We elucidate the asymptotics of the ${L}^{s}$-quantization error induced by a sequence of ${L}^{r}$-optimal $n$-quantizers of a probability distribution $P$ on ${\mathbb{R}}^{d}$ when $s>r$. In particular we show that under natural assumptions, the optimal rate is preserved as long as $s<r+d$ (and for every $s$ in the case of a compactly supported distribution). We derive some applications of these results to the error bounds for quantization based cubature formulae in numerical integration on ${\mathbb{R}}^{d}$ and on the Wiener space.

Keywords: optimal quantization, Zador theorem

@article{PS_2008__12__127_0, author = {Graf, Siegfried and Luschgy, Harald and Pag\`es, Gilles}, title = {Distortion mismatch in the quantization of probability measures}, journal = {ESAIM: Probability and Statistics}, pages = {127--153}, publisher = {EDP-Sciences}, volume = {12}, year = {2008}, doi = {10.1051/ps:2007044}, mrnumber = {2374635}, language = {en}, url = {http://www.numdam.org/articles/10.1051/ps:2007044/} }

TY - JOUR AU - Graf, Siegfried AU - Luschgy, Harald AU - Pagès, Gilles TI - Distortion mismatch in the quantization of probability measures JO - ESAIM: Probability and Statistics PY - 2008 SP - 127 EP - 153 VL - 12 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps:2007044/ DO - 10.1051/ps:2007044 LA - en ID - PS_2008__12__127_0 ER -

%0 Journal Article %A Graf, Siegfried %A Luschgy, Harald %A Pagès, Gilles %T Distortion mismatch in the quantization of probability measures %J ESAIM: Probability and Statistics %D 2008 %P 127-153 %V 12 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps:2007044/ %R 10.1051/ps:2007044 %G en %F PS_2008__12__127_0

Graf, Siegfried; Luschgy, Harald; Pagès, Gilles. Distortion mismatch in the quantization of probability measures. ESAIM: Probability and Statistics, Volume 12 (2008), pp. 127-153. doi : 10.1051/ps:2007044. http://www.numdam.org/articles/10.1051/ps:2007044/

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