Rates of convergence for the empirical quantization error (Q1872278)

Let \(X_1,X_2,\dots\) be \(d\)-dimensional i.i.d random variables with distribution \(P\), and let \(P_k\) denote the empirical measure of \(X_1,\dots,X_k\). The authors provide rates of convergence for the a.s. limiting behaviour of the empirical quantization error, i.e. they prove a.s. bounds, as \(k\to\infty \), for \[ Y_{k,r}(P)=\sup_{n\geq 1}|e_{n,r}(P_k)^r - e_{n,r}(P)^r|\quad\text{ and }\quad Z_{k,r}(P)=\sup|e_{n,r}(P_k) -e_{n,r}(P)|, \] respectively. Here \[ e_{n,r}(P)=\inf\{(E_P\|X-f(X)\|^r)^{1/r}\}\quad \text{ and }\quad e_{n,r}(P_k)=\inf\Bigl\{\Bigl (\frac 1k\sum^k_{i=1}\|X_i-f(X_i)\|^r\Bigr)^{1/r}\Bigr\} \] where the infimum is taken over all measurable maps \(f\) with at most \(n\) values in \(\mathbb{R}^d\), and where \(\|\cdot \|\) denotes any norm in \(\mathbb{R}^d\).