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Renewal in Hawkes processes with self-excitation and inhibition

Part of: Stochastic processes Integral transforms, operational calculus Limit theorems Special processes

Published online by Cambridge University Press:  24 September 2020

Manon Costa Open the ORCID record for Manon Costa [Opens in a new window] ,
Carl Graham ,
Laurence Marsalle  and
Viet Chi Tran
Manon Costa*
Affiliation:
Université Toulouse III
Carl Graham*
Affiliation:
École Polytechnique
Laurence Marsalle*
Affiliation:
Université de Lille
Viet Chi Tran*
Affiliation:
LAMA, Université Gustave Eiffel, UPEM, Université Paris-Est Créteil, CNRS
*
*Postal address: Institut de Mathématiques de Toulouse, UMR 5219; Université de Toulouse, CNRS, UPS IMT, F-31062 Toulouse Cedex 9, France. Email address: manon.costa@math.univ-toulouse.Fr
**Postal address: CMAP, CNRS, École Polytechnique, Institut Polytechnique de Paris, 91128 Palaiseau, France.
***Postal adddress: Université de Lille, CNRS, UMR 8524 - Laboratoire Paul Painlevé, F-59000 Lille, France.
***Postal adddress: Université de Lille, CNRS, UMR 8524 - Laboratoire Paul Painlevé, F-59000 Lille, France.
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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Reynaud-Bouret and Roy (2006) previously proven for nonnegative reproduction functions using a cluster representation no longer valid in our case.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.

Keywords

Point processesself-excitationinhibitionergodic limit theoremsconcentration inequalitiesGalton–Watson treesM/G/∞ queues

MSC classification

Primary: 60G55: Point processes 60F99: None of the above, but in this section
Secondary: 60K05: Renewal theory 60K25: Queueing theory 44A10: Laplace transform
Type
Original Article
Information
Advances in Applied Probability , Volume 52 , Issue 3 , September 2020 , pp. 879 - 915
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Copyright
© Applied Probability Trust 2020

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