@article{lambert_splitting_2013,
Author = {Lambert, Amaury and Trapman, Pieter},
Title = {Splitting trees stopped when the first clock rings and
{Vervaat}'s transformation},
Journal = {Journal of Applied Probability},
Volume = {50},
Number = {1},
Pages = {208--227},
abstract = "We consider a branching population where individuals
have i.i.d.\backslash{} life lengths (not necessarily
exponential) and constant birth rate. We let \$N\_t\$
denote the population size at time \$t\$. We further
assume that all individuals, at birth time, are
equipped with independent exponential clocks with
parameter \$\backslash{}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
\$\backslash{}\{T<\backslash{}infty\backslash{}\}\$,
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
\$\backslash{}\{M\backslash{}not=0\backslash{}\}\$,
where : \$M+1\$ is the number of visits of 0, before
some single independent exponential clock
\$\backslash{}mathbf\{e\}\$ with parameter
\$\backslash{}delta\$ rings, by some specified L\'evy
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
\$\backslash{}mathbf\{e\}\$; \$Y\_M'\$ is the Vervaat
transform of \$Y\_M\$. This identity yields an
explanation for the geometric distribution of \$N\_T\$
\backslash{}cite\{K,T\} and has numerous other
applications. In particular, conditional on
\$\backslash{}\{N\_T=n\backslash{}\}\$, and also on
\$\backslash{}\{N\_T=n, T and residual lifetimes of the \$n\$ alive individuals
at time \$T\$ are i.i.d.\backslash{} 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.",
doi = {10.1239/jap/1363784434},
issn = {0021-9002},
language = {en},
month = mar,
url = {http://projecteuclid.org/euclid.jap/1363784434},
urldate = {2014-08-31},
year = 2013
}