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Teor. Veroyatnost. i Primenen., 1966, Volume 11, Issue 1, Pages 108–119 (Mi tvp570)  

This article is cited in 18 scientific papers (total in 18 papers)

An absolute estimate of the remainder in the central limit theorem

V. M. Zolotarev

Moscow

Abstract: Let $\xi_1,…\xi_n$ be independent random varibles with zero means, variances $\sigma_1,…\sigma_n$ and third absolute moments $\beta_1…\beta_n$. Let us denote
$$ \sigma^2=\sum_j\sigma_j^2,\quad\varepsilon=(\sum_j\beta_j)/\sigma^3, $$
and let $F(x)$ be the distribution function of the sum $\xi_1+…+\xi_n$ and $\Phi(x)$ be the distribution function of the normal $(0,1)$ law. Let further $\varepsilon$ be equal to a fixed positive number and $D(\varepsilon)$ denote the least value for which
$$ \sup_x|F(x\sigma)-\Phi(x)|\le D(\varepsilon)\varepsilon. $$
Estimates of $D(\varepsilon)$ for all $\varepsilon$, $0\le\varepsilon\le0.79$ are obtained and the inequality
$$ \sup_\varepsilon D(\varepsilon)<1.322 $$
is proved.

Full text: PDF file (678 kB)

English version:
Theory of Probability and its Applications, 1966, 11:1, 95–105

Bibliographic databases:

Received: 04.11.1965

Citation: V. M. Zolotarev, “An absolute estimate of the remainder in the central limit theorem”, Teor. Veroyatnost. i Primenen., 11:1 (1966), 108–119; Theory Probab. Appl., 11:1 (1966), 95–105

Citation in format AMSBIB
\Bibitem{Zol66}
\by V.~M.~Zolotarev
\paper An absolute estimate of the remainder in the central limit theorem
\jour Teor. Veroyatnost. i Primenen.
\yr 1966
\vol 11
\issue 1
\pages 108--119
\mathnet{http://mi.mathnet.ru/tvp570}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=198531}
\zmath{https://zbmath.org/?q=an:0154.43602}
\transl
\jour Theory Probab. Appl.
\yr 1966
\vol 11
\issue 1
\pages 95--105
\crossref{https://doi.org/10.1137/1111005}


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  • Теория вероятностей и ее применения Theory of Probability and its Applications
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