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Quantization for probability measures in the Prokhorov metric

TitleQuantization for probability measures in the Prokhorov metric
Publication TypeJournal Article
Year of Publication2009
AuthorsHarald Luschgy, and Siegfried Graf
JournalTheory Probab. Appl.
Volume53
Pagination216-241
Keywordsasymptotic quantization error, empirical measures, entropy, Ky Fan metric, multidimensional quantization, optimal quantizers, Prokhorov metric, quantization dimension
Abstract

For a probability distribution $ P $ on $ \mathbb{R}^d $ and $ n\in 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 $ |\textrm{supp}(Q)|\leq 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 en as $ n\to \infty $.