Exact rate of convergence for the empirical measure of a subordinated process in $p$-Wasserstein distance

We establish exact rates of convergence in the $p$-Wasserstein distance for the empirical measure of a class of non-symmetric jump processes, which are subordinated to a diffusion process on a compact Riemannian manifold. For the quadratic Wasserstein distance, we determine the renormalization limit. We extend the main results of \cite{WW} and \cite{WWZ}. Our method uses two key elements: a Bernstein-type inequality for the subordinated process and the PDE approach established in \cite{AMB}