Limit theorems for Markov processes indexed by continuous time Galton--Watson trees

We study the evolution of a particle system whose genealogy is given by a supercritical continuous time Galton--Watson tree. The particles move independently according to a Markov process and when a branching event occurs, the offspring locations depend on the position of the mother and the number of offspring. We prove a law of large numbers for the empirical measure of individuals alive at time t. This relies on a probabilistic interpretation of its intensity by mean of an auxiliary process. The latter has the same generator as the Markov process along the branches plus additional jumps, associated with branching events of accelerated rate and biased distribution. This comes from the fact that choosing an individual uniformly at time t favors lineages with more branching events and larger offspring number. The central limit theorem is considered on a special case. Several examples are developed, including applications to splitting diffusions, cellular aging, branching L\'{e}vy processes.Comment: Published in at http://dx.doi.org/10.1214/10-AAP757 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Limit theorems for supercritical age-dependent branching processes with neutral immigration

We consider a branching process with Poissonian immigration where individuals have inheritable types. At rate theta, new individuals singly enter the total population and start a new population which evolves like a supercritical, homogeneous, binary Crump-Mode-Jagers process: individuals have i.i.d. lifetimes durations (non necessarily exponential) during which they give birth independently at constant rate b. First, using spine decomposition, we relax previously known assumptions required for a.s. convergence of total population size. Then, we consider three models of structured populations: either all immigrants have a different type, or types are drawn in a discrete spectrum or in a continuous spectrum. In each model, the vector (P_1,P_2,...) of relative abundances of surviving families converges a.s. In the first model, the limit is the GEM distribution with parameter theta/b.Comment: 24 pages, 3 figure

Queuing for an infinite bus line and aging branching process

We study a queueing system with Poisson arrivals on a bus line indexed by integers. The buses move at constant speed to the right and the time of service per customer getting on the bus is fixed. The customers arriving at station i wait for a bus if this latter is less than d\_i stations before, where d\_i is non-decreasing. We determine the asymptotic behavior of a single bus and when two buses eventually coalesce almost surely by coupling arguments. Three regimes appear, two of which leading to a.s. coalescing of the buses.The approach relies on a connection with aged structured branching processes with immigration and varying environment. We need to prove a Kesten Stigum type theorem, i.e. the a.s. convergence of the successive size of the branching process normalized by its mean. The technics developed combines a spine approach for multitype branching process in varying environment and geometric ergodicity along the spine to control the increments of the normalized process

The coalescent point process of branching trees

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 $(A_i; i\ge 1)$, where $A_i$ is the coalescence time between individuals i and i+1. There is a Markov process of point measures $(B_i; i\ge 1)$ 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 $\rho$ 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 $\mathbb{R}_+$ 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 $\varepsilon$.Comment: Published in at http://dx.doi.org/10.1214/11-AAP820 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org