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# Pruning of CRT-sub-trees

Abstract : We study the pruning process developed by Abraham and Delmas (2012) on the discrete Galton-Watson sub-trees of the L\'{e}vy tree which are obtained by considering the minimal sub-tree connecting the root and leaves chosen uniformly at rate $\lambda$, see Duquesne and Le Gall (2002). The tree-valued process, as $\lambda$ increases, has been studied by Duquesne and Winkel (2007). Notice that we have a tree-valued process indexed by two parameters the pruning parameter $\theta$ and the intensity $\lambda$. Our main results are: construction and marginals of the pruning process, representation of the pruning process (forward in time that is as $\theta$ increases) and description of the growing process (backward in time that is as $\theta$ decreases) and distribution of the ascension time (or explosion time of the backward process) as well as the tree at the ascension time. A by-product of our result is that the super-critical L\'{e}vy trees independently introduced by Abraham and Delmas (2012) and Duquesne and Winkel (2007) coincide. This work is also related to the pruning of discrete Galton-Watson trees studied by Abraham, Delmas and He (2012).
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Journal articles
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Cited literature [19 references]

https://hal.archives-ouvertes.fr/hal-00763707
Contributor : Jean-François Delmas Connect in order to contact the contributor
Submitted on : Wednesday, December 12, 2012 - 9:19:20 AM
Last modification on : Tuesday, October 12, 2021 - 5:20:12 PM
Long-term archiving on: : Wednesday, March 13, 2013 - 3:51:49 AM

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subtree-12-12.pdf
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### Identifiers

• HAL Id : hal-00763707, version 1
• ARXIV : 1212.2765

### Citation

Romain Abraham, Jean-François Delmas, Hui He. Pruning of CRT-sub-trees. Stochastic Processes and their Applications, Elsevier, 2015, 125, pp.1569-1604. ⟨hal-00763707⟩

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