Advances in Applied Probability

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

M. Richard

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Abstract

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.

Article information

Source
Adv. in Appl. Probab. Volume 43, Number 1 (2011), 276-300.

Dates
First available: 15 March 2011

Permanent link to this document
http://projecteuclid.org/euclid.aap/1300198523

Digital Object Identifier
doi:10.1239/aap/1300198523

Zentralblatt MATH identifier
05883124

Mathematical Reviews number (MathSciNet)
MR2761158

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60G55: Point processes 92D25: Population dynamics (general) 60J85: Applications of branching processes [See also 92Dxx] 60F15: Strong theorems 92D40: Ecology

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

Citation

Richard, M. Limit theorems for supercritical age-dependent branching processes with neutral immigration. Advances in Applied Probability 43 (2011), no. 1, 276--300. doi:10.1239/aap/1300198523. http://projecteuclid.org/euclid.aap/1300198523.


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