Limit theorems for supercritical age-dependent branching processes with neutral immigration

We consider a branching process with Poissonian immigration where individuals have inheritable types. At rate theta, 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 i.i.d. lifetimes durations (non necessarily exponential) during which they give birth independently at constant rate b. First, using spine decomposition, we relax previously known assumptions required for a.s. convergence of 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 (P_1,P_2,...) of relative abundances of surviving families converges a.s. In the first model, the limit is the GEM distribution with parameter theta/b.