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December 2021 Regenerative properties of the linear Hawkes process with unbounded memory
Carl Graham
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Ann. Appl. Probab. 31(6): 2844-2863 (December 2021). DOI: 10.1214/21-AAP1664

Abstract

Regenerative properties of the linear Hawkes process are proved under minimal assumptions on the transfer function, which may have unbounded support. For this, an original construction of the Hawkes process as a functional of a Poisson point process is derived from the immigration-birth representation, and the independence properties of the Poisson point process are exploited to exhibit regeneration times which are anticipative and not even measurable w.r.t. the Hawkes process. The regeneration time is interpreted as the renewal time at zero of an M/G/ queue, which yields a formula for its Laplace transform. When the transfer function has exponential moments, we stochastically dominate the cluster length by exponential random variables with computable parameters. This provides explicit bounds on the Laplace transform of the regeneration time in terms of simple integrals or of special functions, which yields an explicit negative upper-bound on its abscissa of convergence. The power of the regenerative properties is showcased by being applied to long-time asymptotic results for a class of sliding window statistical estimators, using coupling and sample-path decomposition techniques.

Acknowledgments

Above all, the author wishes to thank Manon Costa, Laurence Marsalle, and Viet Chi Tran for all the work we have done together on their impulse leading to the writing of [12]. Without this work, the present paper would not have existed.

The author wishes to thank the Chaire Modélisation Mathématique et Biodiversité for organizing and supporting many fine meetings on mathematical biology.

He also wishes to thank Bastien Mallein for interesting exchanges on branching random walks and showing him that an implementation of the many-to-one formula leads to the same condition on the tail decay rate of L as in Theorem 4.2.

Lastly, he wishes to thank the referees for their interesting comments which allowed to improve the paper substantially.

Citation

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Carl Graham. "Regenerative properties of the linear Hawkes process with unbounded memory." Ann. Appl. Probab. 31 (6) 2844 - 2863, December 2021. https://doi.org/10.1214/21-AAP1664

Information

Received: 1 January 2020; Revised: 1 September 2020; Published: December 2021
First available in Project Euclid: 13 December 2021

Digital Object Identifier: 10.1214/21-AAP1664

Subjects:
Primary: 60G55
Secondary: 44A10 , 60K05 , 62M09

Keywords: infinite-server queues , Laplace transforms , long-time asymptotics , Poisson cluster processes , regenerative processes

Rights: Copyright © 2021 Institute of Mathematical Statistics

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Vol.31 • No. 6 • December 2021
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