On Errors of Normal Approximation

Abstract

Let $Q_n$ be the distribution of the normalized sum of $n$ independent random vectors with values in $R^k$, and $\Phi$ the standard normal distribution in $R^k$. In this article the error $|\int f d(Q_n - \Phi)|$ is estimated (for essentially) all real-valued functions $f$ on $R^k$ which are integrable with respect to $Q_n$ when $s$th moments are finite, and for which the error may be expected to go to zero. When specialized to known examples, the (main) error bound provides precise rates of convergence.