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Teor. Veroyatnost. i Primenen., 1971, Volume 16, Issue 3, Pages 535–540 (Mi tvp2266)  

Short Communications

On asymptotic expansions in the central limit theorem

F. N. Galstyan

K. Marx Erevan Polytechnic Institute

Abstract: Let $\{X_k\}$ be a sequence of independent identically distributed random variables with $\mathbf EX_1=0$, $\mathbf EX_1^2=1$ and $\mathbf E|X_1|^{m+2}<\infty$ for some integer $m\ge1$. Put
$$ S_n=\sum_{k=1}^nX_k,\quad F_n(x)=\mathbf P\{S_n<x\sqrt n\},\quad f(t)=\mathbf Ee^{itX_1}. $$
Suppose that Cramér's condition (c): $\varlimsup\limits_{|t|\to\infty}|f(t)|<1$ is satisfied. It is known that, in this case, $F_n(x)=G(x)+o(n^{-m/2})$ where
$$ G(x)=\Phi(x)+\frac{e^{-x^2/2}}{\sqrt{2\pi}}\sum_{k=1}^mQ_k(x)n^{-k/2}, $$
$\Phi(x)$ is the normal distribution function, $Q_k(x)$ is a polynomial whose coefficients depend only on the cumulants of $X_1$.
Theorem 1 contains a sufficient condition for convergence of the series
$$ \sum_{n=1}^\infty n^{-1+\frac{m+\delta}2}\sup_x|F_n(x)-G(x)|,\quad0\le\delta<1. $$

Theorem 2 indicates a necessary and sufficient condition for this convergence in the special case of symmetric random variables.

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English version:
Theory of Probability and its Applications, 1971, 16:3, 528–533

Bibliographic databases:

Received: 16.09.1970

Citation: F. N. Galstyan, “On asymptotic expansions in the central limit theorem”, Teor. Veroyatnost. i Primenen., 16:3 (1971), 535–540; Theory Probab. Appl., 16:3 (1971), 528–533

Citation in format AMSBIB
\Bibitem{Gal71}
\by F.~N.~Galstyan
\paper On asymptotic expansions in the central limit theorem
\jour Teor. Veroyatnost. i Primenen.
\yr 1971
\vol 16
\issue 3
\pages 535--540
\mathnet{http://mi.mathnet.ru/tvp2266}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=288810}
\zmath{https://zbmath.org/?q=an:0287.60024}
\transl
\jour Theory Probab. Appl.
\yr 1971
\vol 16
\issue 3
\pages 528--533
\crossref{https://doi.org/10.1137/1116056}


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