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.


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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