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Teoriya Veroyatnostei i ee Primeneniya, 1983, Volume 28, Issue 3, Pages 565–569
(Mi tvp2199)
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This article is cited in 22 scientific papers (total in 22 papers)
Short Communications
A lower bound for the convergence rate in the central limit theorem
V. K. Matskyavichyus Moscow
Abstract:
For every sequence of nonnegative numbers $\varphi(n)\to 0$, $n\to\infty$ there exists a sequence of independent identically distributed random variables $X_1,X_2,\dots$ such that $\mathbf EX_1=0$, $\mathbf DX_1=1$ and for $n\ge n1$
$$
\sup_x|\mathbf P\{n^{-1/2}(X_1+\dots+X_n)<x\}-\Phi(x)|\ge\varphi(n).
$$
The distribution of $X_1$ has the form
$$
\mathbf P\{X_1<x\}=\sum_{k=1}^\infty\lambda_k\Phi(x/\sigma_k);
$$
$\lambda_k$, $\sigma_k$ and $n_1$ are explicit functions of $\{\varphi(n)\}_{n=1}^\infty$.
Received: 28.12.1982
Citation:
V. K. Matskyavichyus, “A lower bound for the convergence rate in the central limit theorem”, Teor. Veroyatnost. i Primenen., 28:3 (1983), 565–569; Theory Probab. Appl., 28:3 (1984), 596–601
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https://www.mathnet.ru/eng/tvp2199 https://www.mathnet.ru/eng/tvp/v28/i3/p565
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Abstract page: | 196 | Full-text PDF : | 90 |
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