Renewal in Hawkes processes with self-excitation and inhibition

TL;DR

This paper extends the analysis of Hawkes processes to include both self-excitation and inhibition by establishing limit theorems and concentration inequalities using renewal techniques, even when the reproduction function can be signed.

Contribution

It introduces a novel approach to analyze Hawkes processes with signed reproduction functions, overcoming limitations of the cluster representation used for nonnegative functions.

Findings

Proves exponential concentration inequalities for signed Hawkes processes.

Establishes the existence of exponential moments for renewal times in M/G/∞ queues.

Extends previous results to cases with inhibition, broadening the applicability of Hawkes process analysis.

Abstract

This paper investigates Hawkes processes on the positive real line exhibiting both self-excitation and inhibition. Each point of this point process impacts its future intensity by the addition of a signed reproduction function. The case of a nonnegative reproduction function corresponds to self-excitation, and has been widely investigated in the literature. In particular, there exists a cluster representation of the Hawkes process which allows to apply results known for Galton-Watson trees. In the present paper, we establish limit theorems for Hawkes process with signed reproduction functions by using renewal techniques. We notably prove exponential concentration inequalities, and thus extend results of Reynaud-Bouret and Roy (2007) which were proved for nonnegative reproduction functions using this cluster representation which is no longer valid in our case. An important step for this is to establish the existence of exponential moments for renewal times of M/G/infinity queues that appear naturally in our problem. These results have their own interest, independently of the original problem for the Hawkes processes.