We study the convergence of centered and normalized sums of independent and identically distributed random elements of the space D of càdlàg functions endowed with Skorokhod's J1 topology, to stable distributions in 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.
"Convergence to stable laws in the space D." J. Appl. Probab. 52 (1) 1 - 17, March 2015. https://doi.org/10.1239/jap/1429282603