publication . Preprint . 2017

A Central Limit Theorem for Fleming-Viot Particle Systems with Hard Killing

Cérou, Frédéric; Delyon, Bernard; Guyader, Arnaud; Rousset, Mathias;
Open Access English
  • Published: 10 Oct 2017
  • Publisher: HAL CCSD
  • Country: France
Abstract
Fleming-Viot type particle systems represent a classical way to approximate the distribution of a Markov process with killing, given that it is still alive at a final deterministic time. In this context, each particle evolves independently according to the law of the underlying Markov process until its killing, and then branches instantaneously on another randomly chosen particle. While the consistency of this algorithm in the large population limit has been recently studied in several articles, our purpose here is to prove Central Limit Theorems under very general assumptions. For this, we only suppose that the particle system does not explode in finite time, a...
Subjects
free text keywords: process with killing, Interacting particle systems, sequential MonteCarlo, 82C22, 82C80, 65C05, 60J25, 60K35, 60K37, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], [ MATH.MATH-PR ] Mathematics [math]/Probability [math.PR], Mathematics - Probability
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Project
MSMATH
Molecular Simulation: modeling, algorithms and mathematical analysis
  • Funder: European Commission (EC)
  • Project Code: 614492
  • Funding stream: FP7 | SP2 | ERC
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2 Main result 5 2.1 Notation and assumptions . . . . . . . . . . . . . . . . . . . . 5 2.2 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Example: Feller process with hard obstacle . . . . . . . . . . . 8

3 Proof 10 3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.2 Well-posedness and non-simultaneity of jumps . . . . . . . . . 11 3.3 Martingale decomposition . . . . . . . . . . . . . . . . . . . . 12 3.4 Quadratic variation estimates . . . . . . . . . . . . . . . . . . 15 3.5 L2-estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.6 Time uniform estimate for pt . . . . . . . . . . . . . . . . . . . 22 3.7 Approximation of the quadratic variation . . . . . . . . . . . . 22 3.8 The asymptotic variance and the convergence . . . . . . . . . 25 3.9 Another formulation of the asymptotic variance . . . . . . . . 27 3.10 Martingale Central Limit Theorem . . . . . . . . . . . . . . . 28

4 Appendix 32 4.1 Preliminary on Feller processes . . . . . . . . . . . . . . . . . 32 4.2 Stopping times and martingales . . . . . . . . . . . . . . . . . 35 4.3 Proof of Lemma 3.1: (A) ⇒ (A') . . . . . . . . . . . . . . . . 35 4.4 Integration rules . . . . . . . . . . . . . . . . . . . . . . . . . . 36

[1] M. Bieniek, K. Burdzy, and S. Finch. Non-extinction of a Fleming-Viot particle model. Probab. Theory Related Fields, 153(1-2):293-332, 2012.

[2] K. Burdzy, R. Holyst, D. Ingerman, and P. March. Configurational transition in a Fleming-Viot-type model and probabilistic interpretation of Laplacian eigenfunctions. Journal of Physics A: Mathematical and General, 29(11):2633, 1996. [OpenAIRE]

[3] F. C´erou, B. Delyon, A. Guyader, and M. Rousset. A Central Limit Theorem for Fleming-Viot Particle Systems. ArXiv e-prints, 2016.

[4] S.N. Ethier and T.G. Kurtz. Markov processes. John Wiley & Sons, Inc., New York, 1986.

[5] I. Grigorescu and M. Kang. Hydrodynamic limit for a Fleming-Viot type system. Stochastic Process. Appl., 110(1):111-143, 2004.

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publication . Preprint . 2017

A Central Limit Theorem for Fleming-Viot Particle Systems with Hard Killing

Cérou, Frédéric; Delyon, Bernard; Guyader, Arnaud; Rousset, Mathias;