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

 Title Universal $L^s$-rate-optimality of $L^r$-optimal quantizers by dilatation and contraction Publication Type Journal Article Year of Publication 2009 Authors Abass Sagna Journal ESAIM: Probability and Statistics Volume 13 Pagination 218-246 Keywords dilatation, empirical measure theorem, Lloyd algorithm, rate-optimal quantizers Abstract We investigate in this paper the properties of some dilatations or contractions of a sequence of -optimal quantizers of an -valued random vector defined in the probability space with distribution . To be precise, we investigate the -quantization rate of sequences when or and . We show that for a wide family of distributions, one may always find parameters such that is -rate-optimal. For the gaussian and the exponential distributions we show the existence of a couple such that also satisfies the so-called -empirical measure theorem. Our conjecture, confirmed by numerical experiments, is that such sequences are asymptotically -optimal. In both cases the sequence is incredibly close to -optimality. However we show (see Rem. 5.4) that this last sequence is not -optimal (e.g. when , ) for the exponential distribution.