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

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_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.