The coalescent in population models with time-inhomogeneous environment. (English) Zbl 1064.92034

Möhle, M.

Stochastic Processes Appl. 97, No. 2, 199-227 (2002).

Summary: The coalescent theory, well developed for the class of exchangeable population models with time-homogeneous reproduction law, is extended to a class of population models with time-inhomogeneous environment, where the population size is allowed to vary deterministically with time and where the distribution of the family sizes is allowed to change from generation to generation. A new class of time-inhomogeneous coalescent limit processes with simultaneous multiple mergers arises. Its distribution can be characterized in terms of product integrals.

MSC:

92D15	Problems related to evolution
60J70	Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
60F17	Functional limit theorems; invariance principles

Keywords:

ancestors; coalescent; diffusion approximation; population genetics; product integral; Stirling numbers

References:

[1]	Billingsley, P., Convergence of Probability Measures (1968), Wiley: Wiley New York · Zbl 0172.21201
[2]	Cannings, C., The latent roots of certain Markov chains arising in genetics: a new approach, I. Haploid models, Adv. in Appl. Probab., 6, 260-290 (1974) · Zbl 0284.60064
[3]	Cannings, C., The latent roots of certain Markov chains arising in genetics: a new approach, II. Further haploid models, Adv. in Appl. Probab., 7, 264-282 (1975) · Zbl 0339.60067
[4]	Donnelly, P., A genealogical approach to variable-population-size models in population genetics, J. Appl. Probab., 23, 283-296 (1986) · Zbl 0635.92008
[5]	Donnelly, P.; Tavaré, S., Coalescents and genealogical structure under neutrality., Annu. Rev. Genet., 29, 401-421 (1995)
[6]	Gill, R. D.; Johansen, S., A survey of product-integration with a view toward application in survival analysis, Ann. Statist., 18, 1501-1555 (1990) · Zbl 0718.60087
[7]	Griffiths, R. C.; Tavaré, S., Sampling theory for neutral alleles in a varying environment, Philos. Trans. Roy. Soc. London B, 344, 403-410 (1994)
[8]	Kingman, J. F.C., On the genealogy of large populations, J. Appl. Probab., 19A, 27-43 (1982) · Zbl 0516.92011
[9]	Kingman, J. F.C., Exchangeability and the evolution of large populations, (Koch, G.; Spizzichino, F., Exchangeability in Probability and Statistics (1982), North-Holland Publishing Company: North-Holland Publishing Company Amsterdam), 97-112 · Zbl 0494.92011
[10]	Kingman, J. F.C., The coalescent, Stochastic Process Appl., 13, 235-248 (1982) · Zbl 0491.60076
[11]	Möhle, M., Robustness results for the coalescent, J. Appl. Probab., 35, 438-447 (1998) · Zbl 0913.60022
[12]	Möhle, M., Weak convergence to the coalescent in neutral population models, J. Appl. Probab., 36, 446-460 (1999) · Zbl 0938.92024
[13]	Möhle, M., Sagitov, S., 1999. A classification of coalescent processes for haploid exchangeable population models. Preprint 10, Department of Mathematical Statistics, Göteborg University, Sweden, Ann. Probab., to appear.; Möhle, M., Sagitov, S., 1999. A classification of coalescent processes for haploid exchangeable population models. Preprint 10, Department of Mathematical Statistics, Göteborg University, Sweden, Ann. Probab., to appear.
[14]	Péano, G., Intégration par séries des équations différentielles linéaires, Math. Ann., 32, 450-456 (1888) · JFM 20.0329.02
[15]	Pitman, J., Coalescents with multiple collisions, Ann. Probab., 27, 1870-1902 (1999) · Zbl 0963.60079
[16]	Sagitov, S., The general coalescent with asynchronous mergers of ancestral lines, J. Appl. Probab., 36, 1116-1125 (1999) · Zbl 0962.92026
[17]	Schweinsberg, J., A necessary and sufficient condition for the \(Λ\)-coalescent to come down from infinity, Electron. Comm. Probab., 5, 1-11 (2000) · Zbl 0953.60072
[18]	Schweinsberg, J., Coalescents with simultaneous multiple collisions, Electron. J. Probab., 5, 1-50 (2000) · Zbl 0959.60065