Asymptotics of symmetric compound Poisson population models. (English) Zbl 1371.60173

Huillet, Thierry; Möhle, Martin

Comb. Probab. Comput. 24, No. 1, 216-253 (2015).

Summary: 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 behaviour of symmetric compound Poisson models is mainly determined by the characteristic value \(\theta{r}\phi'({r}-)\). If \(\theta{r}\phi'({r}-)\geq1\), 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 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 [Probab. Surv. 9, 103–252 (2012; Zbl 1244.60013)] on simply generated trees, conditioned Galton-Watson trees, random allocations and condensation. Several examples of compound Poisson models are provided and analysed.

MSC:

60K35	Interacting random processes; statistical mechanics type models; percolation theory
60J80	Branching processes (Galton-Watson, birth-and-death, etc.)
60F05	Central limit and other weak theorems
60G09	Exchangeability for stochastic processes
92D25	Population dynamics (general)
60J10	Markov chains (discrete-time Markov processes on discrete state spaces)

Citations:

Zbl 1244.60013

References:

[1]	[1]BerestyckiN. and PitmanJ. (2007) Gibbs distributions for random partitions generated by a fragmentation process. J. Statist. Phys.127, 381-418.10.1007/s10955-006-9261-12314353 · Zbl 1126.82013 · doi:10.1007/s10955-006-9261-1
[2]	[2]BillingsleyP. (1999) Convergence of Probability Measures, second edition, Wiley. · Zbl 0172.21201
[3]	[3]BruijnN. G. (1981) Asymptotic Methods in Analysis, Dover. · Zbl 0556.41021
[4]	[4]CharalambidesC. A. and KyriakoussisA. (1985) An asymptotic formula for the exponential polynomials and a central limit theorem for their coefficients. Discrete Math.54259-270.10.1016/0012-365X(85)90110-40790587 · Zbl 0567.33002 · doi:10.1016/0012-365X(85)90110-4
[5]	[5]ComtetL. (1974) Advanced Combinatorics, Reidel. · Zbl 0283.05001
[6]	[6]EldonB. and WakeleyJ. (2006) Coalescent processes when the distribution of offspring number among individuals is highly skewed. Genetics1722621-2633.16452141
[7]	[7]FlajoletP. and SedgewickR. (2009) Analytic Combinatorics, Cambridge University Press.CBO9780511801655 · Zbl 1165.05001
[8]	[8]GrahamR. L., KnuthD. E. and PatashnikO. (1994) Concrete Mathematics: A Foundation for Computer Science, second edition, Addison Wesley.1397498 · Zbl 0836.00001
[9]	[9]HuilletT. and MöhleM. (2011) Population genetics models with skewed fertilities: A forward and backward analysis. Stochastic Models27521-554.10.1080/15326349.2011.593411 · Zbl 1367.92074 · doi:10.1080/15326349.2011.593411
[10]	[10]HuilletT. and MöhleM. (2012) Correction on ‘Population genetics models with skewed fertilities: A forward and backward analysis’. Stochastic Models28527-532.10.1080/15326349.2012.700799 · Zbl 1367.92075 · doi:10.1080/15326349.2012.700799
[11]	[11]HuilletT. and MöhleM. (2013) On the extended Moran model and its relation to coalescents with multiple collisions. Theor. Popul. Biol.875-14.10.1016/j.tpb.2011.09.00422001353 · Zbl 1296.92207 · doi:10.1016/j.tpb.2011.09.004
[12]	[12]IbragimovI. A. and LinnikY. V. (1971) Independent and Stationary Sequences of Random Variables, Wolters-Noordhoff. · Zbl 0219.60027
[13]	[13]JansonS. (2012) Simply generated trees, conditioned Galton-Watson trees, random allocations and condensation. Probab. Surv.9103-252.10.1214/11-PS1882908619 · Zbl 1244.60013 · doi:10.1214/11-PS188
[14]	[14]KarlinS. and McGregorJ. (1964) Direct product branching processes and related Markov chains. Proc. Nat. Acad. Sci. USA51598-602.10.1073/pnas.51.4.598 · Zbl 0129.30504 · doi:10.1073/pnas.51.4.598
[15]	[15]KarlinS. and McGregorJ. (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. · Zbl 0294.60064
[16]	[16]KingmanJ. F. C. (1982) The coalescent. Stoch. Process. Appl.13235-248.10.1016/0304-4149(82)90011-40671034 · Zbl 0491.60076 · doi:10.1016/0304-4149(82)90011-4
[17]	[17]MöhleM. (2000) Total variation distances and rates of convergence for ancestral coalescent processes in exchangeable population models. Adv. Appl. Probab.32983-993.10.1239/aap/1013540343 · Zbl 1002.92015 · doi:10.1239/aap/1013540343
[18]	[18]MöhleM. (2011) Coalescent processes derived from some compound Poisson population models. Electron. Comm. Probab.16567-582.2846651 · Zbl 1367.92105
[19]	[19]MöhleM. and SagitovS. (2001) A classification of coalescent processes for haploid exchangeable population models. Ann. Probab.291547-1562.10.1214/aop/10153457611880231 · Zbl 1013.92029 · doi:10.1214/aop/1015345761
[20]	[20]SagitovS. (2003) Convergence to the coalescent with simultaneous multiple mergers. J. Appl. Probab.40839-854.10.1239/jap/10674360852012671 · Zbl 1052.92044 · doi:10.1239/jap/1067436085
[21]	[21]SchweinsbergJ. (2000) Coalescents with simultaneous multiple collisions. Electron. J. Probab.51-50.1781024 · Zbl 0959.60065