Sharp orders of convergence in the random central limit theorem
Let \(X_ 1\), \(X_ 2\),... be a sequence of i.i.d. random vectors on a probability space (\(\Omega\),\({\mathcal A},P)\) with mean \(\mu\), positive definite covariance matrix V and finite third moments. Put \(S_ n=V^{- }(X_ 1+...+X_ n-n\mu)\) and denote by \(\Phi_{0,I}\) the standard normal distribution on \({\mathbb{R}}^ k.\) Let \(\tau_ n:\Omega \to {\mathbb{N}}\), \(\tau\) :\(\Omega\to [c,\infty)\) be measurable with \(c>0\). The main result of the paper is the following quantitative version of the random central limit theorem: If \[ P\{| \tau_ n/n\tau -1| >\epsilon_ n\}=O(\epsilon_ n^{1/2}) \] for a sequence \(0<\epsilon_ n\to 0\), and the Hausdorff distance of the sigma field \(\sigma\) (\(\tau)\) from \(\sigma (X_ 1\),..., \(X_ n)\) is \(O(n^{- \alpha}(lg n)^{\beta})\) with \(0<\alpha <1/2\), then \[ | P\{\tau_ n^{-}S_{\tau_ n}\in C\}-\Phi_{0,I}(C)| =O(\epsilon_ n^{1/2})+O\quad (n^{-\alpha}(lg n)^{\alpha +\beta}) \] uniformly for all convex subsets C of \({\mathbb{R}}^ k.\) For the case \(\alpha =1/2\) similar estimates are obtained. Examples show, that these convergence orders are optimal.
scientific article; zbMATH DE number 4054694
scientific article