## The Annals of Probability

### The local quantization behavior 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 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 $P$.

#### Article information

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

Dates
First available in Project Euclid: 4 July 2012

https://projecteuclid.org/euclid.aop/1341401149

Digital Object Identifier
doi:10.1214/11-AOP663

Mathematical Reviews number (MathSciNet)
MR2978138

Zentralblatt MATH identifier
1260.60032

#### Citation

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. https://projecteuclid.org/euclid.aop/1341401149

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