Constructive quantization: approximation by empirical measures

In this article, we study the approximation of a probability measure $μ$ on $\mathbb{R}^{d}$ by its empirical measure $\hatμ_{N}$ interpreted as a random quantization. As error criterion we consider an averaged $p$-th moment Wasserstein metric. In the case where $2p<d$, we establish refined upper and lower bounds for the error, a high-resolution formula. Moreover, we provide a universal estimate based on moments, a so-called Pierce type estimate. In particular, we show that quantization by empirical measures is of optimal order under weak assumptions.

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