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Teor. Veroyatnost. i Primenen., 1967, Volume 12, Issue 3, Pages 562–567 (Mi tvp740)  

Short Communications

On a refinemet of the central limit theorem and its global version

Yu. P. Studnev, Yu. I. Ignat

Uzhgorod

Abstract: Let $\{\xi_k\}$ be a sequence of independent random variables with zero means and $\{\sigma_k^{(2)}\}$ be the sequence of their variances. Denote
\begin{gather*} s_n=\frac{\xi_1+…+\xi_n}{B_n},\quad B_n=\sum_{k=1}^n\sigma_k^2,\quad\Phi_n(x)=\mathbf P(s_n<x)
L_n(x)=\frac1{B_n^2}\sum_{k=1}^n\int_{|z|>x}z^2 dF_k(z),\quad\Phi(x)=\frac1{\sqrt{2\pi}}\int_{-\infty}^xe^{-x^2/2} dt. \end{gather*}
The main result of the paper is the following. Theorem. {\it Under Lindeberg's condition in the central limit theorem the inequality
$$ |\Phi_n(x)-\Phi(x)|<C\min\{\frac1{B_n}\int_0^{B_n}L_n(x) dx,\quad\frac{\frac1{|x|B_n}\int_0^{|x|B_n}L_n(x) dx}{1+x^2}\}, $$
holds true where $C$ is an absolute constant}.

Full text: PDF file (321 kB)

English version:
Theory of Probability and its Applications, 1967, 12:3, 508–512

Bibliographic databases:

Received: 10.09.1966

Citation: Yu. P. Studnev, Yu. I. Ignat, “On a refinemet of the central limit theorem and its global version”, Teor. Veroyatnost. i Primenen., 12:3 (1967), 562–567; Theory Probab. Appl., 12:3 (1967), 508–512

Citation in format AMSBIB
\Bibitem{StuIgn67}
\by Yu.~P.~Studnev, Yu.~I.~Ignat
\paper On a~refinemet of the central limit theorem and its global version
\jour Teor. Veroyatnost. i Primenen.
\yr 1967
\vol 12
\issue 3
\pages 562--567
\mathnet{http://mi.mathnet.ru/tvp740}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=215348}
\zmath{https://zbmath.org/?q=an:0178.21101}
\transl
\jour Theory Probab. Appl.
\yr 1967
\vol 12
\issue 3
\pages 508--512
\crossref{https://doi.org/10.1137/1112063}


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