In this article, we study the approximation of a probability measure µ on R d by its empirical measureμN 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 fine upper and lower bounds for the error, a high-resolution formula. Moreover, we provide a universal estimate based on moments, a Pierce type estimate. In particular, we show that quantization by empirical measures is of optimal order under weak assumptions.

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