Splitting trees are those random trees where individuals give birth at a constant rate during a lifetime with general distribution, to i.i.d. copies of themselves. The width process of a splitting tree is then a binary, homogeneous Crump-Mode-Jagers (CMJ) process, and is not Markovian unless the lifetime distribution is exponential (or a Dirac mass at \$\backslashlbrace\backslashinfty\backslashrbrace\$). Here, we allow the birth rate to be infinite, that is, pairs of birth times and life spans of newborns form a Poisson point process along the lifetime of their mother, with possibly infinite intensity measure. A splitting tree is a random (so-called) chronological tree. Each element of a chronological tree is a (so-called) existence point \$(v, \backslashtau)\$ of some individual v (vertex) in a discrete tree where \$\backslashtau\$ is a nonnegative real number called chronological level (time). We introduce a total order on existence points, called linear order, and a mapping \$\backslashphi\$ from the tree into the real line which preserves this order. The inverse of \$\backslashphi\$ is called the exploration process, and the projection of this inverse on chronological levels the contour process. For splitting trees truncated up to level \$\backslashtau\$ , we prove that a thus defined contour process is a L\'evy process reflected below \$\backslashtau\$ and killed upon hitting 0. This allows one to derive properties of (i) splitting trees: conceptual proof of Le Gall-Le Jan's theorem in the finite variation case, exceptional points, coalescent point process and age distribution; (ii) CMJ processes: one-dimensional marginals, conditionings, limit theorems and asymptotic numbers of individuals with infinite versus finite descendances.
@article{lambert_contour_2010,
Author = {Lambert, Amaury},
Title = {The contour of splitting trees is a {Lévy} process},
Journal = {The Annals of Probability},
Volume = {38},
Number = {1},
Pages = {348--395},
abstract = {Splitting trees are those random trees where
individuals give birth at a constant rate during a
lifetime with general distribution, to i.i.d. copies of
themselves. The width process of a splitting tree is
then a binary, homogeneous Crump-Mode-Jagers (CMJ)
process, and is not Markovian unless the lifetime
distribution is exponential (or a Dirac mass at
\$\backslash{}lbrace\backslash{}infty\backslash{}rbrace\$).
Here, we allow the birth rate to be infinite, that is,
pairs of birth times and life spans of newborns form a
Poisson point process along the lifetime of their
mother, with possibly infinite intensity measure. A
splitting tree is a random (so-called) chronological
tree. Each element of a chronological tree is a
(so-called) existence point \$(v, \backslash{}tau)\$ of
some individual v (vertex) in a discrete tree where
\$\backslash{}tau\$ is a nonnegative real number called
chronological level (time). We introduce a total order
on existence points, called linear order, and a mapping
\$\backslash{}phi\$ from the tree into the real line
which preserves this order. The inverse of
\$\backslash{}phi\$ is called the exploration process,
and the projection of this inverse on chronological
levels the contour process. For splitting trees
truncated up to level \$\backslash{}tau\$ , we prove
that a thus defined contour process is a L\'evy process
reflected below \$\backslash{}tau\$ and killed upon
hitting 0. This allows one to derive properties of (i)
splitting trees: conceptual proof of Le Gall-Le Jan's
theorem in the finite variation case, exceptional
points, coalescent point process and age distribution;
(ii) CMJ processes: one-dimensional marginals,
conditionings, limit theorems and asymptotic numbers of
individuals with infinite versus finite descendances. },
doi = {10.1214/09-AOP485},
year = 2010
}
