  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.
