On optimal matching of Gaussian samples III

2019/11/18 by Michel Ledoux, Ledoux, Michel, Jie-Xiang Zhu +1 · 1 citation
Mathematics · #FOS: Mathematics #Geometry and complex manifolds #Markov Chains and Monte Carlo Methods #Point processes and geometric inequalities #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.1911.07579

openalex publication_date 2019/11/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This article is a continuation of the papers [8,9] in which the optimal matching problem, and the related rates of convergence of empirical measures for Gaussian samples are addressed. A further step in both the dimensional and Kantorovich parameters is achieved here, proving that, given X₁, …, X_n independent random variables with common distribution the standard Gaussian measure μ on ℝ^d, d ≥ 3, and μ_n = \frac 1n ∑_i=1ⁿ δ_Xi the associated empirical measure, 𝔼 ( \mathrm W_p^p (μ_n , μ) ) ≈ \frac 1n^p/d for any 1≤ p < d, where \mathrm W_p is the p-th Kantorovich metric. The proof relies on the pde and mass transportation approach developed by L. Ambrosio, F. Stra and D. Trevisan in a compact setting.

Citations

• On the rate of convergence in Wasserstein distance of the empirical\n measure
• One-dimensional empirical measures, order statistics, and Kantorovich transport distances

Cited by

• Convergence of Empirical Measures for i.i.d. samples in W^{-α, p}

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