The local quantization behaviour of absolutely continuous probabilities

Siegried Graf (1) , Harald Luschgy (2) , Gilles Pagès (3)

1 UNIVERSITÄT PASSAU - Fakultät für Mathematik und Informatik
2 FB IV-Mathematik
3 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

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.

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

Domains: Probability [math.PR]

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