Regenerative properties of the linear Hawkes process with unbounded memory
C. D. Graham
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.