Distortion mismatch in the quantization of probability measures

Graf, Siegfried; Luschgy, Harald; Pagès, Gilles

ESAIM: Probability and Statistics, Tome 12 (2008), pp. 127-153.

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 ${ℝ}^d$ 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 ${ℝ}^d$ and on the Wiener space.

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