Asymptotics of Symmetric Compound Poisson Population Models

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 behavior 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 analyzed.