Journal
ADVANCES IN APPLIED PROBABILITY
Volume 52, Issue 3, Pages 879-915Publisher
CAMBRIDGE UNIV PRESS
DOI: 10.1017/apr.2020.19
Keywords
Point processes; self-excitation; inhibition; ergodic limit theorems; concentration inequalities; Galton-Watson trees; M/G/infinity queues
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Funding
- Chair 'Modelisation Mathematique et Biodiversite' of Veolia Environnement
- Ecole Polytechnique
- Museum National d'Histoire Naturelle
- Fondation X
- Labex CEMPI [ANR-11-LABX-0007-01]
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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Reynaud-Bouret and Roy (2006) previously proven for nonnegative reproduction functions using a cluster representation no longer valid in our case.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/infinity queues which appear naturally in our problem. These results possess interest independent of the original problem.
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