arXiv 20 Oct 2026 · Mathematics — Statistics Theory
arXiv:2610.27857 · PDF · Extracted main text
We establish central limit theorems for empirical plug-in estimators of scalar functionals of quadratic optimal transport maps and potentials, including Wasserstein costs and several risk measures constructed from transport-based multivariate quantiles. We study the one-sample problem with a known reference distribution and i.i.d. observations from the target distribution. A natural approach is to combine a limit theorem for the empirical transport map with the functional delta method. We show, however, that in dimensions at least two, empirical transport-map errors admit no nonzero weak limit in $L^2$ under any normalizing scale. Instead, we linearize the functional directly in the target distribution, obtaining an influence-function expansion with a remainder controlled by the transport-map error. The influence function is characterized by a linear elliptic partial differential equation. Under regularity conditions, we use this expansion to prove central limit theorems in low dimensions and establish validity of the empirical bootstrap. We also derive semiparametric efficiency bounds and show that the plug-in estimators attain them under the same conditions.
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The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | González-Sanz, A. and Sheng, S. and Wu, B. and Avella Medina, M (2026) The influence function of transport-based quantiles | 1.000 | 5 | 3 | 100% |
| 2 | González-Sanz, A. and Mordant, G. and Sheng, S (2026) Empirical optimal transport potentials: fast rates and a functional central limit theorem | 0.941 | 6 | 4 | 83% |
| 3 | Manole, T. and Balakrishnan, S. and Niles-Weed, J. and Wasserman, L (2024) Plugin estimation of smooth optimal transport maps | 0.941 | 6 | 3 | 83% |
| 4 | Villani, C (2003) Topics in Optimal Transportation | 0.874 | 6 | 2 | 100% |
| 5 | Sadhu, R. and Goldfeld, Z. and Kato, K (2024) Stability and statistical inference for semidiscrete optimal transport maps | 0.811 | 4 | 2 | 100% |
| 6 | Beirlant, J. and Buitendag, S. and del Barrio, E. and Hallin, M. and… (2020) Center-outward quantiles and the measurement of multivariate risk | 0.737 | 3 | 2 | 100% |
| 7 | Hallin, M. and del Barrio, E. and Cuesta-Albertos, J. and Matrán, C (2021) Distribution and quantile functions, ranks and signs in dimension $d$: A measure transportation approach | 0.737 | 3 | 2 | 100% |
| 8 | van der Vaart, A. W. and Wellner, J. A (1996) Weak Convergence and Empirical Processes: With Applications to Statistics | 0.693 | 6 | 3 | 33% |
| 9 | Manole, T. and Balakrishnan, S. and Niles-Weed, J. and Wasserman, L (2024) Central Limit Theorems for Smooth Optimal Transport Maps | 0.644 | 3 | 2 | 67% |
| 10 | Villani, C (2009) Optimal Transport: Old and New | 0.644 | 3 | 2 | 67% |
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