We consider a branching process with Poissonian immigration where individuals have inheritable types. At rate θ, new individuals singly enter the total population and start a new population which evolves like a supercritical, homogeneous, binary Crump-Mode-Jagers process: individuals have independent and identically distributed lifetime durations (nonnecessarily exponential) during which they give birth independently at a constant rate b. First, using spine decomposition, we relax previously known assumptions required for almost-sure convergence of the total population size. Then, we consider three models of structured populations: either all immigrants have a different type, or types are drawn in a discrete spectrum or in a continuous spectrum. In each model, the vector (P1, P2,...) of relative abundances of surviving families converges almost surely. In the first model, the limit is the GEM distribution with parameter θ / b.
"Limit theorems for supercritical age-dependent branching processes with neutral immigration." Adv. in Appl. Probab. 43 (1) 276 - 300, March 2011. https://doi.org/10.1239/aap/1300198523