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Journal of Theoretical Probability

, Volume 7, Issue 2, pp 211–224 | Cite as

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

  • Vidmantas Bentkus
Article

Abstract

LetF be the distribution function of a sumSn ofn independent centered random variables, Φ denote the standard normal distribution function and ϕ its density. It follows from our results that
$$|F(x\sigma ) - \Phi (x)| \leqslant \varepsilon \left( {\phi (x) + \frac{1}{{6\sqrt {2\pi } }}} \right) + c\varepsilon ^{{\raise0.7ex\hbox{$4$} \!\mathord{\left/ {\vphantom {4 3}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$3$}}} \leqslant \frac{{7\varepsilon }}{{6\sqrt {2\pi } }} + c\varepsilon ^{{\raise0.7ex\hbox{$4$} \!\mathord{\left/ {\vphantom {4 3}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$3$}}} $$
where ε denotes the Lyapunov fraction,c is an absolute constant and σ2=ES n 2 . For symmetric random variables this estimate may be improved.

Key Words

Berry-Esseen estimate constant asymptotic of the constant 

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Copyright information

© Plenum Publishing Corporation 1994

Authors and Affiliations

  • Vidmantas Bentkus
    • 1
  1. 1.Institute of Mathematics and InformaticsVilniusLithuania

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