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Teoriya Veroyatnostei i ee Primeneniya, 1971, Volume 16, Issue 3, Pages 535–540
(Mi tvp2266)
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This article is cited in 3 scientific papers (total in 3 papers)
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.
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
Linking options:
https://www.mathnet.ru/eng/tvp2266 https://www.mathnet.ru/eng/tvp/v16/i3/p535
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