Open Access
May 2020 Characterization of probability distribution convergence in Wasserstein distance by $L^{p}$-quantization error function
Yating Liu, Gilles Pagès
Bernoulli 26(2): 1171-1204 (May 2020). DOI: 10.3150/19-BEJ1146


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.


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Yating Liu. Gilles Pagès. "Characterization of probability distribution convergence in Wasserstein distance by $L^{p}$-quantization error function." Bernoulli 26 (2) 1171 - 1204, May 2020.


Received: 1 June 2018; Revised: 1 July 2019; Published: May 2020
First available in Project Euclid: 31 January 2020

zbMATH: 07166560
MathSciNet: MR4058364
Digital Object Identifier: 10.3150/19-BEJ1146

Keywords: probability distribution characterization , Vector quantization , Voronoï diagram , Wasserstein convergence

Rights: Copyright © 2020 Bernoulli Society for Mathematical Statistics and Probability

Vol.26 • No. 2 • May 2020
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