Renewal in Hawkes processes with self-excitation and inhibition

• Published in: Advances in applied probability Vol. 52; no. 3; pp. 879 - 915
• Main Authors: Costa, Manon, Graham, Carl, Marsalle, Laurence, Tran, Viet Chi
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01.09.2020; Applied Probability Trust
• Subjects: Clusters; Excitation; Immigration; Laplace transforms; Mathematics; Neurosciences; Original Article; Original Articles; Probability; Queuing theory; Random variables; Representations; Statistics; M/G/∞ queues; Point processes; 60G55; 60F99; 44A10; self-excitation; inhibition; 60K05; concentration inequalities; 60K25; Galton–Watson trees; ergodic limit theorems; renewal theory; Galton-Watson trees; ergodic limit theo- rems
• ISSN: 0001-8678; 1475-6064
• DOI: 10.1017/apr.2020.19

We investigate the Hawkes processes on the positive real line exhibiting both self-excitation and inhibition. Each point of such a 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 one to apply known results for Galton–Watson trees. We use renewal techniques to establish limit theorems for Hawkes processes that have reproduction functions which are signed and have bounded support. Notably, we prove exponential concentration inequalities, extending results of Reynaud-Bouret and Roy (2006) previously proven for nonnegative reproduction functions using a cluster representation no longer valid in our case. Importantly, we establish the existence of exponential moments for renewal times of M/G/$\infty$ queues which appear naturally in our problem. These results possess interest independent of the original problem.