Asymptotics of Symmetric Compound Poisson Population Models

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.2010 Mathematics subject classification: Primary 60F05, 60G09, 60J10, 60J80 Secondary 60K35, 92D10, 92D25

Pareto genealogies arising from a Poisson branching evolution model with selection

“…Remark: A similar scaling behavior for c N was recently obtained in Theorem 2.4 of [20], dealing with coalescents arising from compound Poisson discrete reproduction models, in the critical case. Proof: Using ( 18), for small λ, we have…”

“…It is well-known that when α ∈ (0, 1), with π (x) := x −α and π −1 (s) = s −1/α , the positive cumulative rv (20) χ…”

“…Typically indeed, by the Lévy-Khintchine formula, π (x) dx stands for the Lévy measure for the non-negative jumps of χ which is the value at time t = 1 of an infinitely divisible subordinator (χ t ; t ≥ 0) with LST E e −λχ t , [2]. In (20), the terms π −1 (τ n ) are thus the ranked jumps of χ in its Lévy decomposition (see [33] for example).…”

“…It is well-known that when α ∈ (0, 1), with π (x) := x −α and π −1 (s) = s −1/α , the positive cumulative rv (20) χ [27]. When α = 1, the law of χ is the one of a positive Neveu rv with LST E e −λχ = e λ log λ (see [32]); the latter may be viewed as a Lamperti rv in the limit α → 1 + , [18].…”

On the genealogy and coalescence times of Bienaymé–Galton–Watson branching processes

“…In sharp contrast with similar concern for Fisher-Wright like constant population size branching models, [18], [11], this (lower-triangular) transition matrix is timeinhomogeneous and sub-stochastic. In the Fisher-Wright setup, with a very rich development starting from [14], the starting point model is a conditional branching process, introduced in [12] and [13], as a population model with fixed population size, and non-overlapping generations.…”