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

TL;DR

This paper establishes a Central Limit Theorem for the empirical measure of Fleming--Viot particle systems in finite spaces, complementing known Law of Large Numbers results and providing elementary proof techniques.

Contribution

It introduces a CLT for Fleming--Viot systems in finite spaces, extending previous LLN results with a new, elementary proof approach and explicit asymptotic variance expression.

Findings

Proves a CLT for empirical measures of Fleming--Viot systems

Provides explicit formulas for asymptotic variance involving the $ ext{ extpi}$-return process

Extends finite-space Markov chain results to infinite-time setting

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 $\pi$-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 C{\'e}rou, Delyon, Guyader and Rousset [arXiv:1611.00515, arXiv:1709.06771].