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 triggering point causing an impact point after a random delay. We consider the common case of delays being i.i.d random variables and independent of the triggering process. We show the special role, within such a model, of the assumption that the triggering process possesses the order-statistics property. It is in fact revealed that the "offspring" impact process inherits the same property from the parent triggering process. Then, we show that when this property holds, predictions on the future of the impact process are much simplified.