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

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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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} 

• http://mi.mathnet.ru/eng/ssi274
• http://mi.mathnet.ru/eng/ssi/v22/i1/p180

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