Some properties of stationary continuous state branching processes

We consider the genealogical tree of a stationary continuous state branching process with immigration. For a sub-critical stable branching mechanism, we consider the genealogical tree of the extant population at some fixed time and prove that, up to a deterministic time-change, it is distributed as...

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Bibliographic Details
Published in:Stochastic processes and their applications 2021-11, Vol.141, p.309-343
Main Authors: Abraham, Romain, Delmas, Jean-François, He, Hui
Format: Article
Language:English
Subjects:
Ancestral process
Continuous state branching process with immigration
Genealogical tree
Mathematics
Probability
Quasi-stationary distribution
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Summary:We consider the genealogical tree of a stationary continuous state branching process with immigration. For a sub-critical stable branching mechanism, we consider the genealogical tree of the extant population at some fixed time and prove that, up to a deterministic time-change, it is distributed as a continuous-time Galton–Watson process with immigration. We obtain similar results for a critical stable branching mechanism when only looking at immigrants arriving in some fixed time-interval. For a general sub-critical branching mechanism, we consider the number of individuals that give descendants in the extant population. The associated processes (forward or backward in time) are pure-death or pure-birth Markov processes, for which we compute the transition rates.
ISSN:0304-4149
1879-209X
DOI:10.1016/j.spa.2021.07.011