  This paper is devoted to the grazing collision limit of the inelastic Kac
model introduced in [A. Pulvirenti and G. Toscani. J. Statist. Phys.,
114(5-6):1453--1480, 2004], when the equilibrium distribution function is a
heavy-tailed L\'evy-type distribution with infinite variance. We prove that
solutions in an appropriate domain of attraction of the equilibrium
distribution converge to solutions of a Fokker-Planck equation with a
fractional diffusion operator.
