The Annals of Probability

The local quantization behavior of absolutely continuous probabilities

Siegfried Graf, Harald Luschgy, and Gilles Pagès

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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 $P$.

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Ann. Probab., Volume 40, Number 4 (2012), 1795-1828.

First available in Project Euclid: 4 July 2012

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Primary: 60E99: None of the above, but in this section 62H30: Classification and discrimination; cluster analysis [See also 68T10, 91C20] 34A29

Vector quantization probability of Voronoi cells inertia of Voronoi cells


Graf, Siegfried; Luschgy, Harald; Pagès, Gilles. The local quantization behavior of absolutely continuous probabilities. Ann. Probab. 40 (2012), no. 4, 1795--1828. doi:10.1214/11-AOP663.

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