Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

A nonasymptotic theorem for unnormalized Feynman–Kac particle models

F. Cérou, P. Del Moral, and A. Guyader

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Abstract

We present a nonasymptotic theorem for interacting particle approximations of unnormalized Feynman–Kac models. We provide an original stochastic analysis-based on Feynman–Kac semigroup techniques combined with recently developed coalescent tree-based functional representations of particle block distributions. We present some regularity conditions under which the $\mathbb {L}_{2}$-relative error of these weighted particle measures grows linearly with respect to the time horizon yielding what seems to be the first results of this type for this class of unnormalized models. We also illustrate these results in the context of particle absorption models, with a special interest in rare event analysis.

Résumé

Nous présentons un théorème non asymptotique pour les approximation par systèmes de particules en interaction des modèles de Feynman–Kac non normalisés. Nous introduisons une analyse stochastique originale basée sur des techniques de semigroupes de Feynman–Kac, associées avec les représentation, récemment proposées, des distributions de blocks de particules, en terme de développement en arbre de coalescence. Nous présentons des conditions de régularité sous lesquelles l’erreur relative $\mathbb {L}_{2}$ de ces mesures particulaires pondérées croît linéairement par rapport ‘a l’horizon temporel, conduisant ‘a ce qui semble être le premier résultat de ce type pour cette classe de modèles non normalisés. Nous illustrons ces résultats dans le contexte des mesures statiques de Boltzmann–Gibbs et des distributions restreintes, avec un intéret partuculier pour les événements rares.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 47, Number 3 (2011), 629-649.

Dates
First available in Project Euclid: 23 June 2011

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1308834852

Digital Object Identifier
doi:10.1214/10-AIHP358

Mathematical Reviews number (MathSciNet)
MR2841068

Zentralblatt MATH identifier
1233.60047

Subjects
Primary: 47D08: Schrödinger and Feynman-Kac semigroups 60C05: Combinatorial probability 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 65C35: Stochastic particle methods [See also 82C80]
Secondary: 31B10: Integral representations, integral operators, integral equations methods 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 65C05: Monte Carlo methods 92D25: Population dynamics (general)

Keywords
Interacting particle systems Feynman–Kac semigroups Nonasymptotic estimates Genetic algorithms Boltzmann–Gibbs measures Monte Carlo models Rare events

Citation

Cérou, F.; Del Moral, P.; Guyader, A. A nonasymptotic theorem for unnormalized Feynman–Kac particle models. Ann. Inst. H. Poincaré Probab. Statist. 47 (2011), no. 3, 629--649. doi:10.1214/10-AIHP358. https://projecteuclid.org/euclid.aihp/1308834852


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References

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