Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-22T07:02:01.775Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Special processes Markov processes

Published online by Cambridge University Press:  08 September 2017

Erich Baur  and
Jean Bertoin
Erich Baur*
Affiliation:
ENS de Lyon
Jean Bertoin*
Affiliation:
Universität Zürich
*
* Current address: Bern University Of Applied Sciences, Quellgasse 21, 2501 Biel, Switzerland. Email address: [email protected]
** Postal address: Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

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 = pn ≪ 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 pn tending to 0.

Keywords

Branching processmutationpercolationrandom increasing tree

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Type
Research Article
Information
Advances in Applied Probability , Volume 49 , Issue 3 , September 2017 , pp. 877 - 902
Copyright
Copyright © Applied Probability Trust 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Aldous, D. (1989). Probability Approximations Via the Poisson Clumping Heuristic. Springer, New York. CrossRefGoogle Scholar
[2] Alon, N. and Spencer, J. H. (2008). The Probabilistic Method, 3rd edn. John Wiley, Hoboken, NJ. CrossRefGoogle Scholar
[3] Athreya, K. B. and Ney, P. E. (2004). Branching Processes. Dover, Mineola, NY. Google Scholar
[4] Baur, E. (2016). Percolation on random recursive trees. Random Structures Algorithms 48, 655–680. CrossRefGoogle Scholar
[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. CrossRefGoogle Scholar
[6] Bertoin, J. (2014). On the non-Gaussian fluctuations of the giant cluster for percolation on random recursive trees. Electron. J. Prob. 19, 24. Google Scholar
[7] Bertoin, J. (2014). Sizes of the largest clusters for supercritical percolation on random recursive trees. Random Structures Algorithms 44, 29–44. CrossRefGoogle Scholar
[8] Bertoin, J. and Uribe Bravo, G. (2015). Supercritical percolation on large scale-free random trees. Ann. Appl. Prob. 25, 81–103. Google Scholar
[9] Berzunza, G. (2015). Yule processes with rare mutation and their applications to percolation on b-ary trees. Electron. J. Prob. 20, 43. Google Scholar
[10] Billingsley, P. (1999). Convergence of Probability Measures, 2nd edn. John Wiley, New York. CrossRefGoogle Scholar
[11] Bollobás, B. (2001). Random Graphs, 2nd edn. Cambridge University Press. Google Scholar
[12] Dobrow, R. P. and Smythe, R. T. (1996). Poisson approximations for functionals of random trees. Random Structures Algorithms 9, 79–92. 3.0.CO;2-8>CrossRefGoogle Scholar
[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, 319–336. Google Scholar
[14] Goldschmidt, C. and Martin, J. B. (2005). Random recursive trees and the Bolthausen–Sznitman coalescent. Electron. J. Prob. 10, 718–745. CrossRefGoogle Scholar
[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, 28–35. CrossRefGoogle Scholar
[16] Klebaner, F. C. (2005). Introduction to Stochastic Calculus with Applications, 2nd edn. Imperial College Press, London. CrossRefGoogle Scholar
[17] Kuba, M. and Panholzer, A. (2014). Multiple isolation of nodes in recursive trees. Online J. Analysis Comb. 9. Google Scholar
[18] Le Cam, L. (1960). An approximation theorem for the Poisson binomial distribution. Pacific J. Math. 10, 1181–1197. Google Scholar
[19] Meir, A. and Moon, J. W. (1974). Cutting down recursive trees. Math. Biosci. 21, 173–181. CrossRefGoogle Scholar
[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, 35–53. Google Scholar