The Annals of Applied Probability

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

Vincent Bansaye, Jean-François Delmas, Laurence Marsalle, and Viet Chi Tran

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Abstract

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 Lévy processes.

Article information

Source
Ann. Appl. Probab., Volume 21, Number 6 (2011), 2263-2314.

Dates
First available in Project Euclid: 23 November 2011

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1322057322

Digital Object Identifier
doi:10.1214/10-AAP757

Mathematical Reviews number (MathSciNet)
MR2895416

Zentralblatt MATH identifier
1235.60114

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60F17: Functional limit theorems; invariance principles 60F15: Strong theorems 60F05: Central limit and other weak theorems

Keywords
Branching Markov process branching diffusion limit theorems Many-to-One formula size biased reproduction distribution size biased reproduction rate ancestral lineage splitted diffusion

Citation

Bansaye, Vincent; Delmas, Jean-François; Marsalle, Laurence; Tran, Viet Chi. Limit theorems for Markov processes indexed by continuous time Galton–Watson trees. Ann. Appl. Probab. 21 (2011), no. 6, 2263--2314. doi:10.1214/10-AAP757. https://projecteuclid.org/euclid.aoap/1322057322


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