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Sistemy i Sredstva Inform., 2012, Volume 22, Issue 1, Pages 180–204 (Mi ssi274)  

This article is cited in 3 scientific papers (total in 3 papers)

On nonuniform estimates of the rate of convergence in the central limit theorem

M. E. Grigor'eva, S. V. Popov

M. V. Lomonosov Moscow State University, Faculty of Computational Mathematics and Cybernetics

Abstract: It is shown that in the nonuniform analog of the Berry–Esseen inequality $(1+|x|^3)|F_n(xB_n)-\Phi(x)|\le (C/{B_n^3})\sum\limits_{k=1}^n\beta_k$, $n\ge1$, $x\in\mathbb R$, where $F_n(x)$ is the distribution function of the sum of $n$ independent random variables $X_1, …,X_n$ with $E X_k=0$, $E X_k^2=\sigma_k^2$; $\beta_k=E |X_k|^3<\infty$, $k=1,…,n$; $B_n^2=\sigma_1^2+\dotsb+\sigma_n^2$; $\Phi(x)$ is the standard normal distribution function, the absolute constant $C$ satisfies the inequality $C\le 22.2417$.

Keywords: central limit theorem; nonuniform estimate of convergence rate; Berry–Esseen inequality; absolute constant

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Received: 03.06.2012

Citation: M. E. Grigor'eva, S. V. Popov, “On nonuniform estimates of the rate of convergence in the central limit theorem”, Sistemy i Sredstva Inform., 22:1 (2012), 180–204

Citation in format AMSBIB
\Bibitem{GriPop12}
\by M.~E.~Grigor'eva, S.~V.~Popov
\paper On nonuniform estimates of the rate of convergence in the central limit theorem
\jour Sistemy i Sredstva Inform.
\yr 2012
\vol 22
\issue 1
\pages 180--204
\mathnet{http://mi.mathnet.ru/ssi274}


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    See also

    This publication is cited in the following articles:
    1. Grigor'eva M.E., Popov S.V., “An Upper Bound for the Absolute Constant in the Nonuniform Version of the Berry-Esseen Inequalities for Nonidentically Distributed Summands”, Dokl. Math., 86:1 (2012), 524–526  crossref  mathscinet  zmath  isi  elib  scopus
    2. I. Shevtsova, “On the absolute constants in Nagaev–Bikelis-type inequalities”, Inequalities and Extremal Problems in Probability and Statistics: Selected Topics, ed. I. Pinelis, Academic Press Ltd-Elsevier Science Ltd, 2017, 47–102  crossref  mathscinet  isi  scopus
    3. I. Pinelis, “On the nonuniform Berry–Esseen bound”, Inequalities and Extremal Problems in Probability and Statistics: Selected Topics, ed. I. Pinelis, Academic Press Ltd-Elsevier Science Ltd, 2017, 103–138  crossref  mathscinet  isi  scopus
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