Markovian tricks for non-Markovian trees: contour process, extinction and scaling limits

Abstract : In this work, we study a family of non-Markovian trees modeling populations where individuals live and reproduce independently with possibly time-dependent birth-rate and lifetime distribution. To this end, we use the coding process introduced by Lambert. We show that, in our situation, this process is no longer a Lévy process but remains a Feller process and we give a complete characterization of its generator. This allows us to study the model through well-known Markov processes techniques. On one hand, introducing a scale function for such processes allows us to get necessary and sufficient conditions for extinction or non-extinction and to characterize the law of such trees conditioned on these events. On the other hand, using Lyapounov drift techniques , we get another set of, easily checkable, sufficient criteria for extinction or non-extinction and some tail estimates for the tree length. Finally, we also study scaling limits of such random trees and observe that the Bessel tree appears naturally.
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https://hal.archives-ouvertes.fr/hal-01678359
Contributor : Benoît Henry <>
Submitted on : Wednesday, January 24, 2018 - 9:04:10 AM
Last modification on : Friday, July 27, 2018 - 3:12:35 PM
Long-term archiving on : Thursday, May 24, 2018 - 3:31:35 PM

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  • HAL Id : hal-01678359, version 2
  • ARXIV : 1801.03284

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Bertrand Cloez, Benoît Henry. Markovian tricks for non-Markovian trees: contour process, extinction and scaling limits. 2018. ⟨hal-01678359v2⟩

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