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Splitting Trees Stopped when the First Clock Rings and Vervaat's Transformation

Part of: Markov processes Special processes Stochastic processes

Published online by Cambridge University Press:  12 February 2018

Amaury Lambert  and
Pieter Trapman
Amaury Lambert*
Affiliation:
Université Pierre et Marie Curie-Paris 6
Pieter Trapman*
Affiliation:
Stockholm University
*
∗ Postal address: Laboratoire de Probabilités et Modèles Aléatoires, UMR 7599 CNRS, Université Pierre et Marie Curie-Paris 6, case courrier 188, 4 Place Jussieu, F-75252 Paris Cedex 05, France. Email address: amaury.lambert@upmc.fr
∗∗ Postal address: Department of Mathematics, Stockholm University, 106 91 Stockholm, Sweden. Email address: ptrapman@math.su.se
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Abstract

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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.

Keywords

Branching processsplitting treeCrump–Mode–Jagers processcontour processLévy processscale functionresolventage and residual lifetimeundershoot and overshootVervaat's transformationsamplingdetectionepidemiologyprocessor sharing

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 92D10: Genetics 92D25: Population dynamics (general) 92D30: Epidemiology 92D40: Ecology 60J85: Applications of branching processes 60G17: Sample path properties 60G51: Processes with independent increments; L'evy processes 60G55: Point processes 60K15: Markov renewal processes, semi-Markov processes 60K25: Queueing theory
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 1 , March 2013 , pp. 208 - 227
Copyright
Copyright © Applied Probability Trust 2013 

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