The local quantization behavior of absolutely continuous probabilities (Q439886)

Let \(\mathbb R^d\) be endowed with a norm \(\|\cdot\|\) and let \(\operatorname{P}\) be a Borel probability measure on \(\mathbb R^d\) with finite \(r\)-th moment for some \(r>0\). Then its \(n\)-th quantization error w.r.t.~the \(L_r\)-norm is defined by NEWLINE\[NEWLINEe_{n,r}= e_{n,r}(\operatorname{P}):= \inf\left\{\left(\int_{\mathbb R^d} d(x,\alpha)^r \mathrm{d} \operatorname{P}(x)\right)^{1/r} : \alpha\subset\mathbb R^d,\;\mathrm{card}(\alpha)\leq n\right\}. \tag{1}NEWLINE\]NEWLINE Here, as usual, NEWLINE\[NEWLINE d(x,\alpha)=\min_{a\in\alpha}\|x-a\| NEWLINE\]NEWLINE is the distance of \(x\in\mathbb R^d\) to the set \(\alpha\subset\mathbb R^d\). The finite sets \(\alpha\) appearing in (1) are called code books of order \(n\). Code books generate in a natural way a partition \((V_{n,a})_{a\in\alpha}\) of \(\mathbb R^d\) (called Voronoi partition) where the Voronoi cells \(V_{n,a}\) have to obey the nearest neighbor rule.NEWLINENEWLINEAs it is known, for each \(\operatorname{P}\) and each \(n\geq 1\), there are optimal code books \(\alpha_n\) for which the infimum in (1) is attained. The aim of the present paper is to investigate the behavior of the Voronoi cells \(V_{n,a}\) for \(a\in\alpha_n\). Of particular interest are questions about the behavior of NEWLINE\[NEWLINE \operatorname{P}(V_{n,a})\quad\text{and}\quad \int_{V_{n,a}}\|x-a\|^r\mathrm{d} \operatorname{P}(x),\;a\in\alpha_nNEWLINE\]NEWLINE as \(n\to \infty\). For example, an open question for several years is whether or not NEWLINE\[NEWLINE \int_{V_{n,a}}\|x-a\|^r\mathrm{d} \operatorname{P}(x)\sim \frac 1 n e_{n,r}^r,\;a\in\alpha_n, NEWLINE\]NEWLINE as \(n\to\infty\).NEWLINENEWLINEThe aim of the present paper is to prove ``weak'' asymptotics of \(\operatorname{P}(V_{n,a})\) as well as of the above integral whenever \(a\) is chosen in the optimal code book \(\alpha_n\). The probability measure \(\operatorname{P}\) is assumed to be absolutely continuous with respect to the Lebesque measure on \(\mathbb R^d\) and its density and/or its support have to satisfy some additional regularity assumptions. Thereby, the cases in which \(\operatorname{P}\) has compact or non-compact support are treated differently. Then the authors prove that NEWLINE\[NEWLINE \mathbf{P}(V_{n,a})\approx n^{-1}\quad\text{and}\quad \int_{V_{n,a}}\|x-a\|^r\mathrm{d} \operatorname{P}(x)\approx \frac{e_{n,r}^r}{n}\approx n^{-(1+r/d)},\;a\in\alpha_n, NEWLINE\]NEWLINE as \(n\to\infty\) as long as the Voronoi cells \(V_{n,a}\) intersect a fixed compact set in the interior of the support of \(\operatorname{P}\). Although this is a big step to the solution of the general conjecture, many questions about the behavior of the cells \(V_{n,a}\) with \(a\in\alpha_n\) remain unanswered.