×
Möhle, M.

Forward and backward diffusion approximations for haploid exchangeable population models. (English) Zbl 1056.92039

Stochastic Processes Appl. 95, No. 1, 133-149 (2001).
A class of haploid population models is considered with non-overlapping generations and family sizes \(\nu _1,\dots ,\nu _N\) which are exchangeable random variables, \(\nu _i\) being the number of offsprings of the \(i\)-th individual, where the population size \(N=\nu _1+\dots +\nu _N\) is fixed. A criterion for weak convergence in the Skorohod sense is given for proper time and space scales, counting the number of descendants forward in time. The generator of the limit process is constructed by means of the joint moments of the offspring variables \(\nu _1,\dots ,\nu _N\).
In particular, the Wright-Fischer diffusion appears in the limit if and only if the condition \(\lim _{N\to \infty }E((\nu _1-1)^3) /(N\,\text{Var}(\nu _1)) =0\) is satisfied. By duality the convergence results are compared with the limit theorems known for models considered backward in time.
Reviewer: Bohdan Maslowski (Praha)

Cited in 12 Documents

MSC:

92D10 Genetics and epigenetics
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
92D15 Problems related to evolution
60J60 Diffusion processes

Keywords:

diffusion approximation; population genetics; Wright-Fischer diffusion
PDFBibTeX XMLCite
\textit{M. Möhle}, Stochastic Processes Appl. 95, No. 1, 133--149 (2001; Zbl 1056.92039)
Full Text: DOI

References:

[1] Cannings, C., The latent roots of certain Markov chains arising in genetics: a new approach, I. Haploid models, Adv. Appl. Prob., 6, 260-290 (1974) · Zbl 0284.60064
[2] Cannings, C., The latent roots of certain Markov chains arising in genetics: a new approach, II. Further haploid models, Adv. Appl. Prob., 7, 264-282 (1975) · Zbl 0339.60067
[3] Crow, J. F.; Kimura, M., An Introduction to Population Genetics Theory. (1970), Harper & Row: Harper & Row New York · Zbl 0246.92003
[4] Donnelly, P.; Tavaré, S., Coalescents and genealogical structure under neutrality, Annu. Rev. Genet., 29, 401-421 (1995)
[5] Ethier, S. N.; Krone, S. M., Comparing Fleming-Viot and Dawson-Watanabe processes, Stoch. Proc. Appl., 60, 171-190 (1995) · Zbl 0845.60042
[6] Ethier, S. N.; Kurtz, T. G., Markov Processes. Characterization and Convergence. (1986), Wiley: Wiley New York · Zbl 0592.60049
[7] Ethier, S. N.; Nagylaki, T., Diffusion approximations of Markov chains with two time scales and applications to population genetics, Adv. Appl. Prob., 12, 14-49 (1980) · Zbl 0421.60062
[8] Feller, W., An Introduction to Probability Theory and Its Applications, Vol. I (1971), Wiley: Wiley New York · Zbl 0219.60003
[9] Fisher, R. A., On the dominance ratio, Proc. Roy. Soc. Edin., 42, 321-431 (1922)
[10] Hudson, R. R., Gene genealogies and the coalescent process, Oxf. Surv. Evol. Biol., 7, 1-44 (1991)
[11] Kingman, J. F.C., On the genealogy of large populations, J. Appl. Prob., 19A, 27-43 (1982) · Zbl 0516.92011
[12] 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
[13] Kingman, J. F.C., The coalescent, Stoch. Process. Appl., 13, 235-248 (1982) · Zbl 0491.60076
[14] Li, W.H., Fu, Y.X., 1999. Coalescent theory and its application in population genetics. in: Halloran, M.E., Geisser, S. (Eds.), Statistics in Genetics, IMA Volumes in Mathematics and Its Applications, Vol. 112. Springer, New York.; Li, W.H., Fu, Y.X., 1999. Coalescent theory and its application in population genetics. in: Halloran, M.E., Geisser, S. (Eds.), Statistics in Genetics, IMA Volumes in Mathematics and Its Applications, Vol. 112. Springer, New York. · Zbl 0978.62102
[15] Liggett, T. M., Interacting Particle Systems (1985), Springer: Springer Berlin · Zbl 0559.60078
[16] Möhle, M., Robustness results for the coalescent, J. Appl. Prob., 35, 438-447 (1998) · Zbl 0913.60022
[17] Möhle, M., The concept of duality and applications to Markov processes arising in neutral population genetics models, Bernoulli, 5, 761-777 (1999) · Zbl 0942.92020
[18] Möhle, M., Ancestral processes in population genetics—the coalescent, J. Theor. Biol., 204, 629-638 (2000)
[19] Möhle, M., Total variation distances and rates of convergence for ancestral coalescent processes in exchangeable population models, Adv. Appl. Prob., 32, 983-993 (2000) · Zbl 1002.92015
[20] Möhle, M., Sagitov, S., 1999a. A Classification of Coalescent Processes for Haploid Exchangeable Population Models. Chalmers University of Technology, Department of Mathematics, Preprint No. 10. Ann. of Probab., accepted for publication.; Möhle, M., Sagitov, S., 1999a. A Classification of Coalescent Processes for Haploid Exchangeable Population Models. Chalmers University of Technology, Department of Mathematics, Preprint No. 10. Ann. of Probab., accepted for publication.
[21] Möhle, M., Sagitov, S., 1999b. Coalescent Patterns in Diploid Exchangeable Population Models. Berichte zur Stochastik und verwandten Gebieten, Johannes Gutenberg—Universität Mainz, July 1999, ISSN 0177-0098, J. Math. Biol., submitted for publication.; Möhle, M., Sagitov, S., 1999b. Coalescent Patterns in Diploid Exchangeable Population Models. Berichte zur Stochastik und verwandten Gebieten, Johannes Gutenberg—Universität Mainz, July 1999, ISSN 0177-0098, J. Math. Biol., submitted for publication.
[22] Norman, M. F., Markov Processes and Learning Models (1972), Academic: Academic New York · Zbl 0262.92003
[23] Pitman, J., Coalescents with multiple collisions, Ann. Probab., 27, 1870-1902 (1999) · Zbl 0963.60079
[24] Sagitov, S., The general coalescent with asynchronous mergers of ancestral lines, J. Appl. Prob., 36, 1116-1125 (1999) · Zbl 0962.92026
[25] Sato, K., Diffusion processes and a class of Markov chains related to population genetics, Osaka J. Math., 13, 631-659 (1976) · Zbl 0401.60072
[26] Schweinsberg, S., 2000a. A necessary and sufficient condition for the \(Λ\); Schweinsberg, S., 2000a. A necessary and sufficient condition for the \(Λ\) · Zbl 0953.60072
[27] Schweinsberg, S., 2000b. Coalescents with simultaneous multiple collisions. Electron. J. Probab. 5, 1-50.; Schweinsberg, S., 2000b. Coalescents with simultaneous multiple collisions. Electron. J. Probab. 5, 1-50. · Zbl 0959.60065
[28] Trotter, H. F., Approximations of semi-groups of operators, Pacific J. Math., 8, 887-919 (1958) · Zbl 0099.10302
[29] Wright, S., Evolution in Mendelian populations, Genetics, 16, 97-159 (1931)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.