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A central limit theorem for Fleming-Viot particle systems. (English. French summary) Zbl 1447.82021

Fleming-Viot-type particle systems are considered, which consist of independently moving particles that are killed on the boundary of a domain. At the time of death of a particle, another particle branches so that the particles population size in principle might be kept constant. In [M. Bieniek et al., Probab. Theory Relat. Fields 153, No. 1–2, 293–332 (2012; Zbl 1253.60089)], a proof has been given of a generic non-extinction of the Fleming-Viot model. The fact that particle population does not approach the boundary simultaneously in a finite time, in some Lipschitz diomains, has been employed to prove a limit theorem for the empirical distribution of the particle family. An earlier paper [K. Burdzy et al., J. Phys. A, Math. Gen. 29, No. 11, 2633–2642 (1996; Zbl 0901.60054)] has been devoted to the link of Fleming-Viot models with Laplacian eigenfunctions, where the link with the concept of quasi-stationary distributions naturally appears [P. Collet et al., Quasi-stationary distributions. Markov chains, diffusions and dynamical systems. Berlin: Springer (2013; Zbl 1261.60002)]. Accordingly, Fleming-Viot type particle systems may be viewed as representing a classical way to approximate the distribution of a Markov process (not necessarily Brownian motion) with killing, given that it is still alive at a final deterministic time. The model follows a simple algorithm for systems of independently evolving particles. Namely, each particle is presumed to follow to the law of the underlying Markov process until its killing, and then branches instantaneously from the state of another randomly chosen particle. The consistency of this algorithm in the large population limit has been recently studied in several articles. The major purpose of the present paper is to prove central limit theorems under more general assumptions. For this, the key suppositions are that the particle system does not explode in finite time, and that the jump and killing times have atomless distributions. The convergence to a stationary distribution for large particle systems is thereby established. A comparative study is performed with sequential Monte Carlo simulations for discrete time algorithms.

MSC:

82C22 Interacting particle systems in time-dependent statistical mechanics
82M31 Monte Carlo methods applied to problems in statistical mechanics
65C05 Monte Carlo methods
60J25 Continuous-time Markov processes on general state spaces
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60K37 Processes in random environments
60F05 Central limit and other weak theorems
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References:

[1] M. Bieniek, K. Burdzy and S. Finch. Non-extinction of a Fleming-Viot particle model. Probab. Theory Related Fields 153 (1-2) (2012) 293-332. · Zbl 1253.60089 · doi:10.1007/s00440-011-0372-5
[2] K. Burdzy, R. Holyst, D. Ingerman and P. March. Configurational transition in a Fleming-Viot-type model and probabilistic interpretation of Laplacian eigenfunctions. J. Phys. A: Math. Gen. 29 (11) (1996) 2633. · Zbl 0901.60054 · doi:10.1088/0305-4470/29/11/004
[3] F. Cérou, B. Delyon, A. Guyader and M. Rousset. A Central Limit Theorem for Fleming-Viot Particle Systems with Soft Killing, 2016. Available at arXiv:1611.00515v2. · Zbl 1447.82021
[4] F. Cérou, B. Delyon, A. Guyader and M. Rousset. On the Asymptotic Normality of Adaptive Multilevel Splitting. SIAM/ASA J. Uncertain. Quantificat. 7 (1) (2019) 1-30. · Zbl 1412.82029 · doi:10.1137/18M1187477
[5] P. Del Moral. Feynman-Kac Formulae, Genealogical and Interacting Particle Systems with Applications. Springer-Verlag, New York, 2004. · Zbl 1130.60003
[6] S. N. Ethier and T. G. Kurtz. Markov Processes. Wiley, New York, 1986. · Zbl 0592.60049
[7] I. Grigorescu and M. Kang. Hydrodynamic limit for a Fleming-Viot type system. Stochastic Process. Appl. 110 (1) (2004) 111-143. · Zbl 1075.60124 · doi:10.1016/j.spa.2003.10.010
[8] I. Grigorescu and M. Kang. Immortal particle for a catalytic branching process. Probab. Theory Related Fields 153 (1-2) (2012) 333-361. · Zbl 1251.60064 · doi:10.1007/s00440-011-0347-6
[9] J. Jacod and A. N. Shiryaev. Limit Theorems for Stochastic Processes, 288, 2nd edition. Springer-Verlag, Berlin, 2003. · Zbl 1018.60002
[10] O. Kallenberg. Foundations of Modern Probability. Probability and Its Applications. Springer, New York, 2002. · Zbl 0996.60001
[11] J.-U. Löbus. A stationary Fleming-Viot type Brownian particle system. Math. Z. 263 (3) (2009) 541-581. · Zbl 1176.60084 · doi:10.1007/s00209-008-0430-6
[12] P. E. Protter. Stochastic Integration and Differential Equations, 2nd edition. Stochastic Modelling and Applied Probability 21. Springer-Verlag, Berlin, 2005.
[13] M. J. Schervish. Theory of Statistics. Springer Series in Statistics. Springer-Verlag, New York, 1995. · Zbl 0834.62002
[14] D. Villemonais. General approximation method for the distribution of Markov processes conditioned not to be killed. ESAIM Probab. Stat. 18 (2014) 441-467. · Zbl 1310.82032 · doi:10.1051/ps/2013045
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