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

Tony Lelievre 1, 2 Loucas Pillaud-Vivien 3 Julien Reygner 1
2 MATHERIALS - MATHematics for MatERIALS
CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique, Inria de Paris
3 SIERRA - Statistical Machine Learning and Parsimony
DI-ENS - Département d'informatique de l'École normale supérieure, CNRS - Centre National de la Recherche Scientifique, Inria de Paris
Abstract : 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].
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ALEA : Latin American Journal of Probability and Mathematical Statistics, Instituto Nacional de Matemática Pura e Aplicada, 2018, 15, pp.1163-1182. 〈10.30757/ALEA.v15-43〉
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Soumis le : lundi 1 octobre 2018 - 17:48:51
Dernière modification le : vendredi 5 octobre 2018 - 16:15:02

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Tony Lelievre, 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, Instituto Nacional de Matemática Pura e Aplicada, 2018, 15, pp.1163-1182. 〈10.30757/ALEA.v15-43〉. 〈hal-01812120v2〉

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