## Electronic Journal of Probability

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

#### 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
Accepted: 24 July 2016
First available in Project Euclid: 2 September 2016

https://projecteuclid.org/euclid.ejp/1472830615

Digital Object Identifier
doi:10.1214/16-EJP4577

Mathematical Reviews number (MathSciNet)
MR3546390

Zentralblatt MATH identifier
1348.60124

#### 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

#### 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.