  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.
