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Splitting trees stopped when the first clock rings and Vervaat's transformation
Stockholm University, Faculty of Science, Department of Mathematics.
UPMC Univ Paris 06.
2013 (English)In: Journal of Applied Probability, ISSN 0021-9002, E-ISSN 1475-6072, Vol. 50, no 1, 208-227 p.Article in journal (Refereed) Published
Abstract [en]

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}. HereM+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 timeT 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

Place, publisher, year, edition, pages
2013. Vol. 50, no 1, 208-227 p.
Keyword [en]
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
National Category
Probability Theory and Statistics
Research subject
URN: urn:nbn:se:su:diva-89372DOI: 10.1239/jap/1363784434ISI: 000322206200015OAI: diva2:617445
Swedish Research Council, 2010 5873
Available from: 2013-04-23 Created: 2013-04-23 Last updated: 2013-08-20Bibliographically approved

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