Asymptotics for heavy-tailed renewal–reward processes and applications to risk processes and heavy traffic networks

Abstract

Consider a renewal–reward process $S_{N(t)}=\sum_{k=1}^{N(t)}X_{k}$ and let $\{\tau_{n}\}$ be the interarrival times. It is well known that, under regularity conditions, $S_{N(t)}$ is asymptotically Gaussian provided $X_{n}$ and $\tau_{n}$ have finite second moment. However, in modelling risk processes or heavy traffic networks, the assumption of the finiteness of the second moment may not be compatible. Also, the independency of the processes $\{S_{n}\}$ and $\{N(t)\}$ might be not realistic. In this situation, heavy-tailed distributions arise as a proper alternative and dependency between $\tau_{n}$ and the reward $X_{n}$ should be allowed. By making use of the Mallows–Wasserstein distance we derive CLT type results for heavy-tailed renewal–reward dependent processes. Applications to risk processes and heavy traffic networks are exhibited.