A multidimensional analogue of Berry–Esseen's inequality for sets of a bounded diameter

V. M. Zolotarev

Teoriâ veroâtnostej i ee primeneniâ, Tome 11 (1966) no. 3, pp. 507-514

Résumé

In the space $R^n$ functions of hounded variation $L(x)$ and $H(x)$ are considered. Let $P_L$ and $P_H$ be the quasi-measures defined on Borel subsets of $R^n$ by the formulae $$ P_L(B)=\int_B\,dL(x),\quad P_H(B)=\int_B\,dH(x). $$ Let us denote by $\mathfrak A_s$ the class of subsets of $R^n$ with the $(n-l)$-dimensional volumes of their boundaries not greater then $s$ and denote by $\mathfrak B_d$ the class of convex subsets of $R^n$ such that the $(n-l)$-dimensional volumes of their intersections with any hyperplane is not greater then $d$. We construct an upper estimate (analogous to that of Berry–Esseen) of the quantity $$ \Delta=\sup_{A\in\mathfrak G}|P_L(A)-P_H(A)| $$ where $\mathfrak G$ is $\mathfrak A_s$ or $\mathfrak B_d$.