### Computation of moments for the length of the one dimensional ISE support.

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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,...

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 $\left\{\mathcal{G}\right(u\left)\right\}$ by pruning Galton–Watson trees and an analogous process $\left\{{\mathcal{G}}^{*}\left(u\right)\right\}$ 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 $\left\{\mathcal{G}\right(u\left)\right\}$ run until its ascension time has a representation in terms of $\left\{{\mathcal{G}}^{*}\left(u\right)\right\}$. A similar result was obtained by Aldous and...

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