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

2025/12/19 by Gautam Iyer, Iyer, Gautam, Raghavendra Venkatraman +1
Mathematics · #Geometry and complex manifolds #Advanced Harmonic Analysis Research #Point processes and geometric inequalities

paper · doi:10.48550/arxiv.2512.17794

Abstract

Given N i.i.d. samples from a probability measure μ on Rd, we study the rate of convergence of the empirical measure μN → μ in the negative Sobolev space W-α, p. When W-α, p contains point measures (i.e. when αp > (p-1)d), we show E ‖ μN - μ‖W-α, pp ≤ Cd / Np/2 for an explicit dimensional constant Cd, and obtain a Gaussian tail bound. When 0 < αp ≤ d(p-1), we prove a similar result for Gaussian regularizations.

Citations

• A sharp commutator estimate for all Riesz modulated energies
• A short proof on the rate of convergence of the empirical measure for the Wasserstein distance
• Metrizing Weak Convergence with Maximum Mean Discrepancies
• On optimal matching of Gaussian samples III
• Sharp asymptotic and finite-sample rates of convergence of empirical measures in Wasserstein distance
• On the rate of convergence in Wasserstein distance of the empirical\n measure
• Constructive quantization: approximation by empirical measures
• On the mean speed of convergence of empirical and occupation measures in Wasserstein distance

Related

• Convergence of the empirical measure in expected Wasserstein distance: non asymptotic explicit bounds in ℝ^d
• Empirical measures: regularity is a counter-curse to dimensionality
• On the rate of convergence in Wasserstein distance of the empirical\n measure
• Convergence of Adapted Empirical Measures on ℝ^d
• Convergence of random measures in geometric probability