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Teoriya Veroyatnostei i ee Primeneniya, 1978, Volume 23, Issue 4, Pages 744–761
(Mi tvp3109)
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This article is cited in 19 scientific papers (total in 19 papers)
On the accuracy of the remainder term estimation in the central limit theorem
L. V. Rozovskiĭ Leningrad
Abstract:
Let $X_1,\dots$ be a sequence of independent random variables with a common distribution function $V(x)$. Put
\begin{gather*}
F_n(x)=\mathbf P\biggl\{\frac{1}{b_n}(X_1+\dots+X_n)-a_n<x\biggr\},\\
\Phi(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^x e^{t^2/2}\,dt,\quad\Delta(b_n,a_n)=\sup_x|F_n(x)-\Phi(x)|,\\
\Delta_n=\inf_{a_n,b_n}\Delta(b_n,a_n)
\end{gather*}
where $a_n$, $b_n$ ($b_n>0$) are sequences of real numbers.
The paper deals with questions of the accuracy in estimating $|F_n(x)-\Phi(x)|$ when $V(x)$ belongs to the domain of attraction of a normal law. In particular, necessary and sufficient conditions for
$$
\biggl(\sum_{n=1}^{\infty}(g(n)\Delta_n)^s\frac{1}{n}\biggr)^{1/s}<\infty,\qquad 1\le s\le\infty,
$$
are obtained. (Here $g(x)$ is a function which satisfies some conditions.)
Received: 28.02.1977
Citation:
L. V. Rozovskiǐ, “On the accuracy of the remainder term estimation in the central limit theorem”, Teor. Veroyatnost. i Primenen., 23:4 (1978), 744–761; Theory Probab. Appl., 23:4 (1979), 712–730
Linking options:
https://www.mathnet.ru/eng/tvp3109 https://www.mathnet.ru/eng/tvp/v23/i4/p744
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