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

 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:

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}