Renewal in Hawkes processes with self-excitation and inhibition. (September 2020)

• Record Type: Journal Article
• Authors: Costa, Manon; Graham, Carl; Marsalle, Laurence; Tran, Viet Chi
• Is Part Of: Advances in applied probability. Volume 52:Number 3(2020)
• Page Start: 879
• Page End: 915
• Publication Date: 2020-09
• Subjects: Point processes, -- self-excitation, -- inhibition, -- ergodic limit theorems, -- concentration inequalities, -- Galton–Watson trees, -- M/G/∞ queues; 60G55, -- 60F99, -- 60K05, -- 60K25, -- 44A10; Probabilities -- Periodicals; Stochastic models -- Periodicals; Electronic journals; Periodicals; 519.2
• DOI: 10.1017/apr.2020.19
• Languages: English
• ISSNs: 0001-8678

Abstract

Abstract: 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Reynaud-Bouret and Roy (2006) previously proven for nonnegative reproduction functions using a cluster representation no longer valid in our case.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.