Hawkes branching point processes without ancestors

In this article, we prove the existence of critical Hawkes point processes with a finite average intensity, under a heavy-tail condition for the ferti lity rate which is related to a long-range dependence property. Criticality means that the fertility rate integrates to 1, and corresponds to the usua l critical branching process, and, in the context of Hawkes point processes with a finite average intensity, it is equivalent to the absence of ancest ors. 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.