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

Let $r,s>0$. For a given probability measure $P$ on ${\mathbb{R}}^d$, let $({\alpha }_n{)}_{n\ge 1}$ be a sequence of (asymptotically) $L^r(P)$- optimal quantizers. For all $\mu \in {\mathbb{R}}^d$ and for every $\theta >0$, one defines the sequence $({\alpha }_n^{\theta ,\mu }{)}_{n\ge 1}$ by : $\forall n \geq 1, \alpha_n^{\theta, \mu} = \mu + \theta(\alpha_n - \mu) = {\mu + \theta(a- \mu), a \in \alpha_n } $. In this paper, we are interested in the asymptotics of the $L^s$-quantization error induced by the sequence $(\alpha_n^{\theta, \mu})_{n \geq 1}$. We show that for a wide family of distributions, the sequence $({\alpha }_n^{\theta ,\mu }{)}_{n\ge 1}$ is $L^s$-rate-optimal. For the Gaussian and the exponential distributions, one shows how to choose the parameter $\theta$ such that $({\alpha }_n^{\theta ,\mu }{)}_{n\ge 1}$ satisfies the empirical measure theorem and probably be asymptotically $L^s$-optimal.