Quantization for Probability Measures in the Prokhorov Metric

For a probability distribution $P$ on $R^d$ and $n\inN$ consider $e_n=\inf\pi(P,Q)$, where $\pi$ denotes the Prokhorov metric and the infimum is taken over all discrete probabilities $Q$ with $|\mathrm{supp}(Q)|\le n$. We study solutions $Q$ of this minimization problem, stability properties, and consistency of empirical estimators. For some classes of distributions we determine the exact rate of convergence to zero of the $n$th quantization error $e_n$ as $n\to\infty$.