On the rate of convergence in the central limit theorem for weakly dependent random variables
Teoriâ veroâtnostej i ee primeneniâ, Tome 25 (1980) no. 4, pp. 800-818
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Let $X_1,X_2,\dots$ be a stationary sequence of random variables with $\mathbf EX_1=0$,
$\mathbf E|X_1|^3\infty$. Let
\begin{gather*}
\sigma^2_n=\mathbf E\biggl(\sum_{j=1}^n X_j\biggr)^2,\qquad
F_n(x)=\mathbf P\biggl\{\sigma_n^{-1}\sum_{j=1}^n X_j\biggr\},
\\
\Phi(x)=(2\pi)^{-1/2}\int_{-\infty}^x e^{-y^2/2}\,dy,\qquad
\Delta_n=\sup|F_n(x)-\Phi(x)|.
\end{gather*}
We prove that if the sequence $X_n$ satisfies a strong mixing condition and if its mixing
coefficient decreases exponentially then
$$
\Delta_n=O(n^{-1/2}\ln^2n).
$$
For the case of $m$-dependent variables we prove that
$$
\Delta_n=O(m^2n^{-1/2}).
$$
@article{TVP_1980_25_4_a9,
author = {A. N. Tihomirov},
title = {On the rate of convergence in the central limit theorem for weakly dependent random variables},
journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
pages = {800--818},
publisher = {mathdoc},
volume = {25},
number = {4},
year = {1980},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TVP_1980_25_4_a9/}
}
TY - JOUR AU - A. N. Tihomirov TI - On the rate of convergence in the central limit theorem for weakly dependent random variables JO - Teoriâ veroâtnostej i ee primeneniâ PY - 1980 SP - 800 EP - 818 VL - 25 IS - 4 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/TVP_1980_25_4_a9/ LA - ru ID - TVP_1980_25_4_a9 ER -
A. N. Tihomirov. On the rate of convergence in the central limit theorem for weakly dependent random variables. Teoriâ veroâtnostej i ee primeneniâ, Tome 25 (1980) no. 4, pp. 800-818. http://geodesic.mathdoc.fr/item/TVP_1980_25_4_a9/