Convergence to stable laws in the space $D$

TL;DR

This paper investigates how sums of i.i.d. cadlag functions converge to stable laws within the Skorohod space, utilizing regular variation and point process techniques, with applications to renewal-reward empirical processes.

Contribution

It extends convergence results to the space of cadlag functions using regular variation and point process methods, with specific applications to renewal-reward processes.

Findings

Convergence of sums to stable laws in Skorohod space established.

Application demonstrated for renewal-reward empirical processes.

Framework based on regular variation and point process convergence.

Abstract

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.