Characterization of probability distribution convergence in Wasserstein distance by $L^{p}$-quantization error function

Abstract

We establish conditions to characterize probability measures by their L p -quantization error functions in both R d and Hilbert settings. This characterization is two-fold: static (identity of two distributions) and dynamic (convergence for the L p -Wasserstein distance). We first propose a criterion on the qantization level N , valid for any norm on R d and any order p based on a geometrical approach involving the Voronoï diagram. Then, we prove that in the L 2 -case on a (separable) Hilbert space, the condition on the level N can be reduced to N = 2, which is optimal. More quantization based characterization cases on dimension 1 and a discussion of the completeness of a distance defined by the quantization error function can be found in the end of this paper.