Dependence of the Berry-Esseen estimate on the dimension
Denote by \(\{X_ n\}^ a \)sequence of i.i.d. centered random elements of the space \({\mathbb{R}}^ k\) with usual norm and by Y a centered Gaussian random element. Put \(S_ n=n^{-1/2}(X_ 1+...+X_ n)\), \(\beta_ s=E| X_ 1|^ s\). Assume that \(X_ 1\) and Y have identity covariance matrix. For a class \({\mathcal U}\) of convex subsets of \({\mathbb{R}}^ k\) set \[ \Delta_ n=\Delta_ n({\mathcal U})=\sup \{| P(S_ n\in A)-P(Y\in A)|:A\in {\mathcal U}\}. \] Below M denotes an absolute constant not depending on k. Consider the following condition: there exist constants \(a_ k=a_ k({\mathcal U})\), depending only on the class \({\mathcal U}\) and the dimension k such that \(P\{Y\in A^{\epsilon}\setminus A_{\epsilon}\}\leq a_ k\epsilon\) for all \(A\in {\mathcal U}\), \(\epsilon >0\), where \(A^{\epsilon}\) denotes the \(\epsilon\)- neighborhood of the set A and \(A_{\epsilon}={\mathbb{R}}^ k\setminus ({\mathbb{R}}^ k\setminus A)^{\epsilon}\). The author, in this paper, proves the following theorem using the smooth-function technique. Theorem A (Theorem 3): Let us assume that the class \({\mathcal U}\) is closed with respect to transformations and homotheties of the space \({\mathbb{R}}^ k\). Moreover assume \(A^{\epsilon}\), \(A_{\epsilon}\in {\mathcal U}\) if \(A\in {\mathcal U}\), \(\epsilon >0\). Then \(\Delta_ n\leq M(1+a_ k)/\beta_ 3n^{-1/2}.\) The following results are immediate corollaries to Theorem A. Theorem B (Theorem 1): If \({\mathcal U}\) is the class of all convex subsets of the space \({\mathbb{R}}^ k\), then \(\Delta_ n\leq M\beta_ 3k^{1/2}n^{-1/2}.\) Theorem C (Theorem 2): If \({\mathcal U}\) is the class of all balls of the space \({\mathbb{R}}^ k\), then \(\Delta_ n\leq M\beta_ 3n^{-1/2}\).
scientific article; zbMATH DE number 3990499