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)].


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