Hawkes branching point processes without ancestors
- 1 March 2001
- journal article
- Published by Cambridge University Press (CUP) in Journal of Applied Probability
- Vol. 38 (1), 122-135
- https://doi.org/10.1239/jap/996986648
Abstract
In this article, we prove the existence of critical Hawkes point processes with a finite average intensity, under a heavy-tail condition for the fertility rate which is related to a long-range dependence property. Criticality means that the fertility rate integrates to 1, and corresponds to the usual critical branching process, and, in the context of Hawkes point processes with a finite average intensity, it is equivalent to the absence of ancestors. We also prove an ergodic decomposition result for stationary critical Hawkes point processes as a mixture of critical Hawkes point processes, and we give conditions for weak convergence to stationarity of critical Hawkes point processes.Keywords
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