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

Abstract : We consider a branching population where individuals have i.i.d.\ life lengths. We let $N_t$ denote the population size at time $t$. %(called homogeneous, binary Crump--Mode--Jagers process). We further assume that all individuals, at birth time, are equipped with independent exponential clocks with parameter $\delta$. We are interested in the genealogical tree stopped at the first time $T$ when one of those clocks rings. This question has applications in epidemiology, in population genetics, in ecology and in queuing theory. We show that conditional on $\{T<\infty\}$, 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 $\{M\not=0\}$, where : $M+1$ is the number of visits of 0, before some single independent exponential clock $\mathbf{e}$ with parameter $\delta$ rings, by some specified L{é}vy process $Y$ without negative jumps reflected below its supremum; $I_M$ is the infimum of the path $Y_M$ defined as $Y$ killed at its last 0 before $\mathbf{e}$; $Y_M'$ is the Vervaat transform of $Y_M$. This identity yields an explanation for the geometric distribution of $N_T$ \cite{K,T} and has numerous other applications. We also give a more general application to outbreaks of antibiotic-resistant bacteria in the hospital.
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Contributor : Amaury Lambert <>
Submitted on : Friday, October 14, 2011 - 9:26:21 AM
Last modification on : Tuesday, May 14, 2019 - 11:07:49 AM

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  • HAL Id : hal-00632325, version 1
  • ARXIV : 1110.2929


Amaury Lambert, Pieter Trapman. Splitting trees stopped when the first clock rings and Vervaat's transformation. 2011. ⟨hal-00632325⟩



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