Central Limit Theorem for the Solution of the Kac Equation

One proves that the solution of the Kac analogue of Boltzmann\u27s equation can be viewed as probability distribution of a sum of a random number of random variables. It is shown that 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\u27s equation can converge weakly. While this statement may seem to be intuitively clear, to ourknowledge there is no proof of it