ASYMPTOTICS OF SYMMETRIC COMPOUND POISSON POPULATION MODELS

Thierry Huillet (1) , Martin Moehle (2)

1 LPTM - UMR 8089 - Laboratoire de Physique Théorique et Modélisation
2 Mathematisches Institut [Tübingen]

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 theta\in (0,\infty) and a power series phi 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 \theta r\phi'(r-). If \theta r\phi'(r-)\ge 1, then the model is in the domain of attraction of the Kingman coalescent. If \theta r\phi'(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 Lambda-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.

Mots clés: Bell polynomials; coalescent with multiple collisions; compound Poisson model; condensation; Dirichlet model; Kingman coalescent; saddle point method; Wright--Fisher model

Domaines: Probabilités [math.PR]; Bio-Informatique, Biologie Systémique [q-bio.QM]; Bio-informatique [q-bio.QM]

Thierry Huillet, Martin Moehle. ASYMPTOTICS OF SYMMETRIC COMPOUND POISSON POPULATION MODELS. Combinatorics, Probability and Computing, 2015, 24 (Special Issue 01), pp.216-253. ⟨10.1017/S0963548314000431⟩. ⟨hal-00730734⟩