Distortion mismatch in the quantization of probability measures

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 when s > r . In particular we show that under natural assumptions, the optimal rate is preserved as long as s (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 and on the Wiener space.