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

This article is cited in 2 scientific papers (total in 2 papers)

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

I. A. Ibragimov

Leningrad

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}.

Full text: PDF file (1151 kB)

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

Bibliographic databases:

Received: 06.11.1965

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  mathnet  crossref  crossref  mathscinet  isi
    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  crossref  crossref  mathscinet  zmath  isi  elib  scopus
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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