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Teoriya Veroyatnostei i ee Primeneniya, 1978, Volume 23, Issue 1, Pages 109–119
(Mi tvp2979)
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This article is cited in 1 scientific paper (total in 1 paper)
On asymptotic behaviour of the remainder term in the central limit theorem
L. V. Rozovskiĭ Leningrad
Abstract:
Let $X_1,X_2,\dots$ be a sequence of independent identically distributed random variables with zero means and unit variances. Let $k_n$ be a sequence of natural numbers, $k_n\to\infty$, $k_{n+1}/k_n\to 1$ ($n\to\infty$),
$$
F_n(x)=\mathbf P\biggl\{\frac{1}{\sqrt{k_n}}\sum_{k=1}^{k_n}X_k<x\biggr\},\qquad
\Phi(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^x e^{-t^2/2}\,dt.
$$
We study conditions under which
$$
F_n(x)=\Phi(x)+\frac{\Psi(x)+o(1)}{\mu_n}\qquad (n\to\infty)
$$
uniformly in $x$, $-\infty<x<\infty$, where $\mu_n$ is a positive sequence such that $\mu_n\to\infty$,
$\mu_n=o(k_n)$ ($n\to\infty$).
Received: 21.01.1976
Citation:
L. V. Rozovskiǐ, “On asymptotic behaviour of the remainder term in the central limit theorem”, Teor. Veroyatnost. i Primenen., 23:1 (1978), 109–119; Theory Probab. Appl., 23:1 (1978), 106–116
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https://www.mathnet.ru/eng/tvp2979 https://www.mathnet.ru/eng/tvp/v23/i1/p109
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