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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: thierry.huillet@u-cergy.fr)
MARTIN MÖHLE
Affiliation:
Mathematisches Institut, Eberhard Karls Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany (e-mail: martin.moehle@uni-tuebingen.de)
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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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References

[1] Berestycki, N. and Pitman, J. (2007) Gibbs distributions for random partitions generated by a fragmentation process. J. Statist. Phys. 127, 381–418.CrossRefGoogle Scholar
[2] Billingsley, P. (1999) Convergence of Probability Measures, second edition, Wiley.Google Scholar
[3] Bruijn, N. G. (1981) Asymptotic Methods in Analysis, Dover.Google Scholar
[4] Charalambides, C. A. and Kyriakoussis, A. (1985) An asymptotic formula for the exponential polynomials and a central limit theorem for their coefficients. Discrete Math. 54 259–270.Google Scholar
[5] Comtet, L. (1974) Advanced Combinatorics, Reidel.Google Scholar
[6] Eldon, B. and Wakeley, J. (2006) Coalescent processes when the distribution of offspring number among individuals is highly skewed. Genetics 172 2621–2633.Google Scholar
[7] Flajolet, P. and Sedgewick, R. (2009) Analytic Combinatorics, Cambridge University Press.Google Scholar
[8] Graham, R. L., Knuth, D. E. and Patashnik, O. (1994) Concrete Mathematics: A Foundation for Computer Science, second edition, Addison Wesley.Google Scholar
[9] Huillet, T. and Möhle, M. (2011) Population genetics models with skewed fertilities: A forward and backward analysis. Stochastic Models 27 521–554.Google Scholar
[10] Huillet, T. and Möhle, M. (2012) Correction on ‘Population genetics models with skewed fertilities: A forward and backward analysis’. Stochastic Models 28 527–532.Google Scholar
[11] Huillet, T. and Möhle, M. (2013) On the extended Moran model and its relation to coalescents with multiple collisions. Theor. Popul. Biol. 87 5–14.Google Scholar
[12] Ibragimov, I. A. and Linnik, Y. V. (1971) Independent and Stationary Sequences of Random Variables, Wolters-Noordhoff.Google Scholar
[13] Janson, S. (2012) Simply generated trees, conditioned Galton–Watson trees, random allocations and condensation. Probab. Surv. 9 103–252.Google Scholar
[14] Karlin, S. and McGregor, J. (1964) Direct product branching processes and related Markov chains. Proc. Nat. Acad. Sci. USA 51 598–602.CrossRefGoogle ScholarPubMed
[15] Karlin, S. and McGregor, J. (1965) Direct product branching processes and related Markov chains I: Calculations of rates of approach to homozygosity. In Proc. Internat. Res. Sem., Springer, pp. 111–145.Google Scholar
[16] Kingman, J. F. C. (1982) The coalescent. Stoch. Process. Appl. 13 235–248.Google Scholar
[17] Möhle, M. (2000) Total variation distances and rates of convergence for ancestral coalescent processes in exchangeable population models. Adv. Appl. Probab. 32 983–993.Google Scholar
[18] Möhle, M. (2011) Coalescent processes derived from some compound Poisson population models. Electron. Comm. Probab. 16 567–582.Google Scholar
[19] Möhle, M. and Sagitov, S. (2001) A classification of coalescent processes for haploid exchangeable population models. Ann. Probab. 29 1547–1562.Google Scholar
[20] Sagitov, S. (2003) Convergence to the coalescent with simultaneous multiple mergers. J. Appl. Probab. 40 839–854.Google Scholar
[21] Schweinsberg, J. (2000) Coalescents with simultaneous multiple collisions. Electron. J. Probab. 5 1–50.CrossRefGoogle Scholar