Uniform normal approximation orders for families of dominated measures
Let \(X_ n\), \(n\in {\mathbb{N}}\), be i.i.d. random vectors with positive definite covariance matrix V and put \(S^*\!_ n=(nV)^{- 1/2}\sum^{n}_{i=1}(X_ i-E(X_ i)).\) If \(X_ n\in L_ r\), \(\phi \in L_ s\) \((r>1+1/(s-3))\) and if \(f: {\mathbb{R}}^ k\to {\mathbb{R}}\) is a bounded Berry-Esseen function, then there are given approximation orders for \[ A_ n(\phi,f):=E(f\circ^*S_ n\phi)-\Phi_{o,I}(f)E(\phi), \] where \(\Phi_{o,I}\) is the standard normal distribution of \({\mathbb{R}}^ k\). It turns out, that the distance \(d_ n(\phi):=\inf \{E(| \phi - \psi |:\quad \psi \sigma (X_ 1,\ldots,X_ n)-measurable\}\) of \(\phi\) from the space of \(\sigma (X_ 1,\ldots,X_ n)\)-measurable functions plays an essential role for the convergence order of \(A_ n(\phi,f)\). For instance \[ d_ n(\phi)=O(n^{-1/2}(lg n)^{-3/2- \epsilon}\quad implies\quad A_ n(\phi,f)=O(n^{-1/2}), \] \[ d_ n(\phi)=O(n^{-1/2}(lg n)^{-3/2}\quad implies\quad A_ n(\phi,f)=O(n^{-1/2}\log \log n), \] \[ d_ n(\phi)=O(n^{-1/2}(lg n)^{-3/2+\epsilon}\quad implies\quad A_ n(\phi,f)=O(n^{-1/2}(lg n)^{\epsilon}), \] and these convergence rates are optimal. The results can be applied e.g. to obtain convergence rates for \(\sup_{Q\in {\mathcal Q}}\sup_{C\in {\mathcal C}}| Q(^*S_ n\in C)-\phi_{o,I}(C)|\), where \({\mathcal Q}\) is a family of dominated measures and \({\mathcal C}\) is the class of convex measurable sets of \({\mathbb{R}}^ k\).
scientific article; zbMATH DE number 3934110
scientific article