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Wasserstein convergence properties for Gaussian-smoothed empirical measures

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  • Li, Huaiqian
  • Wu, Bingyao

Abstract

We obtain non-asymptotic upper bounds for the expected Gaussian-smoothed p-Wasserstein distance between a probability measure μ on Rd and its empirical counterpart μN, constructed from N independent and identically distributed samples. Our results hold for all 0≤p<∞, with the case p=0 corresponding to the total variation distance. These bounds hold under the assumption that μ has moments of order q>(2p)∨d, improving the latest results in the literature. As a consequence, we also derive Gaussian/exponential-type tail bounds for the smoothed Wasserstein distance. Furthermore, we extend the convergence analysis to dependent data settings, including ρ-mixing sequences and Markov chains satisfying a spectral gap condition.

Suggested Citation

  • Li, Huaiqian & Wu, Bingyao, 2026. "Wasserstein convergence properties for Gaussian-smoothed empirical measures," Statistics & Probability Letters, Elsevier, vol. 238(C).
    • Handle: RePEc:eee:stapro:v:238:y:2026:i:c:s0167715226002117
    DOI: 10.1016/j.spl.2026.110847
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