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

Abstract

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