Rates of convergence for the empirical quantization error

Abstract

For $n, k \in \mathbb{N}$ and $r > 0$ let $e_{n,r}(P_k)^r = \inf 1/k \sum_{i=1}^k ||X_i - f(X_i)||^r$, where the infimum is taken over all measurable maps $f : \mathbb{R}^d \to \mathbb{R}^d$ with $|f(\mathbb{R}^d)| \leq n$ and $X_1, \dots, X_k$ are i.i.d. $\mathbb{R}^d$-valued random variables. We analyse the asymptotic a.s. behaviour of the $n$th empirical quantization error $e_{n,r}(P_k)$.