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Some properties of stationary continuous state branching processes

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  • Abraham, Romain
  • Delmas, Jean-François
  • He, Hui

Abstract

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.

Suggested Citation

  • Abraham, Romain & Delmas, Jean-François & He, Hui, 2021. "Some properties of stationary continuous state branching processes," Stochastic Processes and their Applications, Elsevier, vol. 141(C), pages 309-343.
    • Handle: RePEc:eee:spapps:v:141:y:2021:i:c:p:309-343
    DOI: 10.1016/j.spa.2021.07.011
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    References listed on IDEAS

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    1. Jacka, S. D. & Roberts, G. O., 1997. "On strong forms of weak convergence," Stochastic Processes and their Applications, Elsevier, vol. 67(1), pages 41-53, April.
    2. Schweinsberg, Jason, 2003. "Coalescent processes obtained from supercritical Galton-Watson processes," Stochastic Processes and their Applications, Elsevier, vol. 106(1), pages 107-139, July.
    3. Bingham, N. H., 1976. "Continuous branching processes and spectral positivity," Stochastic Processes and their Applications, Elsevier, vol. 4(3), pages 217-242, August.
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