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We prove regenerative properties for the linear Hawkes process under minimalassumptions on the transfer function, which may have unbounded support. Theseresults are applicable to sliding window statistical estimators. We exploitindependence in the Poisson cluster point process decomposition, and theregeneration times are not stopping times for the Hawkes process. Theregeneration time is interpreted as the renewal time at zero of a M/G/infinityqueue, which yields a formula for its Laplace transform. When the transferfunction admits some exponential moments, we stochastically dominate thecluster length by exponential random variables with parameters expressed interms of these moments. This yields explicit bounds on the Laplace transform ofthe regeneration time in terms of simple integrals or special functionsyielding an explicit negative upper-bound on its abscissa of convergence. Theseregenerative results allow, e.g., to systematically derive long-time asymptoticresults in view of statistical applications. This is illustrated on aconcentration inequality previously obtained with coauthors.

Regenerative properties of the linear hawkes process with unbounded memory