  This paper focuses 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 age pyramid which evolves according to immigration and births.
The virtue if 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
linear 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.
