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

Part of: Limit theorems Markov processes Stochastic processes

Published online by Cambridge University Press:  01 July 2016

M. Richard
M. Richard*
Affiliation:
Université Paris VI
*
∗ Postal address: Laboratoire de Probabilités et Modèles Aléatoires, UMR 7599, Université Paris VI, Case courrier 188, 4 Place Jussieu, 75252 Paris Cedex 05, France. Email address: mathieu.richard@upmc.fr
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Abstract

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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.

Keywords

Splitting treeCrump-Mode-Jagers processspine decompositionimmigrationstructured populationGEM distributionbiogeographyalmost-sure limit theorem

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60G55: Point processes 92D25: Population dynamics (general) 60J85: Applications of branching processes 60F15: Strong theorems 92D40: Ecology
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 43 , Issue 1 , March 2011 , pp. 276 - 300
Copyright
Copyright © Applied Probability Trust 2011 

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