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 Teor. Veroyatnost. i Primenen.: Year: Volume: Issue: Page: Find

 Teor. Veroyatnost. i Primenen., 1966, Volume 11, Issue 4, Pages 632–655 (Mi tvp663)

On the accuracy of Gaussian approximation to the distribution functions of sums of independent random variables

I. A. Ibragimov

Abstract: Let $\{\xi_n\}$ be a sequence of independent identically distributed random variables with a common distribution function (d.f.) $F(x)$. Let us assume that d.f. belongs to the domain of attraction of the Gaussian law. Denote by $F_n(x;A_n,B_n)$ the d.f. of normalized sum $S_n=\frac1{B_n}\sum_1^n\xi_i-A_n$ and let
$$\delta_n=\inf_{A_n,B_n}\sup_x|F_n(x;A_n,B_n)-\Phi(x)|$$
where $\Phi(x)=\frac1{\sqrt{2\pi}}\int_{-\infty}^xe^{-u^2/2} du$.
We investigate in this paper the rate of convergence of $\delta_n$ to 0 and some other related problems. The main results which are also indicative of the other results of the paper are the following theorems.
Theorem 3.1. {\it In order that $\delta_n=O(n^{-\delta/2})$, $0<\delta<1$, it is necessary and sufficient that the following conditions be satisfied}
$$\sigma^2=\int_{-\infty}^\infty x^2 dF(x)<\infty,\eqno(3.2) \int_{|x|>z}x^2 dF(x)=O(|z|^{-\delta}),\quad z\to\infty.\eqno(3.3)$$

Theorem 3.2. {\it In order that $\delta_n=O(n^{-1/2})$ it is necessary and sufficient that conditions (3.1), (3.2) and the following one
$$\int_{-z}^zx^3 dF(x)=O(1),\quad z\to\infty\eqno(3.4)$$
be satisfied}.

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English version:
Theory of Probability and its Applications, 1966, 11:4, 559–579

Bibliographic databases:

Citation: I. A. Ibragimov, “On the accuracy of Gaussian approximation to the distribution functions of sums of independent random variables”, Teor. Veroyatnost. i Primenen., 11:4 (1966), 632–655; Theory Probab. Appl., 11:4 (1966), 559–579

Citation in format AMSBIB
\Bibitem{Ibr66} \by I.~A.~Ibragimov \paper On the accuracy of Gaussian approximation to the distribution functions of sums of independent random variables \jour Teor. Veroyatnost. i Primenen. \yr 1966 \vol 11 \issue 4 \pages 632--655 \mathnet{http://mi.mathnet.ru/tvp663} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=212853} \zmath{https://zbmath.org/?q=an:0161.15207} \transl \jour Theory Probab. Appl. \yr 1966 \vol 11 \issue 4 \pages 559--579 \crossref{https://doi.org/10.1137/1111061} 

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This publication is cited in the following articles:
1. I. G. Shevtsova, “On the asymptotically exact constants in the Berry–Esseen–Katz inequality”, Theory Probab. Appl., 55:2 (2011), 225–252
2. M. A. Lifshits, Ya. Yu. Nikitin, V. V. Petrov, A. Yu. Zaitsev, A. A. Zinger, “Toward the history of the Saint Petersburg school of probability and statistics. I. Limit theorems for sums of independent random variables”, Vestn. St Petersb. Univ. Math., 51:2 (2018), 144–163
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