  We prove regenerative properties for the linear Hawkes process under minimal
assumptions on the transfer function, which may have unbounded support. These
results are applicable to sliding window statistical estimators. We exploit
independence in the Poisson cluster point process decomposition, and the
regeneration times are not stopping times for the Hawkes process. The
regeneration time is interpreted as the renewal time at zero of a M/G/infinity
queue, which yields a formula for its Laplace transform. When the transfer
function admits some exponential moments, we stochastically dominate the
cluster length by exponential random variables with parameters expressed in
terms of these moments. This yields explicit bounds on the Laplace transform of
the regeneration time in terms of simple integrals or special functions
yielding an explicit negative upper-bound on its abscissa of convergence. These
regenerative results allow, e.g., to systematically derive long-time asymptotic
results in view of statistical applications. This is illustrated on a
concentration inequality previously obtained with coauthors.
