# The local quantization behaviour of absolutely continuous probabilities

Abstract : For a large class of absolutely continuous probabilities P it is shown that, for r > 0, for n-optimal L^r(P)-codebooks \alpha_n, and any Voronoi partition V_{n,a} with respect to \alpha_n the local probabilities P(V_{n,a}) satisfy P(V_{a,n}) \approx n^{-1} while the local L^r-quantization errors satis\-fy $\int_{V_{n,a}} \|x-a\|^r dP(x) \approx n^{- (1+ \frac rd)}$ as long as the partition sets V_{n,a} intersect a fixed compact set K in the interior of the support of P.
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Journal articles
Annals of Probability, Institute of Mathematical Statistics, 2012, 40 (4), 1795-1828 ; http://dx.doi.org/10.1214/11-AOP663. <10.1214/11-AOP663>
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https://hal.archives-ouvertes.fr/hal-00521185
Contributor : Gilles Pagès <>
Submitted on : Sunday, September 26, 2010 - 3:48:38 PM
Last modification on : Tuesday, October 11, 2016 - 2:05:09 PM
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### Citation

Siegried Graf, Harald Luschgy, Gilles Pagès. The local quantization behaviour of absolutely continuous probabilities. Annals of Probability, Institute of Mathematical Statistics, 2012, 40 (4), 1795-1828 ; http://dx.doi.org/10.1214/11-AOP663. <10.1214/11-AOP663>. <hal-00521185>

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