• Bernoulli
  • Volume 24, Number 2 (2018), 1576-1612.

On branching process with rare neutral mutation

Airam Blancas and Víctor Rivero

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In this paper, we study the genealogical structure of a Galton–Watson process with neutral mutations. Namely, we extend in two directions the asymptotic results obtained in Bertoin [Stochastic Process. Appl. 120 (2010) 678–697]. In the critical case, we construct the version of the model in Bertoin [Stochastic Process. Appl. 120 (2010) 678–697], conditioned not to be extinct. We establish a version of the limit theorems in Bertoin [Stochastic Process. Appl. 120 (2010) 678–697], when the reproduction law has an infinite variance and it is in the domain of attraction of an $\alpha$-stable distribution, both for the unconditioned process and for the process conditioned to nonextinction. In the latter case, we obtain the convergence (after re-normalization) of the allelic sub-populations towards a tree indexed CSBP with immigration.

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Bernoulli, Volume 24, Number 2 (2018), 1576-1612.

Received: August 2015
Revised: June 2016
First available in Project Euclid: 21 September 2017

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branching process domain of attraction of $\alpha$-stable laws neutral mutations Q-processes regular variation


Blancas, Airam; Rivero, Víctor. On branching process with rare neutral mutation. Bernoulli 24 (2018), no. 2, 1576--1612. doi:10.3150/16-BEJ907.

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  • [1] Athreya, K.B. and Ney, P.E. (1972). Branching Processes. Die Grundlehren der Mathematischen Wissenschaften 196. New York: Springer.
  • [2] Bertoin, J. (1996). Lévy Processes. Cambridge Tracts in Mathematics 121. Cambridge: Cambridge Univ. Press.
  • [3] Bertoin, J. (2010). A limit theorem for trees of alleles in branching processes with rare neutral mutations. Stochastic Process. Appl. 120 678–697.
  • [4] Bingham, N.H., Goldie, C.M. and Teugels, J.L. (1989). Regular Variation. Encyclopedia of Mathematics and Its Applications 27. Cambridge: Cambridge Univ. Press.
  • [5] Chauvin, B. (1986). Arbres et processus de Bellman–Harris. Ann. Inst. Henri Poincaré B, Probab. Stat. 22 209–232.
  • [6] Chung, K.L. and Walsh, J.B. (2005). Markov Processes, Brownian Motion, and Time Symmetry, 2nd ed. Grundlehren der Mathematischen Wissenschaften 249. New York: Springer.
  • [7] Feller, W. (1971). An Introduction to Probability Theory and Its Applications. Vol. II, 2nd ed. New York: Wiley.
  • [8] Jacod, J. and Shiryaev, A.N. (1987). Limit Theorems for Stochastic Processes. Grundlehren der Mathematischen Wissenschaften 288. Berlin: Springer.
  • [9] Joffe, A. (1967). On the Galton–Watson branching process with mean less than one. Ann. Math. Stat. 38 264–266.
  • [10] Kallenberg, O. (2002). Foundations of Modern Probability, 2nd ed. Probability and Its Applications (New York). New York: Springer.
  • [11] Kawazu, K. and Watanabe, S. (1971). Branching processes with immigration and related limit theorems. Teor. Verojatnost. i Primenen. 16 34–51.
  • [12] Lambert, A. (2007). Quasi-stationary distributions and the continuous-state branching process conditioned to be never extinct. Electron. J. Probab. 12 420–446.
  • [13] Lambert, A. (2008). Population dynamics and random genealogies. Stoch. Models 24 45–163.
  • [14] Lamperti, J. and Ney, P. (1968). Conditioned branching processes and their limiting diffusions. Teor. Verojatnost. i Primenen. 13 126–137.
  • [15] Li, Z. (2011). Measure-Valued Branching Markov Processes. Probability and Its Applications (New York). Heidelberg: Springer.
  • [16] Seneta, E. and Vere-Jones, D. (1966). On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states. J. Appl. Probab. 3 403–434.
  • [17] Whitt, W. (2002). Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues. Springer Series in Operations Research. New York: Springer.
  • [18] Yaglom, A.M. (1947). Certain limit theorems of the theory of branching random processes. Dokl. Akad. Nauk SSSR 56 795–798.