CENTRAL LIMIT THEOREM FOR THE SOLUTION OF THE KAC EQUATION

• 成果类型: Article
• 署名作者: Gabetta, Ester; Regazzini, Eugenio
• 署名单位: University of Pavia; Consiglio Nazionale delle Ricerche (CNR); Istituto di Matematica Applicata e Tecnologie Informatiche Enrico Magenes (IMATI-CNR)
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/08-AAP524
• 发表日期: 2008
• 页码: 2320-2336
• 关键词: maxwellian molecules; wild sums; CONVERGENCE; equilibrium

摘要

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 it 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 + delta for some delta 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.