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In this article, we study the approximation of a probability measure $\mu$ on$\mathbb{R}^{d}$ by its empirical measure $\hat{\mu}_{N}$ interpreted as arandom quantization. As error criterion we consider an averaged $p$-th momentWasserstein metric. In the case where $2p<d$, we establish refined upper andlower bounds for the error, a high-resolution formula. Moreover, we provide auniversal estimate based on moments, a so-called Pierce type estimate. Inparticular, we show that quantization by empirical measures is of optimal orderunder weak assumptions.

Constructive quantization: approximation by empirical measures