March 2011 Limit theorems for supercritical age-dependent branching processes with neutral immigration
M. Richard
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Adv. in Appl. Probab. 43(1): 276-300 (March 2011). DOI: 10.1239/aap/1300198523


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.


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M. Richard. "Limit theorems for supercritical age-dependent branching processes with neutral immigration." Adv. in Appl. Probab. 43 (1) 276 - 300, March 2011.


Published: March 2011
First available in Project Euclid: 15 March 2011

zbMATH: 1218.60073
MathSciNet: MR2761158
Digital Object Identifier: 10.1239/aap/1300198523

Primary: 60J80
Secondary: 60F15 , 60G55 , 60J85 , 92D25 , 92D40

Keywords: almost-sure limit theorem , biogeography , Crump-Mode-Jagers process , GEM distribution , immigration , spine decomposition , Splitting tree , structured population

Rights: Copyright © 2011 Applied Probability Trust


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Vol.43 • No. 1 • March 2011
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