Asymptotics of the quantization errors for self-similar probabilities.(English)Zbl 1029.28003

Let $$P$$ be a probability measure on $${\mathbb R}^d$$. For any integer $$n \geq 1$$ and $$r \in [0, \infty]$$, define the $$n$$-th quantization error of order $$r$$ for $$P$$ by $e_{n, r} = \inf \Big\{\exp \int_{{\mathbb R}^d} \log d(x, \alpha) dP(x) \mid \alpha \subset {\mathbb R}^d, \;\text{ card}(\alpha) \leq n \Big\}$ if $$r =0$$; $e_{n,r} = \inf\Big\{ \Big(\int_{{\mathbb R}^d} d (x, \alpha)^r dP(x)\Big)^{1/r}\mid \alpha \subset {\mathbb R}^d \;\text{ card}(\alpha) \leq n \Big\}$ if $$0 < r < \infty$$ and $e_{n, r} = \inf \Big\{ \sup_{x \in \text{ Supp}(P)} d (x, \alpha) \mid \alpha \subset {\mathbb R}^d, \;\text{ card}(\alpha) \leq n \Big\}$ if $$r = \infty$$. In the above, $$d(x, \alpha)$$ denotes the distance between $$x$$ and $$\alpha$$. If the following limit $\lim_{n \to \infty} {{\log n} \over {- \log e_{n, r}}}$ exists, it is defined as the quantization dimension of $$P$$ of order $$r$$ and is denoted by $$D_r(P)$$. In their previous paper [Math. Nachr. 241, 103-109 (2002; Zbl 1010.28009)], the authors showed that if $$P$$ is a self-similar probability associated with a family $$\{S_1, \ldots, S_N\}$$ of contractive similitudes on $${\mathbb R}^d$$ satisfying the open set condition and a probability vector $$p = (p_1, \dots, p_N)$$, then for $$0 < r < \infty$$, the quantization dimension $$D_r$$ of $$P$$ is determined by $\sum_{i=1}^N \big(p_i s_i^r\big)^{{{D_r}\over {r + D_r}}} = 1,$ where $$s_i$$ is the contraction ratio of $$S_i$$. In the paper under review, the authors give a new proof of the above result and extend it for $$r=0$$ and $$r = \infty$$. More systematic information on quantization for probability measures can be found in the monograph of the authors [“Foundations of quantization for probability distributions” Lect. Notes Math. 1730 (2000; Zbl 0951.60003)].

MSC:

 28A80 Fractals 28A78 Hausdorff and packing measures 60E05 Probability distributions: general theory

Citations:

Zbl 1010.28009; Zbl 0951.60003