  We study the convergence of centered and normalized sums of i.i.d. random
elements of the space $\mathcal{D}$ of c{{\'a}}dl{{\'a}}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.
