@article{lambert_coalescent_2013,
Author = {Lambert, Amaury and Popovic, Lea},
Title = {The coalescent point process of branching trees},
Journal = {Ann. Appl. Prob.},
Volume = {23},
Number = {1},
Pages = {99--144},
abstract = {We define a doubly infinite, monotone labeling of
Bienayme-Galton-Watson (BGW) genealogies. The genealogy
of the current generation backwards in time is uniquely
determined by the coalescent point process \$(A\_i;
i\backslash{}ge 1)\$, where \$A\_i\$ is the coalescence
time between individuals i and i+1. There is a Markov
process of point measures \$(B\_i; i\backslash{}ge 1)\$
keeping track of more ancestral relationships, such
that \$A\_i\$ is also the first point mass of \$B\_i\$.
This process of point measures is also closely related
to an inhomogeneous spine decomposition of the lineage
of the first surviving particle in generation h in a
planar BGW tree conditioned to survive h generations.
The decomposition involves a point measure
\$\backslash{}rho\$ storing the number of subtrees on
the right-hand side of the spine. Under appropriate
conditions, we prove convergence of this point measure
to a point measure on \$\backslash{}mathbb\{R\}\_+\$
associated with the limiting continuous-state branching
(CSB) process. We prove the associated invariance
principle for the coalescent point process, after we
discretize the limiting CSB population by considering
only points with coalescence times greater than
\$\backslash{}varepsilon\$. },
doi = {10.1214/11-AAP820},
year = 2013
}