An absolute estimate of the remainder in the central limit theorem

V. M. Zolotarev

Teoriâ veroâtnostej i ee primeneniâ, Tome 11 (1966) no. 1, pp. 108-119

Résumé

Let $\xi_1,\dots\xi_n$ be independent random varibles with zero means, variances $\sigma_1,\dots\sigma_n$ and third absolute moments $\beta_1\dots\beta_n$. Let us denote $$ \sigma^2=\sum_j\sigma_j^2,\quad\varepsilon=\biggl(\sum_j\beta_j\biggr)\biggr/\sigma^3, $$ and let $F(x)$ be the distribution function of the sum $\xi_1+\dots+\xi_n$ and $\Phi(x)$ be the distribution function of the normal $(0,1)$ law. Let further $\varepsilon$ be equal to a fixed positive number and $D(\varepsilon)$ denote the least value for which $$ \sup_x|F(x\sigma)-\Phi(x)|\le D(\varepsilon)\varepsilon. $$ Estimates of $D(\varepsilon)$ for all $\varepsilon$, $0\le\varepsilon\le0.79$ are obtained and the inequality $$ \sup_\varepsilon D(\varepsilon)1.322 $$ is proved.