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

Romain Abraham;  Jean-François Delmas — 2009

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

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,...

Pruning Galton–Watson trees and tree-valued Markov processes

Romain Abraham;  Jean-François Delmas;  Hui He — 2012

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

We present a new pruning procedure on discrete trees by adding marks on the nodes of trees. This procedure allows us to construct and study a tree-valued Markov process { 𝒢 ( u ) } by pruning Galton–Watson trees and an analogous process { 𝒢 * ( u ) } by pruning a critical or subcritical Galton–Watson tree conditioned to be infinite. Under a mild condition on offspring distributions, we show that the process { 𝒢 ( u ) } run until its ascension time has a representation in terms of { 𝒢 * ( u ) } . A similar result was obtained by Aldous and...

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