Asymptotics of Symmetric Compound Poisson Population Models. (8th September 2014)

• Record Type: Journal Article
• Authors: HUILLET, THIERRY; MÖHLE, MARTIN
• Editors: Broutin, Nicolas; Fill, James Allen; Nebel, Markus; Ward, Mark Daniel
• Is Part Of: Combinatorics, probability and computing. Volume 24:Number 1(2015:Jan.)
• Page Start: 216
• Page End: 253
• Publication Date: 2014-09-08
• Subjects: Primary 60F05, -- 60G09, -- 60J10, -- 60J80, -- Secondary 60K35, -- 92D10, -- 92D25; Combinatorial analysis -- Periodicals; Probabilities -- Periodicals; Computer science -- Mathematics -- Periodicals; 511.6
• DOI: 10.1017/S0963548314000431
• Languages: English
• ISSNs: 0963-5483

Abstract

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 θ r φ′( r −). If θ r φ′( r −)≥1, then the model is in the domain of attraction of the Kingman coalescent. If θ r φ′( 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.