## Characterization of probability distribution convergence in Wasserstein distance by $$L^p$$-quantization error function.(English)Zbl 1466.60007

Summary: We establish conditions to characterize probability measures by their $$L^p$$-quantization error functions in both $$\mathbb{R}^d$$ 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 quantization level $$N$$, valid for any norm on $$\mathbb{R}^d$$ 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 in dimension 1 and a discussion of the completeness of a distance defined by the quantization error function can be found at the end of this paper.

### MSC:

 60B10 Convergence of probability measures 60B11 Probability theory on linear topological spaces 60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
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### References:

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