Central Limit Theorem for stationary Fleming–Viot particle systems in finite spaces

Tony Lelièvre (1, 2) , Loucas Pillaud-Vivien (3) , Julien Reygner (1)

1 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique
2 MATHERIALS - MATHematics for MatERIALS
3 SIERRA - Statistical Machine Learning and Parsimony

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 [ESAIM: Probab. Statist., 2003] and Cérou, Delyon, Guyader and Rousset [arXiv:1611.00515, arXiv:1709.06771].

Mots clés: Central Limit Theorem; Fleming–Viot particle system; Stationary distribution.

Domaines: Probabilités [math.PR]

Tony Lelièvre, Loucas Pillaud-Vivien, Julien Reygner. Central Limit Theorem for stationary Fleming–Viot particle systems in finite spaces. ALEA : Latin American Journal of Probability and Mathematical Statistics, 2018, 15, pp.1163-1182. ⟨10.30757/ALEA.v15-43⟩. ⟨hal-01812120v2⟩