An improvement of a convergence rate estimate

V. V. Sazonov

Teoriâ veroâtnostej i ee primeneniâ, Tome 14 (1969) no. 4, pp. 667-678

Résumé

Let $\xi_1,\xi_2,\dots$ be independent random variables equally distributed with a continuous distribution function$F(x)$. Put $$ W_n^2=n\int_{-\infty}^\infty[F_n(x)-F(x)]^2\,dF(x), $$ where $$ F_n(x)=\frac1n\sum_{j=1}^n\delta(x-\xi_j),\quad\delta(x)= \begin{cases} 1,&x>0, \\ 0,&x\le0. \end{cases} $$ Denote by $S(x)$ the distribution function with the characteristic function $$ s(t)=\prod_{j=1}^\infty(1-2it(\pi j)^{-2})^{-1/2}. $$ In [3], it has been shown that $$ \Delta_n=\sup_{x\in R^1}|\mathbf P(W_n^2<x)-S(x)|\underset{n\to\infty}\longrightarrow0 $$ not slowlier than $n^{-1/10}$. In the present paper, we obtain a stronger result: for any $\varepsilon>0$ there exists a $c(\varepsilon)$ such that $$ \Delta_n\le c(\varepsilon)n^{-1/6+\varepsilon},\quad n=1,2,\dots. $$