarithmic factor from Theorem 7; it is also comparable to the result (9) from [20] when T = logO(1)n (though not with as sharp a set of exponents as [20], and one also needs an a”
- Random matrix theory in statistics: A review (2014, background): “In Götze and Tikhomirov (2003, 2004), for both Wigner and Wishart matrices, assuming sufficient number of moments for the entries, it was shown that as long as γ is away from 1, the rate of convergence of expected ESD and the ESD to the limiting ESD is Oðn 1=2Þ.”