A lower bound for the convergence rate in the central limit theorem

V. K. Matskyavichyus

Teoriâ veroâtnostej i ee primeneniâ, Tome 28 (1983) no. 3, pp. 565-569

Résumé

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 $\{\varphi(n)\}_{n=1}^\infty$.