Electronic Journal of Probability

Moments of the frequency spectrum of a splitting tree with neutral Poissonian mutations

Nicolas Champagnat and Benoît Henry

Full-text: Open access

Abstract

We consider a branching population where individuals live and reproduce independently. Their lifetimes are i.i.d. and they give birth at a constant rate $b$. The genealogical tree spanned by this process is called a splitting tree, and the population counting process is a homogeneous, binary Crump-Mode-Jagers process. We suppose that mutations affect individuals independently at a constant rate $\theta $ during their lifetimes, under the infinite-alleles assumption: each new mutation gives a new type, called allele, to his carrier. We study the allele frequency spectrum which is the numbers $A(k,t)$ of types represented by $k$ alive individuals in the population at time $t$. Thanks to a new construction of the coalescent point process describing the genealogy of individuals in the splitting tree, we are able to compute recursively all joint factorial moments of $\left (A(k,t)\right )_{k\geq 1}$. These moments allow us to give an elementary proof of the almost sure convergence of the frequency spectrum in a supercritical splitting tree.

Article information

Source
Electron. J. Probab., Volume 21 (2016), paper no. 53, 34 pp.

Dates
Received: 30 September 2015
Accepted: 24 July 2016
First available in Project Euclid: 2 September 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1472830615

Digital Object Identifier
doi:10.1214/16-EJP4577

Mathematical Reviews number (MathSciNet)
MR3546390

Zentralblatt MATH identifier
1348.60124

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 92D10: Genetics {For genetic algebras, see 17D92} 60J85: Applications of branching processes [See also 92Dxx] 60G51: Processes with independent increments; Lévy processes 60G57: Random measures 60F15: Strong theorems

Keywords
branching process coalescent point process splitting tree Crump–Mode–Jagers process linear birth–death process allelic partition frequency spectrum infinite alleles model Lévy process scale function random measure Palm measure Campbell’s formula

Rights
Creative Commons Attribution 4.0 International License.

Citation

Champagnat, Nicolas; Henry, Benoît. Moments of the frequency spectrum of a splitting tree with neutral Poissonian mutations. Electron. J. Probab. 21 (2016), paper no. 53, 34 pp. doi:10.1214/16-EJP4577. https://projecteuclid.org/euclid.ejp/1472830615


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References

  • [1] Aldous, D. and Popovic, L. (2005). A critical branching process model for biodiversity. Adv. in Appl. Probab. 37, 4, 1094–1115.
  • [2] Athreya, K. B. and Ney, P. E. (1972). Branching processes. Springer-Verlag, New York-Heidelberg. Die Grundlehren der mathematischen Wissenschaften, Band 196.
  • [3] Bertoin, J. (2009). The structure of the allelic partition of the total population for Galton-Watson processes with neutral mutations. Ann. Probab. 37, 4, 1502–1523.
  • [4] Champagnat, N. and Lambert, A. (2012). Splitting trees with neutral Poissonian mutations I: Small families. Stochastic Process. Appl. 122, 3, 1003–1033.
  • [5] Champagnat, N. and Lambert, A. (2013). Splitting trees with neutral Poissonian mutations II: Largest and oldest families. Stochastic Process. Appl. 123, 4, 1368–1414.
  • [6] Champagnat, N., Lambert, A., and Richard, M. (2012). Birth and death processes with neutral mutations. Int. J. Stoch. Anal., Art. ID 569081, 20.
  • [7] Daley, D. J. and Vere-Jones, D. (2008). An introduction to the theory of point processes. Vol. II, Second ed. Probability and its Applications (New York). Springer, New York. General theory and structure.
  • [8] Ewens, W. J. (2004). Mathematical population genetics. I, Second ed. Interdisciplinary Applied Mathematics, Vol. 27. Springer-Verlag, New York. Theoretical introduction.
  • [9] Geiger, J. and Kersting, G. (1997). Depth-first search of random trees, and Poisson point processes. In Classical and modern branching processes (Minneapolis, MN, 1994). IMA Vol. Math. Appl., Vol. 84. Springer, New York, 111–126.
  • [10] Geiger, J. (1996). Size-biased and conditioned random splitting trees. Stochastic Process. Appl. 65, 2, 187–207.
  • [11] Griffiths, R. C. and Pakes, A. G. (1988). An infinite-alleles version of the simple branching process. Adv. in Appl. Probab. 20, 3, 489–524.
  • [12] Benoît Henry. Clts for general branching processes related to splitting trees. Preprint available at arXiv:1509.06583.
  • [13] Jagers, P. (1974). Convergence of general branching processes and functionals thereof. J. Appl. Probability 11, 471–478.
  • [14] Jagers, P. and Nerman, O. (1984). The growth and composition of branching populations. Adv. in Appl. Probab. 16, 2, 221–259.
  • [15] Jagers, P. and Nerman, O. (1984). Limit theorems for sums determined by branching and other exponentially growing processes. Stochastic Process. Appl. 17, 1, 47–71.
  • [16] Kallenberg, O. (1986). Random measures, Fourth ed. Akademie-Verlag, Berlin; Academic Press, Inc., London.
  • [17] Kleiber, C. and Stoyanov, J. (2013). Multivariate distributions and the moment problem. J. Multivariate Anal. 113, 7–18.
  • [18] Kyprianou, A. E. (2014). Fluctuations of Lévy processes with applications, Second ed. Universitext. Springer, Heidelberg. Introductory lectures.
  • [19] Lambert, A. (2010). The contour of splitting trees is a Lévy process. Ann. Probab. 38, 1, 348–395.
  • [20] Meyer, P.-A. (1966). Probability and potentials. Blaisdell Publishing Co. Ginn and Co., Waltham, Mass.-Toronto, Ont.-London.
  • [21] Nerman, O. (1981). On the convergence of supercritical general (C-M-J) branching processes. Z. Wahrsch. Verw. Gebiete 57, 3, 365–395.
  • [22] Mathieu Richard. Arbres, Processus de branchement non Markoviens et processus de Lévy. Thèse de doctorat, Université Pierre et Marie Curie, Paris 6.
  • [23] Taïb, Z. (1990). Branching processes and neutral mutations. In Stochastic modelling in biology (Heidelberg, 1988). World Sci. Publ., Teaneck, NJ, 293–306.