Open Access
Translator Disclaimer
July 2012 The local quantization behavior of absolutely continuous probabilities
Siegfried Graf, Harald Luschgy, Gilles Pagès
Ann. Probab. 40(4): 1795-1828 (July 2012). DOI: 10.1214/11-AOP663

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 satisfy $\int_{V_{n,a}}\|x-a\|^{r}\,dP(x)\approx n^{-(1+r/d)}$ as long as the partition sets $V_{n,a}$ intersect a fixed compact set $K$ in the interior of the support of $P$.

Citation

Download Citation

Siegfried Graf. Harald Luschgy. Gilles Pagès. "The local quantization behavior of absolutely continuous probabilities." Ann. Probab. 40 (4) 1795 - 1828, July 2012. https://doi.org/10.1214/11-AOP663

Information

Published: July 2012
First available in Project Euclid: 4 July 2012

zbMATH: 1260.60032
MathSciNet: MR2978138
Digital Object Identifier: 10.1214/11-AOP663

Subjects:
Primary: 34A29 , 60E99 , 62H30

Keywords: inertia of Voronoi cells , probability of Voronoi cells , Vector quantization

Rights: Copyright © 2012 Institute of Mathematical Statistics

JOURNAL ARTICLE
34 PAGES


SHARE
Vol.40 • No. 4 • July 2012
Back to Top