Geodesic
Quantization for Probability Measures in the Prokhorov Metric
Teoriâ veroâtnostej i ee primeneniâ, Tome 53 (2008) no. 2, pp. 307-335

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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 $n\to\infty$.
Keywords: Ky Fan metric, Prokhorov metric, empirical measures, asymptotic quantization error, entropy
Mots-clés : multidimensional quantization, optimal quantizers, quantization dimension. 
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     author = {S. Graf and H. Luschgy},
     title = {Quantization for {Probability} {Measures} in the {Prokhorov} {Metric}},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {307--335},
     publisher = {mathdoc},
     volume = {53},
     number = {2},
     year = {2008},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TVP_2008_53_2_a5/}
}
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S. Graf; H. Luschgy. Quantization for Probability Measures in the Prokhorov Metric. Teoriâ veroâtnostej i ee primeneniâ, Tome 53 (2008) no. 2, pp. 307-335. http://geodesic.mathdoc.fr/item/TVP_2008_53_2_a5/