Journal of Applied Probability

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

Amaury Lambert and Pieter Trapman

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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

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

First available in Project Euclid: 20 March 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]

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


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.

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