Wasserstein Convergence Rates for Empirical Measures of Subordinated Processes on Noncompact Manifolds

TL;DR

This paper establishes convergence rates of empirical measures for subordinated processes on noncompact Riemannian manifolds, extending previous results and demonstrating sharpness through examples.

Contribution

It provides new Wasserstein convergence rates for empirical measures of subordinated processes on general noncompact manifolds, broadening the scope of prior work.

Findings

Derived explicit convergence rates under Wasserstein distance.

Extended results to more general subordinated processes.

Proved sharpness of the rates with examples.

Abstract

The asymptotic behaviour of empirical measures has been studied extensively. In this paper, we consider empirical measures of given subordinated processes on complete (not necessarily compact) and connected Riemannian manifolds with possibly nonempty boundary. We obtain rates of convergence for empirical measures to the invariant measure of the subordinated process under the Wasserstein distance. The results, established for more general subordinated processes than [arXiv:2107.11568], generalize the recent ones in [Stoch. Proc. Appl. 144(2022), 271--287] and are shown to be sharp by a typical example. The proof is motivated by the aforementioned works.