Berry-Esséen bounds for (absolute) moments of dominated measures (Q1104628)

Let \(X_ n\in {\mathcal L}_ S(P)\), \(n\in {\mathbb{N}}\), be a sequence of i.i.d. random variables with mean 0 and variance 1 and put \(S_ n=n^{- 1/2}\sum^{n}_{i=1}X_ i\). Let g be a density of a p-measure Q with respect to P and denote by \(d_ 1(g, \sigma (X_ 1,...,X_ n))\) the \(\| \|_ 1\)-distance of \(g\in {\mathcal L}_ 1(P)\) from the subspace of all \(\sigma (X_ 1,...,X_ n)\)-measurable functions of \({\mathcal L}_ 1(P)\). The main theorem of the paper yields for instance: (i) If \(s>3\), g is bounded and \(d(g,\sigma(X_ 1,...,X_ n)=O(n^{- 1/2}(\lg n)^{-s/2})\), then \[ | \int | S^*_ n|^ pdQ- (2\pi)^{-1/2}\int | x|^ p e^{-x^ 2/2}dx| =O(n^{- 1/2})\text{ for all } 1\leq p\leq s. \] (ii) If \(s>4\) and \(d(g,\sigma(X_ 1,...,X_ n))=O^{n^{-1/2}}(\log n)^{-(s-1)/2})\), then \[ | \int f\circ S^*_ n dQ-(2\pi)^{-1/2}\int f(x)e^{-x^ 2/2}dx| =O(n^{-1/2}) \] for all p-times differentiable f with bounded p-th derivation, \(p\leq s-1\).