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Wasserstein asymptotics for the empirical measure of fractional Brownian motion on a flat torus

Author

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  • Huesmann, Martin
  • Mattesini, Francesco
  • Trevisan, Dario

Abstract

We establish asymptotic upper and lower bounds for the Wasserstein distance of any order p≥1 between the empirical measure of a fractional Brownian motion on a flat torus and the uniform Lebesgue measure. Our inequalities reveal an interesting interaction between the Hurst index H and the dimension d of the state space, with a “phase-transition” in the rates when d=2+1/H, akin to the Ajtai–Komlós–Tusnády theorem for the optimal matching of i.i.d. points in two-dimensions. Our proof couples PDE’s and probabilistic techniques, and also yields a similar result for discrete-time approximations of the process, as well as a lower bound for the same problem on Rd.

Suggested Citation

  • Huesmann, Martin & Mattesini, Francesco & Trevisan, Dario, 2023. "Wasserstein asymptotics for the empirical measure of fractional Brownian motion on a flat torus," Stochastic Processes and their Applications, Elsevier, vol. 155(C), pages 1-26.
    • Handle: RePEc:eee:spapps:v:155:y:2023:i:c:p:1-26
    DOI: 10.1016/j.spa.2022.09.008
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    References listed on IDEAS

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    1. Wang, Feng-Yu, 2022. "Wasserstein convergence rate for empirical measures on noncompact manifolds," Stochastic Processes and their Applications, Elsevier, vol. 144(C), pages 271-287.
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