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Central limit theorem through expansion of the propagation of chaos for Bird and Nanbu systems
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 25 (2016) no. 4, pp. 829-873.

Les systèmes de Bird et Nanbu sont des systèmes de particules en interaction approchant la solution de l’équation de Boltzmann mollifiée. Ces systèmes vérifient la propagation du chaos. Dans l’esprit de [6, 7, 8], nous utilisons des techniques de couplage pour écrire un développement asymptotique dans la propagation du chaos, en terme du nombre de particules. Ce développement nous permet de démontrer la convergence p.s. de ces systèmes, ainsi qu’un théorème central-limite. Ce théorème central-limite s’applique à la mesure empirique du système. Comme dans [6, 7, 8], ces résultats s’appliquent aux trajectoires des particules sur un intervalle [0;T].

The Bird and Nanbu systems are particle systems used to approximate the solution of the mollified Boltzmann equation. These systems have the propagation of chaos property. Following [6, 7, 8], we use coupling techniques to write a kind of expansion of the error in the propagation of chaos in terms of the number of particles. This expansion enables us to prove the a.s. convergence and the central-limit theorem for these systems. Notably, we obtain a central-limit theorem for the empirical measure of the system. As it is the case in [6, 7, 8], these results apply to the trajectories of particles on an interval [0,T].

Publié le :
DOI : 10.5802/afst.1512
Sylvain Rubenthaler 1

1 Univ. Nice Sophia Antipolis, CNRS, LJAD, UMR 7351, 06100 Nice, France
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     title = {Central limit theorem through expansion of the propagation of chaos for {Bird} and {Nanbu} systems},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
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Sylvain Rubenthaler. Central limit theorem through expansion of the propagation of chaos for Bird and Nanbu systems. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 25 (2016) no. 4, pp. 829-873. doi : 10.5802/afst.1512. https://afst.centre-mersenne.org/articles/10.5802/afst.1512/

[1] Athreya (K. B.) and Ney (P. E.).— Branching processes, Springer-Verlag, New York, Die Grundlehren der mathematischen Wissenschaften, Band 196 (1972).

[2] Dawson (D. A.) and Zheng (X.).— Law of large numbers and central limit theorem for unbounded jump mean-field models, Adv. in Appl. Math. 12, no. 3, p. 293-326 (1991). | DOI | MR | Zbl

[3] Del Moral (P.), Patras (F.), and Rubenthaler (S.).— Tree based functional expansions for Feynman-Kac particle models, Ann. Appl. Probab. 19, no. 2, p. 778-825 (2009). | DOI | MR | Zbl

[4] Del Moral (P.), Patras (F.), and Rubenthaler (S.).— Convergence of u-statistics for interacting particle systems, Journal of Theoretical Probability 24, no. 4, p. 1002-1027 (2011). | DOI | MR | Zbl

[5] Dodge (Y.).— Statistique, second ed., Springer-Verlag, Paris, Dictionnaire encyclopédique. [Encyclopedic dictionary] (2004).

[6] Graham (C.) and Méléard (S.).— Chaos hypothesis for a system interacting through shared resources, Probab. Theory Related Fields 100, no. 2, p. 157-173 (1994). | DOI | MR

[7] Graham (C.) and Méléard (S.).— Stochastic particle approximations for generalized Boltzmann models and convergence estimates, Ann. Probab. 25, no. 1, p. 115-132 (1997). | DOI | MR | Zbl

[8] Graham (C.) and Méléard (S.).— Probabilistic tools and Monte-Carlo approximations for some Boltzmann equations, CEMRACS 1999 (Orsay), ESAIM Proc., vol. 10, Soc. Math. Appl. Indust., Paris, p. 77-126 (1999) (electronic). | DOI | Zbl

[9] Kingman (J. F. C.).— Poisson processes, Oxford Studies in Probability, vol. 3, The Clarendon Press Oxford University Press, New York, Oxford Science Publications (1993). | Zbl

[10] Lee (A. J.).— U-statistics, Statistics: Textbooks and Monographs, vol. 110, Marcel Dekker Inc., New York, Theory and practice (1990).

[11] Méléard (S.).— Convergence of the þuctuations for interacting diffusions with jumps associated with Boltzmann equations, Stochastics Stochastics Rep. 63, no. 3-4, p. 195-225 (1998). | DOI | MR | Zbl

[12] Norris (J. R.).— Markov chains, Cambridge Series in Statistical and Probabilistic Mathematics, vol. 2, Cambridge University Press, Cambridge (1998), Reprint of 1997 original. | Zbl

[13] Nica (A.) and Speicher (R.).— Lectures on the combinatorics of free probability, London Mathematical Society Lecture Note Series, vol. 335, Cambridge University Press, Cambridge (2006). | DOI

[14] de la Pena (V. H.) and Giné (E.).— Decoupling, Probability and its Applications (New York), Springer-Verlag, New York (1999), From dependence to independence, Randomly stopped processes. U-statistics and processes. Martingales and beyond. | DOI | Zbl

[15] Shiga (T.) and Tanaka (H.).— Central limit theorem for a system of Markovian particles with meanfield interactions, Z. Wahrsch. Verw. Gebiete 69, no. 3, p. 439-459 (1985). | DOI | MR | Zbl

[16] Sznitman (A.-S.).— Nonlinear reflecting diffusion process, and the propagation of chaos and fluctuations associated, J. Funct. Anal. 56 (1984), no. 3, p. 311-336. | DOI | MR | Zbl

[17] Sznitman (A.-S.).— A fluctuation result for nonlinear diffusions, Infinite-dimensional analysis and stochastic processes (Bielefeld, 1983), Res. Notes in Math., vol. 124, Pitman, Boston, MA, p. 145-160 (1985).

[18] Uchiyama (K.).— A fluctuation problem associated with the Boltzmann equation for a gas of molecules with a cutoff potential, Japan. J. Math. (N.S.) 9, no. 1, p. 27-53 (1983). | DOI | MR | Zbl

[19] Uchiyama (K.).— Fluctuations of Markovian systems in Kac’s caricature of a Maxwellian gas, J. Math. Soc. Japan 35, no. 3, p. 477-499 (1983). | DOI | MR | Zbl

[20] Uchiyama (K.).— Fluctuations in a Markovian system of pairwise interacting particles, Probab. Theory Related Fields 79, no. 2, p. 289-302 (1988). | DOI | MR | Zbl

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