Convergence to stable laws in the space script D

We study the convergence of centered and normalized sums of independent and identically distributed random elements of the space script D of càdlàg functions endowed with Skorokhod's J₁ topology, to stable distributions in script D. Our results are based on the concept of regular variation on metric spaces and on point process convergence. We provide some applications; in particular, to the empirical process of the renewal-reward process.