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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

 English version: Theory of Probability and its Applications, 1984, 28:3, 596–601

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