# High Dimensional Probability VIOn the Rate of Convergence to the Semi-circular Law

High Dimensional Probability VI: On the Rate of Convergence to the Semi-circular Law [Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$X\;=\;(X_{jk})^n_{j,k=1}$$ \end{document} denote a Hermitian random matrix with entries Xjk, which are independent for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$1\;\leq\;j\;\leq\;k\;\leq\;n$$ \end{document}. We consider the rate of convergence of the empirical spectral distribution function of the matrix X to the semi-circular law assuming that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbf{E}X_{jk}\;=\;0,\;\mathbf{E}X^2_{jk}\;=\;1$$ \end{document} and that the distributions of the matrix elements Xjk have a uniform sub exponential decay in the sense that there exists a constant ϰ> 0 such that for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$1\;\leq\;j\;\leq\;k\;\leq\;n$$ \end{document} and any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$t\;\geq\;1$$ \end{document} we have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathrm{Pr}\left\{|X_{jk}|\;>\;t \right\}\leq\;x^{-1}\;\exp\left\{-t^x\right\}$$ \end{document} By means of a short recursion argument it is shown that the Kolmogorov distance between the empirical spectral distribution of the Wigner matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbf{W}\;=\;\frac{1}{\sqrt{n}}\mathbf{X}$$ \end{document} and the semicircular law is of order \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$O(n^{-1}\;\log^b\;n)$$ \end{document} with some positive constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$b\;>\;0$$ \end{document}] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

# High Dimensional Probability VIOn the Rate of Convergence to the Semi-circular Law

Part of the Progress in Probability Book Series (volume 66)
Editors: Houdré, Christian; Mason, David M.; Rosiński, Jan; Wellner, Jon A.
27 pages

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# References (15)

Publisher
Springer Basel
Copyright
© Springer Basel 2013
ISBN
978-3-0348-0489-9
Pages
139 –165
DOI
10.1007/978-3-0348-0490-5_10
Publisher site
See Chapter on Publisher Site

### Abstract

[Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$X\;=\;(X_{jk})^n_{j,k=1}$$ \end{document} denote a Hermitian random matrix with entries Xjk, which are independent for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$1\;\leq\;j\;\leq\;k\;\leq\;n$$ \end{document}. We consider the rate of convergence of the empirical spectral distribution function of the matrix X to the semi-circular law assuming that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbf{E}X_{jk}\;=\;0,\;\mathbf{E}X^2_{jk}\;=\;1$$ \end{document} and that the distributions of the matrix elements Xjk have a uniform sub exponential decay in the sense that there exists a constant ϰ> 0 such that for any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$1\;\leq\;j\;\leq\;k\;\leq\;n$$ \end{document} and any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$t\;\geq\;1$$ \end{document} we have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathrm{Pr}\left\{|X_{jk}|\;>\;t \right\}\leq\;x^{-1}\;\exp\left\{-t^x\right\}$$ \end{document} By means of a short recursion argument it is shown that the Kolmogorov distance between the empirical spectral distribution of the Wigner matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbf{W}\;=\;\frac{1}{\sqrt{n}}\mathbf{X}$$ \end{document} and the semicircular law is of order \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$O(n^{-1}\;\log^b\;n)$$ \end{document} with some positive constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$b\;>\;0$$ \end{document}]

Published: Apr 1, 2013

Keywords: Spectral distribution function; semi-circular law

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