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Constructive quantization: approximation by empirical measures

Steffen Dereich; Michael Scheutzow; Reik Schottstedt — 2013

Annales de l'I.H.P. Probabilités et statistiques

In this article, we study the approximation of a probability measure $μ$ on $ℝ^{d}$ by its empirical measure ${\hat{μ}}_{N}$ interpreted as a random quantization. As error criterion we consider an averaged $p$ th moment Wasserstein metric. In the case where $2 p l t; d$ , we establish fine upper and lower bounds for the error, a. Moreover, we provide a universal estimate based on moments, a . In particular, we show that quantization by empirical measures is of optimal order under weak assumptions.

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