In this paper we focus on a class of linear Hawkes processes with general immigrants. These are counting processes with shot-noise intensity, including self-excited and externally excited patterns. For such processes, we introduce the concept of the age pyramid which evolves according to immigration and births. The virtue of this approach that combines an intensity process definition and a branching representation is that the population age pyramid keeps track of all past events. This is used to compute new distribution properties for a class of Hawkes processes with general immigrants which generalize the popular exponential fertility function. The pathwise construction of the Hawkes process and its underlying population is also given.
