# Changing the branching mechanism of a continuous state branching process using immigration

Romain Abraham; Jean-François Delmas

Annales de l'I.H.P. Probabilités et statistiques (2009)

- Volume: 45, Issue: 1, page 226-238
- ISSN: 0246-0203

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topAbraham, Romain, and Delmas, Jean-François. "Changing the branching mechanism of a continuous state branching process using immigration." Annales de l'I.H.P. Probabilités et statistiques 45.1 (2009): 226-238. <http://eudml.org/doc/78017>.

@article{Abraham2009,

abstract = {We consider an initial population whose size evolves according to a continuous state branching process. Then we add to this process an immigration (with the same branching mechanism as the initial population), in such a way that the immigration rate is proportional to the whole population size. We prove this continuous state branching process with immigration proportional to its own size is itself a continuous state branching process. By considering the immigration as the apparition of a new type, this construction is a natural way to model neutral mutation. It also provides in some sense a dual construction of the particular pruning at nodes of continuous state branching process introduced by the authors in a previous paper. For a critical or sub-critical quadratic branching mechanism, it is possible to explicitly compute some quantities of interest. For example, we compute the Laplace transform of the size of the initial population conditionally on the non-extinction of the whole population with immigration. We also derive the probability of simultaneous extinction of the initial population and the whole population with immigration.},

author = {Abraham, Romain, Delmas, Jean-François},

journal = {Annales de l'I.H.P. Probabilités et statistiques},

keywords = {continuous state branching processes; immigration process; multitype populations},

language = {eng},

number = {1},

pages = {226-238},

publisher = {Gauthier-Villars},

title = {Changing the branching mechanism of a continuous state branching process using immigration},

url = {http://eudml.org/doc/78017},

volume = {45},

year = {2009},

}

TY - JOUR

AU - Abraham, Romain

AU - Delmas, Jean-François

TI - Changing the branching mechanism of a continuous state branching process using immigration

JO - Annales de l'I.H.P. Probabilités et statistiques

PY - 2009

PB - Gauthier-Villars

VL - 45

IS - 1

SP - 226

EP - 238

AB - We consider an initial population whose size evolves according to a continuous state branching process. Then we add to this process an immigration (with the same branching mechanism as the initial population), in such a way that the immigration rate is proportional to the whole population size. We prove this continuous state branching process with immigration proportional to its own size is itself a continuous state branching process. By considering the immigration as the apparition of a new type, this construction is a natural way to model neutral mutation. It also provides in some sense a dual construction of the particular pruning at nodes of continuous state branching process introduced by the authors in a previous paper. For a critical or sub-critical quadratic branching mechanism, it is possible to explicitly compute some quantities of interest. For example, we compute the Laplace transform of the size of the initial population conditionally on the non-extinction of the whole population with immigration. We also derive the probability of simultaneous extinction of the initial population and the whole population with immigration.

LA - eng

KW - continuous state branching processes; immigration process; multitype populations

UR - http://eudml.org/doc/78017

ER -

## References

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