Time-lagged point processes with the order-statistics property

In many fields of applied probability, one deals with an (observed) impact point process triggered by another (unobservable) point process with each t riggering point causing an impact point after a random delay. We consider t he common case of delays being i.i.d random variables and independent of th e triggering process. We show the special role, within such a model, of the assumption that the triggering process possesses the order-statistics prop erty. It is in fact revealed that the "offspring" impact process inherits t he same property from the parent triggering process. Then, we show that whe n this property holds, predictions on the future of the impact process are much simplified.