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Consider $e_n = \inf \exp \int \log \parallel x - f(x) \parallel dP(x)$, where $p$ is a probability measure on $\real^d$ and the infimum is taken over all measurable maps $f{:}\ \real^d \rightarrow \real^d$ with $| f(\real^d)| \leq n$. We study solutions $f$ of this minimization problem. For absolutely continuous distributions and for self-similar distributions we derive the exact rates of convergence to zero of the $n$th quantization error $e_n$ as $n \rightarrow \infty$. We establish a relationship between the quantization dimension that rules the rates and the Hausdorff dimension of $P$.

The sharp asymptotics for the L 2 -quantization errors of Gaussian measures on a Hilbert space and, in particular, for Gaussian processes is derived. The condition imposed is regular variation of the eigenvalues.

Abstract.We elucidate the asymptotics of the L s -quantization error induced by a sequence of L roptimal 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.Mathematics Subject Classification. 60G15, 60G35, 41A25.