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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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