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 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.
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 . 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.
"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