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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 $2+\delta$ 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.Comment: Published in at http://dx.doi.org/10.1214/08-AAP524 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

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This paper deals with a one-dimensional model for granular materials, which boils down to an inelastic version of the Kac kinetic equation, with inelasticity parameter p > 0. In particular, the paper provides bounds for certain distances -such as specific weighted χ-distances and the Kolmogorov distance -between the solution of that equation and the limit. It is assumed that the even part of the initial datum (which determines the asymptotic properties of the solution) belongs to the domain of normal attraction of a symmetric stable distribution with characteristic exponent α = 2/(1 + p). With such initial data, it turns out that the limit exists and is just the aforementioned stable distribution.A necessary condition for the relaxation to equilibrium is also proved. Some bounds are obtained without introducing any extra-condition. Sharper bounds, of an exponential type, are exhibited in the presence of additional assumptions concerning either the behaviour, near to the origin, of the initial characteristic function, or the behaviour, at infinity, of the initial probability distribution function. already mentioned Pulvirenti and Toscani's paper, it is worth quoting: Bobylev et alKey words and phrases. Central limit theorem, domains of normal attraction, granular materials, Kolmogorov metric, inelastic Kac equation, stable distributions, sums of weighted independent random variables, speed of approach to equilibrium, weighted χ-metrics.

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The Lauricella theory of multiple hypergeometric functions is used to shed some light on certain distributional properties of the mean of a Dirichlet process. This approach leads to several results, which are illustrated here. Among these are a new and more direct procedure for determining the exact form of the distribution of the mean, a correspondence between the distribution of the mean and the parameter of a Dirichlet process, a characterization of the family of Cauchy distributions as the set of the fixed points of this correspondence, and an extension of the Markov-Krein identity. Moreover, an expression of the characteristic function of the mean of a Dirichlet process is obtained by resorting to an integral representation of a confluent form of the fourth Lauricella function. This expression is then employed to prove that the distribution of the mean of a Dirichlet process is symmetric if and only if the parameter of the process is symmetric, and to provide a new expression of the moment generating function of the variance of a Dirichlet process.Comment: Published by the Institute of Mathematical Statistics (http://www.imstat.org) in the Annals of Probability (http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/00911790400000027