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Weak limits for the largest subpopulations in Yule processes with high mutation probabilities

  • Erich Baur (a1) and Jean Bertoin (a2)

We consider a Yule process until the total population reaches size n ≫ 1, and assume that neutral mutations occur with high probability 1 - p (in the sense that each child is a new mutant with probability 1 - p, independently of the other children), where p = p n ≪ 1. We establish a general strategy for obtaining Poisson limit laws and a weak law of large numbers for the number of subpopulations exceeding a given size and apply this to some mutation regimes of particular interest. Finally, we give an application to subcritical Bernoulli bond percolation on random recursive trees with percolation parameter p n tending to 0.

Corresponding author
* Current address: Bern University Of Applied Sciences, Quellgasse 21, 2501 Biel, Switzerland. Email address:
** Postal address: Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland. Email address:
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[1] Aldous, D. (1989). Probability Approximations Via the Poisson Clumping Heuristic. Springer, New York.
[2] Alon, N. and Spencer, J. H. (2008). The Probabilistic Method, 3rd edn. John Wiley, Hoboken, NJ.
[3] Athreya, K. B. and Ney, P. E. (2004). Branching Processes. Dover, Mineola, NY.
[4] Baur, E. (2016). Percolation on random recursive trees. Random Structures Algorithms 48, 655680.
[5] Baur, E. and Bertoin, J. (2015). The fragmentation process of an infinite recursive tree and Ornstein–Uhlenbeck type processes. Electron. J. Prob. 20, 98.
[6] Bertoin, J. (2014). On the non-Gaussian fluctuations of the giant cluster for percolation on random recursive trees. Electron. J. Prob. 19, 24.
[7] Bertoin, J. (2014). Sizes of the largest clusters for supercritical percolation on random recursive trees. Random Structures Algorithms 44, 2944.
[8] Bertoin, J. and Uribe Bravo, G. (2015). Supercritical percolation on large scale-free random trees. Ann. Appl. Prob. 25, 81103.
[9] Berzunza, G. (2015). Yule processes with rare mutation and their applications to percolation on b-ary trees. Electron. J. Prob. 20, 43.
[10] Billingsley, P. (1999). Convergence of Probability Measures, 2nd edn. John Wiley, New York.
[11] Bollobás, B. (2001). Random Graphs, 2nd edn. Cambridge University Press.
[12] Dobrow, R. P. and Smythe, R. T. (1996). Poisson approximations for functionals of random trees. Random Structures Algorithms 9, 7992.
[13] Drmota, M., Iksanov, A., Moehle, M. and Roesler, U. (2009). A limiting distribution for the number of cuts needed to isolate the root of a random recursive tree. Random Structures Algorithms 34, 319336.
[14] Goldschmidt, C. and Martin, J. B. (2005). Random recursive trees and the Bolthausen–Sznitman coalescent. Electron. J. Prob. 10, 718745.
[15] Iksanov, A. and Möhle, M. (2007). A probabilistic proof of a weak limit law for the number of cuts needed to isolate the root of a random recursive tree. Electron. Commun. Prob. 12, 2835.
[16] Klebaner, F. C. (2005). Introduction to Stochastic Calculus with Applications, 2nd edn. Imperial College Press, London.
[17] Kuba, M. and Panholzer, A. (2014). Multiple isolation of nodes in recursive trees. Online J. Analysis Comb. 9.
[18] Le Cam, L. (1960). An approximation theorem for the Poisson binomial distribution. Pacific J. Math. 10, 11811197.
[19] Meir, A. and Moon, J. W. (1974). Cutting down recursive trees. Math. Biosci. 21, 173181.
[20] Möhle, M. (2015). The Mittag–Leffler process and a scaling limit for the block counting process of the Bolthausen–Sznitman coalescent. ALEA Latin Amer. J. Prob. Math. Statist. 12, 3553.
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Advances in Applied Probability
  • ISSN: 0001-8678
  • EISSN: 1475-6064
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