Convergence to stable laws in the space $D$

We study the convergence of centered and normalized sums of i.i.d. random elements of the space $\mathcal{D}$ of c{á}dl{á}g functions endowed with Skorohod's $J_1$ topology, to stable distributions in $\mathcal D$. 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