March 2013 Splitting trees stopped when the first clock rings and Vervaat's transformation
Amaury Lambert, Pieter Trapman
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J. Appl. Probab. 50(1): 208-227 (March 2013). DOI: 10.1239/jap/1363784434


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.


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Amaury Lambert. Pieter Trapman. "Splitting trees stopped when the first clock rings and Vervaat's transformation." J. Appl. Probab. 50 (1) 208 - 227, March 2013.


Published: March 2013
First available in Project Euclid: 20 March 2013

zbMATH: 1277.60140
MathSciNet: MR3076782
Digital Object Identifier: 10.1239/jap/1363784434

Primary: 60J80
Secondary: 60G17 , 60G51 , 60G55 , 60J85 , 60K15 , 60K25 , 92D10 , 92D25 , 92D30 , 92D40

Keywords: age and residual lifetime , branching process , contour process , Crump–Mode–Jagers process , Detection , epidemiology , Lévy process , processor sharing , resolvent , sampling , scale function , Splitting tree , undershoot and overshoot , Vervaat's transformation

Rights: Copyright © 2013 Applied Probability Trust


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Vol.50 • No. 1 • March 2013
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