  A linear Boltzmann equation is interpreted as the forward equation for the
probability density of a Markov process (K(t), Y(t)), where K(t) is a
autonomous reversible jump process, with waiting times between two jumps with
finite expectation value but infinite variance, while Y(t) is an additive
functional of K(t). We prove that under a suitable rescaling the process Y
converges in distribution to a Levy process, stable with index 3/2. Moreover,
the solution of the linear Boltzmann equation converges to the solution of a
fractional diffusion equation.
