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

Martin Huesmann, Francesco Mattesini and Dario Trevisan

Stochastic Processes and their Applications, 2023, vol. 155, issue C, 1-26

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.

Keywords: Fractional Brownian motion; Optimal transport; Empirical measure (search for similar items in EconPapers)
Date: 2023
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Persistent link: https://EconPapers.repec.org/RePEc:eee:spapps:v:155:y:2023:i:c:p:1-26

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DOI: 10.1016/j.spa.2022.09.008

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