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Teor. Veroyatnost. i Primenen., 1976, Volume 21, Issue 1, Pages 107–122 (Mi tvp3278)  

This article is cited in 1 scientific paper (total in 1 paper)

On the accuracy of approximation in the central limit theorem

В. A. Lifšic

Leningrad

Abstract: Let
$$ \Delta_n=\sup_x|\mathbf P(\xi_1+…+\xi_n<x\sqrt n)-\Phi(x)|, $$
where $\xi_1,\xi_2,…$ are independent identically distributed random variables with the distribution function $F(x)$, $\mathbf E|\xi_1|^2=1$, $\mathbf E\xi_1=0$, and where $\Phi$ is the standard normal distribution function.
We investigate necessary and sufficient conditions on $F(x)$ for the following two series to converge:
$$ \sum h(\sqrt n)\frac{1}{n}\Delta_n<\infty,\quad\sum h(\sqrt n)n^{-3/2}\Delta_n<\infty, $$
where
$$ h(y)>0,\qquad h(y)\uparrow,\qquad h(y)/y\downarrow. $$
The case of Chebyshev–Gramer asymptotic expansions is also discussed.

Full text: PDF file (781 kB)

English version:
Theory of Probability and its Applications, 1976, 21:1, 108–124

Bibliographic databases:

Received: 27.08.1974

Citation: В. A. Lifšic, “On the accuracy of approximation in the central limit theorem”, Teor. Veroyatnost. i Primenen., 21:1 (1976), 107–122; Theory Probab. Appl., 21:1 (1976), 108–124

Citation in format AMSBIB
\Bibitem{Lif76}
\by В.~A.~Lif{\v s}ic
\paper On the accuracy of approximation in the central limit theorem
\jour Teor. Veroyatnost. i Primenen.
\yr 1976
\vol 21
\issue 1
\pages 107--122
\mathnet{http://mi.mathnet.ru/tvp3278}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=410860}
\zmath{https://zbmath.org/?q=an:0373.60024}
\transl
\jour Theory Probab. Appl.
\yr 1976
\vol 21
\issue 1
\pages 108--124
\crossref{https://doi.org/10.1137/1121008}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Lifshits M.A. Nikitin Ya.Yu. Petrov V.V. Zaitsev A.Yu. Zinger A.A., “Toward the History of the Saint Petersburg School of Probability and Statistics. i. Limit Theorems For Sums of Independent Random Variables”, Vestnik St. Petersburg Univ. Math., 51:2 (2018), 144–163  crossref  isi
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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