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 $P$.

Published in at http://dx.doi.org/10.1214/11-AOP663 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (1)

• Asymptotics of the maximal radius of an $L^r$-optimal sequence of quantizers

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