Distortion mismatch in the quantization of probability measures

We elucidate the asymptotics of the L-quantization error induced by a sequence of Loptimal n-quantizers of a probability distribution P on R 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 quadrature formulae in numerical integration on R and on the Wiener space.