Asymptotic deviation bounds for cumulative processes

TL;DR

This paper establishes asymptotic deviation bounds for cumulative processes using a Large Deviation Principle, extending previous results by removing dependency restrictions and applying advanced probabilistic techniques.

Contribution

It introduces a generalized LDP-based deviation bound for cumulative processes without dependency constraints, broadening applicability beyond prior models.

Findings

Extended deviation bounds to more general cumulative processes.

Removed dependency restrictions between variables and renewal process.

Applied the results to Hawkes processes with inhibition.

Abstract

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).