Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-22T09:10:41.194Z Has data issue: false hasContentIssue false
  • English
  • Français

The genealogy of branching processes and the age of our most recent common ancestor

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2016

Neil O'Connell
Neil O'Connell*
Affiliation:
Dublin Institute for Advanced Studies
*
* Postal address: DIAS, 10 Burlington Road, Dublin 4, Ireland.
Get access Rights & Permissions [Opens in a new window]

Abstract

We obtain a weak approximation for the reduced family tree in a near-critical Markov branching process when the time interval considered is long; we also extend Yaglom's theorem and the exponential law to this case. These results are then applied to the problem of estimating the age of our most recent common female ancestor, using mitochondrial DNA sequences taken from a sample of contemporary humans.

Keywords

REDUCED BRANCHING PROCESS'NEAR-CRITICALYAGLOM'S THEOREMEXPONENTIAL LIMIT LAWMITOCHONDRIAL DNAEVEEVOLUTIONMUTATION

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60J85: Applications of branching processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 27 , Issue 2 , June 1995 , pp. 418 - 442
Copyright
Copyright © Applied Probability Trust 1995 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

This work constitutes part of the author's Ph.D. thesis, completed at the University of California at Berkeley, and was supported in part by NSF grants MCS90-01710 and DMS91-58583.

References

[1] Athreya, K. and Ney, P. (1972) Branching Processes. Springer-Verlag, New York.Google Scholar
[2] Bühler, W. (1972) The distribution of generations and other aspects of the family structure of branching processes. In Proc. 6th Berkeley Symp. Math. Statist. Prob. 3, 463–480.Google Scholar
[3] Dawson, D. and Perkins, E. (1991) Historical processes. Mem. Amer. Math. Soc. 93(454).Google Scholar
[4] Donnelly, P. (1986) Partition structures, Polya urns, the Ewens' sampling formula, and the ages of alleles. Theoret. Popn. Biol. 30, 271–288.Google Scholar
[5] Durrett, R. (1978) The genealogy of critical branching processes. Stoch. Proc. Appl. 8, 101–116.Google Scholar
[6] Etheridge, A. M. (1992) Conditioned superprocesses and a semilinear heat equation. Preprint.Google Scholar
[7] Etheridge, A. M. and March, P. (1991) A note on superprocesses. Prob. Theory Rel. Fields 89, 141–147.CrossRefGoogle Scholar
[8] Ethier, S. and Kurtz, T. (1986) Markov Processes: Characterisation and Convergence. Wiley, New York.Google Scholar
[9] Feller, W. (1951) Diffusion processes in genetics. In Proc. 2nd Berkeley Symp. Math. Statist. Prob., 227–246.Google Scholar
[10] Felsenstein, J. (1978) Cases in which parsimony or compatibility methods will be positively misleading. Syst. Zool. 27, 401–410.Google Scholar
[11] Fleischmann, K. and Siegmund-Schultze, R. (1977) The genealogy of critical branching processes. Math. Nachr. 79, 233–241.CrossRefGoogle Scholar
[12] Harris, T. E. (1963) The Theory of Branching Processes. Springer-Verlag, Berlin.Google Scholar
[13] Hasegawa, M. and Horai, S. (1990) Time of the deepest root for polymorphism in human mitochondrial DNA. J. Mol. Evol. 32, 37–42.CrossRefGoogle Scholar
[14] Hasegawa, M., Kishino, H., Hayasaka, K. and Horai, S. (1990) Mitochondrial DNA evolution in primates; transition rate has been extremely low in lemur. J. Mol. Evol. 31, 113.Google Scholar
[15] Hoppe, F. M. (1986) Size-biased filtering of Poisson-Dirichlet samples with an application to partition structures in genetics. J. Appl. Prob. 23, 1008–1012.Google Scholar
[16] Jagers, P. (1971) Diffusion approximations for branching processes. Ann. Math. Statist. 42, 2074–2078.CrossRefGoogle Scholar
[17] Jagers, P. (1991) The growth and stabilisation of populations. Statist. Sci. 6, 269–283.Google Scholar
[18] Jagers, P. (1985) Branching Processes with Biological Applications. Wiley, Chichester.Google Scholar
[19] Jagers, P., Nerman, O. and Taib, Z. (1991) When did Joe's great … grandfather live? Or on the timescale of evolution. In Selected Proceedings of the Sheffield Symposium on Applied Probability, ed. Basawa, I. V. and Taylor, R. L.. IMS Lecture Notes–Monograph Series 18.Google Scholar
[20] Jirina, M. (1969) On Feller's branching diffusion process. Casopis. Pest. Mat. 94, 84–90.Google Scholar
[21] Knight, F. (1981) Essentials of Brownian motion and diffusion. Amer. Math. Soc., Math. Surveys 218.Google Scholar
[22] Krishtalka, L. (1989) Dinosaur Plots and Other Intrigues in Natural History. Avon Books, New York.Google Scholar
[23] Lindvall, T. Limit theorems for some functionals of certain Galton-Watson branching processes. Adv. Appl. Prob. 6, 309–321.Google Scholar
[24] Lundstrom, R., Tavare, S. and Ward, R. H. (1992) Estimating substitution rates from molecular data using the coalescent. Proc. Natl. Acad. Sci. USA 89, 5961–5965.CrossRefGoogle ScholarPubMed
[25] O'Connell, N. (1993) Yule process approximation for the skeleton of a branching processes. J. Appl. Prob. 30, 725–729.Google Scholar
[26] Perkins, E. A. (1992) Conditional Dawson-Watanabe processes and Fleming-Viot processes. Preprint.Google Scholar
[27] Perman, M., Pitman, J. W. and Yor, M. (1991) Size-biased sampling of Poisson point processes and excursions. Prob. Theory Rel. Fields. CrossRefGoogle Scholar
[28] Pitman, J. W. (1992) A two-parameter version of Ewens' sampling formula. Preprint.Google Scholar
[29] Pitman, J. W. (1993) Random discrete distributions invariant under size-biased permutation. J. Appl. Prob. To appear.Google Scholar
[30] Pitman, J. W. and Yor, M. (1982) A decomposition of Bessel bridges. Z. Wahrscheinlichkeitsth. 59, 425–457.Google Scholar
[31] Pitman, J. W. and Yor, M. (1990) Arcsine laws and interval partitions derived from a stable subordinator. Technical Report 189, University of California, Berkeley.Google Scholar
[32] Sagitov, S. (1989) The limit behaviour of reduced critical branching processes. Soviet Math. Dokl. 38, 488–491.Google Scholar
[33] Sawyer, S. (1976) Branching diffusion processes in population genetics. Adv. Appl. Prob. 8, 659–689.Google Scholar
[34] Spuhler, J. N. (1988) Evolution of mitochondrial DNA in monkeys, apes and humans. Yearb. Phys. Anthropol. 31, 15–48.CrossRefGoogle Scholar
[35] Stoneking, L., Sherry, S. T., Redd, A. J. and Vigilant, L. (1992) New approaches to dating suggest a recent age for the human mtDNA ancestor. Preprint.CrossRefGoogle Scholar
[36] Taib, Z. Labelled Branching Processes with Applications to Neutral Evolution Theory. PhD thesis, Chalmers University of Technology, Sweden.Google Scholar
[37] Taib, Z. (1991) The most frequent alleles in a branching process with neutral mutations. Preprint.Google Scholar
[38] Tavaré, S. (1984) Line-of-descent and genealogical processes, and their applications in population genetics models. Theoret. Popn Biol. 26, 119–164.CrossRefGoogle ScholarPubMed
[39] Thorne, A. G. and Wolpoff, M. H. (1992) The multiregional evolution of humans. Sci. Amer. April, 76–83.Google Scholar
[40] Vigilant, L. Pennington, R., Harpending, H., Kocher, T. D. and Wilson, A. (1989) Mitochondrial DNA sequences in single hairs from a southern African population. Proc. Natl. Acad. Sci. USA 86, 9350–9354.Google Scholar
[41] Vigilant, L., Stoneking, L., Harpending, H., Hawkes, K. and Wilson, A. (1991) African populations and the evolution of human mitochondrial DNA. Science 253, 1503–1507.Google Scholar
[42] Wakely, J. (1993) Substitute rate variation among sites in hypervariable region 1 of human mitochondrial DNA. Preprint.Google Scholar
[43] Wilson, A. C. and Cann, R. L. (1992) Recent African genesis of humans. Sci. Amer. April, 68–73.Google Scholar
[44] Zubkov, A. (1975) Limiting distributions of the distance to the closest common ancestor. Theory Prob. Appl. 20, 602–612.Google Scholar