Central limit theorem for stationary Fleming-Viot particle systems in finite spaces

We consider the Fleming-Viot particle system associated with a continuous- time Markov chain in a finite space. Assuming irreducibility, it is known that the particle system possesses a unique stationary distribution, under which its empirical measure converges to the quasistationary distribution of the Markov chain. We complement this Law of Large Numbers with a Central Limit Theorem. Our proof essentially relies on elementary computations on the infinitesimal generator of the Fleming-Viot particle system, and involves the so-called π-return process in the expression of the asymptotic variance. Our work can be seen as an infinite-time version, in the setting of finite space Markov chains, of results by Del Moral and Miclo (2003) and Cérou et al. (2016, 2017).