| 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 $\mathbb{R}^d$-valued random vector $X \in L^r(\mathbb{P})$ defined in the probability space $(\Omega,\mathcal{A},\mathbb{P})$ with distribution $\mathbb{P}_{X} = P$. To be precise, we investigate the L s-quantization rate of sequences $\alpha_n^{\theta,\mu} = \mu + \theta(\alpha_n-\mu)=\{\mu + \theta(a-\mu), \ a \in \alpha_n \}$ when $\theta \in \mathbb{R}_{+}^{\star}, \mu \in \mathbb{R}, s \in (0,r)$ or s ∈ (r, +∞) and $X \in L^s(\mathbb{P})$. We show that for a wide family of distributions, one may always find parameters (θ,µ) such that ( α nθ,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.
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