## Journal of Applied Probability

- J. Appl. Probab.
- Volume 50, Number 1 (2013), 208-227.

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

Amaury Lambert and Pieter Trapman

#### 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

**; and**

*e**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.

#### Article information

**Source**

J. Appl. Probab., Volume 50, Number 1 (2013), 208-227.

**Dates**

First available in Project Euclid: 20 March 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.jap/1363784434

**Digital Object Identifier**

doi:10.1239/jap/1363784434

**Mathematical Reviews number (MathSciNet)**

MR3076782

**Zentralblatt MATH identifier**

1277.60140

**Subjects**

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Secondary: 92D10: Genetics {For genetic algebras, see 17D92} 92D25: Population dynamics (general) 92D30: Epidemiology 92D40: Ecology 60J85: Applications of branching processes [See also 92Dxx] 60G17: Sample path properties 60G51: Processes with independent increments; Lévy processes 60G55: Point processes 60K15: Markov renewal processes, semi-Markov processes 60K25: Queueing theory [See also 68M20, 90B22]

**Keywords**

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

#### Citation

Lambert, Amaury; Trapman, Pieter. Splitting trees stopped when the first clock rings and Vervaat's transformation. J. Appl. Probab. 50 (2013), no. 1, 208--227. doi:10.1239/jap/1363784434. https://projecteuclid.org/euclid.jap/1363784434