Journal of Applied Probability

Splitting trees stopped when the first clock rings and Vervaat's transformation

Amaury Lambert and Pieter Trapman

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We consider a branching population where individuals have independent and identically distributed (i.i.d.) life lengths (not necessarily exponential) and constant birth rates. We let Nt denote the population size at time t. We further assume that all individuals, at their birth times, are equipped with independent exponential clocks with parameter δ. We are interested in the genealogical tree stopped at the first time T when one of these clocks rings. This question has applications in epidemiology, population genetics, ecology, and queueing theory. We show that, conditional on {T<∞}, the joint law of (Nt, T, X(T)), where X(T) is the jumping contour process of the tree truncated at time T, is equal to that of (M, -IM, Y'M) conditional on {M≠0}. Here M+1 is the number of visits of 0, before some single, independent exponential clock e with parameter δ rings, by some specified Lévy process Y without negative jumps reflected below its supremum; IM is the infimum of the path YM, which in turn is defined as Y killed at its last visit of 0 before e; and Y'M is the Vervaat transform of YM. This identity yields an explanation for the geometric distribution of NT (see Kitaev (1993) and Trapman and Bootsma (2009)) and has numerous other applications. In particular, conditional on {NT=n}, and also on {NT=n,T<a}, the ages and residual lifetimes of the n alive individuals at time T are i.i.d. and independent of n. We provide explicit formulae for this distribution and give a more general application to outbreaks of antibiotic-resistant bacteria in the hospital.

Article information

Source
J. Appl. Probab., Volume 50, Number 1 (2013), 208-227.

Dates
First available in Project Euclid: 20 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.jap/1363784434

Digital Object Identifier
doi:10.1239/jap/1363784434

Mathematical Reviews number (MathSciNet)
MR3076782

Zentralblatt MATH identifier
1277.60140

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 92D10: Genetics {For genetic algebras, see 17D92} 92D25: Population dynamics (general) 92D30: Epidemiology 92D40: Ecology 60J85: Applications of branching processes [See also 92Dxx] 60G17: Sample path properties 60G51: Processes with independent increments; Lévy processes 60G55: Point processes 60K15: Markov renewal processes, semi-Markov processes 60K25: Queueing theory [See also 68M20, 90B22]

Keywords
Branching process splitting tree Crump–Mode–Jagers process contour process Lévy process scale function resolvent age and residual lifetime undershoot and overshoot Vervaat's transformation sampling detection epidemiology processor sharing

Citation

Lambert, Amaury; Trapman, Pieter. Splitting trees stopped when the first clock rings and Vervaat's transformation. J. Appl. Probab. 50 (2013), no. 1, 208--227. doi:10.1239/jap/1363784434. https://projecteuclid.org/euclid.jap/1363784434


Export citation

References

  • Bertoin, J. (1996). Lévy Processes. Cambridge University Press.
  • Bertoin, J. (1997). Exponential decay and ergodicity of completely asymmetric Lévy processes in a finite interval. Ann. Appl. Prob. 7, 156–169.
  • Bootsma, M. C. J., Wassenberg, M. W. M., Trapman, P. and Bonten, M. J. M. (2011). The nosocomial transmission rate of animal-associated ST398 meticillin-resistant Staphylococcus aureus. J. R. Soc. Interface 8, 578–584.
  • Champagnat, N. and Lambert, A. (2011). Splitting trees with neutral Poissonian mutations II: largest and oldest families. Preprint. Available at http://arxiv.org/abs/1108.4812v1.
  • Champagnat, N. and Lambert, A. (2012). Splitting trees with neutral Poissonian mutations I: small families. Stoch. Process. Appl. 122, 1003–1033.
  • Geiger, J. and Kersting, G. (1997). Depth-first search of random trees, and Poisson point processes. In Classical and Modern Branching Processes (Minneapolis, 1994; IMA Math. Appl. Vol. 84), Springer, New York.
  • Grimmett, G. R. and Stirzaker, D. R. (1992). Probability and Random Processes, 2nd edn. Oxford University Press, New York.
  • Grishechkin, S. (1992). On a relationship between processor-sharing queues and Crump–Mode–Jagers branching processes. Adv. Appl. Prob. 24, 653–698.
  • Kallenberg, O. (2002). Foundations of Modern Probability, 2nd edn. Springer, New York.
  • Kitaev, M. Y. (1993). The M/G/1 processor-sharing model: transient behavior. Queueing Systems 14, 239–273.
  • Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin.
  • Lambert, A. (2009). The allelic partition for coalescent point processes. Markov Process. Relat. Fields 15, 359–386.
  • Lambert, A. (2010). The contour of splitting trees is a Lévy process. Ann. Prob. 38, 348–395.
  • Lambert, A. (2011). Species abundance distributions in neutral models with immigration or mutation and general lifetimes. J. Math. Biol. 63, 57–72.
  • Lambert, A., Simatos, F. and Zwart, B. (2012). Scaling limits via excursion theory: interplay between Crump–Mode–Jagers branching processes and processor-sharing queues. Preprint. Available at http://arxiv.org/abs/1102.5620v2.
  • Trapman, P. and Bootsma, M. C. J. (2009). A useful relationship between epidemiology and queueing theory: the distribution of the number of infectives at the moment of the first detection. Math. Biosci. 219, 15–22.