Asymptotic deviation bounds for cumulative processes

Abstract

The aim of this paper is to prove a Large Deviation Principle (LDP) for cumulative processes also known as coumpound renewal processes. These processes cumulate independent random variables occuring in time interval given by a renewal process. Our result extends the one obtained in [13] in the sense that we impose no specific dependency between the cumulated random variables and the renewal process. The proof is inspired from [13] but deals with additional difficulties due to the general framework that is considered here. In the companion paper [6] we apply this principle to Hawkes processes with inhibition. Under some assumptions Hawkes processes are indeed cumulative processes, but they do not enter the framework of [13].