Constructive quantization: Approximation by empirical measures

In this article, we study the approximation of a probability measure mu on R-d by its empirical measure (mu) over cap (N) interpreted as a random quantization. As error criterion we consider an averaged pth 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.