Optimal quantizers for Radon random vectors in a Banach space

Abstract

For $n\in \mathbb{N},r\in (0,\infty )$ and a Radon random vector X with values in a Banach space E let $e_{n,r}(X,E)=\inf (\mathbb{E}{\min }_{a\in \alpha }\parallel X-a{\parallel }^r{)}^{1/r}$, where the infimum is taken over all subsets $\alpha$ of E with $\mathrm{card}(\alpha )⩽n$ (n-quantizers). We investigate the existence of optimal n-quantizers for this $L^r$-quantization problem, derive their stationarity properties and establish for $L^p$-spaces E the pathwise regularity of stationary quantizers.