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December 2008 Central limit theorem for the solution of the Kac equation
Ester Gabetta, Eugenio Regazzini
Ann. Appl. Probab. 18(6): 2320-2336 (December 2008). DOI: 10.1214/08-AAP524


We prove that the solution of the Kac analogue of Boltzmann’s equation can be viewed as a probability distribution of a sum of a random number of random variables. This fact allows us to study convergence to equilibrium by means of a few classical statements pertaining to the central limit theorem. In particular, a new proof of the convergence to the Maxwellian distribution is provided, with a rate information both under the sole hypothesis that the initial energy is finite and under the additional condition that the initial distribution has finite moment of order 2+δ for some δ in (0, 1]. Moreover, it is proved that finiteness of initial energy is necessary in order that the solution of Kac’s equation can converge weakly. While this statement may seem to be intuitively clear, to our knowledge there is no proof of it as yet.


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Ester Gabetta. Eugenio Regazzini. "Central limit theorem for the solution of the Kac equation." Ann. Appl. Probab. 18 (6) 2320 - 2336, December 2008.


Published: December 2008
First available in Project Euclid: 26 November 2008

zbMATH: 1161.82018
MathSciNet: MR2474538
Digital Object Identifier: 10.1214/08-AAP524

Primary: 60F05, 82C40

Rights: Copyright © 2008 Institute of Mathematical Statistics


Vol.18 • No. 6 • December 2008
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