A certain estimate for the rate of convergence in the central limit theorem with respect to balls in the finite-dimensional space (Q1117564)

[For the entire collection see Zbl 0626.00025.] Let \(X_ 1,X_ 2,..\). be independent, identically distributed random variables (r.v.) with values in the space \(R^ k\). One assumes that these r.v. have zero mean and covariance operator equal to the identity. We denote by P the distribution of the r.v. \(X_ 1\), by \(P_ n\) the distribution of the r.v. \((X_ 1+...+X_ n)n^{-1/2}\), and by \(\Phi\) the standard normal law. One investigates the problem of the estimation of the quantity \[ \rho (P_ n,\Phi,R)=\sup \{| P_ n(S_ r(a))- \Phi (S_ r(a))|:\quad a\in R^ k,\quad r\leq R\}, \] where \(S_ r(a)=S_ r=\{x: | x-a| \leq r\}\) are balls in \(R^ k\).