## The Annals of Probability

### Smaller population size at the MRCA time for stationary branching processes

#### Abstract

We consider an elementary model of random size varying population governed by a stationary continuous-state branching process. We compute the distributions of various variables related to the most recent common ancestor (MRCA): the time to the MRCA, the size of the current population and the size of the population just before the MRCA. In particular we observe a natural mild bottleneck effect as the size of the population just before the MRCA is stochastically smaller than the size of the current population. We also compute the number of individuals involved in the last coalescent event of the genealogical tree, that is, the number of individuals at the time of the MRCA who have descendants in the current population. By studying more precisely the genealogical structure of the population, we get asymptotics for the number of ancestors just before the current time. We give explicit computations in the case of the quadratic branching mechanism. In this case, the size of the population at the MRCA is, in mean, $2/3$ of the size of the current population. We also provide in this case the fluctuations for the renormalized number of ancestors.

#### Article information

Source
Ann. Probab., Volume 40, Number 5 (2012), 2034-2068.

Dates
First available in Project Euclid: 8 October 2012

https://projecteuclid.org/euclid.aop/1349703315

Digital Object Identifier
doi:10.1214/11-AOP668

Mathematical Reviews number (MathSciNet)
MR3025710

Zentralblatt MATH identifier
1275.92076

#### Citation

Chen, Yu-Ting; Delmas, Jean-François. Smaller population size at the MRCA time for stationary branching processes. Ann. Probab. 40 (2012), no. 5, 2034--2068. doi:10.1214/11-AOP668. https://projecteuclid.org/euclid.aop/1349703315

#### References

• [1] Abraham, R. and Delmas, J. F. (2008). A continuum-tree-valued Markov process. Ann. Probab. 40 1167–1211.
• [2] Abraham, R. and Delmas, J.-F. (2009). Williams’ decomposition of the Lévy continuum random tree and simultaneous extinction probability for populations with neutral mutations. Stochastic Process. Appl. 119 1124–1143.
• [3] Aldous, D. (1991). The continuum random tree. II. An overview. In Stochastic Analysis (Durham, 1990). London Mathematical Society Lecture Note Series 167 23–70. Cambridge Univ. Press, Cambridge.
• [4] Aldous, D. (1993). The continuum random tree. III. Ann. Probab. 21 248–289.
• [5] Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Die Grundlehren der Mathematischen Wissenschaften 196 Springer, New York.
• [6] Berestycki, J., Berestycki, N. and Limic, V. (2010). The $\Lambda$-coalescent speed of coming down from infinity. Ann. Probab. 38 207–233.
• [7] Berestycki, J., Kyprianou, A. E. and Murillo, A. (2009). The prolific backbone for supercritical superdiffusions. Available at ArXiv:0912.4736.
• [8] Bertoin, J. (1996). Lévy Processes. Cambridge Tracts in Mathematics 121. Cambridge Univ. Press, Cambridge.
• [9] Bertoin, J. and Le Gall, J.-F. (2003). Stochastic flows associated to coalescent processes. Probab. Theory Related Fields 126 261–288.
• [10] Bertoin, J. and Le Gall, J.-F. (2005). Stochastic flows associated to coalescent processes. II. Stochastic differential equations. Ann. Inst. Henri Poincaré Probab. Stat. 41 307–333.
• [11] Bertoin, J. and Le Gall, J.-F. (2006). Stochastic flows associated to coalescent processes. III. Limit theorems. Illinois J. Math. 50 147–181 (electronic).
• [12] Birkner, M., Blath, J., Capaldo, M., Etheridge, A., Möhle, M., Schweinsberg, J. and Wakolbinger, A. (2005). Alpha-stable branching and beta-coalescents. Electron. J. Probab. 10 303–325 (electronic).
• [13] Dawson, D. A. and Perkins, E. A. (1991). Historical processes. Mem. Amer. Math. Soc. 93 iv+179.
• [14] Donnelly, P. and Kurtz, T. G. (1996). A countable representation of the Fleming–Viot measure-valued diffusion. Ann. Probab. 24 698–742.
• [15] Donnelly, P. and Kurtz, T. G. (1999). Particle representations for measure-valued population models. Ann. Probab. 27 166–205.
• [16] Duquesne, T. and Le Gall, J.-F. (2002). Random Trees, Lévy Processes and Spatial Branching Processes. Astérisque 281.
• [17] Duquesne, T. and Le Gall, J.-F. (2005). Probabilistic and fractal aspects of Lévy trees. Probab. Theory Related Fields 131 553–603.
• [18] Etheridge, A. and March, P. (1991). A note on superprocesses. Probab. Theory Related Fields 89 141–147.
• [19] Etheridge, A. M. and Williams, D. R. E. (2003). A decomposition of the $(1+\beta)$-superprocess conditioned on survival. Proc. Roy. Soc. Edinburgh Sect. A 133 829–847.
• [20] Evans, S. N. (1993). Two representations of a conditioned superprocess. Proc. Roy. Soc. Edinburgh Sect. A 123 959–971.
• [21] Evans, S. N. and Perkins, E. (1990). Measure-valued Markov branching processes conditioned on nonextinction. Israel J. Math. 71 329–337.
• [22] Evans, S. N. and Ralph, P. L. (2010). Dynamics of the time to the most recent common ancestor in a large branching population. Ann. Appl. Probab. 20 1–25.
• [23] Fisher, R. A. (1930). The Genetical Theory of Natural Selection. Clarendon, Oxford.
• [24] Fleming, W. H. and Viot, M. (1979). Some measure-valued Markov processes in population genetics theory. Indiana Univ. Math. J. 28 817–843.
• [25] Galton, F. and Watson, H. W. (1874). On the probability of the extinction of families. J. Roy. Anthropol. Inst. 4 138–144.
• [26] Grey, D. R. (1974). Asymptotic behaviour of continuous time, continuous state-space branching processes. J. Appl. Probab. 11 669–677.
• [27] Jagers, P. and Sagitov, S. (2004). Convergence to the coalescent in populations of substantially varying size. J. Appl. Probab. 41 368–378.
• [28] Jiřina, M. (1958). Stochastic branching processes with continuous state space. Czechoslovak Math. J. 83 292–313.
• [29] Kaj, I. and Krone, S. M. (2003). The coalescent process in a population with stochastically varying size. J. Appl. Probab. 40 33–48.
• [30] Kawazu, K. and Watanabe, S. (1971). Branching processes with immigration and related limit theorems. Teor. Verojatnost. i Primenen. 16 34–51.
• [31] Kingman, J. F. C. (1982). The coalescent. Stochastic Process. Appl. 13 235–248.
• [32] Lambert, A. (2003). Coalescence times for the branching process. Adv. in Appl. Probab. 35 1071–1089.
• [33] Lambert, A. (2007). Quasi-stationary distributions and the continuous-state branching process conditioned to be never extinct. Electron. J. Probab. 12 420–446.
• [34] Le Gall, J.-F. (2006). Random real trees. Ann. Fac. Sci. Toulouse Math. (6) 15 35–62.
• [35] Le Gall, J.-F. and Le Jan, Y. (1998). Branching processes in Lévy processes: Laplace functionals of snakes and superprocesses. Ann. Probab. 26 1407–1432.
• [36] Le Gall, J.-F. and Le Jan, Y. (1998). Branching processes in Lévy processes: The exploration process. Ann. Probab. 26 213–252.
• [37] Li, Z. (2011). Measure-Valued Branching Markov Processes. Springer, Heidelberg.
• [38] Li, Z.-H. (2000). Asymptotic behaviour of continuous time and state branching processes. Austral. Math. Soc. Lect. Ser. 68 68–84.
• [39] Limic, V. (2010). On the speed of coming down from infinity for $\Xi$-coalescent processes. Electron. J. Probab. 15 217–240.
• [40] Möhle, M. (2002). The coalescent in population models with time-inhomogeneous environment. Stochastic Process. Appl. 97 199–227.
• [41] Moran, P. A. P. (1958). Random processes in genetics. Math. Proc. Cambridge Philos. Soc. 54 60–71.
• [42] Mukherjea, A., Rao, M. and Suen, S. (2006). A note on moment generating functions. Statist. Probab. Lett. 76 1185–1189.
• [43] Perkins, E. A. (1992). Conditional Dawson–Watanabe processes and Fleming–Viot processes. In Seminar on Stochastic Processes, 1991 (Los Angeles, CA, 1991). Progress in Probability 29 143–156. Birkhäuser, Boston, MA.
• [44] Pinsky, M. A. (1972). Limit theorems for continuous state branching processes with immigration. Bull. Amer. Math. Soc. (N.S.) 78 242–244.
• [45] Pitman, J. (1999). Coalescents with multiple collisions. Ann. Probab. 27 1870–1902.
• [46] Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 293. Springer, Berlin.
• [47] Roelly-Coppoletta, S. and Rouault, A. (1989). Processus de Dawson–Watanabe conditionné par le futur lointain. C. R. Acad. Sci. Paris Sér. I Math. 309 867–872.
• [48] Sagitov, S. (1999). The general coalescent with asynchronous mergers of ancestral lines. J. Appl. Probab. 36 1116–1125.
• [49] Wright, S. (1931). Evolution in Mendelian populations. Genetics 16 97–159.