Universal L^s-rate-optimality of L^r-optimal quantizers by dilatation and contraction

Laboratoire de Probabilités et Modèles Aléatoires, UMR 7599, Université Pierre et Marie Curie, Case 188, 4 place Jussieu, 75252 Cedex 05, Paris, France; sagna@ccr.jussieu.fr

Received: 12 July 2007
Revised: 20 February 2008

Abstract

We investigate in this paper the properties of some dilatations or contractions of a sequence (α_n)_n≥1 of L^r-optimal quantizers of an -valued random vector defined in the probability space with distribution . To be precise, we investigate the L^s-quantization rate of sequences when or s ∈ (r, +∞) and . We show that for a wide family of distributions, one may always find parameters (θ,µ) such that (α_n^θ,µ)_n≥1 is L^s-rate-optimal. For the Gaussian and the exponential distributions we show the existence of a couple (θ*,µ*) such that (α^θ*,µ*)_n≥1 also satisfies the so-called L^s-empirical measure theorem. Our conjecture, confirmed by numerical experiments, is that such sequences are asymptotically L^s-optimal. In both cases the sequence (α^θ*,µ*)_n≥1 is incredibly close to L^s-optimality. However we show (see Rem. 5.4) that this last sequence is not L^s-optimal (e.g. when s = 2, r = 1) for the exponential distribution.