Convergence rate of optimal quantization grids and application to empirical measure

Abstract

We study the convergence rate of the optimal quantization for a probability measure sequence $({\mu }_n{)}_{n\in {\mathbb{N}}^{\ast }}$ on ${\mathbb{R}}^d$ converging in the Wasserstein distance in two aspects: the first one is the convergence rate of optimal quantizer $x^{(n)}\in ({\mathbb{R}}^d{)}^K$ of ${\mu }_n$ at level $K$; the other one is the convergence rate of the distortion function valued at $x^{(n)}$, called the "performance" of $x^{(n)}$. Moreover, we also study the mean performance of the optimal quantization for the empirical measure of a distribution $\mu$ with finite second moment but possibly unbounded support. As an application, we show that the mean performance for the empirical measure of the multidimensional normal distribution $\mathcal{N}(m,\Sigma )$ and of distributions with hyper-exponential tails behave like $\mathcal{O}(\frac{\log n}{\sqrt{n}})$. This extends the results from [BDL08] obtained for compactly supported distribution. We also derive an upper bound which is sharper in the quantization level $K$ but suboptimal in $n$ by applying results in [FG15].