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

Richard, M.

Adv. in Appl. Probab., Tome 43 (2011) no. 1, p. 276-300 / Harvested from Project Euclid

Résumé

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.

Détail

Publié le : 2011-03-15
Classification:  Splitting tree,  Crump-Mode-Jagers process,  spine decomposition,  immigration,  structured population,  GEM distribution,  biogeography,  almost-sure limit theorem,  60J80,  60G55,  92D25,  60J85,  60F15,  92D40