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

  • Amaury Lambert (a1) and Pieter Trapman (a2)

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 N t 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 (N t , T, X (T)), where X (T) is the jumping contour process of the tree truncated at time T, is equal to that of (M, -I M , 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; I M is the infimum of the path Y M , which in turn is defined as Y killed at its last visit of 0 before e ; and Y′ M is the Vervaat transform of Y M . This identity yields an explanation for the geometric distribution of N T (see Kitaev (1993) and Trapman and Bootsma (2009)) and has numerous other applications. In particular, conditional on {N T =n}, and also on {N T =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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Copyright

Corresponding author

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

References

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[2] Bertoin, J. (1997). Exponential decay and ergodicity of completely asymmetric Lévy processes in a finite interval. Ann. Appl. Prob. 7, 156169.
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[5] Champagnat, N. and Lambert, A. (2012). Splitting trees with neutral Poissonian mutations I: small families. Stoch. Process. Appl. 122, 10031033.
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[13] Lambert, A. (2010). The contour of splitting trees is a Lévy process. Ann. Prob. 38, 348395.
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[16] 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, 1522.

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

  • Amaury Lambert (a1) and Pieter Trapman (a2)

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