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On the Height and Length of the Ancestral Recombination Graph

Part of: Stochastic processes Markov processes

Published online by Cambridge University Press:  14 July 2016

Etienne Pardoux  and
Majid Salamat
Etienne Pardoux*
Affiliation:
Université de Provence
Majid Salamat*
Affiliation:
Université de Provence and Sharif University of Technology
*
∗ Postal address: LATP, UMR-CNRS 6632, Centre de Mathématiques et d'Informatique, 39 rue F. Joliot-Curie, F-13453, Marseille cedex 13, France.
∗ Postal address: LATP, UMR-CNRS 6632, Centre de Mathématiques et d'Informatique, 39 rue F. Joliot-Curie, F-13453, Marseille cedex 13, France.
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Abstract

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The goal of this paper is to provide formulae for the expectation and variance of the height and length of the ancestral recombination graph (ARG). While the formula for the expectation of the height is known (see, e.g. Krone and Neuhauser (1997)), the other formulae seem to be new. We obtain in particular (see Theorem 4.1) a very simple formula which expresses the expectation of the length of the ARG as a linear combination of the expectations of both the length of the coalescent tree and the height of the ARG. Finally, we study the speed at which the ARG comes down from infinity.

Keywords

Wright–Fisher modelcoalescentrecombinationancestral recombination graph

MSC classification

Secondary: 60J27: Continuous-time Markov processes on discrete state spaces 60G51: Processes with independent increments; L'evy processes 92D10: Genetics

Information

Type
Research Article
Information
Journal of Applied Probability , Volume 46 , Issue 3 , September 2009 , pp. 669 - 689
Copyright
Copyright © Applied Probability Trust 2009 

References

[1] Aldous, D. J. (1999). Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists. Bernoulli 5, 3–48.Google Scholar
[2] Cannings, C. (1974). The latent roots of certain Markov chains arising in genetics: a new approach. I. Haploid models. Adv. Appl. Prob. 6, 260–290.Google Scholar
[3] Donnelly, P. (1991). Weak convergence to a Markov chain with an entrance boundary: ancestral processes in population genetics. Ann. Prob. 19, 1102–1117.Google Scholar
[4] Griffiths, R. C. and Marjoram, P. (1996). Ancestral inference from samples of DNA sequences with recombination. J. Comput. Biol. 3, 479–502.CrossRefGoogle ScholarPubMed
[5] Griffiths, R. C. and Marjoram, P. (1997). An ancestral recombination graph. In Progress in Population Genetics and Human Evolution (IMA Vol. Math. Appl. 87), eds Donnelly, P. and Tavaré, S., Springer, New York, pp. 257–270.Google Scholar
[6] Hein, J., Schierup, M. and Wiuf, C. (2004). Gene Genealogies, Variation and Evolution: A Primer in Coalescent Theory. Oxford University Press.Google Scholar
[7] Kingman, J. F. C. (1982). {The coalescent.} Stoch. Process. Appl. 13, 235–248.CrossRefGoogle Scholar
[8] Krone, S. M. and Neuhauser, C. (1997). {Ancestral processes with selection.} Theoret. Pop. Biol. 51, 210–237.Google Scholar
[9] Moran, P. A. (1958). {A general theory of the distribution of gene frequencies. II. Non-overlapping generations.} Proc. R. Soc. London B 149, 113–116.Google Scholar
[10] Slater, L. J. (1966). Generalized Hypergeometric Functions. Cambridge University Press.Google Scholar
[11] Stanley, R. P. (1999). Enumerative Combinatorics, Vol. 2. Cambridge University Press.CrossRefGoogle Scholar
[12] Tavaré, S. (2004). Ancestral inference in population genetics. In Lectures on Probability Theory and Statistics (Lecture Notes Math. 1837), Springer, Berlin, pp. 1–188.Google Scholar