Quantization for Probability Measures in the Prohorov Metric

Abstract

For a probability distribution P on ${\bf R}^d$ and $n\in{\bf N}$ consider $e_n = \inf \pi (P,Q)$, where $\pi$ denotes the Prokhorov metric and the infimum is taken over all discrete probabilities Q with $|\mbox{supp}(Q) | \le n$. 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 nth quantization error $e_n$ as $n \rightarrow\infty$.