On the accuracy of approximation in the central limit theorem
Teoriâ veroâtnostej i ee primeneniâ, Tome 21 (1976) no. 1, pp. 107-122
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Let
$$
\Delta_n=\sup_x|\mathbf P(\xi_1+\dots+\xi_n\sqrt n)-\Phi(x)|,
$$
where $\xi_1,\xi_2,\dots$ are independent identically distributed random variables with the distribution function $F(x)$, $\mathbf E|\xi_1|^2=1$, $\mathbf E\xi_1=0$, and where $\Phi$ is the standard normal distribution function.
We investigate necessary and sufficient conditions on $F(x)$ for the following two series to converge:
$$
\sum h(\sqrt n)\frac{1}{n}\Delta_n\infty,\quad\sum h(\sqrt n)n^{-3/2}\Delta_n\infty,
$$
where
$$
h(y)>0,\qquad h(y)\uparrow,\qquad h(y)/y\downarrow.
$$
The case of Chebyshev–Gramer asymptotic expansions is also discussed.
@article{TVP_1976_21_1_a7,
author = {{\CYRV}. A. Lif\v{s}ic},
title = {On the accuracy of approximation in the central limit theorem},
journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
pages = {107--122},
publisher = {mathdoc},
volume = {21},
number = {1},
year = {1976},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TVP_1976_21_1_a7/}
}
В. A. Lifšic. On the accuracy of approximation in the central limit theorem. Teoriâ veroâtnostej i ee primeneniâ, Tome 21 (1976) no. 1, pp. 107-122. http://geodesic.mathdoc.fr/item/TVP_1976_21_1_a7/