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Asymptotics of Symmetric Compound Poisson Population Models

Published online by Cambridge University Press:  08 September 2014

THIERRY HUILLET  and
MARTIN MÖHLE
THIERRY HUILLET
Affiliation:
Laboratoire de Physique Théorique et Modélisation, CNRS-UMR 8089 and Université de Cergy-Pontoise, 2 Avenue Adolphe Chauvin, 95302 Cergy-Pontoise, France (e-mail: [email protected])
MARTIN MÖHLE
Affiliation:
Mathematisches Institut, Eberhard Karls Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany (e-mail: [email protected])
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Abstract

Compound Poisson population models are particular conditional branching process models. A formula for the transition probabilities of the backward process for general compound Poisson models is verified. Symmetric compound Poisson models are defined in terms of a parameter θ ∈ (0, ∞) and a power series φ with positive radius r of convergence. It is shown that the asymptotic behaviour of symmetric compound Poisson models is mainly determined by the characteristic value θrφ′(r−). If θrφ′(r−)≥1, then the model is in the domain of attraction of the Kingman coalescent. If θrφ′(r−) < 1, then under mild regularity conditions a condensation phenomenon occurs which forces the model to be in the domain of attraction of a discrete-time Dirac coalescent. The proofs are partly based on the analytic saddle point method. They draw heavily from local limit theorems and from results of S. Janson on simply generated trees, conditioned Galton-Watson trees, random allocations and condensation. Several examples of compound Poisson models are provided and analysed.

Keywords

Primary 60F0560G0960J1060J80Secondary 60K3592D1092D25
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 24 , Special Issue 1: Honouring the Memory of Philippe Flajolet - Part 3 , January 2015 , pp. 216 - 253
Copyright
Copyright © Cambridge University Press 2014 

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