On the asymptotical behavior of the constant in the Berry-Esseen inequality期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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On the asymptotical behavior of the constant in the Berry-Esseen inequality

Authors:

Vidmantas Bentkus

Institution:

1. Institute of Mathematics and Informatics, Akademijos 4, 2600, Vilnius, Lithuania

Abstract:

LetF be the distribution function of a sumS _n ofn independent centered random variables, PHgr

denote the standard normal distribution function and phiv

its density. It follows from our results that

$\|F(x\sigma ) - \Phi (x)\| \leqslant \varepsilon \left( {\phi (x) + \frac{1}{{6\sqrt {2\pi } }}} \right) + c\varepsilon ^{{\raise0.7ex\hbox{$4$} \!\mathord{\left/ {\vphantom {4 3}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$3$}}} \leqslant \frac{{7\varepsilon }}{{6\sqrt {2\pi } }} + c\varepsilon ^{{\raise0.7ex\hbox{$4$} \!\mathord{\left/ {\vphantom {4 3}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$3$}}}$

Keywords:	Berry-Esseen estimate constant asymptotic of the constant
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