  The aim of this paper is to get asymptotic deviation bounds via a Large
Deviation Principle (LDP) for cumulative processes also known as compound
renewal processes or renewal-reward processes. These processes cumulate
independent random variables occurring in time interval given by a renewal
process. Our result extends the one obtained in Lefevere et al. (2011) in the
sense that we impose no specific dependency between the cumulated random
variables and the renewal process and the proof uses Mariani et al. (2014). In
the companion paper Cattiaux-Costa-Colombani (2021) 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 Lefevere et
al. (2011).
